Slides

Optimal Decisions of the Producer and Distributor Optimal Decisions of the Producer and Distributor

in a Fresh Product Supply Chain Involving Long in a Fresh Product Supply Chain Involving Long

Distance TransportationDistance Transportation

1 The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

2 Tsinghua University, Beijing 100084, China3 Amazon.com / The University of Texas at Austin

2023年4月13日

Xiaoqiang Cai1, Jian Chen2, Yongbo Xiao2, Xiaolin Xu1, Gang Yu3

2

Outline of Presentation

1. Introduction2. Related Literature3. Problem Definition4. Optimal Decentralized Decisions5. Optimal Centralized Decisions6. Incentives to Facilitate Coordination7. Numerical Evaluation8. Summary and Future Study

3

1. Introduction

• Motivation: Real practice of the producer and distribution of fresh products– Production: highly characterized by geographical location– Distribution: to distant market and usually involves long

distance transportation• Features of such fresh product supply chain:

– The supply chain of such products usually involves long distance transportation, which contains a high risk of unexpected delays(Delays may often be caused by disruptive variations of weather, equipment breakdowns, transport network congestion, etc.)

– Because of the perishable nature of the product and the likeliness of transportation delays, it is almost inevitable that certain degree of decay and/or deterioration will have taken place when the product arrives at the distant market.

– The market demand for the product depends on both the freshness level of the product and selling price.(price discount bases on freshness and effective supply)

– Any product will become obsolete if it is unsold before a certain time.

4

1. Introduction

• Examples:– Seafood imported from overseas countries to a local

market (China’s exports of fishery and seafood products totaled more than $4.35 billion and vegetables reached $3.05billion in 2005)

– Agriculture imported from one country to another (in 2003, U.S. exports of agriculture accounted for 8.6% of its GDP)

– Vegetables/Fruit grown in southern provinces and sold to the northern regions (In California, more than 485,000 truckloads of fresh fruits and vegetables travel 100 to 3100 miles to reach their destinations)

5

1. Introduction

•The challenging problems:– How to determine the proper decisions to maximize the expected

profits or minimize the possible risks in these situations?– Could an incentive mechanism be introduced to motivate the

producer and the distributor to act in a coordinated way (so that the risks involved might be shared and every one will be better off)?

•The main results:– The optimal decisions for both the producer and the distributor in

the decentralized system, to optimize their respective objectives;– The optimal decisions in the centralized system, to optimize their

joint objective;– A transportation-time dependent, price-discount sharing

mechanism and a compensation scheme, which can ensure that both parties take the coordinated decisions and are all better off by the coordination;

– Managerial insights, uncovered from both theoretical results and computational studies.

6

1. Introduction

• The general model studied in this work:– A distributor procures a kind of fresh product (e.g.,

live seafood, fresh vegetable/fruit, etc.) from a producer.

– The product ordered has to undergo a long distance transportation before it reaches the market.

– Transaction between the producer and the distributor: Free-on-Board (FOB);

– Risks faced by the distributor:• random fluctuations of the market demand • decay and/or deteriorate of the product during long

distance transportation– The actual market demand depends on:

• freshness level• selling price

7

1. Introduction

• The decisions:– The producer’s decision: wholesales price, based on its

impact to the order quantity of the distributor.– The distributor’s decision: order quantity and selling

price, based on • the price of the producer,• the likely loss in long distance transportation, • the freshness level of the product, • the possible market demand.

– Decentralize, uncoordinated decisions: Both parties (the producer and the distributor) optimize their individual objectives respectively.

– Centralized, coordinated decisions: The two parties optimize their joint objective, under perhaps certain incentive scheme (contract).

8

2. Related Literature

• Inventory management of perishable products: Nahmias (1982), Raafat (1991), Goyal and Giri (2001), Rajan et al. (1992), etc.– The majority of the literature has mainly dealt with

quantity loss, our study considers both the quantity loss and quality drop.(Rajan et al. (1992) study both value drop and quantity decrease. They focus, however, on a model with deterministic demand)

• Stochastic lead times: Eppen et. al., 1988; Lau and Zhao, 1993; Bookbinder and Cakanyildirim, 1999, etc.– Most study multi-period models and focus on the

influence of the stochastic lead time upon the inventory/safety inventory

– Our model considers the impacts of stochastic lead time(transport time) on the effective inventory and demand simultaneously.

9

2. Related Literature•Supply chain coordination with contracts(e.g., buy back, quantity

flexibility): Chen (2003), Cachon (2003), etc. Our study will propose a price-discount sharing mechanism (which is also time-dependent) together with a compensation scheme to coordinate the producer and the distributor, in the situation where the demand depends on the price and the product freshness.

•Risk aversion in supply chain contracting: Agrawal and Seshadri (2000), Weng (1999), Gan et al. (2005), etc.

Our model also uses “downside penalty” to reflect the risk-averse attitude on the uncertain transportation delays.

•Recently, Ray et al. (2005) have studied a problem to determine the optimal pricing and stocking decisions, in which the market demand is stochastic and price-sensitive, and the delivery times are random.

Our model is distinguished by the considerations of perishable product, risk aversion towards the random transportation time, and a multiplicative demand function with the random factor following a general distribution

10

3. Problem Definition

• We study a single-period FOB (Free on Board) model– The distributor in charge of transportation

– Transportation time: T ~ G(t), g(t) on [a,b]– Fresh duration of the product:

Producer Distributor MarketLong Distance Transportation

Wholesale price: w

Production Cost: cm

Order quantity: q Selling Price: p

Transportation Cost: ct

],[ ba

11


• Assumptions:

– Product Deterioration: For q units of product loaded to the vehicle, the amount becomes qm(t) after a period of time t, where 0<m(t) ≤1.

– Product Decay: The freshness level of the product is a function θ(t) defined on [0,1]: θ(t) = 1 if t≤τ and 0≤θ(t) < 1 otherwise.

– Market demand: (i) is the random factor with CDF F(x) and PDF f(x) (ii) k is the price elasticity, with k>1– No shortage cost, no salvage value– Information is symmetrical

1 )(),( 0 ktpytpD k ，

12


• The sequence of events:

t3T+ t30

(1) The producer sets a wholesale price

(2) The distributor determines the order quantity and places an order

(3) Production begins

(4) Finished products are loaded onto the transport vessel

(5) Products arrive at the distant market, the distributor sets a retailing price

(6) Demands are realized and satisfied

t1 t2 time

Production Transportation

13


• The producer’s objective:– Setting a wholesale price w to maximize the profit

• The distributor’s objective:– In stage 2(T+t3), given that the order arrives at

time t, determining a selling price p to maximize the expected profit

– In stage 1(t1), determining the order quantity q to maximize the expected utility (downside penalty factor )

where )],|),(()},|),(({[),|),((),( tqtqptqtqpEtqtqptqU ddtdd

qcwqtqmtpDEptqp td )}}(),,({min{),|(

qcww mm )()(

0

)},({:)( tqUEqU dtd

14

4. Optimal Decentralized Decisions

• The distributor’s optimal decision in stage 2– Objective function

– let z0 be the solution of the following equation:

dtddddddd qcwqtmqtpDEptqp )}}(),,({min{),|(

)](1[)()1(0

zFzdxxxfkz

15


• Theorem 1.Suppose that the distributor orders a quantity q and the product arrives at time t. Then the optimal selling price p*(q, t) of the distributor is:

(10)

• Remark 1:Note that (t) is the product's level of freshness when it reaches the market at time t. It follows from (10) that, the higher the level of freshness, the higher the selling price. Note also that qm(t) is the marketable quantity of the product (effective supply) when it reaches the market at time t. It is easy to see from equation (10) that, the smaller the effective supply, the higher the price.

16

4. Optimal Decentralized Decisions• Remark 1:

It is interesting to analyze the relationship between the optimal selling price p*(q; t) and the transportation time t. A general intuition may suggest that the price may decrease if a transportation delay occurs (t is longer) because the product may have become less fresh. The key here is, nevertheless, the selling price also depends on the effective supply - If a transportation delay occurs, the marketable quantity qm(t) may have dropped and thus the effective supply to the market is reduced.

Our result shows that whether p*(q, t) decreases or increases in t depends on the function (t)/m(t), which we call the “quality-quantity” ratio. If the quantity decreases quickly (a small m(t)) but the quality drops slowly (a large (t)), then the optimal selling price may be even higher than the “normal” optimal selling price (y0z0/q)1-k with a fully fresh product.

17


• Remark 1:

Suppose that the actual transportation time is t and the order quantity of the distributor is q. Equation (10) gives a close-loop formula to determine the optimal price: When the product arrives at t, the distributor can observe “the level of freshness” and “the effective supply” and then set his selling price.

18


• Theorem 2. For any given wholesale price w of the producer, the distributor's optimal order quantity should be:

– where

The corresponding optimal expected utility value is:

k

tt

d tSEKcw

zFzyq

}])({[)(1

00

00*

},)()({ /11/10

kkt tmtEK .)()()( /11/1

0kk tmtKtS

19

• Remark 2:

q* decreases in the producer's wholesale price w and the unit transportation cost cT ;

We can also show that:

where

Should the transportation lead time be zero (or the product not be perishable), the distributor would have ordered:

Therefore, q*<q0. That is, the possibility of deterioration and decay discourages the distributor from ordering more.


20

4. Optimal Decentralized Decisions• Theorem 3. The producer’s optimal wholesale

price is as follows:

• Remark 3: – w* is greater than cm because k>1, which

guarantees that the producer always earns a positive profit; and

– w* is decreasing in k, implying that the producer should decrease his wholesale price if the demand is more price-sensitive.

1*

k

kccw mT

21


• Corollary 1. In the scenario where there is a lack of coordination between the producer and distributor:– (i) the producer’s optimal profit is

– (ii) the distributor’s his expected profit is

where

k

mt

mtm K

cc

zF

k

k

k

cczy

10

00* )(11

1

*

1

0* )(111 dmtd qccK

K

k

k

k

k

.1})({: 001 KtSEKK t

22


• Remarks:– Without coordination, the producer’s profit might be

much lower than that of the distributor:

– β is increasing in ρ(represents the degree of risk aversion). This means that the producer's share of profit decreases as the distributor becomes more conservative towards the transportation risk.

Whether the producer should coordinate with the distributor to share his transportation risk, so that everyone will be better off ?

111

:1

0*

*

K

K

k

kk

m

d

23

5. Optimal Centralized Decisions

• Theorem 4. In the centralized supply chain, the optimal decisions are as follows:– Optimal order quantity:

– Optimal selling price, where t is the arrival time of the product at the market:

– The corresponding expected profit and expected utility are

k

mtc K

cc

zFzyq

1

000

* )(1

k

ccc

tmq

ztytqp

/1

*00**

)(

)(),(

24


• Theorem 4. The optimal centralized order quantity is always larger than that in the decentralized system:

25


• Profit loss of competition

• Observation: The profit loss could be very substantial if there is no coordination, in particular when the market demand is very price sensitive and the distributor is very risk averse.

• How to motivate the two parties to coordinate?

1

1

0 11

1

1

11

K

K

k

kk

k

k

k

k

*

**

1:c

dm

0%10%20%30%40%50%60%70%

1 2 3 4 5Price-Elasticity

Pro

fit L

oss

(%)

Lower Bound

Upper Bound

26

6. Incentives to Facilitate Coordination

• We have known that qc* > qd

*. That is, the distributor will

order less when there is no coordination.

• Since the producer will be benefited by the coordination, it is imperative for the producer to provide enough incentives for the distributor to order more.

• One mechanism is the following: The producer sets a basic price w0(cM) and offers the distributor a compensation

when a transportation delay actually occurs.

– However, this mechanism will not be acceptable by the producer. This can be shown as follows, by using a counter-example with = 0: It follows from Theorem 2 that the distributor's optimal order quantity is given as

– To achieve , the relation must hold;

27

The price-discount sharing contract suggests:

where w(p, t) is the wholesale price which depends on the transportation time and the actual selling price of the distributor, and is a constant taking value in (0, 1).

It is equivalent to

where w0(t) is the base or gross wholesale price equal to , and p0 is the “list price” of the product in the market.


• The proposed Time-Dependent Price-Discount Sharing (TDPDS) scheme:– A price-discount sharing contract, under which the producer

shares a certain portion of the price discount that the distributor has to mark due to deterioration and decay of the product;

– A compensation contract (similar to the traditional buy-back arrangement), under which the producer compensates the distributor a certain amount for any unsold unit of the product.

])([),( mtm ccptmctpw

28


The producer compensates the distributor for every unsold unit of v, which is proportional to the distributor’s selling price

• Theorem 4. TDPDS together with the compensation contract v =αp will induce the distributor to order the quantity qc

* for any 0<α<1.

• Remarks:– We can show that the producer and the distributor's

shares of profit will be αПc* and (1-α)Пc

*, respectively.

– To ensure that each party's profit with coordination is not less than that before, we can derive a lower bound and upper bound respectively:

– The final choice of α depends on the chain member’s bargaining powers, but all such choices ensure that both parties will be better off by coordinating with each other.

**** /1,/ cdcm

pv

29

7. Numerical Evaluation

• Some computational experiments to evaluate the effects of the following parameters on the optimal decisions, in order to uncover certain managerial insights that are not obvious in the theoretical results:– The fresh duration – The downside penalty factor – The price elasticity k

30


• Impact of the freshness duration

*

*

m

d

*

**

1c

dm

*w *dq

*m

*d

*cq

*c

Decentralized Centralized

2.00 2.50 1.47 2.21 5.03 5.89 10.07 2.28 28.05%

2.25 2.50 1.57 2.35 5.30 6.27 10.60 2.25 27.82%

2.50 2.50 1.66 2.49 5.55 6.64 11.10 2.23 27.56%

2.75 2.50 1.75 2.63 5.79 7.02 11.58 2.20 27.28%

3.00 2.50 1.85 2.77 6.01 7.39 12.02 2.17 26.97%

3.25 2.50 1.94 2.90 6.21 7.74 12.43 2.14 26.64%

3.50 2.50 2.02 3.03 6.40 8.08 12.79 2.11 26.32%

3.75 2.50 2.10 3.15 6.55 8.39 13.11 2.08 25.99%

4.00 2.50 2.17 3.25 6.69 8.67 13.37 2.06 25.69%

4.25 2.50 2.23 3.34 6.79 8.90 13.59 2.03 25.42%

4.50 2.50 2.27 3.41 6.87 9.09 13.74 2.02 25.20%

4.75 2.50 2.30 3.45 6.92 9.20 13.84 2.00 25.06%

5.00 2.50 2.31 3.46 6.93 9.24 13.87 2.00 25.03%

31


• Observation I:– Note that τ is an index on the perishability of the

product. As τ decreases, the quantity and quality losses of the product increases.

– Table 1 shows that if the product is less perishable, the distributor's share of profit in the decentralized scenario decreases. This may be partially caused by the bargaining power of the producer which becomes stronger when the product is strong.

– The profit loss when there is no coordination, , strictly decreases inτ. This implies that cooperation between the producer and the distributor is especially profitable when the product is highly perishable.

32


• Impact of the risk-averse attitude

37.25%3.929.053.074.521.150.772.505.00

36.31%3.659.253.384.621.270.842.504.50

35.31%3.409.433.694.721.390.922.504.00

34.26%3.189.594.034.801.511.012.503.50

33.15%2.979.734.374.861.641.092.503.00

31.98%2.789.854.734.921.771.182.502.50

30.75%2.609.945.104.971.911.282.502.00

29.44%2.4310.025.495.012.061.372.501.50

28.05%2.2810.075.895.032.211.472.501.00

26.57%2.1310.106.315.052.371.582.500.50

CentralizedDecentralized

*

*

m

d

*

**

1c

dm

*w *

dq*m

*d

*cq

*c

33


• Observation II:– When increases from 0.50 to 5.00, the

distributor's expected profit has a decrease of only 10.50%, whereas the producer has a profit decrease of 51.27%.

– Therefore, the producer should cooperate with the distributor, especially when the distributor is quite risk-averse.

– The profit loss because of competition, , is also increasing in . This means the supply chain members should coordinate when they are risk-averse.

34


• Impact of the price-elasticity

Decentralized Centralized

1.25 7.00 0.72 4.34 24.00 5.41 35.88 5.53 7.48 21.02%

1.50 4.00 1.21 3.63 12.15 6.28 21.04 3.35 5.20 25.02%

1.75 3.00 1.46 2.91 7.67 6.42 14.48 2.63 4.41 26.90%

2.00 2.50 1.47 2.21 5.03 5.89 10.07 2.28 4.00 28.05%

2.25 2.20 1.44 1.73 3.57 5.39 7.45 2.07 3.75 28.87%

2.50 2.00 1.36 1.36 2.63 4.88 5.66 1.94 3.59 29.52%

2.75 1.86 1.25 1.07 1.97 4.33 4.35 1.84 3.47 30.06%

3.00 1.75 1.11 0.83 1.48 3.75 3.33 1.78 3.37 30.54%

k *

*

m

d

*

*

d

c

q

q

*

**

1c

dm

*w *

dq*m

*d

*cq

*c

35


• Observation III:– As the market demand becomes more sensitive to

the selling price, the producer's optimal wholesale price strictly decreases in the absence of coordination.

– The profits for both the producer and the distributor do strictly decrease in k. Hence both of them should prefer a less price-sensitive market demand.

– The profit loss due to competition is increasing in k. Therefore, if the market is very sensitive to the price, the coordination of the producer and the distributor is more beneficial.

36

8. Summary and Future Study

• Our work is to address a class of supply chain management problems where fresh products are procured via long distance transportation.

• Our model is distinguished by the following features:– Uncertain transportation delays;– Decay and deterioration of the product during

transportation, which may cause reduction in both quality (freshness level) and quantity, respectively;

– Random market demand, sensitive to the selling price as well as the freshness level of the product.

37


• Our contributions include:– Optimal decisions for both the manufacturer and

the distributor if they are not coordinated;– Optimal decisions when they are coordinated;– A time-dependent, price-discount sharing

mechanism and a compensation scheme, to induce coordination; and

– Managerial insights, uncovered from both theoretical results and computational studies.

38


• Coordination is more important when the following three scenarios prevail:– the product has a very short fresh duration;– the supply chain members are risk averse towards

transportation risk; and– the demand is sensitive to sales price.

39


• Many directions for further research:– The CIF (Cost Insurance and Freight) business

model, where the producer bears the transportation cost and risk;

– Three-party coordination: consider the optimal decisions of the third-party logistics provider;

– Multiple freshness levels: consider the joint pricing of multiple freshness levels;

– Multiple distributors: consider the competition between the competitive distributors;

– Multiple markets: a local market and an export market, multiple export markets (capacity allocation should be considered);

– etc.

40

.:. Thanks~

Any Questions?

Documents

Slides