Solution Methods for the Periodic Petrol Station

TJER 2013, Vol. 10, No. 2, 69-77

Solution Methods for the Periodic Petrol StationReplenishment Problem

C Triki

Mechanical and Industrial Engineering Department, College of Engineering, Sultan Qaboos University, PO Box 33, PC123, Al-Khoud, Muscat - Sultanate of Oman

Received 19 June 2012; accepted 20 February 2013

Abstract: Abstract: In this paper we introduce the Periodic Petrol Station Replenishment Problem(PPSRP) over a T-day planning horizon and describe four heuristic methods for its solution. Even thoughall the proposed heuristics belong to the common partitioning-then-routing paradigm, they differ inassigning the stations to each day of the horizon. The resulting daily routing problems are then solvedexactly until achieving optimalization. Moreover, an improvement procedure is also developed with theaim of ensuring a better quality solution. Our heuristics are tested and compared in two real-life cases,and our computational results show encouraging improvements with respect to a human planning solu-tion

Keywords: Petrol delivery, Periodic constraints, Vehicle routing problem

__________________________________________*Corresponding author’s e-mail: [email protected]

1. Introduction

The problem of planning petrol delivery to distri-bution stations is well-recognized in the operationsresearch literature under the subject name petrol sta-tion replenishment problem (PSRP). It consists ofsimultaneously making several decisions, includingdetermining the minimum number of trucks required,assigning the stations to the available trucks, defininga feasible route for each truck, settling whether drivers'work overtime is required, etc. The objective to beachieved is usually defined as the minimization of thetraveled distance by the trucks to serve all of the dis-tribution stations.

Most of the studies presented in the literature solvethis problem over a time period of one single day,ignoring the possibility that solving the PSRP over a T-day planning horizon may yield delivery savings forthe company (Cornillier et al. 2012). However, mostof the PSRPs are characterized by the fact that the sta-tions should not be served at each day of the T-dayplanning horizon but rather through a specified num-ber of times. This would introduce a further character-ization of the PSRP; namely, the periodic nature of theapplication. Specifically, each station (i) must be

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served (ri) on certain days within the time horizon, andthese service days are assigned to i by selecting one ofthe feasible combinations of ri service days with theobjective of minimizing the total distance traveled bythe delivery trucks.

In this paper, we define a new variant of the abovedescribed problem that we will denote as the periodicpetrol station replenishment problem (PPSRP). Wepropose a novel three-phase heuristic method for itssolution. Our approaches consist of first assigning theservice combinations by solving different integer pro-gramming models, then defining the routing for eachday and, finally, using a local improvement techniqueto further reduce the delivery distance. A real-lifeapplication was chosen for testing and assessing theperformance of our solution methods. For some of theproposed heuristics, satisfactory results were obtainedwith respect to the solution currently adopted by thecompany.

The paper is organized as follows: the next section,section 2, will be devoted to surveying the relatedworks proposed in the literature and highlighting theoriginal contribution of this paper. Section 3 will for-mally define the PPSRP and describe the periodicityconstraints to be implemented. Section 4 will be dedi-cated to the development of our heuristic approachesto solve the PPSRP. The computational performanceof the heuristics will be discussed in section 5 and, finally, some concluding remarks and future develop-ments will be drawn in section 6.

2. Literature Review

The problem introduced in this paper finds its rootin two different classes of distribution problems thatare broadly studied in the literature; namely, the PSRPand the periodic vehicle routing problem (VRP). Fromone perspective, the PPSRP is a natural extension ofthe PSRP as it prolongs the planning horizon to covera T-day period. The 1-day variant of the problem hasbeen subjected to extensive research activity, and sev-eral mathematical models and numerical methods havebeen proposed for its solution. A limited list of theseworks includes a collection of papers by (Cornillier etal. (2008a, 2008b, 2009, 2012) and the contributionsof (Brown and Graves 1981; Brown et al. 1987; BenAbdelaziz et al. 2002; Rizzoli et al. 2003; Ng et al.2008; Surjandari et al. 2011 and Boctor et al. 2012).Only a few papers have considered the multi-periodaspect in the context of PSRP, but none of them includ-ed explicitly the periodic nature of the problem.

Taqa Allah et al. (2000) specifically have extendedthe planning horizon of the PSRP to the multi-periodcase and have proposed several heuristic methods forthe solution of the so-called MPSRP. In their paper,they examined a single depot and an unlimited homo-geneous truck fleet.

Malepart et al. (2003) also tackled the multi-visitpetrol station replenishment problem (MPSRP) in areal-life context and included a case in which the dis-tribution company would decide for some stations thequantity to be delivered at each visit.

Cornillier et al. (2008b) penned the most recentwork, which considered a multi-period horizon andproposed a heuristic that maximizes the profit, whichis calculated as the difference between the deliveryrevenue and the sum of regular and overtime drivers'costs.

Thus, to the best of our knowledge this paper rep-resents a first tentative step in tackling a PSRP whichis characterized by the periodic service requirementsof the petrol stations.

On the other hand, the PPSRP can find its roots inthe literature of the periodic VRP that attracted theinterest of many researchers over the previous twodecades (Gaudioso and Paletta 1992; Chao et al. 1995;Cordeau et al. 1997). The two problems have manysimilarities, but the main difference between the twoconsists of the fact that the PPSRP should includeadditional constraints (for example, those related tothe drivers' shifts' lengths).

3. Problem Description

The objective of the PPSRP consists of finding aminimum distance routing to serve all the stationswhen considering an extended planning horizon of T-days.

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Solution Methods for the Periodic Petrol Station Replenishment Problem

The resulting network model can be expressed interms of combinatorial optimization having two sets ofdecision variables. The first set defines the routingsolution and the corresponding truck assignment, (ie. itwill give clear insight on which truck will serve whichstations in which day in order to satisfy all thedemand). The second set of variables will choose foreach station the best periodicity combination that willminimize the overall delivery distance. The integermodel with all its constraints is cumbersome and rep-

resents only slight differences with respect to themathematical model reported (Cornillier et al. 2008b).For this reason and for brevity we avoid reporting thewhole model in this paper and we focus in the sequelon the innovative heuristic methods developed for itssolution.

4. Heuristic Algorithms

Since PPSRP instances are generally characterizedby a high number of variables and constraints duemainly to the extended time horizon and also to thelarge number of stations to be served, we have chosento solve the problem heuristically. We propose herefour different heuristics all based on the idea of split-ting the task of assigning the periodicity combinationsto the stations from that of performing the routingrather than dealing with both the tasks simultaneously(Fig. 2). The routing task then is carried out by solv-ing the resulting VRPs-one for each day of the plan-ning horizon. Finally, all of the heuristics are conclud-ed by running an improvement procedure that is com-mon to all four approaches.

4.1 Combination Assignment ProcedureThe procedure of assigning feasible combinations

to the stations is carried out by solving exactly, usingany general purpose software, one of the integer pro-gramming models proposed below. The notation to beused through this section is the following:

Figure 1. An example of a region to be served (left) and its corresponding connected network (right). Thestations correspond to the nodes of the graph and the roads are the undirected edges

Table 1. Periodic sequence combinations for T = 6

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4.1.1. Heuristic 1: Minimum Daily DemandThe idea here is to minimize the total demand to be

served on each single day of the planning horizon. Inthis way, the workload will be as balanced as possibleand, consequently, the number of vehicles will be min-imized on a daily basis. While this procedure has theadvantage of being easy to understand and implement,it has the disadvantage of not taking into account thegeographical position of the stations. The proposedinteger program has yir as the decision variables thatdefine the periodicity combination to be assigned toeach station, as follows:

yir is 1 if combination r is assigned to station i, 0otherwise.

Moreover, by denoting by Dt the total quantity ofpetrol to be served on day t, the mathematical modelcan be expressed as:

(1)

(2)

(3)

(4)

The objective function (1) minimizes the highestquantity of petrol to be served over all the days of theplanning horizon. It is expressed here as a min-maxobjective function but it can be easily transformed to astandard minimization problem by using conventionaloperations research techniques. Constraints (2) forcethe model to choose only one periodic combinationamong all the feasible combinations. Constraints (3)restrict the daily quantity (Dt) to be delivered to matchat least the total demand to be served on day t. Finally,constraints (4) impose the domain of the decision vari-ables.

4.1.2. Heuristic 2: Minimum Daily StationsThe idea of this heuristic is to minimize the per day

total number of stations to be served. Even in this case,the attempt aimed at balancing the workload wasexpressed in terms of the number of stations and, con-sequently, at minimizing the number of vehicles to beused on a daily basis. This heuristics shares the sameadvantage as its predecessor of being easy to under-stand and implement and, moreover, ensures a bal-anced workload for the trucks over the T days of thehorizon. Both the heuristics also share the same disad-vantage of ignoring the geographical position of thestations.

By using the same binary variables yir as definedpreviously and by introducing a new variable Wntdefining the number of stations to be visited duringday t, the mathematical model is expressed as:

(5)

(6)

(7)

(8)

(9)

The objective function (5) makes use of the newvariable Wnt in order to minimize the highest numberof stations to be served over all the days of the plan-ning horizon. Constraints (6) assigns as before onlyone periodic combination to each station. Constraint(7) imposes logic relations between the new variableWnt and the assignment variables yir on a daily basis,and the domain of the decision variables is definedthrough constraints (8) and (9).

4.1.3. Heuristic 3: Depot Minimum DistanceThe aim of this heuristic is to solve a relaxation of

a PVRP-like process in order to minimize the total dis-tance between the depot (node 0) and all the stations tobe served. Even though the resulting model is a sim-plified version of the PVRP, it will maintain some ofthe complexity but on the other hand it has the advan-tage of making use of more information since it takesinto consideration the geographical position of the sta-tions with respect to the depot.

The integer model has two sets of variables. Besides,the binary variables yir as previously used we need tointroduce also the following binary variables:

x0it is 1 if station i is assigned to the depot (node 0)on day t, and 0 otherwise.

n1,...,i1y

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The optimization model can be summarized as fol-lows:

(10)

(11)

(12)

(13)

(14)

(15)

The objective function (10) minimizes the total dis-tance of all the stations with respect to the depot overthe days of the planning horizon. Constraints (11) hasthe same role as in the previous models, whereas con-straint (12) imposes logic relations between the twosets of variables x0it and yir. The inequalities of (13)are capacity constraints that ensure, for each day t, therespect of the trucks' capacity and, finally, the integri-ty of the decision variables that are defined by con-straints (14) and (15). It should be noted that thismodel is similar to that in (1) through (4). The maindifference between the two problems lies in the objec-tive function (1), which becomes (10).

4.1.4. Heuristic 4: Minimum Distance ClustersThis heuristic is an extension of the previous one,

allowing the minimization of the total distance of thestations with respect to virtual centers, with one foreach day of the horizon, rather than considering onlythe depot. This clustering strategy allows better use ofthe information about the distances between stationsand, hopefully, generates more compact sets of sta-tions from a geographical viewpoint. The obvious dis-advantage is the complexity of the resulting modelinvolving a large number of binary variables.

For simplicity, the virtual centers are chosen hereamong a set of specific positions that coincide withsome of the stations, so that the topology of the net-work remains unaffected. The integer model makesuse of the variables yir and, moreover, introduces thefollowing two sets of binary variables:

xijt is 1 if station i is assigned to the virtual cen-tre j in day t, and 0 otherwise

zjt is 1 if virtual centre j is assigned to day t, and 0 otherwise

The integer model corresponding to heuristic 4 is asfollows:

(16)

(17)

(18)

(19)

Figure 2. The four heuristic solution approaches

Heuristic 1

Heuristic 2

Heuristic 3

Heuristic 4

Minimum Daily DemandModel

Minimum Daily StationsModel

Minimum Daily DistanceModel

Minimum Daily ClustersModel

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(20)

(21)

(22)

(23)

Similar to the previous heuristic, the objective func-tion (16) minimizes the total distance of all the stationswith respect to the T virtual centres (one for each dayof the planning horizon). All the other constraintshave the same meaning and role as in the previousheuristic except for some slight adjustments due topresence of T virtual centres and the introduction ofthe new set of variables zjt. We should mention that thenotation may be further simplified in the above modelby merging the indexes j and t since they represent thesame set of horizon days and have a one-to-one rela-tionship but we preferred, for the sake of clarity, to dis-tinguish between the temporal index t and the spatialindex j.

Finally, we should note that the last two heuristicsseem to be more challenging computationally sincethey include the geographical aspect of the problemand, thus, involve additional decision variables withrespect to the former two methods. Possible higherquality solutions to be yielded by the latter heuristicsmay be paid by longer computational time when solv-ing large-scale problems. This time-quality trade offmay be very useful when solving several instanceswith different sizes.

4.2. Solving the VRPs: Routing ProcedureOnce the hard task of assigning the stations to the

days of the planning horizon while keeping in mindthe periodicity constraints on each station has beensolved, we need to define the routes for each day. Therouting procedure takes as input the sets of stationscorresponding to each day (ie. the output of any of themodels of subsection 4.1) and generates a set of routesfor the available trucks. This requires nothing morethan solving independently one T standard VRP foreach day of the horizon. Since each resulting VRP isrelatively small with respect to the original PPSRP, wedecided to solve these T problems exactly by using astandard VRP model and a general purpose optimiza-tion software package.

4.3. Getting a Better Solution: Improvement Procedure

The final step of our approach consisted of tryingto improve the quality of the solution by further mini-mizing the overall distance through the application of

a local search technique. Theoretically, room for itsimprovement should be available since the assignmentof the stations to the T days has been done heuristical-ly without any guarantee to get the best feasible solu-tion. Starting from a set of routes assigned to the avail-able trucks for every day of the planning horizon, thetechnique employed here is based on switching anytwo stations assigned to two different service days andchecking if any improvement is taking place. Whilechecking the improvement in the total distance at eachswitching station, we should check also the feasibilityof the solution expressed in terms of service periodic-ity of each station and the capacity of the correspon-ding trucks. This procedure is easy to understand butrequires considerable effort for its implementation,since no standard software is available to achieve thistask.

5. Computational Results

This section is dedicated to the application of ourheuristic methods to solving a real-life test problem,and to analyzing and comparing their performances.All the models introduced above were implementedusing the LINGO commercial package, Version 8.0(LINDO Systems, Inc., Chicago, Ill., USA) as model-ing language that has been interfaced with the state-of-the-art solver CPLEX, Version 12.1 (IBM ILOG,Gentilly, France) for the solution of the resulting inte-ger programs. On the other hand, the improvementprocedure required specific programming skills and,thus, the language JAVA has been chosen for its imple-mentation. A computer with an Intel Core 2 DuoProcessor 2.53 GHz and 4 GB of RAM was used torun all the resulting codes.

5.1. Real-life Test ProblemTo our knowledge, there are no standard test prob-

lems available as a benchmark for the PPSRP, since webelieve it is the first time this problem is being tackled.For this reason, we will compare the quality of ourheuristics' solutions from one side with respect to thesolution currently being adopted by the delivery com-pany, and from the other with respect to the best feasi-ble solution that can be reached by the exact solver.

This case study consisted of a regional companylocated in the south of Italy having a central petroldepot from which a heterogeneous fleet of trucks startsdaily trips to serve 38 petrol stations located in differ-ent places throughout the regional territory. The plan-ning horizon for this problem is a week consisting of 6work days (T = 6). The weekly demand for petrolvaries from one station to another requiring in somecases only one delivery visit; however, most depotsrequire a multi-visit service fractioned into 2 or 3deliveries per week. The available trucks have differ-

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ent capacities, allowing the model to choose for eachroute the truck that best fits the model's constraints.

Concerning the drivers, generally speaking thecompany has enough drivers available during eachweekday except for the sixth working day (ie. the firstweekend day) in which the number of drivers is limit-ed because of the unavailability of most of them towork. This limitation has been considered in our mod-els by dropping, whenever is possible, the sequencecombinations that involve day 6. For example, whenri = 2, we have considered only the set Ci = {(1,4..2,5)} as feasible combinations and we disregarded thecombination (3, 6) in order to take into account thedrivers' unavailability. We kept, however, both thesequences Ci = {(1,3,5. .2,4,6)} for the stations requir-ing 3 visits per week.

The integrated PPSRP model corresponding to ourapplication resulted in a large-scale problem having84,740 constraints and 84,701 variables. The state-of-the-art CPLEX software package was not able to yieldan exact solution for the problem but generated at least

a feasible upper bound solution of 13,975 km/week.Indeed, after more than 30 CPU hours and a hugenumber of iterations, the software presented a faultymemory error due to the large number of sub problemsgenerated during the branch-and-bound method.

We also collected data for a different aspect of thetest problem in order to ascertain information aboutthe performance of our heuristic procedures. The dif-ferent aspect refers to another working week with dif-ferent demands for petrol and also different frequen-cies of periodicity.

5.2. Results and DiscussionsWe started by assessing the solution quality of each

approach (including the exact method) before and afterapplying the improvement procedure and the corre-sponding CPU time. The results collected in Table 2,show, from one side, the superiority of heuristic 4,both in its solution quality and CPU time, with respectto the other heuristics and also with respect to the fea-sible upper bound obtained by CPLEX. From the

Figure 3. Comparison between our solutions and the company’s solution

Figure 4. Savings obtained with respect to company’s solution

other side, the positive effect of the improvement pro-cedure on all the methods, except heuristic 4, is clear.This means that the solution generated by heuristic 4 isprobably close enough to the optimal solution so thatno margin was left for improvement.

The second set of experiments deals with the assess-ment of our approaches with respect to the solutioncurrently adopted by the company. Such a solution hasnot been obtained through the use of any decision sup-port tool but just as the result of a long experience ofthe company's human operators in managing not onlythe loading/unloading operations but also the trans-portation phase. For the data corresponding to theweek we took into consideration, the actual distancerun by the trucks for serving all 38 stations was 14,820kms. It is evident from Table 3 that all our approachesimprove the human operator solution and, particularly,heuristic 4 will reach, after improvement, a relativedistance saving of 17.7% with respect to the compa-

ny's decisions. These same results are also depicted inFigs. 3 and 4.

Finally, for the sake of completeness, we tested ourheuristic procedures on a different set of data for thesame distribution network. This kind of sensitivityanalysis experiment has the aim of checking therobustness of the obtained solutions, and specially ver-ifying the conclusions related to heuristic 4. This newset of data has been purposely collected for anotherweek belonging to a different season and with a differ-ent demand pattern with respect to the previous data.The two sets differ only by the amount of petrol to beserved to each station and, consequently, by their fre-quency of service.

The results reported in Table 4 show that these newresults substantially confirm the trend observed in theprevious test (except for heuristic 1). More specifical-ly, heuristic 4 once again outperforms all the otherapproaches. Moreover, even in this case the improve-

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Table 2. Solution costs (in terms of distance) and CPU times

Table 3. Comparison between our solutions and the company’s solution

Table 4. Comparison of the solutions for different test data

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ment procedure has no effect on the initial solutionobtained, confirming our intuitive conclusion thatheuristic 4 generates a nearly optimal solution.

From another point of view, the results reported inthe final column of Table 4 confirm the improvementthat three of our heuristics can achieve with respect tothe solution adopted by the company for that week.

6. Conclusions

In this paper, we presented and discussed thePPSRP. We proposed four heuristic approaches basedon three phases each: the assignment of periodicitycombinations to each station, the construction oftrucks' delivery routes, and, finally, the routes'improvement phase. The performance of the proposedheuristics was tested on two different instances of areal-life periodic problem. Both instances showed thesuperiority of heuristic 4 in relation to the others,reaching an average improvement of more than 15% interms of traveled distance, with respect to the solutionsactually adopted by the company while keeping theexecution time within reasonable limits. Future devel-opments in this direction should consist of solvinganother more challenging real-life problem on anational scale for a petrol distribution company oper-ating in Oman. Besides the difficulties that will beintroduced by the problem's size, the new applicationalso might be characterized by further complexitiessuch as uncertainty related to the drivers' availabilityand regulatory restrictions on the citizenship of thedrivers to accomplish some routes.

Acknowledgment

Thanks are due to the anonymous referees for theirvaluable comments, which improved the quality of thepaper.

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Solution Methods for the Periodic Petrol Station