An exact solution approach for vehicle routing and scheduling problems with soft time windows

Transportation Research Part E 45 (2009) 960–977

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

An exact solution approach for vehicle routing and schedulingproblems with soft time windows

A.G. Qureshi *, E. Taniguchi, T. YamadaDepartment of Urban Management, Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8540, Japan

a r t i c l e i n f o

Article history:Received 4 July 2008Received in revised form 3 February 2009Accepted 10 April 2009

Keywords:City logisticsVehicle routing with soft time windowsColumn generation

1366-5545/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.tre.2009.04.007

* Corresponding author. Tel.: +81 75 383 3231; faE-mail address: [email protected]

a b s t r a c t

A new column generation based exact optimization approach for the vehicle routing andscheduling problem with semi soft time windows (VRPSSTW) is presented. Elementaryshortest path problem with resource constraints and late arrival penalties is solved as asubproblem, which rises from the Dantzig–Wolfe decomposition method. Exact solutionsof VRPSSTW and hard time windows variant are compared on Solomon’s benchmarkinstances as well as on an instance based on Tokyo road network. It was found that theVRPSSTW solution results in fewer routes thus overall costs are reduced and late arrivalpenalties contribute only a small fraction to total cost.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

City logistics is a branch of urban management systems that deals with the typical problems relating to urban freight trans-port, such as traffic congestion, loading and unloading on street and environmental emissions (Ortuzar and Willumsen, 1995).City logistics-related measures, for example, route optimization, optimal location of logistics terminals and depots, load factorcontrols and cooperative delivery systems are useful to mitigate these problems. The vehicle routing and scheduling problem(VRP) can be used as a principal tool for evaluating many types of city logistics schemes (Duin et al., 2007). The VRP is a wellknown NP-hard problem which consists of determining a set of optimum routes covering all the demands of a given set ofcustomers without violating the capacity of vehicles. The vehicle routing and scheduling problem with time windows(VRPTW) is an extension of the VRP, where another constraint is added requiring the start of service at each customer i withina pre-specified time window [ai, bi]. With the vehicle routing and scheduling problem with hard time windows (VRPHTW),delivery of goods outside the time windows is not allowed at all. The VRPHTW is an important variant of the VRP and has beenthe centre of VRP-related research, though it often lacks the practicality found in real world. A more practical variant of theVRPTW is the vehicle routing and scheduling problem with soft time windows (VRPSTW), where deliveries are still possibleoutside the time windows with some penalty costs. Unlike the VRPHTW, most of the research on the VRPSTW has adoptedheuristics approaches (for example see, Hashimoto et al., 2006; Taillard et al., 1997; Taniguchi et al., 2001).

Figs. 1 and 2 show the penalty cost functions of the VRPHTW and the VRPSTW, respectively. A very high penalty cost out-side the time windows is used to model the case of hard time windows. In contrast, arriving earlier and waiting for start ofservice at any customer is considered as early arrival penalty in the VRPSTW, which is usually much smaller than the latearrival penalty as shown in Fig. 2. All the exact approaches allow waiting at no cost in solutions of the VRPHTW as indicatedin Fig. 3. Furthermore the exact solution techniques for the VRPHTW rely on the triangular inequality as represented in Eq.

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Fig. 1. Penalty function for the VRPHTW.

Time [ [ ] ] ai’ ai bi bi’

Penalty Cost

Fig. 2. Penalty function for the VRPSTW.

Time [ ] ai bi

Penalty Cost

Fig. 3. Penalty function for the VRPHTW (waiting allowed at no cost).

A.G. Qureshi et al. / Transportation Research Part E 45 (2009) 960–977 961

(1), where cij represents the cost of servicing customer j after customer i in the solution, using arc (i, j) and h represents someintermediate customer location.

cij 6 cih þ chj ð1Þ

Presence of the early arrival penalty constraint in the VRPSTW does not always follow the triangular inequality, making itdifficult to develop an exact approach for it on the lines of available exact techniques for the VRPHTW. The VRPSTW can besolved exactly using complete enumeration or using the service start time as a continuous variable (Tagmouti et al., 2007). Bothof these techniques, however, require very long computation time, limiting the maximum size of instances that can be handled.Thus both the VRPHTW and the VRPSTW are placed on the two extremes of two important scales of practicality and technicallyfeasible cost structure. The VRPHTW presents the best cost structure to be exploited by the exact optimization techniques suchas Lagrangian relaxation and column generation, but it lacks the practicality found in real world logistics problems. Contrary tothat, the VRPSTW provides efficient mapping of the real logistics choices as far as the economical use of resources is concerned,but its cost structure is too complex such that no exact optimization technique has been developed for this variant so far.

This paper considers the best parts of both VRPTW variants in an effort to develop an exact optimization approach usingcharacteristics of the soft time windows. The early arrival penalty constraint in the VRPSTW is relaxed by allowing waiting atno cost to formulate a new VRPTW variant. This sort of problem can be called the vehicle routing and scheduling problem withsemi soft time windows (VRPSSTW), which only takes into account late arrival time penalties (see Fig. 4). This type of timewindow is very important from a practical point of view for logistics managers, as it offers more economical use of resourcesat minimum delays in delivery due to the penalties for late arrival. Moreover, the cost structure in the VRPSSTW follows thebasic exact optimization assumption of cost triangular inequality (Eq. (1)). A new branch and price algorithm is presented forthe VRPSSTW by extending the column generation solution of the VRPHTW. Dantzig–Wolfe decomposition of the VRPSSTWresults in a new subproblem, namely the elementary shortest path problem with resource constraint and late arrival penalties

Time

Penalty Cost

[ ] ] ai bi bi’

Fig. 4. Penalty function for the VRPSSTW.

962 A.G. Qureshi et al. / Transportation Research Part E 45 (2009) 960–977

(ESPPRCLAP). New path extension rules were developed to modify available labeling algorithms for the elementary shortestpath problem with resource constraint (ESSPRC) to deal with time dependent late arrival penalties. A branching scheme forthe VRPSSTW is also presented to obtain integer solutions in the branch and price algorithm. Finally, exact solutions of theVRPSSTW are compared to those of the VRPHTW using Solomon’s benchmark instances as well as a practical logistics instancebased on the Tokyo Metropolitan road network. Appendix A provides a list of abbreviations used in this paper.

2. Review of related research

After the earlier work of Kolen et al. (1987) using dynamic programming, the literature on exact solution techniques forthe VRPHTW has been increasing. Most of the research has been along two optimal approaches, namely Lagrangian relaxa-tion (Fisher et al., 1997; Kallehauge et al., 2006; Kohl and Madsen, 1997) and Dantzig–Wolfe decomposition (commonlyknown as column generation) (Chabrier, 2006; Desrochers et al., 1992; Feillet et al., 2004; Irnich and Villeneuve, 2003; Kohlet al., 1999). Dantzig–Wolfe decomposition of the VRPTW results in the set partitioning master problem and an elementaryshortest path problem with resource constraints (ESPPRC) as its subproblem. While the master problem remains the same,many researchers have worked with various shortest path variations as subproblems in their column generation schemes forthe VRPHTW. For instance, Desrochers et al. (1992) presented the first column generation based approach for the VRPHTW,using 2-cycle elimination while solving the relaxed shortest path subproblem. Irnich and Villeneuve (2003) used a relaxedshortest path subproblem with k-cycle elimination (with k P 3) in their column generation scheme for the VRPHTW. Kohlet al. (1999) introduced 2-path cuts in the master problem to improve the lower bound at branching nodes, while still work-ing with the same sub-problem as in Desrochers et al. (1992). Recently, Feillet et al. (2004) and Chabrier (2006) have usedthe ESPPRC as the subproblem in a very efficient Dantzig–Wolfe decomposition based approach for VRPHTW; whereas Righ-ini and Salani (2006) have used a bounded bi-directional dynamic program to solve the ESPPRC subproblem to reduce com-putation time. An exhaustive review of the VRPHTW can be found in Cordeau et al. (2002).

As far as research on the exact solution using soft time windows is concerned, much research has focused on the scheduleoptimization of a fixed path such as Sexton and Bodin (1985) and Dumas et al. (1990). The earliest work on the shortest pathproblem with soft time windows (STW) is due to Ioachim et al. (1998). These considered the STW by minimizing a cost func-tion at the vertices that depend on the service start time which is strictly within the time windows. The optimum service starttime is found by allowing waiting within time windows. Ioachim et al. (1998) used only negative slopes to minimize the addi-tional node cost that depended on the service start time within user specified time windows. Recently, Tagmouti et al. (2007)have presented a column generation scheme for the arc routing problem, using an improved version of shortest path algo-rithm presented by Ioachim et al. (1998) as the subproblem. In their work, vehicles are not allowed to wait along their route;whereas the service cost for each arc is defined as a time-dependent function that contains both positive and negative slopes.

On the contrary, in this paper, the STW has been considered by relaxing the latest possible arrival time bi to b0i as shown inFig. 4. A vehicle can start service even if it arrives after bi (but not after b0i) by paying a late arrival penalty cost. Furthermore,waiting is only allowed if the vehicle arrives earlier than the service start time ai, i.e. no waiting is allowed within the relaxedtime windows ½ai; b

0i�. Using these STW settings and Dantzig–Wolfe decomposition, the VRPSSTW is decomposed into a set

partitioning master problem and an elementary shortest path problem with resource constraint and late arrival penalties(ESPPRCLAP) as a subproblem. Furthermore, exact solutions of the VRPHTW and the VRPSSTW are compared with each otherfor the first time. To our knowledge, no exact solution algorithm for VRPSSTW is available for such STW settings. However,Taillard et al. (1997) and Gendreau et al. (1999) have used similar semi soft time windows settings in their tabu search heu-ristics, arguing for their practicality and a trade-off between fleet size and service quality to customers (i.e. delivery withintime windows). Gendreau et al. (1999) have cited a courier delivery system, whereas pizza home deliveries can be taken asanother real life application of the VRPSSTW. The VRPHTW often results in more vehicles compared to those required to car-ry the customers’ demands in order to strictly restrict the service start within the time windows. On the other hand, theVRPSSTW aims at finding the best balance between the cost of required vehicles (fleet size) and the late arrival penaltieswhich relates to the reduced service quality.

Unlike our column generation scheme that iteratively adds the feasible routes of marginal negative cost from thesubproblem to the set partitioning master problem, Calvete et al. (2007) exploited goal programming to enumerate all


the feasible routes in the first stage and then used the set partitioning problem to solve the VRP with soft time windows witha heterogeneous fleet and multiple objectives. In their work, two sets of problems are presented; in the first set, the cus-tomer’s desired service time is modeled by soft time windows, which are smaller than global hard time windows; whereas,the other set consists of Solomon’s R1-type instances with time dependent penalties. They only provided the solution time,with no data on the cost of the problem provided. Their methodology depends on many real life constraints to limit the firststage enumeration. In a similar approach, Fagerholt (2001) solved a ship-scheduling problem with soft time windows. Thetraveling salesman problem with capacity, hard time windows and precedence constraint (TSP-CHTWPC) was used to enu-merate all feasible routes and then their schedules were optimized using soft time windows, before using a set partitioningproblem. Alvarenga et al. (2007) presented a very effective hybrid approach for the VRPHTW, which utilizes the set partition-ing problem to find an optimum heuristic solution based on the routes generated by a GA heuristic. A similar approach whichconsidered various forms of fitness values was presented in Alvarenga and Mateus (2004).

Much research has used soft time windows in the heuristics solutions with the idea to reduce number of vehicles or over-all solution cost (e.g. Balakrishnan, 1993; Qureshi and Hanaoka, 2005; Taniguchi and Thompson, 2002) including some prac-tical applications (Ando and Taniguchi, 2006). Recently, Hashimoto et al. (2006) used a local search algorithm and a dynamicprogramming based approach in a route first schedule second algorithm to solve Solomon’s benchmarks instances with softtime windows, soft travel time and soft capacity constraints. The optimum service start time for each customer was found asa subproblem after the route had been optimized using a local search algorithm. A detailed overview on the VRPSTW and itssolution techniques can be found in Taniguchi et al. (2001). Excellent review of the heuristics methods used for solving theVRPHTW and the VRPSTW can be found in Braysy and Gendreau (2005a,b).

3. Model formulation

The VRPSSTW is defined on a directed graph G = (V, A). The vertex set V includes the depot vertex 0 and set of customersC = {1, 2, . . ., n}. The arc set A consists of all feasible arcs (i, j), i, j 2 V. Both cost cij as well as time tij are associated with each arc(i, j) 2 A. Time tij includes the travel time on arc (i, j) and the service time at vertex i, and a fixed vehicle utilization cost isadded to all outgoing arcs from the depot, i.e. in c0j, j 2 C. With every vertex of V there is an associated demand di, withd0 = 0, and a time window [ai, bi] representing the earliest and the latest possible service start times.

This study incorporates the semi soft time windows constraint by extending the latest possible service start time bi to b0i asshown in Fig. 4. To obtain the maximum limit b0i, a maximum late arrival penalty equivalent to the cost of a dedicated singlevehicle route only serving the concerned vertex was used (see Eq. (2)); however, smaller values of b0i were also investigated inthis paper based on arbitrary limits. Taking cl as the unit late arrival penalty cost, the maximum limit of b0i can be defined as:

b0i ¼min b0 � ti0; bi þðc0i þ ci0Þ

cl

� �ð2Þ

Feasible arcs follow the inequality ai þ tij 6 b0j in the VRPSSTW as compared to the arc feasibility condition of the VRPHTWi.e. ai + tij 6 bj. This condition increases the number of feasible links in the VRPSSTW network as compared to the VRPHTWnetwork. A set of identical vehicles (represented by K) with capacity q stationed at the depot, is available to service custom-ers’ demands. Furthermore, sjk defines the service start time at a vertex j 2 C by a vehicle k 2 K. For all arcs (i, j) 2 A, exceptarcs which violate the inequality, ai þ tij 6 b0j, the modified time dependent travel cost, c0ijk is defined as a function of sjk.

c0ijk ¼cij; if sjk 6 bj

cij þ clðsjk � bjÞ; if sjk > bj

�ð3Þ

Now the VRPSSTW can be mathematically formulated as

minXk2K

Xði;jÞ2A

c0ijkxijk ð4Þ

s:t:Xk2K

Xj2V

xijk ¼ 1 8i 2 C ð5ÞXi2C

di

Xj2V

xijk 6 q 8k 2 K ð6ÞXj2V

x0jk ¼ 1 8k 2 K ð7ÞXi2V

xihk �Xj2V

xhjk ¼ 0 8h 2 C; 8k 2 K ð8ÞXi2V

xi0k ¼ 1 8k 2 K ð9Þ

sik þ tij � sjk 6 ð1� xijkÞM 8ði; jÞ 2 A; 8k 2 K ð10Þai 6 sik 6 b0i 8i 2 V ; 8k 2 K ð11Þxijk 2 f0;1g 8ði; jÞ 2 A; 8k 2 K ð12Þ


The model contains two decision variables: sjk determines the service start time at customer j as well as the travel cost ofarc (i, j), and xijk represents whether arc (i, j) is used in the solution (xijk = 1) or not (xijk = 0). The objective function (4) min-imizes the total cost including the fixed cost for vehicle utilization and the travel cost on the arcs with late arrival penaltycosts. Constraint (5) ensures that every customer is only serviced once, while constraint (6) is the capacity constraint thatkeeps the cumulative demands along a vehicle’s route within its capacity. Constraints (7)–(9) are flow conservation con-straints, specifying that every route shall start from and end at depot; while, on the route, if a vehicle travels to a customerlocation h it must also travel from it. Constraint (10) is a time window constraint specifying that if a vehicle travels from i to j,service at j can not start earlier than that at i. Here, M is a large constant. Constraint (11) represents the relaxed time win-dows for the VRPSSTW and restricts the service start time at all vertices within their relaxed time windows ½ai; b

0i�. Finally,

constraint (12) ensures the integrality of the flow variables xijk.

4. Column generation (branch and price)

Column generation or Dantzig–Wolfe decomposition provides a flexible framework that can accommodate complex con-straints and time dependent costs (Fisher, 1995). It decomposes the VRPSSTW problem, Eqs. (4)–(12) into a set partitioningmaster problem and an elementary shortest path problem with resource constraints and late arrival penalties (ESPPRCLAP) asits subproblem. The ESPPRCLAP gives the feasible shortest path subjected to constraints (6)–(12). The objective function (4) isa non-linear function; however, the same decomposition methodology can be adopted for exact solution of the non-linearoptimization problem (4)–(12) as that used for a variant of the capacitated arc routing problem (CARP) (Tagmouti et al.,2007). The master problem remains linear while the non-linearity is handled at the subproblem level using dynamic program-ming. The master problem, which consists of selecting a set of feasible paths of minimum cost, is mathematically described as:

minXp2P

cpyp ð13Þ

s:t:Xp2P

aipyp ¼ 1 8i 2 C ð14Þ

yp 2 f0;1g 8p 2 P ð15Þ
where P is the set of all feasible paths. The variable yp takes value 1 if the path p 2 P, is selected and 0 otherwise. The cost ofpath p is denoted by cp, and aip represents the number of times path p serves customer i. The set covering master problem issolved by replacing constraint (14) with (16), as linear programming relaxation of set covering type master problem is morestable than set partitioning type (Desrochers et al., 1992) X
p2P

aipyp P 1 8i 2 C ð16Þ

Fig. 5 shows complete flow chart of the column generation algorithm for the VRPSSTW.

4.1. Elementary shortest path problem with resource constraints and late arrival penalties (ESPPRCLAP)

By assuming all vehicles are identical, the ESPPRCLAP subproblem is solved on the same network as the VRPSSTW, withthe arc costs being reduced by the corresponding dual variables (prices) generated in the master problem using

c0ij ¼ c0ij � pi 8i 2 V ð17Þ

As no initial cardinality constraints are added for the depot (restraining the number of vehicles to be used) in the formu-lation of the master problem presented in the previous section, initially the dual variable at the depot is set as zero (i.e.p0 = 0). Later on the value of p0 is based on an additional constraint for the depot, which is added to the master problemwhen branching decision is taken on number of vehicles. The ESPPRCLAP can be formulated as:

minXði;jÞ2A

c0ijxij ð18Þ

s:t:Xi2C

di

Xj2V

xij 6 q ð19ÞXj2V

x0j ¼ 1 ð20ÞXi2V

xih �Xj2V

xhj ¼ 0 8h 2 C ð21ÞXi2V

xi0 ¼ 1 ð22Þ

si þ tij � sj 6 ð1� xijÞMij 8ði; jÞ 2 A ð23Þai 6 si 6 b0i 8i 2 V ð24Þxij 2 f0;1g 8ði; jÞ 2 A ð25Þ

InitializationOne vehicle for each

customer, Prices= π = cp

Master Problem LPOptimizations, Prices

ESPPRCLAPFeasible columns of

negative reduced cost

Label FiltrationLP is expended

Branching on kor xij

Cost of most promising

negative cost col. > Tolerance

Prices are same as previous

iteration

IntegerSolution

Terminate

IntegerSolution

Reduced Cost cij = cij - π

Yes

Yes

Yes

No

No

No

Yes

No

Fig. 5. Flow chart of column generation.


To incorporate semi soft time windows, the reduced arc cost c0ij is calculated by using c0ij as per Eq. (3). The ESPPRC is a NP-hardproblem in the strong sense (Dror, 1994), but there exists some pseudo-polynomial dynamic programming labeling algorithmfor solving it. As the ESPPRCLAP is more general and can be reduced to the ESPPRC by setting very high unit late arrival penaltycost cl (such as cl =1), it is also a NP-hard problem. The ESPPRCLAP algorithm is based on the template-labeling algorithm forshortest path problem described in Irnich and Villeneuve (2003) that incorporates the idea of permanent labels presented inDesrochers and Soumis (1998). To implement the elementary condition |V| extra resources were used as defined in Feilletet al. (2004). Basic knowledge about labeling algorithms is described in Ahuja et al. (1993) while excellent reviews of the shortestpath problem with resource constraints (SPPRC) can be found in Irnich and Desaulniers (2005) and Irnich and Villeneuve (2003).

In the labeling algorithms for various variants of shortest path problem, the dominance rule and path extension step formthe two main building blocks (Irnich and Villeneuve 2003). As long as the cost function and consumption of resources followa non-decreasing function, the dominance rules developed for any ESPPRC variant remain the same (Larsen, 1999). In thecase of the ESPPRCLAP, waiting up to ai is allowed at no cost if the vehicle arrives before ai and the cost matrix still followsthe triangular inequality (Eq. (1)), because the penalty function is non-decreasing, the same dominance rules are also appli-cable for the ESPPRCLAP. To obtain the labeling algorithm for ESPPRCLAP, new path extension rules were defined to incor-porate the time dependent late arrival penalties. The algorithm can be described in the following steps, and Fig. 6 presentsthe corresponding flow chart.

1. DefineL = {res(L), t(L), q(L), c(L), vis(L), pred(L), S(L), Label no.}

where

res(L) = resident vertex of label Lt(L) = time resource of label, which shows the service start time at res(L)

Path Extension

Li Lj

Initialization Label, L0

Sets L and Ui ; add L0 to L

L = Φ

Dominance Rule

Li survive

j = depot

| Un+1 | > 500

Yes

Add Li to Ui

Yes

Add Lj to Un+1

Yes

Delete LiNo

No

Label Selection LiTerminate

Yes

Add Lj to L

No

No

Fig. 6. Flow chart of the ESPPRCLAP.


q(L) = demand resource of the label, which shows cumulative demand along the path ending at res(L)c(L) = cost of the label, which shows the cost of the path ending at res(L)vis(L) = visited vertex resource vector of the label with |V| entries of 1 for all unreachable vertices and 0 for all reachableverticespred(L) = immediate predecessor vertex of res(L) on path p(L), represented by the label no. of the label with res(L) = pre-decessor vertexS(L) = total number of unreachable vertices from res(L)Label no. = a label counter assigning a unique number to each label

Also define L = {L} be the set of all unprocessed labels, and U(i) = set of useful labels for every i 2 V.

2. Initialization
L = {L0}, where L0 is the label created at the origin depot, and described as:L0 = {0, 0, 0, 0, I1�|V|, 0, 1, 1}, where I1�|V| = single row identity matrix with |V| columnsU(i) = U, for every i 2 V, where U = empty set.
3. Repeat steps 4, 5 and 6 until L = U4. Label selection

Take the label L0 with t(L0) that is min. from the set of unprocessed labels L and let i = res(L0),Set L(i) = {L 2 L: res(L) = i and t(L) < t(L0) + min.(tij)}.Set L ¼ L n LðiÞ.

5. Dominance rule
Apply the dominance rule to all labels in L(i) using U(i) U L(i).Let U0(i) = {L(i) | which survive the dominance rule}.Set U(i) = U(i) U U0(i).
6. Path extension
Outer loop: For each label L 2 U0(i).
Inner Loop: For all vertices j having entry 0 in vis(L), extend L to all j.If j = n + 1, i.e. dummy destination depot (a copy of depot vertex),

Add L to U(n + 1).Else, add L to L.

7. Path Filtration


The shortest path will be the one represented by the label with least cost among all labels L 2 U(n + 1); to obtain the pathit represents, link back the labels following the pred(L).

4.1.1. Dominance ruleA labeling algorithm generates new states or labels from previously generated labels, dominance rules are implemented

to avoid proliferation of labels. Next, the dominance rules are described which are same as the ESPPRC dominance rules de-scribed in Feillet et al. (2004). Consider two labels L1 and L2 both having the same resident node res(L1) = res(L2). To verifythat whether the label L1 dominates the other label L2, following dominance criteria (Eqs. (26)–(30)) are validated in the gi-ven sequence:

SðL1Þ 6 SðL2Þ ð26ÞtðL1Þ 6 tðL2Þ ð27ÞqðL1Þ 6 qðL2Þ ð28ÞvisðL1Þ 6 v isðL2Þ ð29ÞcðL1Þ 6 cðL2Þ ð30Þ

If all the abovementioned conditions hold, then L2 is deleted as any future extensions of L2 are covered by extensions of L1

(Eq. (29)) at less resource consumption (Eqs. (27) and (28)) and at less cost (Eq. (30)). Note that S(L) is not used in the dom-inance rule, actually it helps to accelerate it as condition (29) is only possible if condition (26) is satisfied.

4.1.2. Path extensionDuring the path extension step, an existing state at vertex i is extended to all new possible states by updating the resource

consumptions and cost. If a state Li with res(Li) = i is to be extended to Lj with res(Lj) = j using arc (i, j) following rules are usedto update associated resources and cost:

tðLjÞ ¼max :½tðLiÞ þ tij; aj� ð31ÞqðLjÞ ¼ qðLiÞ þ dj ð32Þ

cðLjÞ ¼cðLiÞ þ cij; if tðLjÞ 6 bj

cðLiÞ þ cij þ clðtðLjÞ � bjÞ; if tðLjÞ > bj

�ð33Þ

A newly generated label inherits the visited vertex resource vector from its processor label. This vector is further updated byassigning a value of 1 for the vertices h having a 0 entry in vis(Li) and which become unreachable from new resident vertex jdue to violation of any of the following equations:

tðLjÞ þ tjh < b0h ð34ÞqðLjÞ þ dh < q ð35Þ

Finally, the total number of unreachable nodes S(Lj) in the new state is calculated as the sum of vis(Lj) and Label no.(Li) is setas the immediate predecessor in pred(Lj). New path extension rules were defined for the ESPPRCLAP to incorporate the var-iable costs (due to possible late arrival penalty) and in vis(Li) update rule to consider relaxed time windows ½ai; b

0i� as shown in

Eqs. (33) and (34), respectively.

4.1.3. Label filtrationA label represents a path by maintaining a link to previous labels. To extract that path in terms of an ordered sequence of

customers the label is sequentially linked back to its predecessor label. In the column generation case, only those labelsbelonging to destination depot vertex with a negative reduced cost are linked backed to get a negative reduced cost columnto be added to the master LP problem. To facilitate an easy filtration process every label contains an entry pred(L), which isthe label number of its predecessor label. Let a path p be extracted from a label L 2 U(n + 1) which has a negative reduce cost,then the first entry in the path is added as (n + 1) i.e. res(L). Next, the process searches a label L0 2 U(i), i 2 C with the labelnumber equal to the pred(L) and the path is extended to res(L0). The same process is repeated after assigning L = L0 until thecomplete path is obtained in the reverse order. The original cost cp of path p based on the actual cost matrix cij is calculatedand added to the master LP problem with an additional column representing the path p, i.e. setting the value of aip = 1 forevery customer i served on path p.

4.2. Branching scheme

The branching strategy used in this study is an extension of that used in Kohl et al. (1999). However, the difference is inthe use of time dependent costs on links in the present solution that depend on the start time of service at the end vertex.When the column generation procedure terminates due to its stopping conditions with a non-integer solution, the integersolution is obtained using a branch and price scheme. If the solution contains a fractional number of vehicles k, the firstbranch is created by adding a covering constraint as given by Eq. (36) for the depot in the master LP problem, where dke rep-resents the integer obtained by rounding up k. In the other branch the value of k is rounded down and Eq. (37) is added


instead of Eq. (36). Branching on arc flow variables1 (xij) was then used to obtain an integer solution if number of vehiclesused in the solution is integer but flow variables are fractional for the arcs used in the solution. Branching decisions weremade at the subproblem level and the branching arc was chosen on the basis of Eq. (38).

1 The(Gelinagenerat

2 Conrespectcardinarespect

where

from th

dual va

returns

covered

Xp2P

a0pyp P dke ð36ÞXp2P

a0pyp 6 bkc ð37Þ

maxðc0ij �minfxij;1� xijgÞ ð38Þ

Note, that the cost c0ij includes a time dependent penalty cost in deciding the arc for branching. Once the branching arc(i, j) was decided, two child nodes were created, one with xij = 1, and the other with xij = 0. Depth first approach was usedto explore the branch and price tree; whereas, the branch with xij = 1 is explored first.

4.3. Lower bound

At every iteration of the column generation, linear relaxation solution of the restricted master LP problem gives an upperbound on the optimum value at the current branch and price node. A lower bound was also calculated at every column gen-eration iteration based on Eq. (39), which is based on the work of Vanderbeck (1994).2 Zu

LPðePÞ is the current optimum fromthe restricted LP master problem over generated columns eP , at node u; whereas, mu(p) is the minimum value of negativereduced cost column from the subproblem. The dual vector p contains dual variable for the depot vertex as well

LBuðpÞ ¼ ZuLPðePÞ þ vuðpÞ ð39Þ

Figs. 7 and 8 show the convergence of upper and lower bound found by Eq. (39), for a few of the test problems taken fromSolomon’s benchmark instances for the VRPSSTW. As the lower bound does not increase monotonously, the previous highestis taken as the current lower bound (Vanderbeck, 1994). At branching points, the upper bound jumps a little being followedby the lower bound. Sometimes a considerably larger jump (as between iterations 8 and 9 in Fig. 7) is observed when branch-ing is based on the number of vehicles (especially at K P dke, where k is the current fractional number of vehicles). Some-times the subproblem (ESSPRCLAP) is not solved to optimality rather it is forced to stop after generating a fixed number offinal state labels at the destination depot vertex (set to 500 in our application) to speed up the column generation algorithm.Therefore, Fig. 8a shows lower bounds from 8th iteration as before this the subproblem was forced to stop before reachingoptimality; whereas breaks in the lower bound curve in Fig. 8b show branch and bound nodes. From these figures, it can beobserved that the integrality gap is not dependent on the number of customers, as R101-50 was solved to optimality at theroot node while R101-25 needed some branching in the VRPSSTW with 10 min time windows relaxation. The convergence ofupper and lower bound can be different (see Fig. 8a and b) in same problem with different time windows relaxation makingthem two completely different instances. The master LP problem is started with columns representing one route for eachcustomer providing a very large Global Upper Bound (GUB). At every branching node, the column generation stops with al-most the same upper and lower bound, at this stage the present optimum Zu

LPðePÞ becomes the lower bound if the solution isnot integer. However if the solution is integer it may be marked as GUB if it is less than previous GUB. When all the branchingnodes are explored the current GUB gives the optimum integer solution.

other branching options for the VRPTW could be Ryan–Foster branching scheme (Larsen, 1999) and the branching scheme based on resource windowss et al., 1995). The branching strategy based on the binary decisions on flow variables is used in this paper as it is easier to implement in columnion (Cordeau et al., 2002).sidering the additional initial cardinality constraints of Eq. (i) and Eq. (ii), where K0 is the lower bound on number of vehicles and L0 represents the

ive upper bound, Vanderbeck (1994) describe a way to calculate a lower bound Eq. (iii). The same equation is used at every iteration even if thelity conditions have changed from K0 to K and/or L0 to L at any branching node u, i.e. minimum and maximum number of vehicles is bounded by K and L,ively.X

p2P

yp P K0 ðiÞ

Xp2P

yp 6 L0 ðiiÞ

LBuðl; t;piÞ ¼ ZuLPðeP � l0K0 � t0L0Þ þminðK0ðl; t;piÞ þ l0 þ t0; L0ðmuðl; t;piÞ þ l0 þ t0ÞÞ ðiiiÞ

l0 and t0 represent the dual variables for the initial cardinality constraints and mu(l, t, pi) is the minimum value of negative reduced cost column

e subproblem. In absence of initial cardinality conditions or setting initial cardinality conditions unrestrictive (such as L =1), the corresponding

riables will be zero. The LP optimality condition i.e. LBuðl; t;piÞ ¼ ZLPu ðePÞ reduces to stopping criterion of column generation i.e. the subproblem

no negative reduced cost column (mu(l, t , pi) = 0). A trivial initial cardinality condition is used in this paper that requires the depot vertex to be

at least once, equivalent to K0 = 1, therefore the lower bound equation reduces to Eq. (39).

Upper Bound vs Lower Bound R101-25 (VRPSSTW)

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Fig. 7. Upper and lower bound convergence for one of the 25 customer instances (VRPSSTW with 10 min of time windows relaxation).


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Fig. 8a. Upper and lower bound convergence for one of the 50-customer instances (VRPSSTW with 10 min of time windows relaxation).


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Fig. 8b. Upper and lower bound convergence for one of the 50-customer instances (VRPSSTW with 20 min of time windows relaxation).


5. Test problems

Solomon’s benchmark instances (Solomon, 1987) were used to test the performance of the algorithms and relative advan-tages of semi soft time windows over hard time windows with waiting allowed at no cost. Solomon’s benchmark instances

Table 1Demand and hypothetical time windows data in Tokyo road network based instance.

Vertex Demand (kg) ai (min) bi (min) b0i (min)

1 0 510 1440 14402 117 615 625 6353 100 574 584 5944 83 550 560 5705 133 520 530 5406 83 538 548 5587 133 529 539 5498 133 530 540 5509 300 556 566 576

10 133 595 605 61511 50 540 550 56012 133 580 590 60013 150 585 595 60514 167 525 535 54515 83 592 602 61216 150 540 550 56017 133 625 635 64518 133 610 620 63019 83 625 635 64520 117 583 593 60321 117 550 560 57022 150 547 557 56723 100 590 600 61024 83 570 580 59025 83 605 615 62526 133 601 611 62127 200 565 575 58528 117 615 625 63529 200 565 575 58530 133 560 570 58031 183 635 645 65532 133 575 585 59533 100 605 615 62534 117 645 655 66535 117 510 520 53036 100 626 636 64637 133 515 525 53538 117 635 645 65539 150 619 629 639


have 100 customers but smaller instances can be generated by taking the first few customers. The cost cij of the arc (i, j) isbased on the Euclidian distances between customers i and j; whereas the corresponding travel time tij includes the Euclidiandistance and the service time at customer i. Mainly R1-type instances with few RC1-type instances have been used to dem-onstrate the effects of utilizing time dependent late arrival penalties in the VRPTW formulation. Cumulative demand of cus-tomers divided by the capacity of vehicle provides a trivial lower bound on the number of vehicles to be used in a testinstance. For R1-type instances, this lower bound assigns 2, 4 and 8 vehicles to 25, 50 and 100 customer instances, respec-tively. The number of vehicles in hard time windows optimal solutions is much larger than this trivial bound and thus vehi-cle utilization effect of hard time windows is rather pronounced in this type of instances. On the contrary, for the C1-type,hard time windows optimal solutions assign 3, 5 and 10 vehicles to 25, 50 and 100 customers, respectively which are same asthe trivial bounds. RC1-type contains mixed clustered and randomly located customers with wider time windows than theR1-type. The number of vehicle used in their hard time windows optimum solution, is not much larger than that required bytrivial capacity constraint. We tried to solve 25- and 50-customer instances taken from a few of the R1-type and RC1-typeinstances.

Along with Solomon’s benchmark instances one practical instance was also solved. The instance was derived fromTokyo Metropolitan Road Network data. Customer locations are represented by the outlets of a chain of conveniencestores with their demands being known. Time windows were generated randomly with a width of 10 min, and a fixedservice time of 10 min is considered at every customer vertex. Table 1 shows actual demand and hypothetical timewindows data for the convenience stores and depot. Fig. 9 shows the study area along with customers’ locations;depot is located at vertex 1. Data input in the model is of the form of an origin destination matrix which is obtainedfrom link based road network data. A shortest path was found from the depot to all customers, between each pair ofcustomers and from customers to the depot, thus making an unsymmetrical travel time matrix, using Dijkstra’s algo-rithm.

Fig. 9. Test instance’s road network and customer location.


6. Results and discussions

The algorithms were implemented in MATLAB, and were run on a computer with 2.41 GHz AMD Athlon with 64 � 2 dualcore processors with 2 GB of RAM. Results of the presented column generation exact solution scheme for the VRPSSTW arecompared with the exact solution results of the VRPHTW. Same column generation algorithm settings have been used forhard time windows as for the semi soft time windows variant such as same rules for earlier termination of the subproblem(after generating 500 labels at destination depot). The difference was the subproblem; the ESPPRC was solved for theVRPHTW and for the VRPSSTW the subproblem was ESPPRCLAP, which included the late arrival penalties and consideredthe fixed cost of vehicle utilization in path costs cp. For the practical instance vehicle operating cost (VOC) was taken as14.02 ¥/min and a fixed cost of 10417.50 ¥/vehicle. Furthermore, the slope of late arrival penalty curve (i.e. the unit late ar-rival penalty cost cl) was set to five times that of the VOC. These parameter settings were based on an interview survey oflogistics firms in Japan. A scaled cost matrix was used in programs taking VOC = 1 (travel cost = travel time), all costs werealso scaled to same level.

6.1. Performance evaluation of VRPSSTW on Solomon’s benchmark instances

6.1.1. Maximum time windows relaxationThe limit b0i at every vertex i can be defined as per Eq. (2), which also ensures that if the path travels the arc (i, n + 1) it

should still remains feasible i.e. it obeys [an+1, bn+1] where n + 1 represents the sink vertex. Note, that time window at thesink vertex is not extended i.e. a feasible route still follows the global time window constraint or the maximum scheduling

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R101-25

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Fig. 10. Comparison of computation time, cost and number of labels per subproblem in the VRPSSTW and in the VRPHTW.


time constraint. The above feasibility condition produces very large relaxed time windows, and the complexity of the label-ing algorithm depends on the width of time windows (Desrochers et al., 1992), therefore, it was also tried to relax the latestpossible service start times in steps of 10 units. We could only solve 25 customer instances using time windows relaxationlimit as per Eq. (2). Fig. 10 shows the comparison between time, cost and number of labels generated per subproblem iter-ation, for various levels of time windows relaxations for few 25 customer instances in the VRPSSTW and the VRPHTW. Theratio of number of labels and computation time of the VRPSSTW to the VRPHTW increased rapidly for maximum time win-dows relaxation found as per Eq. (2), even for these small 25 customers instances. The solutions for larger instances (50 cus-tomers) with maximum time windows relaxation could not be obtained due to excessive computation time. While timewindows relaxations of 10 or 20 units present better choices as both number of labels and computation time ratios are con-siderably less than those in maximum relaxation.

The VRPSSTW is a relaxation of the VRPHTW, and thus cost savings depend on the fact that how severely hard time win-dows constrain the problem. Narrow time windows (tightly constrained problems) will result in a better relaxed solution ascan be seen in Fig. 10c. Here, the cost decreases with an increase in the time windows relaxations for tightly constrained(small time windows width) instances such as R101-25, and R102-25 and to some extent in R105-25. Both R101-25 andR102-25 have 100% and 72% customers with binding time windows of 10 min interval, whereas all customers in R105-25have time windows of 30 min interval. On the other hand, very wide time windows hardly constrain the problem and thus,the relaxed solution, at most, would lead to same solution cost as the original, but with more computation requirements ascan be seen in the case of R104-25. Only 7 out of 25 customers have binding time windows in R104-25; whereas, all remain-ing customers have very wide time windows almost similar to the time windows of depot. R103-25, which has 52% custom-ers with time windows of 10 min interval, presents some interesting results as all three relaxations resulted in same costalthough number of labels and computation time ratio followed the normal trends. With a late arrival time of only 0.2 units(causing late arrival penalty of only 1 unit) it was possible to save one vehicle route in the optimum solution of the VRPSSTW.

6.1.2. Time windows relaxation by 10 minTable 2 summarizes the results for the VRPSSTW for the 10 min time windows relaxation settings. Table 3 gives the cor-

responding summary for the VRPHTW. Column 2 in Table 2 gives the number of feasible arcs; as discussed earlier, network


size (number of feasible arcs) for VRPSSTW is considerably larger than VRPHTW. Column 3 specifies the number of branchand bound nodes. Column 4 gives LP lower bound obtained at the root node of the branch and bound tree when set parti-tioning LP stops due to its stopping criteria i.e. either subproblem returns no negative reduced cost column or if the simplexmultipliers (prices) are same as previous iteration. Column 5 gives the optimum integer solution obtained at the end of col-umn generation algorithm; whereas Column 6 shows the solution cost which includes the fixed vehicle cost, travel cost ofthe used arcs as well as the late arrival penalty. Column 7 shows the number of used vehicles (number of routes). In theVRPSSTW, relaxing the time windows and allowing late arrival causes a late arrival penalty shown in Column 8. The late ar-rival penalty is less for instances having large time windows such as in R103-25 and R104-25. Column 9 reports the waitingtime. Columns 10 and 11 show the number of iterations and number of columns in the master LP problem for the columngeneration algorithm when optimum solution is found. Column 12 gives the average number of labels generated in each runof the ESPPRCLAP in the root node.

A comparison of the above tables shows that the VRPSSTW is much harder to solve than the VRPHTW. Many 25 customerinstances were solved to integrality at root node in the VRPHTW along with a few larger instances; on the other hand manysmaller 25 customers’ instances required branching in the VRPSSTW solutions. In addition, integrality gap between the LPlower bound (LB1) and the final optimum solution (IP-opt) is much larger in the VRPSSTW than the corresponding gapsin the VRPHTW. In general, the VRPSSTW case resulted in higher number of branch and bound nodes and number of columngeneration iteration explaining partly, the high computation time ratio of the VRPSSTW to that of the VRPHTW (Tables 4 and6, Column 7).

Table 4 provides detailed comparison of the VRPHTW with the VRPSSTW with 10 min relaxed time windows. Column 2describes the percentage increase in the number of feasible arcs in the VRPSSTW network to those in the VRPHTW network.Column 3 shows the percentage reduction in overall cost (travel plus fixed vehicle cost) in the VRPSSTW as compared to theVRPHTW. Column 4 gives the decrease in waiting time, Column 5 reports the reduction in number of vehicles, and Column 6shows the ratio of number of labels generated while solving the ESPPRCLAP vs. the ESPPRC. Finally, ratio of computation timeof the VRPSSTW to computation time of the VRPHTW is shown in Column 7.

The main contribution of adding late arrival penalties to the VRPTW was to reduce vehicle utilization and to save overallcost of the solution. Out of all the tested instances, almost all (except R104-25) showed a reduction in total cost of the solu-tions as well as reduction in the number of vehicles. When time windows relaxation was caped at a maximum of 10 min, as

Table 3Summary of exact solutions of the VRPHTW using column generation.

Instance Arcs BB LB1 IP-opt Cost K WT It. Column Lab.(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

R101-25-HTW 224 1 617.100 617.1 6561.1 8 208.5 2 225 523R101-50-HTW 812 1 1044.000 1044.0 9960.0 12 418.5 4 450 2437R102-25-HTW 412 1 547.100 547.1 5748.1 7 318.9 3 325 3171R102-50-HTW 1503 1 909.000 909.0 9082.0 11 306.5 10 1039 12,982R103-25-HTW 524 1 454.600 454.6 4169.6 5 167.4 8 704 7114R103-50-HTW 1856 4 769.233 772.9 7459.9 9 196.2 39 2350 43,241R104-25-HTW 579 1 416.900 416.9 3388.9 4 105.0 6 625 12,551R105-25-HTW 300 1 530.500 530.5 4988.5 6 96.9 4 360 1306R105-50-HTW 1067 12 892.120 899.3 7586.3 9 127.7 47 1159 6333RC101-25-HTW 276 8 406.625 461.1 3433.1 4 43.5 27 759 2009RC101-50-HTW 832 22 850.020 944.0 6888.0 8 111.3 90 3313 5329RC105-25-HTW 370 1 411.300 411.3 3383.3 4 106.6 7 686 4133RC105-50-HTW 1241 10 761.558 855.3 6799.3 8 203.1 61 2757 13,926

Table 2Summary of exact solutions of the VRPSSTW using column generation with 10 min time windows relaxation.

Instance Arcs BB LB1 IP-opt Cost K LAP WT It. Column Lab.(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

R101-25-SSTW 267 2 5553.900 5841.4 5841.4 7 35.0 184.3 9 333 941R101-50-SSTW 943 1 7952.500 7952.5 7952.5 9 199.5 92.5 10 1038 4760R102-25-SSTW 433 2 4559.425 4985.5 4985.5 6 12.0 306.5 14 865 6924R102-50-SSTW 1594 6 6501.033 6909.7 6909.7 8 79.5 102.7 76 5481 51,981R103-25-SSTW 531 1 3444.200 3444.2 3444.2 4 1.0 104.8 12 1225 23,060R103-50-SSTW 2032 6 5501.987 5998.7 5998.7 7 9.0 130.0 107 7563 222,191R104-25-SSTW 581 2 2996.856 3388.9 3388.9 4 0.0 105.0 25 2200 71,837R105-25-SSTW 338 2 4249.000 4249.0 4249.0 5 11.0 79.7 10 691 3019R105-50-SSTW 1222 11 6421.570 6890.9 6890.9 8 20.5 58.5 82 3358 20,143RC101-25-SSTW 305 1 2665.000 2665.0 2665.0 3 69.0 7.9 9 781 2386RC101-50-SSTW 957 15 5506.780 6349.7 6349.7 7 239.0 54.7 101 5756 9846RC105-25-SSTW 415 1 2631.100 2631.1 2631.1 3 40.0 33.4 23 2074 6589RC105-50-SSTW 1391 24 4850.513 6300.0 6300.0 7 126.0 49.3 208 11,930 29,191


many as three vehicles were saved in R101-50, and R102-50. On average about 1.31 vehicles were saved per tested instance(average over 13 tested instances). Cost savings were as much as 23.92% in R102-50, whereas average for all tested instancewas 14.54%. Relaxing the time windows and allowing late arrivals, adds a cost component of late arrival penalties (LAP) tothe total solution cost but it is interesting to note that it contributes only a small fraction in the total solution cost. The max-imum being 3.76% in RC101-50 and averaging at only 1.16% of the total solution cost in the VRPSSTW. The increase in net-work size varied from only 0.35% for R104-25 to 19.20% for R101-25. With this increase in network size, average number oflabels generated in each run of the ESPPRCLAP was significantly higher than the average number of labels generated in asingle run of the ESPPRC. Column 6 in Table 4 shows that as much as 5.72 times more labels were generated in a singlerun of the ESPPRCLAP than the ESPPRC while attempting to solve R104-25 instance. This increase in the number of labelsper subproblem along with the larger size of branch and bound tree and higher number of column generation iterations, con-tributed to higher computation times in the VRPSSTW that soared up significantly when compared with the VRPHTW. An-other reason could be the use of a large fixed cost in the cost of all outgoing arcs from the depot that provided very largeupper bounds on the prices or dual variables’ values, which required more computation time and column generation itera-tions to reach stable optimum values of the dual variables (Desaulniers, 2007). Furthermore, higher prices in the column gen-eration lead to difficult subproblems requiring more time to solve an iteration of subproblem (Kallehauge et al., 2006).

Table 6Comparison of exact solutions of the VRPHTW and the VRPSSTW using column generation with 20 min time windows relaxation.

Instance Network increase (%) Decrease in cost (%) Decrease in waiting time (%) Vehicle saved Labels ratio Time ratio(1) (2) (3) (4) (5) (6) (7)

R101-25 33.93 �25.94 �70.79 3 3.32 11.98R101-50 31.77 �22.41 �88.72 4 4.00 39.80R102-25 9.47 �28.89 �93.20 3 2.83 15.52R102-50 12.04 �29.79 �85.51 4 5.76 68.71R103-25 2.29 �17.40 �37.40 1 3.77 18.97R103-50 12.34 �26.97 �79.26 3 5.03 60.56R104-25 1.04 0.00 �1.43 0 7.97 75.66R105-25 24.33 �24.92 �72.45 2 2.71 7.05R105-50 28.21 �14.97 �89.90 2 4.61 10.78

Table 5Summary of exact solutions of the VRPSSTW using column generation with 20 min time windows relaxation.


R101-25-SSTW 300 3 4794.167 4859.3 4859.3 5 592.5 60.9 8 413 1735R101-50-SSTW 1070 8 7662.656 7728.4 7728.4 8 801.5 47.2 53 1616 9754R102-25-SSTW 451 2 4047.825 4087.4 4087.4 4 506.5 21.7 17 1539 8979R102-50-SSTW 1684 3 6306.157 6376.3 6376.3 7 289.0 44.4 57 4709 74,752R103-25-SSTW 536 1 3444.200 3444.2 3444.2 4 1.0 104.8 13 1325 26,789R103-50-SSTW 2085 8 5399.400 5448.2 5448.2 6 174.5 40.7 115 9019 217,319R104-25-SSTW 585 2 2849.579 3388.9 3388.9 4 0.0 103.5 27 2414 100,081R105-25-SSTW 373 1 3745.200 3745.2 3745.2 4 284.5 26.7 7 633 3543R105-50-SSTW 1368 16 6406.839 6450.5 6450.5 7 312.0 12.9 63 3083 29,209

Table 4Comparison of exact solutions of the VRPHTW and the VRPSSTW using column generation with 10 min time windows relaxation.

Instance Network increase (%) Decrease in cost (%) Decrease in waiting time (%) Vehicle saved Labels ratio Time ratio(1) (2) (3) (4) (5) (6) (7)

R101-25 19.20 �10.97 �11.61 1 1.80 5.67R101-50 16.13 �20.16 �77.90 3 1.95 7.03R102-25 5.10 �13.27 �3.89 1 2.18 6.21R102-50 6.05 �23.92 �66.49 3 4.00 46.21R103-25 1.34 �17.40 �37.40 1 3.24 11.58R103-50 9.48 �19.59 �33.74 2 5.14 52.71R104-25 0.35 0.00 0.00 0 5.72 40.64R105-25 12.67 �14.82 �17.75 1 2.31 6.54R105-50 14.53 �9.17 �54.19 1 3.18 7.28RC101-25 10.51 �22.37 �81.84 1 1.19 0.56RC101-50 15.02 �7.82 �50.85 1 1.85 1.97RC105-25 12.16 �22.23 �68.67 1 1.59 5.87RC105-50 12.09 �7.34 �75.73 1 2.10 4.32

Table 7Summary of exact solutions of the VRPSSTW and the VRPHTW.


TD1-38-SSTW 782 12 3173.253 3255.7 3255.7 4 88 19.8 113 2348 30,380TD1-38-HTW 707 11 172.300 176.800 3891.8 5 0 119.5 81 278 7920


6.1.3. Time windows relaxation by 20 minTable 5 provides the solution details for the VRPSSTW with 20 min time windows relaxation, and Table 6 provides its

comparison with the VRPHTW solution. The column headings are same for Tables 5 and 6 as in Tables 2 and 4, respectively.Compared with 10 min time windows relaxation, it provides even better comparative figures when compared with theVRPHTW. The number of vehicles saved per tested instance increased from 1.31 to 2.44 vehicles per instance with a max-imum savings of four vehicles in R101-50 and R102-50. Average savings in cost per tested instances improved from 14.54% to21.25% with the highest individual reduction being 29.79% for R102-50 instance. Once again the contribution of LAP in totalcost remained low and averaged 6.13% of total solution costs. Both the network size and the ratio of labels increased quiterapidly too, resulting in very high computation time ratios, as much as 7.97 for R104-25 compared with 5.72, the highest in10 min time windows relaxation for the same instance. Even though this ratio seems to be high, and at first glance reducesthe effectiveness of the VRPSSTW, it should be kept in mind that with massive increase in available computation power theexact solutions of the VRPHTW are found within seconds. Thus, a figure of 21.25% represents a highly significant cost reduc-tion per tested instance even with higher computation time ratios.

Although reducing the waiting time was not used as an objective for either the VRPSSTW or the VRPHTW, the VRPSSTWresulted in considerably less waiting time (Column 4, Tables 4 and 6). The average reduction was found to be 44.62% and68.74% per tested instances in the VRPSSTW with 10 and 20 min time windows relaxations, respectively. This reductionin waiting time is meaningful in two ways. First, it saves additional cost as, to a minimum, waiting time results in additionalvehicle utilization hours and increase in vehicle operating cost. Secondly, it helps reduce traffic related problems such as on-street parking and congestion due to waiting delivery vehicles on street sides. Most of the times delivery vehicles wait nearthe next client location to be served in engine on state, thus not only causing on-street parking problem but continuouslycontributing environmentally non-friendly exhausts as well. Therefore, a reduction in waiting time not only helps logisticsfirms, it helps the cause of the city administration and general public (residents of the area of service) by reducing environ-mentally non-friendly emissions.

6.2. Performance evaluation of VRPSSTW on real life instance

Table 7 gives the solution details for the VRPHTW and the VRPSSTW with 10 min time windows relaxation for the realisticinstance. Column definitions are same as the corresponding columns in Tables 2 and 5. Once again the number of arcs,branch and bound nodes, subproblem iterations, number of columns in LP and labels generated per subproblem are higherin the VRPSSTW compared to the VRPHTW, making it comparatively difficult to solve. The solution cost including fixed vehi-cle cost, travel cost and any late arrival penalty cost converted to the original VOC of 14.02 ¥/min was 45647.50 ¥ in theVRPSSTW compared to 54566.20 ¥ in the VRPHTW, showing cost savings of 16.34%. In the VRPSSTW solution, four vehicleswere required compared to five vehicles in the VRPHTW solution.

Similar to Solomon’s test instances, the VRPSSTW resulted in massive reduction (83.4%) in waiting time in this realistictest instance, as compared to the VRPHTW solution. It shows that the reduction in waiting time is not just problem specific assimilar results were also reported by Qureshi et al. (2007), when they used the presented STW settings in their heuristicanalysis of the VRPTW. Once again LAP were only a small fraction (2.70% only) of the total solution cost in the VRPSSTW.

7. Conclusions

In this paper, time dependent late arrival penalties have been considered in the formulation of the VRPTW, which resultedin the formulation of the VRPSSTW. Column generation technique available for the VRPHTW was extended for the VRPSSTWby solving a new ESPPRCLAP subproblem instead of the ESPPRC. New path extension rules were defined for cost calculationand for update of visiting resource vector to obtain a labeling algorithm for the ESPPRCLAP. As mentioned in the literaturereview, so far only heuristics (such as Gendreau et al., 1999; Taillard et al., 1997) have been developed for the VRPSSTW.Therefore, the main contribution of the paper is the development of an exact solution approach for the VRPSSTW. The exactoptimization approach presented in this paper has significant advantages over the heuristics approaches used in the litera-ture, such as, exact evaluation of many characteristics of the soft time windows. This paper also presented such an exactevaluation, where the cost saving characteristics of soft time windows were assessed using the exact solutions of theVRPSSTW and the VRPHTW. Furthermore, the presented approach gives the lower bound exact solution for the VRPSSTWthat can not be further improved; whereas, the available heuristics do not guarantee the lower bound solution nor they pro-vide any information about the obtained approximate solution that how close it is to the optimal solution. Therefore, the


new exact approach will help increase the confidence of logistics managers in the VRPSSTW solutions. The exact VRPSSTWsolutions can be used to calculate the errors (between optimal and approximate solutions) and to calibrate the heuristics onsmaller instances before the heuristics are used to solve large scale instances.

An application of the VRPSSTW model and the developed column generation exact solution was also presented on a real-istic logistics instance based on the Tokyo Metropolitan road network. The VRPSSTW was able to significantly reduce theoverall cost, waiting time and the number of vehicles used. On the other hand number of labels per subproblem iterationwas very high in VRPSSTW as compared to VRPHTW and thus it required more computation time. More subproblem itera-tions and bigger branch and bound tree were the other factors for higher computation times.

Therefore, at this stage the presented exact solution approach for the VRPSSTW is only recommended to smaller size prac-tical instances. Reduced waiting time and less number of vehicles, not only effect the overall cost; these can be helpful inreducing traffic related problems such as on-street parking, congestion and other environment related issues. The only stake-holders that might lose would be the clients, but late arrival penalties as well as some proportion of the saved cost can beused to offer good concessionary packages to clients as well for any inconvenience caused by the late arrival. Moreover, fewand not all time windows can be relaxed to satisfy clients with very strict time windows or the ones which are very impor-tant to logistics firms.

It is important to note that the VRPSSTW is a relaxation of the VRPHTW, and thus cost savings depend on the fact thathow severely hard time windows constrain the problem. Narrow time windows (tightly constrained problems) will resultin better relaxed solution, while very wide time windows hardly constrain the problem. Thus, the relaxed solution, at most,would lead to same solution cost as the original, but with more computation requirements as was seen in the VRPSSTW solu-tion of R104-25. Moreover, the VRPSSTW results depend on the slope of the late arrival penalty curve (i.e. the unit late arrivalpenalty cl). A value of five times VOC is used in this paper, which was based on an interview survey of logistics firms in Japan,other values can also be tried to gauge the sensitivity of the VRPSSTW results.

TABLE A1. LIST of Abbreviations

Abbreviation
Full name
ESPPRC
Elementary Shortest Path Problem with Resource Constraint ESPPRCLAP Elementary Shortest Path Problem with Resource Constraint and Late Arrival Penalties GUB Global Upper Bound LAP Late Arrival Penalties SPPRC Shortest Path Problem with Resource Constraint STW Soft Time Windows VOC Vehicle Operating Cost VRP Vehicle Routing and scheduling Problem VRPTW Vehicle Routing and scheduling Problem with Time Windows VRPHTW Vehicle Routing and scheduling Problem with Hard Time Windows VRPSTW Vehicle Routing and scheduling with Soft Time Windows VRPSSTW Vehicle Routing and scheduling Problem with Semi Soft Time Windows TSP-CHTWPC Traveling Salesman Problem with Capacity, Hard Time Windows and Precedence Control
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An exact solution approach for vehicle routing and scheduling problems with soft time windows