downloads.hindawi.comdownloads.hindawi.com/journals/mpe/2020/5620834.pdf · 1 day ago · (Yanetal.[7]).Forexample,DiDi’sSpringFestivalservice fee and the daily price adjustment

Research ArticleDynamic Pricing of Ride-Hailing Platforms considering ServiceQuality and Supply Capacity under Demand Fluctuation

Zhongmiao Sun Qi Xu and Baoli Shi

Glorious Sun School of Business and Management Donghua University 200051 Shanghai China

Correspondence should be addressed to Qi Xu xuqidhueducn

Received 7 April 2020 Accepted 28 May 2020 Published 16 July 2020

Academic Editor Alexander Paz

Copyright copy 2020 Zhongmiao Sun et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Increasing attention is being paid to the pricing decisions of ride-hailing platforms ese platforms usually face market demandfluctuation and reflect supply and demand imbalances Unlike existing studies we focus on the optimal dynamic pricing of theplatforms under imbalance between supply and demand caused by market fluctuation Dynamic models are constructed based onthe state change of supply and demand by using optimal control theory with the aim of maximizing the platformrsquos total profit Weobtain the optimal trajectories of price supply and demand under three ride demand situations e effects of some keyparameters on pricing decisions such as coefficient of demand fluctuation service quality and fixed commission rate areexamined We find the optimal dynamic price can improve the match of supply-demand in ride-hailing market and enhance therevenue of platform

1 Introduction

As an innovation of the mobile Internet era ride-hailingplatforms (eg Uber Lyft and DiDi) have played a sig-nificant role in improving vehicle use efficiency increasingtransportation service supply facilitating taxiing and pro-moting employment As such ride-hailing platforms havebeen rapidly popularized and have subverted the traditionaltaxi market completing millions of trips every day (Ma et al[1]) By July 2018 Uber had completed over 10 billion ride-hailing transactions globally and was active in over 80countries and 700 cities (Uber [2]) e total market value ofthe global ride-hailing industry is projected to grow to $285billion by 2030 (MarketWatch [3]) which will bring hugeeconomic benefits to ride-hailing platforms Ride-hailingplatforms rely on freelance drivers who decide when and forhow long to work (Hu and Zhou [4]) e platforms usemobile Internet to integrate the online and offline functionsmatch the demand of riders with the supply of driversreduce the empty load rate of drivers save the cost ofcommunication between drivers and riders and improve theconvenience and efficiency of travel (Cramer and Krueger

[5]) Compared with public transport the benefits of ride-hailing are convenience time-saving and comfort whichhas won the favor of some ride-hailing passengers In ad-dition when the weather is bad time is tight and travellocation is remote users can conveniently use a mobilephone platform to access a ride-hailing service

However since 2018 the supply of vehicles in China hasbeen insufficient and there has been difficulty in finding aride-hailing service in some areas and during certain periods(Xinhua news [6])e reason for the difficulty is mainly dueto fluctuations in demand and the mismatch between supplyand demand Demand during the peak period increasesrapidly and supply cannot keep up however during thenormal period supply is excessive (Hu and Zhou [4]) Forexample in big cities such as Beijing and Shanghai queuingis necessary during the morning and evening peak hours andeven more so during holidays e ride-hailing platformrsquosapplication usually shows that there are more than a dozenpeople in the queue with an average wait of 20 minutes orthat the driver is far away etc (Xinhua news [6]) To solvethese difficulties some ride-hailing platforms try to dy-namically adjust the price to match the supply and demand

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5620834 26 pageshttpsdoiorg10115520205620834

(Yan et al [7]) For example DiDirsquos Spring Festival servicefee and the daily price adjustment mechanism for themorning peak are based on supply and demand Passengerswilling to pay higher service fees are sent to motivateddrivers who give up their holiday break is arrangement isconvenient for the passengers but also gives the drivers areasonable and legitimate return it is a win-win scenario(Cachon et al [8])

Compared with traditional taxi companies ride-hailingplatforms have their own unique characteristics For ex-ample ride-hailing passengers can quickly obtain ride in-formation (eg the waiting time the number of people inline and even the number of drivers around them) andcoordinate with nearby drivers in real time whereas tra-ditional taxi passengers usually wait at the side of the road orbook in advance In addition the drivers of ride-hailingplatforms are relatively free to choose when and where theywork e fluctuation of ride-hailing demand based onspecific reasons was related to more people wanting to usethe service (eg time of day price and service quality) andthe supply of services usually be influenced by factors suchwhether there are drivers available geographic areascommissions etc Due to the influence of weather factors(eg sunny rainy and snowy weather) and special occasions(eg after a concert) there can be a mismatch in supply anddemand (Hall et al [9] Cachon et al [8] Yan et al [7]Afeche et al [10] Hu and Zhou [4])

We are motivated by the dynamic pricing problems thatarise when ride-hailing platforms face market demandfluctuation and supply-demand mismatch Although therehave been studies on pricing strategies many authors (egFeng et al [11] Yan et al [7] Hu and Zhou [12] Sun et al[13] Bimpikis et al [14]) have not designed their models totake into account the characteristic of demand fluctuationover time ey also do not consider the supply functionwhich is sensitive to dynamic wages in relation to timeWhen rider demand increases but there is not enough supplyto meet the demand (Cachon et al [8]) platforms imple-ment ldquodemand rationingrdquo by delaying the service extendingthe rider wait time or even losing the rider bookings edifficulty in hailing a car in some regions and during certainperiods affects the consumersrsquo travel experience When riderdemand decreases supply can exceed the demand (Cachonet al [8]) In such cases the platform resorts to capacityrationing that is to idle available transportation capacityerefore to maximize the profits it is important to de-termine the platformsrsquo optimal pricing under market en-vironments of unbalanced supply and demand caused byfluctuations in rider demand and thus reduce the riderbooking delay and underutilized transportation capacity

is study considers a ride-hailing platform in themarket e platform crowdsources drivers who have idlecar resources and provides shared transport for riders Weincorporate the rider demand fluctuation into the commondemand function (Desiraju and Moorthy [15]) and use thecoefficient of demand fluctuation to describe different de-mand conditions Unlike the previous studies (eg Cachonet al [8] Hu and Zhou [4] Liu et al [16]) we assume riderdemand is an exponential function of time and is sensitive to

price and service quality is means rider demand changesdynamically over time

As a service enterprise the ride-hailing platform canprovide ride supply-demand matching services that can bereflected by service quality e service quality usually refersto passenger satisfaction with ride-hailing services such aspassengersrsquo ride efficiency platformsrsquo supervision and useexperience of APP In reality the passengers are concernednot only with the price but also with the service qualityHowever service quality is closely related to service costwhich affects the platformrsquos pricing decisionserefore it isnecessary to study the impact of service quality on platformpricing decisions transaction volumes and profits Fur-thermore the fixed commission rate is the percentage of thefare that the platform pays to the drivers after they completeeach ride (Cachon et al [8] and Hu and Zhou [4]) It directlyaffects not only the platformrsquos revenue from each transactionbut also the driversrsquo participation through their incomeerefore we take into account the service quality and thefixed commission rate of the ride-hailing platforms theopportunity cost when supply exceeds demand and thebooking delay cost when supply fails to meet demand intime We thus study three different demand situationsdecrease in demand increase in demand and stable de-mand Optimal control theory has been used in the literatureon dynamic pricing (eg Lu et al [17] Herbon andKhmelnitsky [18] and Feng et al [19])us we use optimalcontrol theory to construct models and study the dynamicpricing of ride-hailing platforms with different market de-mand situations in a finite service time [0 T] An analysis ofkey parameters reveals further insights into the effect of thecoefficient of demand fluctuation service quality and fixedcommission rate on the platformrsquos pricing decisions Wefind that although the peak congestion period can be re-duced by reducing service quality andor increasing the fixedcommission rate the reduction in the service quality maynot be desirable due to the loss of peak demand

We investigate the impact of key parameters on the totaltransaction volume and the total profit in a finite servicetime in which the total profit is a concave function of thefixed commission rate and service quality To the best of ourknowledge few studies have investigated dynamic pricingaccording to the characteristics of demand fluctuation in theactual operation of ride-hailing platforms Our study is thefirst to assume demand is an exponential function of timeand to use optimal control theory to examine the pricingproblem of ride-hailing platforms We address the followingresearch questions

(1) How do the ride-hailing platforms develop effectivedynamic pricing strategies in the face of demand-supply imbalances under different market demandsituations (decreased demand increased demandand stable demand) and how do the trajectories ofthe platformsrsquo supply rate and demand rate changeunder the influence of optimal dynamic prices

(2) If the opportunity cost is not considered how doesthe platform adjust its optimal pricing strategy ifmarket demand decreases If a ceiling price

2 Mathematical Problems in Engineering

constraint is considered how do the optimal pricingstrategies change if market demand increases

(3) How do the coefficient of demand fluctuation ser-vice quality and fixed commission rate affect pricingdecisions transaction volumes and profits underdifferent market demand situations

2 Literature Review

21 Two-Sided Markets ere are a number of studies ontwo-sided markets in various industries such as the creditcard industry (Rochet and Tirole [20 21] Armstrong [22])the telecomsmarket (Waverman [23]) online gaming (Parkerand Van Alstyne [24]) e-commerce (Gaudeul and Jullien[25]) media (Anderson and Gabszewicz [26] Kind andStahler [27]) and transportation (Furuhata et al [28] Wanget al [29] Djavadian and Chow [30]) Rochet and Tirole [20]Gaudeul and Jullien [31] and Armstrong [22] have donepioneering research on two-sided markets Rochet and Tirole[21] defined two-sided markets ldquoA market is two-sided if theplatform can affect the volume of transactions by changingthe price share between two groupsrdquo Most of the literatureconsiders network externalities in pricing strategies (egRochet and Tirole [20 21] Parker and Van Alstyne [24]Wang et al [29] Djavadian and Chow [30]) Some studiesfocus on competition (eg Rochet and Tirole [20] Armstrong[22] Kind and Stahler [27])

However unlike most two-sided markets ride-hailingplatforms have pricing power and need to match the supplywith the demand in a timely manner In our study ratherthan focusing on equilibrium and the impact of networkexternalities we examine the dynamic setting in which aride-hailing platform sets price and wage over time underdifferent market demand situations Specifically we refer tothe description by Hu and Zhou [4] in which the platformrsquosactual transaction volume at time t is the minimum of riderdemand and supply capacity at is if in a period there areD riders and S drivers at time t in a neighborhood the actualtransaction volume at time t is close to minS D

22 Pricing Problems Optimal pricing problems have beenextensively studied in the revenue management and pricingliterature (Kolisch and Zatta [32]) Most revenue manage-ment studies usually consider a fixed supply side and price-sensitive demand (Petruzzi and Dada [33]) In contrast weconsider a time-varying supply side that is sensitive to thewage offered by the ride-hailing platform

Our study is related to the dynamic pricing in operationmanagement Over the past two decades dynamic pricinghas attracted considerable attention from both industry andacademia (Cao et al [34]) Gallego and van Ryzin [35] firstconnected dynamic pricing with revenue management andapplied the intensity control theory system to the dynamicpricing of perishable goods and later extended their researchto multiproduct situations (Gallego and van Ryzin [36])Smith and Achabal [37] posited that the sales rate is relatedto seasonal factors of price and the level of residual

inventory they established an optimized model based on thepricing problem of retail products at the end of the quarterSome authors consider the consumer strategic behavior (egBesanko andWinston [38] Anderson andWilson [39] Avivand Pazgal [40] Levin et al [41]) In the recent years therehave been many studies on dynamic pricing (eg Hu et al[42] Varella et al [43] Ajorlou et al [44] Feng et al [19]Chen and Chen [45]) especially regarding perishable items

Lu et al [17] introduced a perishable item inventorysystem with limited replenishment capacity and achieved anoptimal joint dynamic pricing and replenishment strategy bysolving the optimal problem using Pontryaginrsquos maximumprinciple Herbon and Khmelnitsky [18] used optimalcontrol theory to study the optimal dynamic pricing andoptimal replenishment strategies of storable perishableitems Feng [19] studied the dynamic optimizationmodel forthe maximization of total profit per unit time of perishableproducts with dynamic pricing and quality investment theydetermined an optimal joint dynamic pricing quality in-vestment and replenishment strategy based on Pontryaginrsquosmaximum principle Related studies include Chen et alrsquos[46] and Satorsquos study [47] ese authors used optimalcontrol theory (Lu et al [17] Herbon and Khmelnitsky [18]Feng [19]) to construct an equation of state change andobtain an optimal dynamic price solution according toPontryaginrsquos maximum principle We construct an equationof state change based on supply and demand and study theride-hailing platformsrsquo dynamic pricing of ride-hailingplatforms using optimal control theory to maximize the totalprofit in a period

23 Ride-Hailing Platforms e service operation of ride-hailing platforms based on the concept of ldquoshared trans-portrdquo has increasingly attracted the scholars to do researchin the field of operation management At present researchon the service operation of ride-hailing platforms focuses onthe pricing strategy and on the matching supply with de-mand To examine platform pricing strategy Cachon et al[8] designed five contracts by adjusting prices and com-missions of ride-hailing platforms and found that the op-timal contract substantially increases the platformrsquos profitrelative to contracts with a fixed price or fixed wage (orboth) and although surge pricing is not optimal it generallyachieves optimal profit Zha et al [48] studied the peakpricing effect on ride-hailing platforms under different as-sumptions of labor supply behavior Yan et al [7] used datafrom Uber to show that by jointly optimizing the dynamicpricing and dynamic waiting price variability can bemitigated while increasing the capacity utilization tripthroughput and welfare Guda and Subramanian [49]considered the strategic interactions among ride-hailingplatform drivers when deciding to move across regions andfound that even in areas where driver supply exceeds de-mand peak pricing can be profitable Bimpikis et al [14]explored the spatial price discrimination in the context of aride-hailing platform that serves a network of locations Sunet al [13] were the first to take into account both ride details

Mathematical Problems in Engineering 3

and driver locations and assumed that drivers and customersmaximize the utility to determine the optimal pricingstrategy for online ride-hailing platforms Liu et al [16]studied the pricing decisions of a profit-maximizing plat-form by considering the providerrsquos threshold participatingquantity value-added service (VAS) and matching abilityWu et al [50] considered spatial differentiation and networkexternality in the pricing mechanism of an online car-hailingplatform and analyzed the game competition among plat-form users and the impact on the platformrsquos optimal pricingAlthough the literature has demonstrated and studied thepricing strategies of ride-hailing platforms at peak demandthese studies have not incorporated the characteristics ofdemand fluctuation and uncertain supply into the modelsnor have they explored how service quality and the plat-formrsquos fixed commission rates affect the pricing decisionstransaction volume and profit

On matching the supply with demand many re-searchers modeled the matching process between driversand customers as an unobservable queue e arrival ofpassengers is regarded as a Poisson process and drivers areregarded as servers in the queuing system (Bai et al [51]Taylor [52] Hu and Zhou [4]) with queuing theory used tostudy the supply and demand matching Feng et al [11]compared the efficiency of online supply and demandmatching with the average waiting time of traditional taxihailing Hu and Zhou [12] considered an intermediaryrsquosproblem of dynamically matching demand and supply ofheterogeneous types in a periodic review In research onother aspects of ride-hailing platforms Ma et al [1] studiedthe optimal scheduling that would enable drivers to chooseto accept the platformrsquos scheduling rather than drive toanother area or wait for higher prices Afeche et al [10]found that drivers make strategic decisions on whether andwhere to provide services according to soaring price andincome differences

24 Academic Contribution Our paper differs from theabove studies in two aspects First although it is common tostudy pricing strategies in different industries using optimalcontrol theory few researchers have used this method tostudy the pricing strategies of ride-hailing platforms Weintroduce optimal control theory to study the pricingproblems of ride-hailing platforms us we extend theexisting research methods to ride-hailing platforms Secondwe consider the characteristics of demand fluctuation intothemodels in view of the practices of such platforms as DiDiand Uber that face the ride-hailing demand vary dynamicallywith time at different periods e optimal dynamic pricetrajectory of ride-hailing platforms are investigated and wefind out the key time points that affect the platform opti-mization pricing under different circumstances so as tomake optimal pricing decisions for different ride time pe-riods be more practical guiding In addition we examine theimpacts of platform service quality fixed commission rateand market demand fluctuation on the pricing decisionstransaction volume and profit under different marketsituations

3 Problem Description andModel Assumptions

Ride-hailing platforms match riders and drivers by coor-dinating the supply and demand of ride services Drivers canfreely choose to receive bookings and complete offlinetransportation services Once the ride is complete ridersusually pay fares through third-party payment platforms(eg Alipay or WeChat) on their mobile phones and thedrivers receive a corresponding commission from the ride-hailing platform e operational process of ride-hailingplatforms is shown in Figure 1

We consider a ride-hailing market consisting of a ride-hailing platform a number of drivers and a number ofriders In a finite service time [0 T] the platformrsquos demandrate is D(t) and the platformrsquos supply rate is S(t) eplatform charges the riders price P(t) based on the supplyand demand situation at time t e platform pays com-mission W(t) to the drivers who use their personal vehiclesand provide the transportation serviceWhen the supply rateis higher than the demand rate the platform bears unitopportunity cost c When the supply rate cannot meet thedemand rate the platform bears unit booking delay cost hNote that opportunity cost c or booking delay cost h areincurred per time unit Such a ride-hailing platform is il-lustrated in Figure 2

e assumptions used to formulate the problem are asfollows (the list of notations is in Table 1)

(1) It is assumed that the supply and demand at time 0are in balance under the dynamic pricing policy thatis D(0) S(0)

(2) In the Uber setting the ldquosurge multiplierrdquo multipliedby the base fare (as a whole corresponding to theprice in our model) together with the travel time anddistance determines the trip fare For simplicity ourmodel assumes away the spatial dimension (egCachon et al [8] Hu and Zhou [4])

(3) Referring to Liu et al [16] the demand rate for ride-hailing platforms is sensitive to price and servicequality and is assumed to be a linear function of theprice

D(P t) α(t) minus βP(t) + cq (1)

where α(t) represents the initial demand of themarket in a certain region βgt 0 indicates the pricesensitivity factor and cgt 0 is the service sensitivityfactor In addition D(P t)≧ 0 at the ride service time[0 T] otherwise this model would be uninteresting

(4) We assume the following supply rate function

S(W t) s middot (W(t) minus ε) (2)

where W(t) represents the commission for driversW(t) r middotP(t) (Cachon et al [8] Hu and Zhou [4]) ε isthe minimum participation cost per time unit of adriver which indicates a driver is willing to accept aride if the commission W(t) exceeds ε (Sun et al [13])


(ie W(t)gt ε) and s is the wage sensitivity factorwhich reflects the sensitivity of supply to changes incommission (Hu and Zhou [4])

According to the demand rate and supply rate modelsabove the platformrsquos total transaction volume is asfollows

ψ(P t) 1113946T

0min(D(P t) S(W t))dt (3)

(5) e ride-hailing platformrsquos unit cost function is asfollows

C(t) W(t) + ηq2 (4)

where ηgt 0 is the platformrsquos service cost parameter qis the platformrsquos service quality and ηq2 has beenwidely used (eg Moorthy [53] Motta [54] Ha et al[55])

(6) During the ride service time [0 T] when D(P t)≦S(W t) the ride demand decreases and the supplycapacity of the platform is larger than the demandLet v(t) represent the platformrsquos excess supply ca-pacity and v(0) 0 When D(P t)≧ S(W t) the ridedemand increases and thus some ride requirementscannot be met in time Let u(t) represent the quantityof delayed bookings and u(0) 0

(7) To describe the effect of market demand fluctuationwe assume that the initial demand of the market in acertain region over time is as follows

α(t) αeminus at

(5)

where αgt 0 is the size of the rider demandmarket at t 0or a 0 Referring to Maihami and Kamalabadi [56] andGhoreishi et al [57] αeminus at describes the fluctuation for theinitial ride demand in a certain region (ie decreasingincreasing and stable) e coefficient of demand fluc-tuation a represents a decreasing market demand if agt 0as shown in Figure 3 which is consistent with the realscenario of decreasing market demand (eg when thedemand decreases after rush hour) If alt 0 the marketdemand increases as shown in Figure 4 which is con-sistent with the real scenario of increasing market de-mand (eg when the demand increases during rush hourduring rainysnowy weather or at the end of a largecommunity event) If a 0 it represents the stable marketdemand which is between decreasing market demandand increasing market demand

4 Model Analysis

In Sections 41 42 and 43 we present the optimal pricingstrategies in a finite service time [0 T] under three ridemarket situations decreasing demand increasing demandand stable demand Based on three optimal pricing strate-gies we obtain the platformrsquos supply rate demand rate totaltransaction volume and total profit

41 Optimal Pricing Strategies with Decreasing Market De-mand (agt 0) Decreasing market demand indicates thatpeoplersquos demand for transportation is decreasing within a

Ride-hailing platform

Driver

Driver

DriverRider Rider

Rider

Ride-hailing service request Match drivers

Fares Commissions

Offline transportation

Figure 1 Operational process of ride-hailing platforms

Booking delay cost

D (t)Ride-hailing

platform

S (t)

W (t)P (t)

c

h

Opportunity cost

Riderdemandmarket

Driversupplymarket

Figure 2 Model of a ride-hailing platform

Table 1 NotationsIndexv e situation in decreasing market demandu e situation in increasing market demandb e situation in stable market demandt Time t isin [0 T]Parametersa e coefficient of demand fluctuationα e size of rider demand market at t 0 or a 0β e price sensitivity factorc e service sensitivity factors e wage sensitivity factorr efixed commission rate 0lt rlt 1η e service cost parameterq Service qualityc e unit opportunity costh e unit booking delay costε Minimum participation cost per time unit of a driverP e ceiling priceP(t) e ride-hailing service priceW(t) e commission (wage) for driversC(t) e unit cost functionα(t) e initial demand of the market at time tD(P t) e demand rate functionS(W t) e supply rate functionv(t) e excess supply capacity of the platformu(t) e quantity of delayed bookings of the platformψ(P t) e total transaction volume of the platformπ(P t) e total profit of the platform


period [0 T] such as during weekday nonrush hours (seeFigure 3) erefore the balance of the supply and demandof the ride-hailing platform at time t 0 will be brokenExcess supply capacity (v(t)≧ 0) or capacity rationingoccurs during this period (Cachon et al [8]) Based on theproblem description and the model assumptions in Section3 we set price Pv(t) of the ride-hailing service as the controlvariable e optimal pricing model can be constructedwith optimal control theory when the platformrsquos supplycapacity is in excess As in Joslashrgensen and Kort [58] theinventory of physical products is the state variable and theequation of state change is based on the inventory re-plenishment rate and the demand rate In our setup thestate of the ride-hailing platform at time [0 T] is describedby the accumulated excess supply capacity rate v(t) Weconstruct the state change equation of excess supply ca-pacity based on the platformrsquos supply rate S(t) and demandrate D(t)

v(t) Sv Wv t( 1113857 minus Dv Pv t( 1113857 (6)

e accumulated excess supply capacity at time t is

v(t) v(0) + 1113946t

0Sv Wv τ( 1113857 minus Dv Pv τ( 1113857( 1113857dτ (7)

e ride-hailing platformrsquos objective is to find the op-timal dynamic pricing and ride supply to maximize the totalprofit over the ride service time [0 T]e objective functionis given as follows

πv Pv t( 1113857 maxPv(t)

1113946T

0Dv Pv t( 1113857 middot Pv(t) minus Dv Pv t( 11138571113858

middot Cv(t) minus c middot v(t)1113859dt

maxPv(t)

1113946T

0Dv Pv t( 1113857 middot Pv(t) minus Wv(t) minus ηq

21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0

Sv Wv τ( 1113857 minus Dv Pv τ( 1113857( 1113857dτ⎛⎜⎜⎝ ⎞⎟⎟⎠⎤⎥⎥⎥⎥⎥⎥⎦dt

maxPv(t)

1113946T


21113872 11138731113960

minus c(T minus t) middot Sv Wv t( 1113857 minus Dv Pv t( 1113857( 11138571113859dt

(8)

with the following constraints

v(t) Sv Wv t( 1113857 minus Dv Pv t( 1113857

s middot Wv(t) minus ε( 1113857 minus αeminus at

+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)

To solve the optimization model above the Hamiltonequation satisfied by a certain value function is constructedas

Hv v(t) Pv(t) λv(t) t( 1113857

Dv Pv t( 1113857 middot Pv(t) minus Wv(t) minus ηq2

1113872 1113873 minus c(T minus t)

middot Sv Wv t( 1113857 minus Dv Pv t( 1113857( 1113857

+ λv(t) middot sWv(t) minus αeminus at

+ βPv(t) minus cq1113872 1113873

αeminus at

minus βPv(t) + cq1113872 1113873 middot Pv(t) minus Wv(t) minus ηq2

1113872 1113873

+ λv(t) minus c(T minus t)( 1113857

middot s middot Wv(t) minus ε( 1113857 minus αeminus at

+ βPv(t) minus cq1113872 1113873

(10)

where the Lagrange multiplier λv(t) is the costate variableassociated with the state variables S(t) and D(t) and λv(t)

denotes the shadow price of the excess supply capacityus we can state the solution for the scenario in which

the market ride demand decreases

Lemma 1 With decreasing market demand the platformrsquosprofit πv(Pv t) is strictly concave in service price Pv(t) andthe optimal dynamic price solution Plowastv (t) can maximize theplatformrsquos profit πv(Pv t)

e proof is in Appendixe maximum principle states that the necessary con-

ditions for Plowastv (t) with corresponding state trajectory vlowast(t)to be an optimal control are the continuous and piecewise

α (t)

α (t) = 400endash004t



100

150

200

250

300

350

400

450

5 10 15 20 25 300t

Figure 3 Decreasing market demand

α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t

Figure 4 Increasing market demand


differentiable function λv(t) such that the following conditionshold According to the necessary conditions of Pontryaginrsquosmaximum principle we obtain the following costate equations

v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)

From (9) and (10) the optimal dynamic price and theshadow price λlowastv (t) can be obtained as follows

Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)

minusα(sr minus β) middot 1 minus eminus aT( 1113857

2βTa(sr + β)+

v(T)

T(sr + β)

(12)

λlowastv (t) ct +(1 minus r) middot (sr minus β) middot αeminus aT minus aT(cq + sε) minus α( 1113857

Ta(sr + β)2

+2β(1 minus r) middot v(T)

T(sr + β)2minus

βηq2

sr + β

(13)

Based on Lemma 1 the optimal pricing control problemsare given in eorem 1

Theorem 1 During the ride service time [0 T] when themarket demand is decreasing the optimal trajectory of theplatformrsquos dynamic price Plowastv (t) is

Plowastv (t)

α2β

middot eminus at

+c(sr + β)

β(1 minus r)middot t +

cq + sε(sr + β)

minusα(sr minus β)

2β(sr + β)

(14)

e optimal trajectory of the shadow price λlowastv (t) is

λlowastv (t) c middot t +(1 minus r) middot (2βα minus (cq + sε) middot (sr minus β))

(sr + β)2

minusα(1 minus r) + βηq2( 1113857

sr + β+ cT

(15)

e optimal trajectories of the platformrsquos supply rateSlowastv (t) and demand rate D lowastv (t) over time are as follows

Slowastv (t)

srα2β

middot eminus at

+src(sr +β)


s(rcq minus βε)(sr +β)

minussrα(sr minus β)

2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)

(1 minus r)middot t +


+α(sr minus β)

2(sr +β)

(17)

Remark 1 In this section we set the constraint v(t) ≧ 0 in(9) which represents the scenario in which the platformrsquos

demand is not higher than the supply when the market ridedemand decreases in a finite service time [0 T] Given theoptimal solution of eorem 1 the parameters satisfy theconditions as follows (a) β≧ sr (b) if βlt sr2c(sr + β)2 ge aα(sr minus β) middot (1 minus r)

e proof is in Appendixeorem 1 indicates that the ride-hailing platformrsquos

optimal dynamic price shadow price supply rate and de-mand rate all change dynamically with time t in the rideservice time [0 T] Furthermore the shadow price λlowastv (t) is amonotonic increasing function of time t and its slope givesthe unit opportunity cost c As the shadow price representsthe influence of constraint conditions on the objectivefunction through equation (15) we know that the changerate of the platformrsquos excess supply capacity has an in-creasing impact on its profit rate with the passage of time

Corollary 1 During the ride service time [0 T] when themarket demand decreases we conclude the following

(a) Plowastv (t) is a convex function of time t(b) Assuming c 0 there is a threshold Tth When

0≦Tth≦T Plowastv (t) decreases first and then remainsstable or unchanged and when TltTth Plowastv (t)

monotonically decreases(c) Assuming cne 0 we have a critical point

t1 minus (1a)ln(2c(sr + β)aα(1 minus r)) When 0≦t1≦T Plowastv (t) decreases first and then increases andPlowastv (min) Plowastv (t1) and when Tlt t1 Plowastv (t) mono-tonically decreases

e proof is in AppendixCorollary 1 shows the optimal pricing strategies when a

ride-hailing platform faces decreasing market demandduring the ride service time [0 T] First the platformrsquosoptimal price decreases monotonically over time On the onehand the platform reduces the optimal price Plowastv (t) overtime both to alleviate the weakening market demand and tostimulate market demand as much as possible to maximizethe utilization of supply capacity On the other hand theplatform adopts a fixed commission contract (eg Uber) topay Wlowastv (t) for drivers where Wlowastv (t) r middot Plowastv (t) us re-ducing the service price Plowastv (t) means correspondingly re-ducing the commission Wlowastv (t) of drivers who join theplatform to provide ride service e platform can avoidexcessive supply capacity during the period of decreasingdemand to reduce the waste of idle transportation resources

Second with the passage of time if the platform doesnot consider the opportunity cost (c 0) the optimal priceeventually tends to be stable or unchanged over timewhich ensures basic income for the platform In contrastif the platform considers the opportunity cost (c gt 0) theoptimal price first decreases to a certain level Plowastv (min)

Plowastv (t1) and then increases over time When the marketdemand decreases to an extent such that the accumulationof opportunity cost has an increasing impact on theplatformrsquos profit the platform must raise the price tomaximize profit


Corollary 2 During the ride service time [0 T] marketdemand decreases Be platformrsquos total transaction volumeψ lowastv is

ψ lowastv (t) α 1 minus eminus aT( 1113857

2aminus

cT2 middot (sr + β)

2(1 minus r)

+ T middots(rcq + βε) + α(sr minus β)

2(sr + β)

(18)

e platformrsquos total profit π lowastv is

π lowastv 1113946T

pDlowastv (t) middot (1 minus r) middot P

lowastv (t) minus ηq

21113872 11138731113960

minus c middot 1113946t

0Slowastv (τ) minus D

lowastv (τ)( 1113857dτ1113891dt

(19)

e proof is in Appendix

42 Optimal Pricing Strategies with Increasing Market De-mand (alt 0) During peak ride periods such as during rushhour or rainysnowy weather and at the end of large-scalecommunity activities the market demand for rides surgese increasing rate of demand is shown in Figure 4 At thistime the balance of the supply and demand of the ride-hailingplatform at time t 0 might be broken Demand can exceedsupply capacity at the ride service time [0 T] (Cachon et al[8]) and thus demand rationing can occur Ride bookings willbe delayed (u(t)≧ 0) In this situation if the platform dy-namically adjusts price according to the change in marketdemand it is conducive to greater profits and better matchingof supply with demand In practice price raising strategies areused when the platform faces high demand Uber calls thisstrategy ldquosurge pricingrdquo and Lyft calls it ldquoprime timerdquo

Based on the problem description and the model as-sumptions in Section 3 let price Pu(t) of the ride-hailingservice be the control variable and the volume of delayedbookings be the state variable Now we can construct theoptimal pricing model using optimal control theory (Her-bon and Khmelnitsky [18] Joslashrgensen and Kort [58]) Firstthe state change equation of the quantity of delayed bookingsis constructed based on the platformrsquos supply rate S(t) anddemand rate D(t)

u(t) Du Pu t( 1113857 minus Su Wu t( 1113857 (20)

us the quantity of delayed bookings at time t is

u(t) u(0) + 1113946t

0(D(P τ) minus S(P τ))dτ (21)

During the ride service time [0 T] the platform facesincreasing market demand As the actual supply capacitycannot meet the demand in time the platformrsquos actualtransaction volume at time t is min(Du(Pu t)

Su(Wu t)) Su(Wu t) e ride-hailing platformrsquos objec-tive is to maximize the total profit in the ride service time [0T] e objective function is given by

πu Pu t( 1113857 maxPu(t)

1113946T

0Su Wu t( 1113857 middot Pu(t) minus Su Wu t( 1113857 middot Cu(t)1113858

minus h middot u(t)]dt

maxPu(t)

1113946T

0Su Wu t( 1113857 middot Pu(t) minus Wu(t) minus ηq

21113872 11138731113960

minus h 1113946t

0Du Pu τ( 1113857 minus Su Wu τ( 1113857( 1113857dτ1113888 11138891113891dt

maxPu(t)

1113946T

01113946

T

0

Su Wu t( 1113857 middot Pu(t) minus Wu(t) minus ηq2

1113872 11138731113960

minus h(T minus t) middot Du Pu t( 1113857 minus S Wu t( 1113857( 11138571113859dt

(22)

e constraints of (22) are as follows

u(t) Du Pu t( 1113857 minus Su Wu t( 1113857

αeminus at

minus βPu(t) + cq minus s middot Wu(t) minus ε( 1113857

u(t)ge 0

u(0) 0

alt 0

(23)

We seek the platformrsquos optimal price in the case ofincreasing demand We introduce the Lagrange multiplierλu(t) and construct the Hamilton function as follows

Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873

minus h(T minus t) middot Du Pu t( 1113857 minus Su Wu t( 1113857( 1113857

+ λu(t) middot αeminus at

minus βPu(t) + cq minus s middot Wu(t) minus ε( 11138571113872 1113873

sWu(t)( 1113857 middot Pu(t) minus Wu(t) minus ηq2

1113872 1113873 + λu(t) minus h(T minus t)( 1113857middot

αeminus at


(24)

Lemma 2 In increasing market demand the platformrsquosprofit πu(Pu t) is strictly convex in service price Pu(t) Bereis no maximum point Pu(t) that can maximize the platformrsquosprofit πu(Pu t)

e proof is in AppendixLemma 2 shows that this model cannot use Pontryaginrsquos

principle otherwise a price solution can be obtained tominimize the platformrsquos profit However we should find theoptimal dynamic price to maximize the platformrsquos profitduring the ride service time [0 T] We consider two cases (i)the platform is not constrained by the ceiling price(Pu(t)≦P P⟶+infin) and (ii) the platform is constrainedby the ceiling price (Pu(t)≦P P is a constant) which ismotivated by the reality (eg the government wants to avoidthe malicious price increase of ride-hailing platforms insome regions that is regulatory constraints) Respectivelywe propose eorems 2 and 3


Theorem 2 In the case of increasing market demand duringthe ride service time [0 T] for case (i) the optimal dynamicprice of the platform Plowastu (t) is

Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)

e trajectories of the platformrsquos supply rate Slowastu (t) anddemand rate Dlowastu (t) over time are as follows

Slowastu (t)

srαβ + sr

middot eminus at

+s middot (rcq minus βε)

β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)

e proof is in Appendixeorem 2 indicates that the platformrsquos optimal dynamic

price supply rate and demand rate all change dynamicallywith time t in the ride service time [0 T] According toeorem 2 we can obtain some properties of the optimalsolution as in Corollary 3 and the platformrsquos total trans-action volume and total profit as in Corollary 4

Corollary 3 During the ride service time [0 T] based onBeorem 2 the optimal solution for the ride-hailing platformhas the following properties

(a) Plowastu (t) is a convex function of time t and Plowastu (t) in-creases monotonically

(b) Slowastu (t) Dlowastu (t) and u(t) 0 so the platform does nothave the booking delay cost at time t

e proof is in AppendixWhen the ride-hailing platform faces increasing market

demand during the ride service time [0T] we consider that theplatform is not constrained by the ceiling price (Plowastu (t)≦PP⟶+infin) erefore there are enough drivers wanting tojoin the platform to provide ride serviceus the platform canacquire a stable and continuous supply capacity by increasingservice price Corollary 3 indicates that the optimal price Plowastu (t)

increases monotonically with the passage of time in the case ofincreasing market demand us the platform can continu-ously obtainmore supply capacityMoreover the optimal pricePlowastu (t) canmaximally incentivize drivers to join the platform toprovide transportation service which in turn can dynamicallyaffect the supply capacity and reduce the delay in bookingserefore the platformrsquos supply and demand reach a balanceand the delayed bookings reduce to zero

Corollary 4 During the ride service time [0 T] based onBeorem 2 the platformrsquos total transaction volume ψt

u is

ψ lowastu (t) srα 1 minus eminus aT( 1113857

a(β + sr)+

s middot (rcq minus βε)Tβ + sr

(28)

Be platformrsquos total profit π lowastu is

π lowastu 1113946T

0Slowastu (t) middot (1 minus r) middot P

lowastu (t) minus ηq

21113872 1113873dt (29)

Be proof is in Appendix

Theorem 3 In case (ii) when the market demand increasesduring the ride service time [0 T] there exists a critical pointt2 minus (1a)ln(P middot (sr + β) minus (cq + sε)α) If T≦ t2 the opti-mal dynamic price Plowastu (t) is calculated as in equation (25) If0≦ t2≦T and Pmax

u Plowastu (t2) P thus Plowastu (t) is as follows

Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

e proof is in Appendixeorem 3 indicates that when the ride-hailing platform

faces increasing market demand and the service price is limitedby the ceiling price during the ride service time [0 T] theplatformrsquos optimal price increases monotonically over timewhich is consistent with Corollary 3 Once the optimal pricereaches the ceiling price the platform will achieve maximumsupply capacity and the optimal pricewill not change over time

According to eorem 3 we can obtain the trajectoriesof the platformrsquos supply rate and demand rate over time asin Corollary 5

Corollary 5 During the ride service time [0 T] based onBeorem 3 the following can be obtained

(a) If T≦ t2 Slowastu (t) is equal to equation (26) Dlowastu (t) isequal to equation (27) and Slowastu (t) Dlowastu (t)

(b) If 0≦ t2≦T Slowastu (t) is

Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859

sr middot P minus sε t isin t2 T( 1113859

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Dlowastu (t) is as follows and Slowastu (t)≦Dulowast(t)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859

αeminus at minus βP + cq t isin t2 T( 1113859

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)

e proof is in AppendixCorollary 5 shows how the platformrsquos supply rate and

demand rate change based on eorem 3 At first the plat-formrsquos supply capacity increases monotonically under theeffect of the optimal price with its supply rate equal to thedemand rate to maximize profit As time passes due to theconstraint of the ceiling price the platform can acquire limitedsupply capacity from drivers However as the market ridedemand increases continuously the platform will maintainmaximum supply capacity to meet ride demand in time andcorrespondingly reduce the booking delay e platform mustallow for some booking delay cost because delayed bookings


are unavoidable which is consistent with the real scenario thatduring peak traffic periods passengers have to wait for rides

Corollary 6 During the ride service time [0 T] based onBeorem 3 and Corollary 5 the platformrsquos total transactionvolume and total profit meet

(a) If T≦ t2 the total transaction volume ψ lowastu is equal toequation (28) and the total profit π lowastu is equal toequation (29)

(b) If 0≦ t2≦T ψ lowastu is given by

ψ lowastu (t) sr(α + cq + sε minus P(sr + β))

a(β + sr)

minus1a

middot lnP middot (sr + β) minus (cq + sε)

α1113888 1113889

middots middot (rcq minus βε)

β + srminus (srP minus sε)P1113888 1113889 +(srP minus sε)T

(33)

e platformrsquos total profit π lowastu is given by

π lowastu 1113946t2



21113872 1113873dt

+ 1113946T

t2

1113946

T

t2

Slowastu (t) middot (1 minus r) middot P minus ηq

21113872 11138731113960

minus h middot 1113946t

t2

Dlowastu (τ) minus S

lowastu (τ)( 1113857dτ1113891dt

(34)


43 Optimal Pricing Strategy with Stable Market Demand(a 0) In this section we consider the scenario in which themarket demand is stable that is a scenario between a highdemand state and a low demand state with no fluctuationduring the period In practice sometimes the number ofriders who need to take a trip is almost equal to the numberof the drivers in a certain area erefore in this case thesupply and demand balance of the ride-hailing platform attime 0 remains the same and there is no opportunity cost orbooking delay cost during the ride service time [0 T] issituation can be modeled as follows

e objective function is given by

πb Pb t( 1113857 maxPb(t)

1113946T

01113946

T

0

Db Pbt( 1113857 middot Pb(t) minus Db Pbt( 1113857 middot Cb(t)1113858 1113859dt

maxPb(t)

1113946T

0Db(Pt) middot Pb(t) minus Wb(t) minus ηq

21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)

Theorem 4 During the ride service time [0 T] the marketdemand is stableBe platformrsquos optimal dynamic pricePlowastb (t) is

Plowastb (t)

αsr + β

+cq + sεsr + β

(37)

e trajectories of the platformrsquos supply rate Slowastb (t) anddemand rate D lowastb (t) over time are

Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)

e proof is in AppendixAccording to eorem 4 the optimal price does not

change dynamically with time t when the market demand isstable during the ride service time [0 T]e optimal price isalso a constant and the platform retains a certain number ofdrivers to provide transport and balance the market demandto ensure smooth operation

Based on the optimal solution of eorem 4 the plat-formrsquos total transaction volume and total profit can beobtained as in Corollary 7

Corollary 7 During the ride service time [0 T] when themarket demand is stable the platformrsquos total transactionvolume ψ lowastb is

ψ lowastb (t) sT middot(rα + rcq minus βε)

(sr + β) (40)

e platformrsquos total profit π lowastb is

π lowastb 1113946T

0Dlowastb (t) middot (1 minus r) middot P

lowastb (t) minus ηq

21113872 1113873dt (41)

e proof is in AppendixNext according to the modeling and the analysis above

we summarize in Table 2 the properties of the optimaldynamic prices in the three scenarios where uarr indicates thevariable is increasing over time⟶ indicates the variable isstable or unchanged over time and darr indicates the variable isdecreasing over time

44 Impact of Key Parameters on the Optimal PricingStrategies Some key parametersmdashsuch as the coefficient ofdemand fluctuation a service quality q and fixed com-mission rate rmdashhave an important impact on the platformrsquospricing decisions In this section we analyze these param-etersrsquo impact on the platformrsquos optimal pricing strategiesbased on the optimal solutions in Sections 41 42 and 43us we try to answer the following questions how does theplatform dynamically adjust its pricing decisions based onthe changes in the degree of market demand fluctuationHow do the platformrsquos service quality and fixed commissioncontract affect pricing decisions under different marketdemand situations We divide our analysis into three sec-tions In Section 441 we study the impact of a In Section


422 we analyze the impact of q In Section 443 we discussthe impact of r In Section 44 for the convenience ofanalysis and discussion we consider the general cases inwhich we assume cgt 0 P is a constant and t1 t2ltTFurthermore as Plowastu (t) P when t isin [t2 T] we only considert isin [0 t2] in the case of increasing market demand

441 Coefficient of Demand Fluctuation In this section weanalyze how parameter a affects the platformrsquos pricing de-cision under both scenarios of decreasing and increasingfluctuating market demand Usually the degree of marketdemand fluctuation varies in different regions for examplethere is more ride-hailing demand in certain central businessdistricts of Shanghai Of course even in the same region thedegree of market demand fluctuation at different timesmight vary such as during rainy weather erefore tomaximize the profit the platform adjusts its optimal pricingstrategies dynamically according to the different degrees ofmarket demand fluctuation For example Uberrsquos currentpricing engine generates tens of millions of price decisions atthe level of geographic units every minute across the world

Proposition 1 During the ride service time [0 T] the impactof a on the optimal pricing strategies as follows

(a) In decreasing market demand Plowastv (t) decreases whena increases Bere is a critical point alowast

(2ce(sr + β)α(1 minus r)) If a≦ alowast t1 increases when aincreases If agt alowast t1 decreases when a increases

(b) In increasing market demand during the ride servicetime [0 t2] when a decreases Plowastu (t) increases and t2decreases

According to Proposition 1 the influence of a on theplatformrsquos pricing decisions can be summed up as followsthe faster the decrease in market demand the lower theoptimal price If the coefficient of demand fluctuation a issmaller than the critical value alowast and the platformrsquos pricereaches Pmin

v the time point t1 moves backward when aincreases However if the coefficient of demand fluctuationa is higher than the critical value alowast and the platformreaches Pmin

v the time point t1 moves forward when aincreases e faster the market demand increases thehigher the optimal price when the platformrsquos price is not atits ceiling price and the time point t2 moves forward whenthe platformrsquos price reaches Pmax

u

442 Service Quality e platformrsquos service quality in-cludes passenger satisfaction with the ride service For ex-ample Chinarsquos DiDi platform usually adjusts the matchingdistance between passengers and surrounding drivers basedon the demand situation in a certain area which influencesthe time it takes for the driver to pick up the passenger andthus affects customer satisfaction Passengers are concernednot only with the price but also with the quality of theservice Passenger satisfaction is related to the platformrsquosservice quality However from the platformrsquos perspectiveservice quality is closely related to service cost which affectspricing decisions erefore it is important to discuss theservice quality and determine how the service quality affectsthe optimal pricing strategies is section analyzes howservice quality parameter q affects the platformrsquos pricingdecisions under three market situations decreasing demandincreasing demand and stable demand

Proposition 2 During the ride service time [0 T] the impactof service quality q on the optimal pricing strategies is asfollows

(a) In the three market demand situations (ie agt 0alt 0 a 0) and t isin [0 t2] when market demand in-creases the optimal price Plowast(t) increases when qincreases For each unit of increase in q Plowast(t) in-creases by c(β + sr)

(b) In decreasing market demand t1 remains unchangedwhen q decreases

(c) In increasing market demand t2 increases when qdecreases

Proposition 2 indicates that regardless of any marketdemand situation the platformrsquos optimal prices rise as servicequality increases In reality when service quality improves itis because the platform usually has invested more human andtechnical resources which inevitably increase operation costsAs a result service prices rise accordingly to maximize theplatformrsquos profit In addition when q decreases the time pointt1 is unchanged when the platformrsquos price reaches Pmin

v butthe time point t2 moves backward when the platformrsquos pricereaches Pmax

u In other words the service period t2 to T rep-resents the peak congestion period During this time theplatform might correspondingly reduce the peak congestionperiod if q is appropriately reduced to decrease part of the ridedemand and thus reduce the time spent in traffic jams

443 Fixed Commission Rate e fixed commission rate isthe percentage of the fare that the platform pays to thedrivers after they complete each ride Setting a fixed com-mission contract not only directly affects the platformrsquosrevenue from each transaction but also affects the driversrsquoincome In reality the platformrsquos fixed commission ratemight change for example Didi Chuxingrsquos fixed commis-sion rate varies from approximately 07 to 08 us tomaximize the platformrsquos profits it is necessary to adjust theoptimal pricing strategies by setting different commissionrates In this section we analyze how the fixed commissionrate parameter r affects the platformrsquos pricing decisions

Table 2 e optimal dynamic price changes with t

Market demandsituations Cases Trsquos conditions Plowast(t)

agt 0c 0 T≦Tth darr

TgtTth darr ⟶

cne 0 T≦ t1 darrTgt t1 darr uarr

alt 0P⟶ +infin uarr

P is a constant T≦ t2 uarrTgt t2 uarr ⟶

a 0 ⟶


under three market demand situations For the convenienceof analysis we assume rhmeans the platform adopts a higherfixed commission rate and rl means the platform adopts alower fixed commission rate and rhgt rl

Proposition 3 During the ride service time [0 T] rh is thehigher fixed commission rate rl is the lower fixed commissionrate and rhgt rl Be platform adjusts its optimal pricingstrategies with different r

(a) In decreasing market demand we have a critical pointt3 (βs(α + cq + sε) middot (1 minus r)2c(s + β) middot (sr + β)2)when tlt t3 Plowastv (rh t)ltPlowastv (rl t) When t t3Plowastv (rh t3) Plowastv (rl t3) When tgt t3Plowastv (rh t3)gtPlowastv (rl t3) Additionally we havet1(rh)lt t1(rl)

(b) In increasing market demand during the ride servicetime [0 t2] Plowastu (rh t)ltPlowastu (rl t) t2(rh)gt t2(rl)

(c) In stable market demand Plowastb (rh t)ltPlowastb (rl t)

According to Proposition 3 the effects of r on the plat-formrsquos pricing decisions can be summarized as follows theoptimal price is higher when the platform reduces the fixedcommission rate r is result is intuitive because the decreasein r leads to less supply capacity whereas higher prices canchange the situation However for decreasing market de-mand once demand drops to a certain level the platform hasto increase r and thus the optimal price is higher eplatform faces deteriorating market demand but once r in-creases it leads to lower revenue for each transaction thusthe platform raises the optimal price to ensure revenue Inaddition when r increases the time point t1 moves forwardwhen the platformrsquos price reaches Pmin

v but the time point t2moves backward when the platform reaches Pmax

u In otherwords as the service period t2 to T represents the peakcongestion period the platform can correspondingly reducethe peak congestion period by appropriately improving r

Tables 3 and 4 summarize the impact of the key pa-rameters on pricing decisions based on the analysis aboveand Propositions 1ndash3

5 Extended Model

From a long cycle perspective the ride demand may firstincrease and then decrease or first decrease and then increaseso the benchmark model may not be intuitive In this sectionwe construct a general model to extend our benchmarkmodelin Section 4 which can provide reference for the ride-hailingplatform enterprises when the situation in which there isincreasing or decreasing fluctuation of market demand (egfirst increasing and then decreasing) In addition we find thecharacteristics of the optimal dynamic price trajectory in thegeneral model are consistent with those of the benchmarkmodel that is the other forms of requirement functions willnot change the correctness of the conclusion

To facilitate the construction of the general model weadjusted the assumptions of the benchmark model as fol-lows 1 assumption 7 is removed which means that theinitial demand of the market (ie α(t)) in the demand

function can be non-monotonic 2 from a long cycle per-spective changes in the supply and demand state of theplatform will be difficult to control and model so we as-sumed that all ride requests can be satisfied eventually (evenif the ride bookings are delayed during peak demand) at timet that is we consider the demand rate to be fully satisfied attime t where there is no booking delay or opportunity cost(in this section the authors focus on studying the effect ofthe non-monotonic demand fluctuation on optimal pricingproblem of the ride-hailing platform without consideringthe supply-side characteristics of the platforms e plat-formrsquos opportunity cost and booking delay cost can beviewed as endogenously determined by setting the pricewhich does not affect the analysis in this section)

e ride-hailing platformrsquos objective is to find the optimaldynamic price trajectory when the market demand fluctuationis uncertain to maximize the total profit over the ride servicetime of [0 T] e objective function is given as follows

π(P t) maxP(t)

1113946T

0[D(P t) middot P(t) minus D(P t) middot C(t)]dt

maxP(t)

1113946T

0(α(t) minus βP(t) + cq) middot P(t) minus W(t) minus ηq

21113872 11138731113960 1113961dt

(42)

where α(t) changes over time is uncertain it can be non-monotonic With the support of big data and artificial in-telligence technology (eg DiDi Brain that is an artificialintelligence brain based on traffic data is established by DiDiin the cloud) the platform can predict the fluctuation of theride demand market in a certain service region based onhistorical data

Observe that in the general model the solution of theoptimal dynamic price trajectory of the ride-hailing platformcan be regarded as a functional extremal problemAccording to equation (42) the optimal control variationalmethod is used to solve the extremal value denoted asfollows

F (α(t) minus βP(t) + cq) middot P(t) minus W(t) minus ηq2

1113872 1113873 (43)

Lemma 3 During the ride service time [0 T] with uncertainmarket demand fluctuation the platformrsquos profit π(P t) is

Table 3 e impact of the key parameters on optimal prices

Optimal price auarr (agt 0) adarr (alt 0) quarrrdarr

tlt t3 t t3 tgt t3Plowastv (t) darr mdash uarr uarr ⟶ darrPlowastu (t) mdash uarr uarr uarr uarr uarrPlowastb (t) mdash mdash uarr uarr uarr uarr

Table 4 e impact of the key parameters on critical points

Critical pointauarr (agt 0)

adarr (alt 0) qdarr ruarra≦ alowast agt alowast

t1 uarr darr mdash ⟶ darrt2 mdash mdash darr uarr uarr


strictly concave in price P(t) and the optimal dynamic pricesolution Plowast(t) can maximize the platformrsquos profit π(P t)

Proof From equation (43) the second partial derivative of Fwith respect to P(t) can be calculated as follows

d2FdP2 minus 2β(1 minus r) (44)

Since βgt 0 0lt rlt 1 then by equation (44) we have(d2FdP2)lt 0 which indicates that π(P t) is a concavefunction of P(t) Obviously we can get Lemma 3

us Lemma 3 is provedIn order to maximize the profit with uncertain market

demand fluctuation according to the necessary conditionfor an extremum of the functional we have the Eulerequation that is (zFzP(t)) minus (ddt)(zFzPprime(t)) 0 Sincethere is no Pprime(t) term in equation (43) the Euler equationcan be simplified to (dFdP(t)) 0 By solving this first-order condition eorem 5 can be obtained

Theorem 5 During the ride service time [0 T] with un-certain market demand fluctuation the ride-hailing plat-formrsquos optimal dynamic price trajectory Plowast(t) is

Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)

where if we consider the minimum wage constraint (ieW(t)gt ε) and the price ceiling P the optimal dynamic pricetrajectory satisfies εrltPlowast(t)≦P

eorem 5 indicates that the optimal dynamic pricetrajectory of ride-hailing platform changes dynamically withtime t Intuitively the characteristics of the optimal dynamicprice trajectory Plowast(t) changing with time are positivelycorrelated with the fluctuation of potential market demandα(t) with time erefore the optimal dynamic price willchange dynamically with the uncertain fluctuation of marketdemand during the ride service time [0 T] For examplewhen the market demand is decreasing in a short period oftime the optimal dynamic price trajectory Plowast(t) decreasesmonotonically over time so as to alleviate the weakeningmarket demand and to stimulate market demand as much aspossible to maximize the utilization of supply capacitywhich is consistent with the benchmarkmodel of Section 41In the same way when the market demand is increasing in ashort period of time the optimal dynamic price trajectoryPlowast(t) increases monotonically over time which is consistentwith the benchmark model of Section 42 As we can see theother forms of requirement functions (eg the marketdemand rate fluctuations are not driven by the exponentialprocess) will not change the correctness of the conclusion

6 Numerical Analysis

In this section we conduct numerical studies to verify theconclusions obtained in the previous sections We also drawsome new conclusions by analyzing the impact of the keyparameters on the platformrsquos total transaction volume and

total profit under different market demand situations Forsimplicity we assume the driverrsquos minimum participation costper time unit ε 0is assumption does not affect the resultsAs in [8] and [16] for the numerical analysis we first providethe parameters (Table 5) used in the different scenarios

61 Optimal Solutions In Sections 41 42 and 43 wepresent the optimal solutions for the pricing decisions ofthree market demand situations decreasing increasing andstable e three situations are examined in this section Inaddition to the fixed parameters in Table 5 the other pa-rameters are q 30 r 07 and a 003 minus 003 0 a 003indicates that the platform faces decreasing market demanda -003 indicates that the platform faces increasing marketdemand and a 0 indicates that the platform faces stablemarket demand

First we analyze the scenario of decreasing marketdemand during the ride service time [0 T] According toeorem 1 Figure 5(a) shows the optimal price trajectorywhen the platformrsquos demand decreases first and then in-creases is result is consistent with the general case of cne 0in Corollary 1

Here the corresponding optimal dynamic price is

Plowastv (t) 50e

minus 003t+ 08t minus 625 t isin [0 30] (46)

Figure 5(b) shows the trajectories of demand rate andsupply rate under the influence of optimal prices InFigure 5(b) area N indicates that the platform reduces thesupply capacity to appropriately avoid too many driversaccepting bookings on the platform area M indicates thatthe optimal price alleviates the attenuation of market de-mand to some extent and area Q indicates that the platformdoes not achieve a balance of supply and demand underprofit maximization or that the platform is not aiming toeliminate excess supply capacity From Figure 5(c) we seethat the platformrsquos reduced profit rate with time cannotoffset the impact of weakening market demand on theplatformrsquos profit even with the price increase in the laterservice period

Next we look at the second scenariomdashincreasing marketdemand during the ride service time [0 T]is research onlysimulates the case of the platform being constrained by theceiling price (Pu(t)≦P P is a constant) Figure 6(a) showsthat the platformrsquos optimal price increases first and thenremains unchanged which is consistent with the analysis ofeorem 3 e corresponding optimal dynamic price is

Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)

Figure 6(b) shows the trajectories of demand rate andsupply rate under the influence of optimal prices InFigure 6(b) area X indicates that although the increase inprice decreases the demand it can satisfy the demand ofpeople who are more anxious for a ride service area Yindicates that the price increase greatly increases the plat-formrsquos supply capacity which can satisfy the ride demands ofmore people in that period and area Z indicates that the


platform faces the peak congestion period because of theceiling price From Figure 6(c) we see that the platformrsquosprofit rate first increases and then stabilizes

Finally we examine the scenario of stable market de-mand during the ride service time [0 T] As shown ineorem 4 and Figures 7(a) and 7(b) the trajectories of theoptimal price demand rate and supply rate are unchangedover time Here the corresponding optimal dynamic price is

Plowastb (t) 4375 t isin [0 30] (48)

In addition Figure 7(c) shows that the platformrsquos profitrate remains unchanged over time

62 Key Parameters

621 Coefficient of Demand Fluctuation (a) In addition tothe fixed parameters in Table 5 other parameters are q 30r 07 and a 0025 003 004 minus 0025 minus 003 minus 004 a

0025 003 004 indicates the different degrees of de-creasing market demand and a minus 0025 minus 003 minus 004indicates the different degrees of increasing market demandIn decreasing market demand a greater value a represents afaster decrease for market demand In increasing marketdemand a smaller value a represents a faster increase formarket demand Figure 8(a) shows the curves of optimaldynamic prices with different a values when market demand

Table 5 Tested parameter values

Parameter T α β c s η C h P

Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)

Figure 5 (a) Optimal price trajectory (b) the trajectories of demand and supply and (c) the trajectory of the profit rate in decreasingmarketdemand (agt 0)


is decreasing It is shown from Figure 8(a) that for a greatervalue a the platformrsquos price is relatively lower and the timepoint t1 will move backward since alt alowast 0043 in thisscenario which is consistent with Proposition 1 (a)Figure 8(b) shows the curves of optimal dynamic prices withdifferent a values when market demand increases InProposition 1 (b) it is shown from Figure 8(b) that for asmaller value a the platformrsquos price is relatively higher andthe time point t2 moves forward which means the platformwill face a longer peak congestion period

From Table 6 we observe the following

(i) With decreasing market demand in the rideservice time [0 T] the platformrsquos total transactionvolume and total profit both decrease when aincreases In other words a faster decrease inmarket demand results in lower transaction vol-ume and profit

(ii) With increasing market demand in the ride servicetime [0 T] the platformrsquos total transaction volume

and total profit both increase when a decreases Inother words a faster increase in market demandresults in relatively higher transaction volume andprofit

622 Service Quality (q) In addition to the fixed param-eters in Table 5 other parameters are q 10 30 50r 07 and a 003 minus 003 0 q 10 30 50 indicatesthe platformmay adopt different service qualities Figure 9shows that the optimal prices increase with an increase inq regardless of market demand situations when theplatform is not constrained by the ceiling price which isconsistent with Proposition 2 (a) In addition by ob-serving Figure 9(a) the time point t1 is unchanged with adifferent q value which is consistent with Proposition 3(b) As depicted in Proposition 2 (c) Figure 9(b) showsthat the time point t2 will move backward when q de-creases and t2 is pushed backward meaning the peakcongestion period is reduced

t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)

Peak congestion period

t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)

Figure 6 (a) Optimal price trajectory (b) the trajectories of demand and supply and (c) the trajectory of the profit rate in increasing marketdemand (alt 0)



(i) Regardless of market demand the platformrsquos totaltransaction volume increases when q increases

(ii) For the same service quality the total transactionvolume under increasing market demand is higherthan under stable market demand and the volumeunder stable market demand is higher than underdecreasing market demand at isψ lowastu (q)gtψ lowastb (q)gtψ lowastv (q)

(iii) Regardless of market demand the platformrsquos totalprofit increases first and then decreases when qincreases e total profit is the largest when q 40under decreasing market demand that isπ lowastv (qlowastv 40)≧ π lowastv (q) e total profit is the largestwhen q 30 under stable market demand that isπ lowastb (qlowastb 30)≧ π lowastb (q) e total profit is the largestwhen q 20 under increasing market demand thatis π lowastu (qlowastu 20)≧ π lowastu (q) Furthermore we find the

service quality of the total profit maximized underdecreasing market demand is higher than understable market demand and the service quality understable market demand is higher than under in-creasing market demand at isqlowastv 40ge qlowastb 30gt qlowastu 20

(iv) For the same service quality the total profit underincreasing market demand is higher than understable market demand and the profit under stablemarket demand is higher than under decreasingmarket demand at is π lowastu (q)gt π lowastb (q)gt π lowastv (q)

623 Fixed Commission Rate (r) In addition to the fixedparameters in Table 5 other parameters are q 30 r 0507 and a 003 minus 003 0 Hu and Zhou [4] obtained theoptimal fixed commission rate of 06063 We set r 05 07to compare the scenarios when the platform adopts differentcommission rates As shown in Proposition 3 (a) for

5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)

Figure 7 (a) Optimal price trajectory (b) the trajectories of demand and supply and (c) the trajectory of the profit rate in decreasingmarketdemand (a 0)


a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)

a = ndash0025a = ndash003a = ndash004

t2 (a = ndash004)t2 (a = ndash003)t2 (a = ndash0025)

2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)

Figure 8 Optimal dynamic prices with different a values under different market situations (a) agt 0 (b) alt 0

Table 6e impact of a values on the platformrsquos total transaction volume and total profit under decreasing and increasing market demand

a ψ lowastv (times103) π lowastv (times104) a ψ lowastu (times103) π lowastu (times104)

0015 474 598 minus 0015 877 1305002 442 527 minus 002 993 17710025 413 467 minus 0025 1064 2051003 387 417 minus 003 1110 22360035 362 374 minus 0035 1144 2366004 340 337 minus 004 1169 2461

t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued


decreasing market demand Figure 10(a) shows that Plowastv (07t)ltPlowastv (05 t) when tlt t3 Plowastv (07 t)Plowastv (05 t) when t t3and Plowastv (07 t)gtPlowastv (05 t) when tgt t3 and the time point t1moves forward when rmoves from 07 to 05 the time pointt1 of r 05 has actually moved out of T in our simulationsFor increasing market demand as depicted in Proposition 3(b) Figure 10(b) shows that Plowastu (07 t)ltPlowastu (05 t) and thetime point t2 is pushed backward when r from 05 to 07 thisimplies the peak congestion period is reduced For stablemarket demand Figure 10(c) shows that Plowastb (07 t)ltP

lowastb (05

t) which is consistent with Proposition 3 (c)From Table 8 we observe the following

(i) With decreasing market demand the platformrsquostotal transaction volume increases first and thendecreases when r increases When the total trans-action volume is maximized r is 065 Howeverunder increasing market demand and stable marketdemand the total transaction volume increaseswhen r increases

(ii) For the same r the total transaction volume underincreasing market demand is higher than understable market demand and the volume under stable

market demand is higher than under decreasingmarket demand at is ψ lowastu (r)gtψ lowastb (r)gtψ lowastv (r)

(iii) Regardless of market demand the platformrsquos totalprofit increases first and then decreases when rincreases e total profit is the largest when r 035under decreasing market demand that isπ lowastv (rlowastv 035)≧ π lowastv (r) e total profit is the largest

when r 025 under stable market demand that isπ lowastb (rlowastb 025)≧ π lowastb (r) e total profit is the largest

when r 04 under increasing market demand thatis π lowastu (rlowastu 04)≧ π lowastu (r) Moreover we find thatwhen the total profit is maximized r varies underdifferent market demand situations at isrlowastv 035ne rlowastb 025ne rlowastu 04

(iv) For the same r the total profit under increasingmarket demand is higher than under stable demandand the profit under stable market demand is higherthan under decreasing market demand at isπ lowastu (r)gt π lowastb (r)gt π lowastv (r)

From observation (iii) we note that the platform ob-viously cannot set a low rate of return for its own profitmaximization in realitymdashthat is rlowastb 025 rlowastv 035 or

q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)

Figure 9 Optimal dynamic prices with different q under different market situations (a) agt 0 (b) alt 0 (c) a 0

Table 7e impact of q on the platformrsquos total transaction volume and total profit under decreasing increasing and stable market demand

q ψ lowastv (times103) ψ lowastb (times10

3) ψ lowastu (times103) π lowastv (times10

4) π lowastb (times104) π lowastu (times10

4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122


rlowastu 04mdashbecause the platform must consider the basicincome of drivers and motivate their participation In ourstudy because we do not focus on the platformrsquos incentives

for drivers we find that the optimal r is relatively smallHowever this has no impact on our analysis of the rela-tionship between r and the platformrsquos profit

t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)

Figure 10 Optimal dynamic prices with different r values under different market situations (a) agt 0 (b) alt 0 (c) a 0

Table 8e impact of r on the platformrsquos total transaction volume and total profit under decreasing increasing and stable market demand

r ψ lowastv (times103) ψ lowastb (times10



4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894


7 Conclusions Managerial Insights andFuture Research

is paper examines the dynamic pricing problems of ride-hailing platforms under the imbalance between the supplyand the demand caused by the market demand fluctuationUnlike existing demand models which usually consider theeffect of price we combine the effects of price and time ondemand We model demand as an exponential function oftime introducing the coefficient of demand fluctuation torepresent three different market demand situations de-creasing demand increasing demand and stable demandWe use optimal control theory to construct our modelsbased on the state change in supply and demand with theaim of maximizing the platformrsquos total profit in a finiteservice time

71 Concluding Remarks Due to the fluctuations of pas-senger demand ride-hailing platforms often face undesir-able situations of insufficient supply or insufficient demandDetermining how tomatch passenger supply and demand byoptimally setting platform prices is an important issue Inthis study we contribute to the literature on demandfunction and supply function (eg Cachon et al [8] Hu andZhou [4] Liu et al [16]) In considering the scenarios of bothdecreasing and increasing passenger demand we investigatethe dynamic pricing policies of a ride-hailing platform andobtain the optimal pricing strategies under different marketdemand situations Some key parameters such as the co-efficient of demand fluctuation service quality and fixedcommission rate are examined so we can better understandhow they affect pricing decisions e main conclusions areas follows

(1) e optimal prices vary dynamically over time indifferent cases under different market demand sit-uations For decreasing market demand with thepassage of time when we do not consider oppor-tunity cost the optimal price decreases first and thenremains unchanged when we consider the oppor-tunity cost the optimal price decreases first and thenincreases For increasing market demand with thepassage of time when we do not consider the ceilingprice constraint the optimal price increases whenwe consider the ceiling price constraint the optimalprice increases first and then remains unchangedFor stable market demand the optimal price doesnot change over time

(2) e optimal prices dynamically affect the platformrsquossupply capacity and demand For decreasing marketdemand on the one hand the platform reduces theexcess supply capacity through the optimal price toavoid too many drivers accepting bookings on theother hand to some extent the optimal price alsoalleviates the weakening market demand For in-creasing market demand the platform greatly im-proves the supply capacity through the optimal pricewhich can satisfy more people in time the increase in

the optimal price decreases the demand but satisfiespassengers who are more anxious for the ride serviceFor stable market demand the platformrsquos supplycapacity and demand do not change over time underthe optimal price

(3) e platform should adjust its pricing decisionsaccording to different key parameters (a) In terms ofthe impact of the coefficient of demand fluctuationfor decreasing market demand when market de-mand decreases faster the optimal price is relativelylower When the coefficient of demand fluctuation issmaller than the critical value the time point of thelowest price becomes larger However when thecoefficient is higher than the critical value the timepoint becomes smaller For decreasing market de-mand when market demand increases faster theoptimal price is relatively higher and thus the timepoint of the ceiling price is smaller (b) In terms ofthe impact of service quality regardless of marketdemand the optimal price should be raised whenservice quality is improved When service qualitydeteriorates the time point of the lowest price isunchanged but that of the ceiling price is larger (c)In terms of the impact of the fixed commission rateregardless of market demand the optimal priceshould be raised when the platform reduces the fixedcommission rate When the platform adopts a higherfixed commission rate the time point of the lowestprice is smaller but that of the ceiling price is larger

(4) e numerical results show that the key parametershave some effect on the platformrsquos total transactionvolume and total profit (a) In terms of the impact ofthe coefficient of demand fluctuation for decreasingmarket demand when market demand decreasesfaster the total transaction volume and total profitare smaller For increasing market demand whenmarket demand increases faster the total transactionvolume and total profit are larger (b) In terms of theimpact of service quality regardless of market de-mand the total transaction volume is larger whenservice quality is improved and the total profit isconcave in service quality e optimal servicequality when the total profit is largest under differentmarket demand scenarios is different For the sameservice quality the total transaction volume and totalprofit under increasing market demand are higherthan under stable market demand and the volumeand profit under stable market demand are higherthan under decreasing market demand (c) In termsof the impact of the fixed commission rate regardlessof market demand the total transaction volume islarger when the platform adopts a higher fixedcommission rate and the total profit is concave in thefixed commission rate e optimal profit-maxi-mizing commission rate depends on the marketdemand (scenario) However when the platformadopts the same fixed commission rate under dif-ferent market demand scenarios the total


transaction volume and total profit under increasingmarket demand are higher than under stable marketdemand and the volume and profit under stablemarket demand are higher than under decreasingmarket demand

72Managerial Insights Based on the above conclusions weoffer several important managerial insights

(1) is paper can guide ride-hailing platformmanagers inpricing decisions when the platform faces differentmarket demand situations decreasing demand in-creasing demand and stable demand For example DidiChuxing is considering dynamically adjusting pricesbased on supply and demand within certain serviceregions Even when dynamic pricing is limited ourresearch can be a reference for a platformrsquos static pricingdecisions for example in regard to the range of staticprices based on different regionsrsquo market demand

(2) Ride-hailing platforms should set different pricingdecisions according to the degree of market demandfluctuation in different service regions For exampleDidi Chuxing tends to have more pronounced pricechanges in some of Shanghairsquos CBD areas andsubway stations

(3) e fixed commission rate design can affect theplatformrsquos pricing decisions e platform managersshould have a flexible commission strategy accordingto the different service regions and different marketdemands to thus reduce costs while providing betterservice and maximizing revenue

(4) All platform companies should have social respon-sibility and a ride-hailing platform is no exceptionSome of the ride-hailing platformrsquos social respon-sibilities are to reduce traffic congestion make travelbetter and help users improve the travel efficiency(eg Didi Chuxing and Meituan Zhong et al [59])From this perspective the platform can changeservice quality or improve its fixed commission rateto achieve the purpose of improving traffic condi-tions when market demand is increasing For ex-ample the platform reduces the service quality bychanging the maximum matching distance betweenthe passenger and the driver which can reducecongestion passengers appropriately and improvethe travel efficiency of the remaining passengersthrough long-distance driver scheduling

(5) As some new ride-hailing platforms enter themarket they tend to pursue the maximization oftransaction volume in the early stages to compete formarket share For example the Meituan ride-hailingplatform aiming to maximize its transaction vol-ume competes with Didi Chuxing for market shareOur study might be of interest to these new plat-forms Managers can improve the service qualityandor the commission rate to maximize thetransaction volume under different market demandsituations

73FutureResearch Our work has some limitations and canbe extended in several ways First one might examine morerealistic scenarios with our models For example it would bereasonable to consider the driversrsquo loss avoidance behaviorunder decreasing market demand or to consider the pas-sengersrsquo booking cancellation behavior under increasingmarket demand Also it would be more applicable in sce-narios considering the supply-demand uncertainty andother initial states Second for simplicity we ignore thespatial dimension that is important to a platformrsquos pricingdecisions Future work can use the distance factor to in-vestigate a platformrsquos pricing strategies under market de-mand fluctuation ird future research can analyze theoptimal static prices (including surge price) under differentmarket demand situations when dynamic pricing is con-strained It would be interesting to compare the optimalstatic prices to the optimal dynamic prices Fourth futuremodels can introduce the driversrsquo incentive constraints Forexample it would be valuable to consider the driversrsquoparticipation constraint and incentive compatible constraintthrough the lens of principal-agent theory Fifth as morethan one platform compete for both demand and supply inreality future research can examine how two or moreplatforms compete under the different market demandscenarios of this study

Appendix

Proof of Lemma 1 e second partial derivative of Hv withrespect to Pv(t) can be calculated as

z2Hv

zP2v

minus 2β middot (1 minus r) (A1)

Since βgt 0 0lt rlt 1 and then by (A1) we havez2HvzP2

v lt 0 which indicates that πv(Pv t) is a concavefunction of Pv(t) Obviously we can get Lemma 1

Proof of Beorem 1 Since the supply and demand balance ofthe ride-hailing platform at time 0 then by (1) and (2) weknow s(W(0) minus τ) α minus βP(0) + cq

We obtain

P(0) α + cq + sε

sr + β (A2)

By equation (7) the optimal price at time 0 can beobtained

Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)

Next by (A2) and (A3) we can get

v(T) cT2(sr + β)2

2β(1 minus r)+α(sr minus β) 1 minus eminus aT( 1113857

2βa+αT

2minus

srαT

2β

(A4)


en by substituting v(T) into (12) and (13) respec-tively we can get equations (14) and (15) next substitutingequation (14) into (1) and (2) respectively equations (16)and (17) can be obtained

us eorem 1 is proved

Proof of remark in Beorem 1 By equations (16) and (17) ofeorem 1 the excess supply capacity of the platform at timet can be obtained as follows

vlowast(t) 1113946

t

0Slowastv (t) minus D

lowastv (t)( 1113857dt (A5)

e first derivative of vlowast(t) with respect to t can becalculated as

dvlowast(t)

dt Slowastv (t) minus D

lowastv (t) (A6)

Since vlowast(t) 0 in order to satisfy vlowast(t)≧ 0 we only need(dvlowast(t)dt)ge 0 and we use f(t) (dvlowast(t)dt) ge 0 byequations (16) and (17) of eorem 1 f(t) can be obtainedas follows

f(t) α(sr minus β)

2βmiddot e

minus at+

c(sr + β)2

β(1 minus r)middot t minus

α(sr minus β)

2β (A7)

e first derivative of f(t) with respect to t can becalculated as

fprime(t) minusaα(sr minus β)

2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)

Since f(0) 0 in order to satisfy f(t) ≧ 0 we only needfprime(t)≧ 0 When β≧ sr obviously we know fprime(t)≧ 0 by(A8) us the constraint v(t)≧ 0 in (9) can be satisfied

If βlt sr tge minus (1a)ln(2c(sr + β)2aα(sr minus β) middot (1 minus r))

can be obtained by making fprime(t)≧0 if 2c(sr +β)2geaα(sr minus β) middot (1 minus r) then minus (1a)ln(2c(sr +β)2aα(sr minus β) middot

(1 minus r))le0 furthermore tge0ge minus (1a)ln(2c(sr +β)2aα(sr minus β) middot (1 minus r)) so vlowast(t)≧0 can be satisfied during theride service time [0 T] us the constraint v(t)≧0 in (9)can be satisfied

us remark of eorem 1 is proved

Proof of Corollary 1 By equation (13) ofeorem 1 the firstderivative and second derivative of Plowastv (t) with respect to tcan be calculated respectively as follows

dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)

Since αgt 0 βgt 0 by (A10) we know dP2lowastv(t)dt2

Plowastv (t) is strictly convex in time tus Corollary 1 (a) is provedFor Corollary 1 (b) we assume c 0 since agt 0 by (A9)

we know limt⟶+infin

(dPlowastv (t)dt) limt⟶+infin

((minus aα2β) middot eminus at) 0

there is a threshold Tth limTth⟶ +infin

(dPlowastv (t)dt)

limTth⟶ +infin

(minus (aα2β) middot eminus at) 0 When 0≦Tth≦T during the

ride service time [0 Tth] since (dPlowastv (t)dt)lt 0 Plowastv (t) de-creases monotonously however during the ride service time[Tth T] since (dPlowastv (t)dt) 0 Plowastv (t) stabilizes or remainsunchangedWhen TltTth during the ride service time [0 T]since (dPlowastv (t)dt)lt 0 Plowastv (t) decreases monotonously

us Corollary 1 (b) is provedFor Corollary 1 (c) we assume cne0 t1 is obtained from

(dPlowastv (t)dt) 0 ie t1 minus (1a)ln(2c(sr + β)aα(1 minus r))we have known Plowastv (t) is strictly convex in time t from Cor-ollary 1 (a) When 0≦ t1≦T there exists an extreme point t1and Pmin

v Plowastv (t1) in the ride service time [0 T] so Plowastv (t)

decreases first and then increases When Tlt t1 (dPlowastv (t)dt)lt0 Plowastv (t) decreases monotonously in service time [0 T]

us Corollary 1 (c) is proved

Proof of Corollary 2 By equations (15)ndash(17) of eorem 1we substitute Slowastv (t) and Dlowastv (t) into (3) and ψ lowastv can becalculated en we substitute Plowastv (t) Slowastv (t) and Dlowastv (t) into(8) and π lowastv can be calculated

us Corollary 2 is proved

Proof of Lemma 2 e second partial derivative of Hu withrespect to Pu(t) can be calculated as

z2Hu

zP2u

2sr middot (1 minus r) (A11)

Since sgt 0 0lt rlt 1 by (A11) we have z2HuzP2u gt 0

which indicates that πu(Pu t) is a convex function of Pu(t)Obviously we can get Lemma 2

Proof of Beorem 2 Because of the constraints u(t)≧ 0 andu(0) 0 in (20) we only need uprime(t)ge 0 By (1) and (2)(A12) can be obtained by making D(P t)≧ S(W t) asfollows

P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)

By (A12) we know Pmaxu as follows

Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)

Since πu(Pu t) is strictly convex in Pu(t) from Lemma 2Pmax

u (t) is the optimal price point ie Pu(t) Pmaxu (t)

Next substituting Pu(t) into (1) and (2) respectivelyequations (26) and (27) can be obtained


Proof of Corollary 3 By equation (25) ofeorem 2 the firstderivative and second derivative of Pu(t) with respect to tcan be calculated respectively as follows

dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)


Since alt 0 αgt 0 βgt 0 sgt 0 0lt rlt 1 by (A14) and(A15) we know (dPlowastu (t)dt)gt 0 and (dP2u

lowast (t)dt2)gt 0obviously Corollary 3 (a) can be proved

By equations (26) and (27) of eorem 2 we knowSlowastu (t) D lowastu (t) Also by u(0) 0 and (21) it will be easy toget u(t) 0 so order delay cost hmiddotu(t) 0 us Corollary 3(b) is proved

Proof of Corollary 4 By equations (25)ndash(27) of eorem 2we substitute S lowastu (t) and Dlowastu (t) into (3) and ψ lowastu can becalculated en we substitute Plowastu (t) Slowastu (t) and Dlowastu (t) into(22) and π lowastu can be calculated


Proof of Beorem 3 Since Pu(t)≦P by equation (22) ofeorem 2 we can gettle minus (1a)ln(P middot (sr + β) minus (cq + sε)α) and t2 can be ob-tained as follows

t2 minus1aln

P middot (sr + β) minus (cq + sε)α

1113888 1113889 (A16)

If T≦ t2 since the platformrsquos service price is not con-strained by P in the ride service time [0 T] Pu(t) is equal toequation (22) of eorem 2 if 0≦ t2≦T when t isin [0 t2Pu(t)

is equal to equation (22) ofeorem 2 however when t isin (t2T] the optimal price will increase to the ceiling price P andthe platform will maintain P so that the platform can use themaximum supply capacity to provide transport


Proof of Corollary 5 If T≦ t2 we substitute Pu(t) into (1)and (2) respectively based on eorem 3 and Corollary 5(a) can be obtained

If 0≦ t2≦T similarly we substitute Pu(t) into (1) and(2) respectively based on eorem 3 equations (31) and(32) can be obtained Also since Slowastu (t) Dlowastu (t) when tisin[0t2] and Slowastu (t) Slowastu (t2) Dlowastu (t2)ltDlowastu (t) when t isin (t2 T]Slowastu (t)≦Dlowastu (t) if 0≦ t2≦T

us Corollary 5 (b) is proved

Proof of Corollary 6 If T≦ t2 we substitute Slowastu (t) and Dlowastu (t)

into (3) based on Corollary 5 ψ lowastu can be calculated and wesubstitute Plowastu (t) Slowastu (t) and Dlowastu (t) into (22) based oneorem 3 and Corollary 5 π lowastu can be calculated

us Corollary 6 (a) is provedIf 0≦ t2≦T similarly Corollary 6 (b) can be

obtained

Proof of Beorem 4 Because of the supply and demandbalance of the ride-hailing platform at time 0 then by (1) and(2) we know P(0) P(0) (α + cq minus sεsr + β) ie (A2)Also the supply and demand balance of the ride-hailingplatform at time 0 will remain as the market demand isstable otherwise the constraint v(t) u(t) 0 of (29) willnot be satisfied Obviously Plowastb (t) P(0) Next we sub-stitute Plowastb (t) into (1) and (2) respectively and equations(31) and (32) can be obtained


Proof of Corollary 7 By equations (37)ndash(39) of eorem 4we substitute Slowastb (t) and Dlowastb (t) into (3) and ψ lowastb can becalculated en we substitute Plowastb (t) Slowastb (t) and Dlowastb (t) into(35) and π lowastb can be calculated


Proof of Proposition 1 First for the market demand de-creases (ie agt 0) by calculating dPlowastv (t)da we find that

dPlowastv (t)

da minus

α middot t

2βmiddot e

minus at le 0a (A17)

For the critical point t1 we know Pminv Plowastv (t) by

calculating dt1da we getdt1

da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)

e critical point alowast can be obtained from dt1da 0that is

alowast

2ce(sr + β)

α(1 minus r) (A19)

Combining equation (A18) with equation (A19) it willbe easy to get the following if a≦ alowast dt1dage 0 if agt alowastdt1dalt 0

en for the market demand increases (ie alt 0) whent isin [0 t2] by calculating dPlowastu (t)da we find that

dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)

For the critical point t2 we know Pmaxu Plowastu (t2) P by

calculating dt2da we get

dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)

Obviously according to the derivation above we can getProposition 1

us Proposition 1 is proved

Proof of Proposition 2 By calculating dPlowastv (t)dq anddPlowastu (t)dq and when t isin [0 t2] and dPlowastb (t)dq we find that

dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)

For the critical point t1 when market demand decreasesobviously by calculating dt1dq we get

dt1

dq 0 (A23)

For the critical point t2 when market demand increaseswe know Pmax

u Plowastu (t2) P by calculating dt2dq we get

dt2

dq1a

middotαc

P(sr + β) minus (cq + sε)lt 0 (A24)




Proof of Proposition 3 First for the market demand de-creases (ie agt 0) by calculating dPlowastv (t)dr we get

dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)

ere exists a critical point t3 which can be obtainedfrom dPlowastv (t)dr 0 that is

t3 βs(α + cq + sε) middot (1 minus r)2

c(s + β) middot (sr + β)2 (A26)

For the critical point t1 we know Pminv Plowastv (t1) by

calculating dt1dr we getdt1

dr minus

1a

middots(1 minus r) + r(sr + β)

(sr + β) middot (1 minus r)lt 0 (A27)

Second for the market demand increases (ie alt 0)when t isin [0 t2] by calculating dPlowastu (t)dr we find that

dPlowastu (t)

dr minus

s middot eminus at + cq + sε( 1113857

(β + sr)2lt 0 (A28)


calculating dt2dr we get

dt2

dr minus

1a

middotαsP

P(sr + β) minus (cq + sε)gt 0 (A29)

Finally we analyze the relation of the optimal price withrespect to r when the market demand is stable (ie a 0) Bycalculating dPlowastb (t)dr we find that

dPlowastb (t)

dr minus

s middot (α + cq + sε)(β + sr)2

lt 0 (A30)



Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was partially supported by the NationalNatural Science Foundation of China (Grant nos 71572033and 71832001)

References

[1] H Ma F Fang and D C Parkes ldquoSpatio-temporal pricing forridesharing platformsrdquo 2018 httpsarxivorgabs180104015

[2] Uber 10 billion trips 2018 httpswwwubercomnewsroom10-billion

[3] MarketWatch Ride-Hailing Industry Expected to GrowEightfold to $285 Billion By2030 MarketWatch New YorkNY USA 2017 httpwwwmarketwatchcomstoryride-hailing-industry-expected-to-grow-eightfold-to-285-billion-by-2030-2017-05-24

[4] M Hu and Y Zhou Price Wage and Fixed Commission inOn-Demand matching Working Paper University of TorontoToronto Canada 2019 httpsssrncomabstract2949513

[5] J Cramer and A B Krueger ldquoDisruptive change in the taxibusiness the case of Uberrdquo American Economic Reviewvol 106 no 5 pp 177ndash182 2016

[6] Xinhua News Agency How Does ldquoRide-Hailingrdquo Get Betterand Better Xinhua News Agency Beijing China 2019

[7] C W Yan H Zhu N Korolko and D Woodard ldquoDynamicpricing and matching in ride-hailing platformsrdquo Naval Re-search Logistics vol 66 no 8 pp 1ndash20 2019 httpsssrncomabstract3258234

[8] G P Cachon K M Daniels and R Lobel ldquoe role of surgepricing on a service platform with self-scheduling capacityrdquoManufacturing amp Service Operations Management vol 19no 3 pp 368ndash384 2017

[9] J Hall K Cory andN ChrisBeEffects of Uberrsquos Surge PricingA Case Study e University of Chicago Booth School ofBusiness Chicago IL USA 2015 httpswwwvaluewalkcom201509uber-surge-pricingutm_sourcemailchimpamputm_mediumemailamputm_campaignEMAIL_DAILYamputm_contentquick_link

[10] P Afeche L Zhe and M Costis ldquoRide-hailing networks withstrategic drivers the impact of platform control capabilitieson performancerdquo University of Toronto Toronto Canada2018 httpsssrncomabstract3120544

[11] G Feng G Kong and Z WangWe are on the Way Analysisof On-Demand Ride-Hailing Systems University of Minne-sota Minneapolis Minnesota 2018 httpsssrncomabstract2960991

[12] M Hu and Y Zhou Dynamic Type Matching Rotman Schoolof Management Toronto Canada 2019 httpsssrncomabstract2592622

[13] L Sun R H Teunter M Z Babai and G Hua ldquoOptimalpricing for ride-sourcing platformsrdquo European Journal ofOperational Research vol 278 no 3 pp 783ndash795 2019

[14] K Bimpikis O Candogan and D Saban ldquoSpatial pricing inride-sharing networksrdquo Operations Research vol 67 no 3pp 744ndash769 2019

[15] R Desiraju and S Moorthy ldquoManaging a distribution channelunder asymmetric information with performance require-mentsrdquo Management Science vol 43 no 12 pp 1628ndash16441997

[16] W Liu X Yan W Wei and D Xie ldquoPricing decisions forservice platform with providerrsquos threshold participatingquantity value-added service and matching abilityrdquo Trans-portation Research Part E Logistics and Transportation Re-view vol 122 pp 410ndash432 2019

[17] L Lu J Zhang and W Tang ldquoOptimal dynamic pricing andreplenishment policy for perishable items with inventory-level-dependent demandrdquo International Journal of SystemsScience vol 47 no 6 pp 1480ndash1494 2016

[18] A Herbon and E Khmelnitsky ldquoOptimal dynamic pricingand ordering of a perishable product under additive effects ofprice and time on demandrdquo European Journal of OperationalResearch vol 260 no 2 pp 546ndash556 2017

[19] L Feng ldquoDynamic pricing quality investment and replen-ishment model for perishable itemsrdquo International


Transactions in Operational Research vol 26 no 4pp 1558ndash1575 2019

[20] J-C Rochet and J Tirole ldquoPlatform competition in two-sidedmarketsrdquo Journal of the European Economic Associationvol 1 no 4 pp 990ndash1029 2003

[21] J Rochet and J Tirole ldquoTwo-sided markets an overviewrdquo inProceedings of the IDEI Working Papers pp 1ndash44 January2004 httpwebmitedu14271wwwrochet_tirolepdf

[22] M Armstrong ldquoCompetition in two-sided marketsrdquo BeRAND Journal of Economics vol 37 no 3 pp 668ndash691 2005

[23] L Waverman ldquoTwo-sided telecom markets and the unin-tended consequences of business strategyrdquo Competition PolicyInternational vol 3 no 1 2007

[24] G G Parker and M W van Alstyne ldquoTwo-sided networkeffects a theory of information product designrdquoManagementScience vol 51 no 10 pp 1494ndash1504 2005

[25] A Gaudeul and B Jullien ldquoE-commerce two-sided marketsand info-mediationrdquo in Internet and Digital EconomicsPrinciples Methods and Applications pp 268ndash290 Cam-bridge University Press New York NY USA 2007

[26] S P Anderson and J J Gabszewicz ldquoChapter 18 the mediaand advertising a tale of two-sided marketsrdquoHandbook of theEconomics of Art and Culture vol 1 pp 567ndash614 2006

[27] H J Kind and F Stahler ldquoMarket shares in two-sided mediaindustriesrdquo Journal of Institutional and Beoretical Eco-nomics vol 166 no 2 pp 205ndash211 2010

[28] M Furuhata M Dessouky F Ordontildeez M-E BrunetXWang and S Koenig ldquoRidesharing the state-of-the-art andfuture directionsrdquo Transportation Research Part B Method-ological vol 57 pp 28ndash46 2013

[29] X Wang F He H Yang and H Oliver Gao ldquoPricingstrategies for a taxi-hailing platformrdquo Transportation ResearchPart E Logistics and Transportation Review vol 93pp 212ndash231 2016

[30] S Djavadian and J Y J Chow ldquoAn agent-based day-to-dayadjustment process for modeling rsquoMobility as a Servicersquo with atwo-sided flexible transport marketrdquo Transportation ResearchPart B Methodological vol 104 pp 36ndash57 2017

[31] B Caillaud and B Jullien ldquoChicken amp egg competitionamong intermediation service providersrdquo Be RAND Journalof Economics vol 34 no 2 pp 309ndash328 2003

[32] R Kolisch and D Zatta ldquoImplementation of revenue man-agement in the process industry of north America andEuroperdquo Journal of Revenue and Pricing Management vol 11no 2 pp 191ndash209 2012

[33] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a review with extensionsrdquo Operations Researchvol 47 no 2 pp 183ndash194 1999

[34] P Cao N Zhao and J Wu ldquoDynamic pricing with Bayesiandemand learning and reference price effectrdquo EuropeanJournal of Operational Research vol 279 no 2 pp 540ndash5562019

[35] G Gallego and G van Ryzin ldquoOptimal dynamic pricing ofinventories with stochastic demand over finite horizonsrdquoManagement Science vol 40 no 8 pp 999ndash1020 1994

[36] G Gallego and G van Ryzin ldquoA multiproduct dynamicpricing problem and its applications to network yield man-agementrdquoOperations Research vol 45 no 1 pp 24ndash41 1997

[37] S A Smith and D D Achabal ldquoClearance pricing and in-ventory policies for retail chainsrdquo Management Sciencevol 44 no 3 pp 285ndash300 1998

[38] D Besanko andW LWinston ldquoOptimal price skimming by amonopolist facing rational consumersrdquoManagement Sciencevol 36 no 5 pp 555ndash567 1990

[39] C K Anderson and J G Wilson ldquoWait or buy the strategicconsumer pricing and profit implicationsrdquo Journal of theOperational Research Society vol 54 no 3 pp 299ndash306 2003

[40] Y Aviv and A Pazgal ldquoOptimal pricing of seasonal productsin the presence of forward-looking consumersrdquoManufacturing amp Service Operations Management vol 10no 3 pp 339ndash359 2008

[41] Y Levin J Mcgill and M Nediak ldquoOptimal dynamic pricingof perishable items by a monopolist facing strategic con-sumersrdquo Production and Operations Management vol 19no 1 pp 40ndash60 2010

[42] Y Hu J Li and L Ran ldquoDynamic pricing for airline revenuemanagement under passenger mental accountingrdquo Mathe-matical Problems in Engineering vol 2015 Article ID 8364348 pages 2015

[43] R R Varella J Frazatildeo and A V M Oliveira ldquoDynamicpricing and market segmentation responses to low-costcarrier entryrdquo Transportation Research Part E Logistics andTransportation Review vol 98 pp 151ndash170 2017

[44] A Ajorlou A Jadbabaie and A Kakhbod ldquoDynamic pricingin social networks the word-of-mouth effectrdquo ManagementScience vol 64 no 2 pp 971ndash979 2018

[45] M Chen and Z-L Chen ldquoUncertain about your travel planLock it and decide later dynamic pricing with a fare-lockoptionrdquo Transportation Research Part E Logistics andTransportation Review vol 125 pp 1ndash26 2019

[46] J Chen M Dong Y Rong and L Yang ldquoDynamic pricing fordeteriorating products with menu costrdquoOmega vol 75 no 3pp 13ndash26 2018

[47] K Sato ldquoPrice trends and dynamic pricing in perishableproduct market consisting of superior and inferior firmsrdquoEuropean Journal of Operational Research vol 274 no 1pp 214ndash226 2019

[48] L Zha Y Yin and Y Du ldquoSurge pricing and labor supply inthe ride-sourcing marketrdquo Transportation Research Procediavol 23 pp 2ndash21 2017

[49] H Guda and U Subramanian ldquoYour Uber is arrivingmanaging on-demand workers through surge pricing forecastcommunication and worker incentivesrdquo Management Sci-ence vol 65 no 5 pp 1949ndash2443 2019

[50] T Wu M Zhang X Tian S Wang and G Hua ldquoSpatialdifferentiation and network externality in pricing mechanismof online car hailing platformrdquo International Journal ofProduction Economics vol 219 no 1 pp 275ndash283 2020

[51] J Bai K C So and C S Tang ldquoCoordinating supply anddemand on an on-demand service platform with impatientcustomersrdquo Manufacturing amp Service Operations Manage-ment vol 21 no 3 pp 479ndash711 2019

[52] T A Taylor ldquoOn-demand service platformsrdquo Manufacturingamp Service Operations Management vol 20 no 4 pp 704ndash7202017

[53] K S Moorthy ldquoProduct and price competition in a duopolyrdquoMarketing Science vol 7 no 2 pp 141ndash168 1998

[54] M Motta ldquoEndogenous quality choice price vs quantitycompetitionrdquo Be Journal of Industrial Economics vol 41no 2 pp 113ndash131 1993

[55] A Ha X Long and J Nasiry ldquoQuality in supply chain en-croachmentrdquo Manufacturing amp Service Operations Manage-ment vol 18 no 2 pp 280ndash298 2016

[56] R Maihami and I N Kamalabadi ldquoJoint pricing and in-ventory control for non-instantaneous deteriorating itemswith partial backlogging and time and price dependent de-mandrdquo International Journal of Production Economicsvol 136 no 1 pp 116ndash122 2012


[57] M Ghoreishi G-W Weber and A Mirzazadeh ldquoAn in-ventorymodel for non-instantaneous deteriorating items withpartial backlogging permissible delay in payments inflation-and selling price-dependent demand and customer returnsrdquoAnnals of Operations Research vol 226 no 1 pp 221ndash2382015

[58] S Joslashrgensen and P M Kort ldquoOptimal pricing and inventorypolicies centralized and decentralized decision makingrdquoEuropean Journal of Operational Research vol 138 no 3pp 578ndash600 2002

[59] Y Zhong Z Lin Y-W Zhou T C E Cheng and X LinldquoMatching supply and demand on ride-sharing platformswith permanent agents and competitionrdquo InternationalJournal of Production Economics vol 218 pp 363ndash374 2019


(Yan et al [7]) For example DiDirsquos Spring Festival servicefee and the daily price adjustment mechanism for themorning peak are based on supply and demand Passengerswilling to pay higher service fees are sent to motivateddrivers who give up their holiday break is arrangement isconvenient for the passengers but also gives the drivers areasonable and legitimate return it is a win-win scenario(Cachon et al [8])

Compared with traditional taxi companies ride-hailingplatforms have their own unique characteristics For ex-ample ride-hailing passengers can quickly obtain ride in-formation (eg the waiting time the number of people inline and even the number of drivers around them) andcoordinate with nearby drivers in real time whereas tra-ditional taxi passengers usually wait at the side of the road orbook in advance In addition the drivers of ride-hailingplatforms are relatively free to choose when and where theywork e fluctuation of ride-hailing demand based onspecific reasons was related to more people wanting to usethe service (eg time of day price and service quality) andthe supply of services usually be influenced by factors suchwhether there are drivers available geographic areascommissions etc Due to the influence of weather factors(eg sunny rainy and snowy weather) and special occasions(eg after a concert) there can be a mismatch in supply anddemand (Hall et al [9] Cachon et al [8] Yan et al [7]Afeche et al [10] Hu and Zhou [4])

We are motivated by the dynamic pricing problems thatarise when ride-hailing platforms face market demandfluctuation and supply-demand mismatch Although therehave been studies on pricing strategies many authors (egFeng et al [11] Yan et al [7] Hu and Zhou [12] Sun et al[13] Bimpikis et al [14]) have not designed their models totake into account the characteristic of demand fluctuationover time ey also do not consider the supply functionwhich is sensitive to dynamic wages in relation to timeWhen rider demand increases but there is not enough supplyto meet the demand (Cachon et al [8]) platforms imple-ment ldquodemand rationingrdquo by delaying the service extendingthe rider wait time or even losing the rider bookings edifficulty in hailing a car in some regions and during certainperiods affects the consumersrsquo travel experience When riderdemand decreases supply can exceed the demand (Cachonet al [8]) In such cases the platform resorts to capacityrationing that is to idle available transportation capacityerefore to maximize the profits it is important to de-termine the platformsrsquo optimal pricing under market en-vironments of unbalanced supply and demand caused byfluctuations in rider demand and thus reduce the riderbooking delay and underutilized transportation capacity

is study considers a ride-hailing platform in themarket e platform crowdsources drivers who have idlecar resources and provides shared transport for riders Weincorporate the rider demand fluctuation into the commondemand function (Desiraju and Moorthy [15]) and use thecoefficient of demand fluctuation to describe different de-mand conditions Unlike the previous studies (eg Cachonet al [8] Hu and Zhou [4] Liu et al [16]) we assume riderdemand is an exponential function of time and is sensitive to

price and service quality is means rider demand changesdynamically over time

As a service enterprise the ride-hailing platform canprovide ride supply-demand matching services that can bereflected by service quality e service quality usually refersto passenger satisfaction with ride-hailing services such aspassengersrsquo ride efficiency platformsrsquo supervision and useexperience of APP In reality the passengers are concernednot only with the price but also with the service qualityHowever service quality is closely related to service costwhich affects the platformrsquos pricing decisionserefore it isnecessary to study the impact of service quality on platformpricing decisions transaction volumes and profits Fur-thermore the fixed commission rate is the percentage of thefare that the platform pays to the drivers after they completeeach ride (Cachon et al [8] and Hu and Zhou [4]) It directlyaffects not only the platformrsquos revenue from each transactionbut also the driversrsquo participation through their incomeerefore we take into account the service quality and thefixed commission rate of the ride-hailing platforms theopportunity cost when supply exceeds demand and thebooking delay cost when supply fails to meet demand intime We thus study three different demand situationsdecrease in demand increase in demand and stable de-mand Optimal control theory has been used in the literatureon dynamic pricing (eg Lu et al [17] Herbon andKhmelnitsky [18] and Feng et al [19])us we use optimalcontrol theory to construct models and study the dynamicpricing of ride-hailing platforms with different market de-mand situations in a finite service time [0 T] An analysis ofkey parameters reveals further insights into the effect of thecoefficient of demand fluctuation service quality and fixedcommission rate on the platformrsquos pricing decisions Wefind that although the peak congestion period can be re-duced by reducing service quality andor increasing the fixedcommission rate the reduction in the service quality maynot be desirable due to the loss of peak demand

We investigate the impact of key parameters on the totaltransaction volume and the total profit in a finite servicetime in which the total profit is a concave function of thefixed commission rate and service quality To the best of ourknowledge few studies have investigated dynamic pricingaccording to the characteristics of demand fluctuation in theactual operation of ride-hailing platforms Our study is thefirst to assume demand is an exponential function of timeand to use optimal control theory to examine the pricingproblem of ride-hailing platforms We address the followingresearch questions

(1) How do the ride-hailing platforms develop effectivedynamic pricing strategies in the face of demand-supply imbalances under different market demandsituations (decreased demand increased demandand stable demand) and how do the trajectories ofthe platformsrsquo supply rate and demand rate changeunder the influence of optimal dynamic prices

(2) If the opportunity cost is not considered how doesthe platform adjust its optimal pricing strategy ifmarket demand decreases If a ceiling price




2 Literature Review



























ψ(P t) 1113946T



C(t) W(t) + ηq2 (4)




α(t) αeminus at

(5)


4 Model Analysis




Driver

Driver

DriverRider Rider

Rider


Fares Commissions



Booking delay cost

D (t)Ride-hailing

platform

S (t)

W (t)P (t)

c

h

Opportunity cost

Riderdemandmarket

Driversupplymarket







v(t) v(0) + 1113946t




1113946T



maxPv(t)

1113946T


21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0


maxPv(t)

1113946T


21113872 11138731113960


(8)




+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)


Hv v(t) Pv(t) λv(t) t( 1113857


1113872 1113873 minus c(T minus t)



+ βPv(t) minus cq1113872 1113873

αeminus at


1113872 1113873



+ βPv(t) minus cq1113872 1113873

(10)







α (t)




100

150

200

250

300

350

400

450

5 10 15 20 25 300t


α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t




v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


































































2 Literature Review



























ψ(P t) 1113946T



C(t) W(t) + ηq2 (4)




α(t) αeminus at

(5)


4 Model Analysis




Driver

Driver

DriverRider Rider

Rider


Fares Commissions



Booking delay cost

D (t)Ride-hailing

platform

S (t)

W (t)P (t)

c

h

Opportunity cost

Riderdemandmarket

Driversupplymarket







v(t) v(0) + 1113946t




1113946T



maxPv(t)

1113946T


21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0


maxPv(t)

1113946T


21113872 11138731113960


(8)




+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)


Hv v(t) Pv(t) λv(t) t( 1113857


1113872 1113873 minus c(T minus t)



+ βPv(t) minus cq1113872 1113873

αeminus at


1113872 1113873



+ βPv(t) minus cq1113872 1113873

(10)







α (t)




100

150

200

250

300

350

400

450

5 10 15 20 25 300t


α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t




v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


















































































ψ(P t) 1113946T



C(t) W(t) + ηq2 (4)




α(t) αeminus at

(5)


4 Model Analysis




Driver

Driver

DriverRider Rider

Rider


Fares Commissions



Booking delay cost

D (t)Ride-hailing

platform

S (t)

W (t)P (t)

c

h

Opportunity cost

Riderdemandmarket

Driversupplymarket







v(t) v(0) + 1113946t




1113946T



maxPv(t)

1113946T


21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0


maxPv(t)

1113946T


21113872 11138731113960


(8)




+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)


Hv v(t) Pv(t) λv(t) t( 1113857


1113872 1113873 minus c(T minus t)



+ βPv(t) minus cq1113872 1113873

αeminus at


1113872 1113873



+ βPv(t) minus cq1113872 1113873

(10)







α (t)




100

150

200

250

300

350

400

450

5 10 15 20 25 300t


α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t




v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


































































ψ(P t) 1113946T



C(t) W(t) + ηq2 (4)




α(t) αeminus at

(5)


4 Model Analysis




Driver

Driver

DriverRider Rider

Rider


Fares Commissions



Booking delay cost

D (t)Ride-hailing

platform

S (t)

W (t)P (t)

c

h

Opportunity cost

Riderdemandmarket

Driversupplymarket







v(t) v(0) + 1113946t




1113946T



maxPv(t)

1113946T


21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0


maxPv(t)

1113946T


21113872 11138731113960


(8)




+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)


Hv v(t) Pv(t) λv(t) t( 1113857


1113872 1113873 minus c(T minus t)



+ βPv(t) minus cq1113872 1113873

αeminus at


1113872 1113873



+ βPv(t) minus cq1113872 1113873

(10)







α (t)




100

150

200

250

300

350

400

450

5 10 15 20 25 300t


α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t




v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References



































































v(t) v(0) + 1113946t




1113946T



maxPv(t)

1113946T


21113872 1113873⎡⎣

minus c 1113946T

01113946

t

0


maxPv(t)

1113946T


21113872 11138731113960


(8)




+ βPv(t) minus cq

v(t) ge 0

v(0) 0

agt 0

(9)


Hv v(t) Pv(t) λv(t) t( 1113857


1113872 1113873 minus c(T minus t)



+ βPv(t) minus cq1113872 1113873

αeminus at


1113872 1113873



+ βPv(t) minus cq1113872 1113873

(10)







α (t)




100

150

200

250

300

350

400

450

5 10 15 20 25 300t


α (t) α (t) = 400e004t

α (t) = 400e002t

α (t) = 400e0t

200

400

600

800

1000

1200

1400

5 10 15 20 25 300t




v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References

































































v(t) zHv

zλv

λv(t) minuszHv

zv

zHv

zPv

0

(11)


Plowastv (t)

α2β

eminus at

+c(sr + β)

β(1 minus r)t +

cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(12)


Ta(sr + β)2


T(sr + β)2minus

βηq2

sr + β

(13)



Plowastv (t)

α2β

middot eminus at

+c(sr + β)


cq + sε(sr + β)


2β(sr + β)

(14)



(sr + β)2


sr + β+ cT

(15)


Slowastv (t)

srα2β

middot eminus at

+src(sr +β)




2β(sr +β)

(16)

Dlowastv (t)

α2

middot eminus at

minusc(sr +β)



+α(sr minus β)

2(sr +β)

(17)

















2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


































































2aminus


2(1 minus r)


2(sr + β)

(18)


π lowastv 1113946T



21113872 11138731113960



lowastv (τ)( 1113857dτ1113891dt

(19)






u(t) u(0) + 1113946t





1113946T



maxPu(t)

1113946T


21113872 11138731113960

minus h 1113946t


maxPu(t)

1113946T

01113946

T

0


1113872 11138731113960


(22)



αeminus at


u(t)ge 0

u(0) 0

alt 0

(23)


Hu u(t) Pu(t) λu(t) t( 1113857


1113872 1113873






αeminus at


(24)






Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References

































































Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(25)


Slowastu (t)

srαβ + sr

middot eminus at


β + sr (26)

Dlowastu (t)

srαβ + sr

middot eminus at


β + sr (27)










u is


a(β + sr)+


(28)


π lowastu 1113946T



21113872 1113873dt (29)




Plowastu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

t isin 0 t21113858 1113859

P t isin t2 T( 1113859

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)







Slowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)


Dlowastu (t)

srαβ + sr

middot eminus at


β + sr t isin 0 t21113858 1113859


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(32)









a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References





































































a(β + sr)

minus1a


α1113888 1113889



(33)





21113872 1113873dt

+ 1113946T

t2

1113946

T

t2


21113872 11138731113960


t2


lowastu (τ)( 1113857dτ1113891dt

(34)





1113946T

01113946

T

0


maxPb(t)

1113946T


21113872 11138731113960 1113961dt

(35)

e constraints are

v(t) u(t) 0

a 0(36)


Plowastb (t)

αsr + β

+cq + sεsr + β

(37)


Slowastb (t)

srαsr + β


sr + β (38)

Dlowastb (t)

srαsr + β


sr + β (39)






(sr + β) (40)


π lowastb 1113946T



21113872 1113873dt (41)














u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References









































































u













TgtTth darr ⟶




a 0 ⟶











5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References









































































5 Extended Model





π(P t) maxP(t)

1113946T


maxP(t)

1113946T


21113872 11138731113960 1113961dt

(42)




1113872 1113873 (43)

















Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References







































































Plowast(t)

α(t) + cq

2β+

ηq2

2(1 minus r) (45)









Plowastv (t) 50e




Plowastu (t)

4167e003t + 2086 t isin [0 2086]

80 t isin (2086 30]1113896 (47)





Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


































































Plowastb (t) 4375 t isin [0 30] (48)


62 Key Parameters





Value 30 400 4 1 8 0001 01 01 80

Plowast v (t) t1

5 10 15 20 25 300t

37

38

39

40

41

42

43

44

(a)

Slowast

v (Plowast

v (t) t)Dlowast

v (Plowast

v (t) t)

S v an

d D

v

M

N

Q

Sv (P (0) t)Dv (P (0) t)

5 10 15 20 25 300t

0

50

100

150

200

250

300

(b)

5 10 15 20 25 300t

0

500

1000

1500

2000

2500

3000

3500

Profi

t rat

e

(c)









t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References






































































t2

10 15 20 25 305t

Plowast u (t)

30

40

50

60

70

80

90

100

(a)


t2

X

Y

Z

5 10 15 20 25 300t

S u an

d D

u

200

300

400

500

600

700

800

900

Slowast

u (Plowast

u (t) t)Dlowast

u (Plowast

u (t) t)Su (P (0) t)Du (P (0) t)

(b)

5 10 15 20 25 300t

3000

4000

5000

6000

7000

8000

9000

10000

11000

Profi

t rat

e

(c)










5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References







































































5 10 15 20 25 300t

Plowast b (t)

40

41

42

43

44

45

46

47

48

(a)

5 10 15 20 25 300t

S b an

d D

b

150

200

250

300

350

Slowast

b (Plowast

b (t) t)Dlowast

b (Plowast

b (t) t)

(b)

5 10 15 20 25 300t

3000

3050

3100

3150

3200

Profi

t rat

e

(c)



a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References
































































a = 0025a = 003a = 004

t1 (a = 004)t1 (a = 003)t1 (a = 0025)

5 10 15 20 25 300t

Plowast v (t)

28

30

32

34

36

38

40

42

44

46

(a)



2510 15 20 300 5t

Plowast u (t)

40

45

50

55

60

65

70

75

80

85

(b)





t1

q = 10q = 30q = 50

Plowast v (t)

32

34

36

38

40

42

44

46

48

5 10 15 20 25 300t

(a)

q = 10q = 30q = 50

t2 (q = 10)

t2 (q = 30)t2 (q = 50)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

(b)

Figure 9 Continued



lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References

































































lowastb (05










q = 10q = 30q = 50

Plowast b (t)

5 10 15 20 25 300t

40

42

44

46

48

50

(c)






4)

5 360 709 1092 384 895 221210 369 718 1098 396 912 222615 378 726 1104 407 926 223420 387 735 1110 417 935 223625 395 744 1116 424 941 223230 404 753 1122 429 943 222335 413 761 1128 431 942 220740 422 770 1134 432 936 218545 430 779 1140 429 925 215750 439 788 1146 424 911 2122




t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References


































































t3

t1 (r = 07)

Plowast v (t)

35

40

45

50

5 10 15 20 25 300t

r = 07r = 05

(a)

t2 (r = 07)

t2 (r = 05)

Plowast u (t)

5 10 15 20 25 300t

40

50

60

70

80

90

r = 07r = 05

(b)

Plowast b(t)

5 10 15 20 25 300t

35

40

45

50

55

60

r = 07r = 05

(c)






4)

02 124 360 383 646 2146 2415025 180 420 475 793 2188 279103 227 473 564 860 2152 3060035 267 519 648 873 2062 322604 302 560 728 850 1938 3298045 330 597 803 803 1790 328305 354 630 874 741 1629 3191055 372 660 940 668 1459 303106 384 687 1001 589 1285 2813065 390 712 1058 505 1109 254507 387 735 1110 417 935 2236075 372 756 1158 322 764 1894



















Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References

















































































Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References








































































Appendix


z2Hv

zP2v





We obtain

P(0) α + cq + sε

sr + β (A2)


Plowastv (0)

α2β

+cq + sεsr + β

minuscT(sr + β)

2β(1 minus r)


2βTa(sr + β)+

v(T)

T(sr + β)

(A3)


v(T) cT2(sr + β)2


2βa+αT

2minus

srαT

2β

(A4)





vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References



































































vlowast(t) 1113946

t




dvlowast(t)


lowastv (t) (A6)



2βmiddot e

minus at+

c(sr + β)2


α(sr minus β)

2β (A7)



2βmiddot e

minus at+

c(sr + β)2

β(1 minus r) (A8)







dPlowastv (t)

dt

minus aα2β

middot eminus at

+c(sr + β)

β(1 minus r) (A9)

dP2lowastv(t)

dt2

a2α2β

middot eminus at

(A10)







(dPlowastv (t)dt)

limTth⟶ +infin











z2Hu

zP2u





P(t)leα

β + srmiddot e

minus at+

cq + sεβ + sr

(A12)


Pmaxu (t)

αβ + sr

middot eminus at

+cq + sεβ + sr

(A13)






dPlowastu (t)

dt

minus aαβ + sr

middot eminus at

(A14)

dP2lowastu(t)

dt2

a2αβ + sr

middot eminus at

(A15)








t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References






































































t2 minus1aln


1113888 1113889 (A16)










obtained






dPlowastv (t)

da minus

α middot t

2βmiddot e




da

1a2 middot 1 + ln

2c(sr + β)

aα(1 minus r)1113888 11138891113888 1113889 (A18)


alowast

2ce(sr + β)

α(1 minus r) (A19)



dPlowastu (t)

da minus

α middot t

β + srmiddot e

minus at le 0 (A20)



dt2

da

1a2 middot ln


1113888 1113889gt 0 (A21)




dPlowastv (t)

dqdPlowastu (t)

dqdPlowastb (t)

dq

c

β + srgt 0 (A22)


dt1

dq 0 (A23)



dt2

dq1a

middotαc






dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References

































































dPlowastv (t)

dr

c(s + β) middot t

β(1 minus r)2minus

s(α + cq + sε)(sr + β)2

(A25)






dr minus

1a




dPlowastu (t)

dr minus


(β + sr)2lt 0 (A28)



dt2

dr minus

1a

middotαsP



dPlowastb (t)

dr minus


lt 0 (A30)



Data Availability




Acknowledgments


References















































































































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