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presents the conclusions and outlines future areas of research. Problem background The ITP deals with the management of two different aspects of retail chain logistics, inventory costs and transportation costs, which have both received lots of attention in the logistics literature. Here we will describe how they have b... |
2.1 The Vehicle Routing Problem The Vehicle Routing Problem (VRP) consists on finding an optimal set of delivery routes from a depot to a set of customers to serve |
(Toth and Vigo,, 2001 . The routes must start and finish in the depot and each customer must be served by one and only one vehicle , which means a customer cannot be contained in more than one route. Different versions of the problem have slightly different objectives or ways to define optimality: it can refer to findi... |
Amongst these variants of the problem the most popular is the capacitated VRP, or CVRP, which refers to the fact that each delivery vehicle has a limited capacity . Also popular is the VRP with time windows , or VRPTW, in which each customer must be served during a specified time interval or window. |
Another variant of the VRP relevant to our problem is the periodic vehicle routing problem (PVRP), which appears when customers have established a predetermined delivery frequency and a combination of admissible delivery days within the planning horizon. The objective is to minimise the total duration of the routes, wh... |
(Cordeau et al.,, 1997 and (Toth and Vigo,, 2001 for a general description of the periodic VRP. In this paper we are concerned with the CVRP, simply referred to as VRP. The only limitation will be in the capacity of the vehicles used, and not in their number. We will also consider that the fleet is homogeneous, i.e. on... |
For simplicity reasons we will consider that the customers (shops in the retail chain in our case) have no time windows, i.e. the deliveries can take place at any time. However, the hours a driver can work are limited by regulations and this has to be taken into account in the time needed per delivery, as is the unload... |
Finally, we will consider that the transport cost only includes the cost per kilometer; there is no cost attached to either the time of use of the vehicle or to the number of units delivered, nor there is a fixed cost per vehicle. |
2.2 Inventory and transportation management One step beyond the VRP is the problem of optimising simultaneously the costs of inventory and transportation. In Esparcia-Alcázar et al., 2007a, the reader can find a review of different approaches employed in the literature. |
Amongst these, one of the most relevant to our case is the inventory routing problem (IRP), which arises when a vendor delivers a single product and implements a Vendor Inventory Management (VIM) |
(Çetinkaya and Lee,, 2000 policy with its clients, so that the vendor decides the delivery (time and quantity) in order to prevent the clients from running out of stock while minimising transportation and inventory holding costs (Campbell and Savelsbergh,, 2004 . However, retail chains cannot be addressed in this way, ... |
The work presented here focuses on the Inventory and Transportation Problem (ITP), which was first described in (Cardós and García-Sabater,, 2006 |
with the aim of addressing the case of retail chains. Thus, the ITP can be viewed as a generalisation of the IRP to the multiproduct case. Additionally, it can also be viewed as a variant of the PVRP described previously that includes inventory costs |
and a set of delivery frequencies instead of a unique delivery frequency for each shop. The main feature that differentiates the ITP from other similar supply chain management ones addressed in the literature |
(Çetinkaya and Lee,, 2000 ; dos Santos Coelho and Lopes,, 2006 ; Federgruen and Zipkin,, 1984 ; Sindhuchao et al.,, 2005 ; Viswanathan and Mathur,, 1997 |
is that we have to decide on the frequency of delivery to each shop, which determines the size of the deliveries. The inventory costs can then be calculated accordingly, assuming a commonly-employed periodic review stock policy for the retail chain shops. Besides, for a given delivery frequency, expressed in terms of n... |
Because a pattern assumes a given frequency, the problem is limited to obtaining the optimal patterns (one per shop) and set of routes (one set per day). The optimum is defined as a combination of patterns and routes that minimises the total cost, which is calculated as the sum of the individual inventory costs per sho... |
The operational constraints at the shop level are imposed by the business logic and can be listed as follows (Esparcia-Alcázar et al.,, 2009 |
1. A periodic review stock policy is applied for the shop items. This means that the decision of whether to deliver to a particular shop is taken centrally and not at the shop. As a consequence, stockout is allowed. |
2. Shops have a limited stock capacity. 3. The retail chain tries to fulfil backorders in as few days as possible, so there is a lower bound for delivery frequency, which depends on the target client service level. |
4. The expected stock reduction between replenishments (deliveries) cannot be too high in order to avoid two problems: (a) the unappealing empty-shelves aspect of the shop just before the replenishment; and (b) replenishment orders too large to be placed on the shelves by the shop personnel in a short time compatible w... |
5. Conversely, the expected stock reduction must be high enough to perform an efficient allocation of the replenishment order. 6. |
As a consequence of points to above, not all frequencies are admissible for all shops; in general, very high and very low frequencies are undesirable. For instance, frequency 1 (one delivery per week) is not applicable to most shops. |
7. Sales are not uniformly distributed over the time horizon (week), tending to increase over the weekend. Hence, in order to match deliveries to sales, only a given number of delivery patterns are allowed for every feasible frequency. For instance, a frequency-2 pattern such as (Mon, Fri) is admissible, while another ... |
8. Although we are dealing with thousands of items, the load is containerised; hence, the size of the deliveries is expressed as an integer, representing the number of roll-containers. |
Thus, to summarise, our task involves finding: The optimal set of patterns [MATH] with which all shops can be served. A pattern [MATH] represents a set of days in which a shop is served which implies a delivery frequency (expressed as number of days a week) for the shop |
The optimal routes for each day of the working week, by solving the VRP for the shops allocated to that day by the corresponding pattern |
2.3 Objective functions in EVITA EVITA operates in two levels: the lower one deals exclusively with the transportation costs per day and the top one incorporates these and the inventory costs into the total costs. |
In the single objective case, the optimum is defined as the solution that minimises the total cost, given by the function [EQUATION] |
In the multiple objective approach, we will deal with two separate cost functions: [EQUATION] However, we will still use the total cost defined for the single objective case to compare the results between multi and monoobjective solutions. The reason is that the company is interested in spending less, irrespective of w... |
Inventory costs are computed from the patterns for each shop by taking into account the associated delivery frequency and looking up the inventory cost per shop in the corresponding table. An example of the latter is given in Table |
For instance, let us assume that shop N was assigned a pattern of frequency 4; we would look up in the table the inventory cost for the shop at that frequency, which is 286€. Proceeding in the same way with all shops and adding up the results we would obtain the total inventory cost. |
Transport costs are obtained by solving the VRP with one of the algorithms under study. The demands (size of deliveries) of each shop would also be taken from Table . In the example above, for shop N at frequency 4 the delivery size is 2 roll containers. |
Problem data is freely available from our group website: The top level: Evolutionary Algorithm The top level in EVITA is an evolutionary algorithm in which a population of individuals (candidate solutions) undergoes evolution following Darwinian principles. Each individual is a set of patterns |
[MATH] represented as a vector of length equal to the number of shops to serve ( [MATH] ), [EQUATION] and whose components [MATH] [MATH] are integers representing a particular delivery pattern, where [MATH] is the number of days in the working week. |
The relationship between patterns and delivery days is made at bit level. Each pattern is formed by [MATH] bits and each day corresponds to one bit: 1 means that the store is visited that day and 0 that it is not. In our case the working week has 5 days (Monday to Friday) so [MATH] . Hence patterns are coded by the rig... |
The total number of possible patterns is [MATH] ; in our case [MATH] , so patterns range from 1 to 31 or, alternatively, from 00001 to 11111 (obviously pattern 00000, i.e. not delivering any day, is not admissible). However, as was stated in Subsection 2.2 , not all patterns are suitable for all shops. Hence, [MATH] |
must be contained in the set of admissible patterns, [MATH] [MATH] adm . The elements in [MATH] adm are given in Table A point to note is that although the binary representation of the patterns is convenient in order to figure out what delivery days are associated to a pattern and also for calculating its corresponding... |
The details of the top level evolutionary algorithm are given in Table . The pseudo-code for the evaluation function is given in Algorithm |
Procedure Evaluate input: Chromosome [MATH] problem data tables [MATH] output: Fitness [Calculate inventory cost] InventoryCost = 0 |
for [MATH] to [MATH] Look up frequency [MATH] for pattern [MATH] Look up cost [MATH] for shop [MATH] and frequency [MATH] [MATH] |
[Calculate transportation cost] [MATH] [MATH] repeat Identify shops to be served on [MATH] Run VRP solver to get [MATH] [MATH] [MATH] |
until [MATH] [MATH] if multiobjective return [MATH] [Calculate total cost] [MATH] return [MATH] end procedure; Algorithm 1 Evaluation function. |
Lower level: Solving the VRP The calculation of the fitness of an individual requires an algorithm to solve the VRP (a VRPsolver ). In this work three algorithms have been tested for this purpose, namely: |
CWLS , the classical Clarke and Wright’s algorithm (Clarke and Wright,, 1964 enhanced with local search, which is presented in subsection 4.1 |
ACO , a bioinspired ant colony optimisation algorithm (Dorigo and Stutzle,, 2004 , described in subsection 4.2 , and CWTS , tabu search (Glover and Kochenberger,, 2002 seeded with a solution obtained by Clarke and Wright algorithm, as described in subsection 4.3 |
In (Esparcia-Alcázar et al.,, 2009 we also employed a fourth algorithm as VRPsolver , evolutionary computation. However, the results obtained there were not very promising, which is why it has been omitted in this study. |
4.1 Clarke and Wright’s algorithm Clarke and Wright’s algorithm (Clarke and Wright,, 1964 is based on the concept of saving , which is the reduction in the traveled length achieved when combining two routes. We employed the parallel version of the algorithm, which works with all routes simultaneously. |
Due to the fact that the solutions are not guaranteed to be locally optimal with respect to simple neighbourhood definitions, it is almost always profitable to apply local search to attempt and improve each constructed solution. For this purpose we designed a simple and fast local search method, which consists on perfo... |
This combination of C&W’s algorithm with local search is what we have termed CWLS, the pseudo-code for both can be found in Algorithms and |
4.2 Ant Colony Optimisation Some ant systems have been applied to the VRP (see for instance (Coltorti and Rizzoli,, 2007 ; Gendreau et al.,, 2002 ) with various degrees of success. Ant algorithms are derived from the observation of the self-organized behavior of real ants (Dorigo and Stutzle,, 2004 . The main idea is t... |
Each artificial ant builds a solution by choosing probabilistically the next node to move to among those it has not visited yet. The choice is biased by pheromone trails previously deposited on the graph by other ants and some heuristic function. Also, each ant is given a limited form of memory in which it can store th... |
(Dorigo and Stutzle,, 2004 In this work we employed a variant of ACO described by Xiang-pei et al., ( 2006 which differs from the original ACO algorithm in three aspects: (1) the way the pheromone matrix is updated, (2) the transition function and (3) that |
[MATH] -interchanges are used instead of local search. Two ways of updating the pheromone matrix [MATH] are defined: local updating and a posteriori updating |
(i.e. taking place after all ants have built their solutions). The former consists of adding [MATH] to each element [MATH] of the pheromone matrix, with [MATH] being the distance between shops [MATH] and [MATH] . The latter is given by Equation |
. Here the best path built in iteration [MATH] receives a reinforcement while the worst path is reset to the initial pheromone value, [MATH] |
[EQUATION] where [MATH] is the minimum cost obtained in iteration [MATH] and the value of [MATH] is automatically corrected in each iteration, as follows, |
[EQUATION] The second difference with respect to the original ACO is the transition function. In our ACO an ant located at shop [MATH] will select as its next shop [MATH] the one given by Equation |
, with probability [MATH] [EQUATION] where [MATH] is the list of shops not visited yet, [MATH] is the pheromone matrix, [MATH] is the heuristic function, |
[EQUATION] and finally, [EQUATION] corresponds to the concept of saving used in Clarke & Wright’s algorithm, with shop 0 being the depot . Alternatively, the next shop |
[MATH] will be uniformly selected at random from [MATH] The parameters [MATH] [MATH] and [MATH] (whose sum does not necessarily equal 1) measure the relative importance of each component. The probability value [MATH] is dynamically adjusted at runtime in a similar way as for [MATH] , following Equation |
[EQUATION] Finally, instead of using local search in the closest neighbours as in conventional ACO, we use [MATH] -interchanges, a concept borrowed from the CWLS algorithm described earlier. |
The pseudo-code for the ACO algorithm employed here is given in Algorithm , its transition function is shown in Algorithm and the parameters used are given in Table |
4.3 Tabu Search Tabu Search (TS) is a metaheuristic introduced by Glover and Kochenberger in order to allow Local Search (LS) methods to overcome local optima (Glover and Kochenberger,, 2002 . The basic principle of TS is to pursue LS whenever a local optimum is found by allowing non-improving moves; cycling back to pr... |
(Gendreau,, 1999 which last for a period given by their tabu tenure . The main TS loop is given in Algorithm To obtain the best neighbour of the current solution we must move in the solution’s neighbourhood, avoiding moving into older solutions and returning the best of all new solutions. This solution may be worse tha... |
The way to obtain the best neighbour is described in Algorithm . Table lists the relevant information about the algorithm. In order to improve over the TS algorithm used in |
(Esparcia-Alcázar et al.,, 2009 , we considered seeding the TS with a good solution. The solution chosen as a starting point was one obtained by the C&W algorithm. Statistical analysis carried out shows that the seeded algorithm obtains significantly better results than when the initial solution is obtained at random. ... |
Experiments This section is devoted to present the data we have used in the problem (subsection 5.1 ) and the experimental procedure we have followed (in the next subsection, |
5.2 ). 5.1 Problem data As explained above, we will employ a number of geographical layouts available on the web. We have selected our instances so as to achieve the maximum representation on three categories: |
size , given by the number of shops, distribution . We consider two kinds of distributions: uniform and in clusters , corresponding to shops that are scattered more or less uniformly on the map or grouped in clusters and, |
eccentricity . This represents the distance between the depot and the geographical centre of the distribution of shops. The coordinates of the geographical centre are calculated as follows: |
[EQUATION] An instance with low eccentricity (in practise, less than 25) would have the depot centered in the middle of the shops while in another with high eccentricity (above 40) most shops would be located on one side of the depot. |
We chose ten instances with different levels of each category, see Table It must be noted that we are only using the spatial location and not other restrictions given in the bibliography, such as the number of vehicles or the shop demand values. As pointed out earlier, a main characteristic of our problem is that the l... |
We also added a list of admissible patterns , which are given in Table , and inventory costs , an example of which is given in Table . The inventory, demand and admissible patterns data were obtained from Druni SA, a major regional Spanish drugstore chain. |
Finally, we have used the vehicle data given in Table 5.2 Experimental procedure Our aim is, on the one hand, to evaluate whether the multiobjective approach yields better results than the single objective one employed in previous work. On the other, we aim to verify if the conclusions reached in (Esparcia-Alcázar et a... |
For this purpose, we have tested the selected VRP algorithms described above on each one of the ten instances selected, each with a different geographic layout. We performed 10 runs per VRP algorithm and instance with a termination criterion in all cases of 100 generations. The motivation for such a small number of run... |
The results were evaluated on two fronts: quantitatively for the total costs obtained, and qualitatively for the computational time taken in the runs. The latter is important when considering a possible commercial application of the EVITA methodology. |
Results and analysis We carried out Kruskal-Wallis tests for the total costs yielded by the best individuals for all runs and problem instances, both in single and multiobjective. In the multiobjective case, we define the best individual as that member of the final Pareto front yielding the lowest total cost (as define... |
and show the resulting boxplots for the ten instances studied. The boxplot consists of a box and whisker plot for each algorithm. The box has lines at the lower quartile, median, and upper quartile values. The whiskers are lines extending from each end of the box to show the extent of the rest of the data. Outliers are... |
The conclusions that can be reached after analysis of the tests are as follows: The single objective approach yields the best performance in all instances. |
Considering separately the multi and single objective runs, in general there are no significant differences between CWLS and CWTS, although at first sight it would seem that CWLS performs better than CWTS on instances A32, A69, A80, P100 and X200 and vice versa on A33, B35, B45 and B67. On instance B68 there are no dif... |
ACO is significantly worse in all cases except B35. Furthermore, it is the method that scales worse, getting worse results as the size of the problem increases. |
The case of instance B35 is unique in the sense that monoobjective CWTS is significantly better than CWLS and does not differ significantly from ACO, both for single and multiobjective. |
In order to be able to compare results between the different instances we normalised the fitness values by defining the relative percentage deviation [MATH] , given by the following expression: |
[EQUATION] where [MATH] is the fitness value obtained by an algorithm configuration on a given instance. The [MATH] is, therefore, the average percentage increase over the lower bound for each instance, |
[MATH] . In our case, the lower bound is the best result obtained for that instance across all algorithm configurations. With the [MATH] results of all the runs for all VRP solvers we ran the tests again; the results are shown in Figure |
split into two groups: uniform distribution of shops and distribution in clusters . The conclusions in this case are similar. In both groups CWLS and CWTS perform better than ACO and, at first sight, CWLS is better than CWTS for group uniform and vice versa for group |
clusters . Further, considering each VRP solver separately, the single objective approach is better than the multiobjective one. |
Figure portrays the comparison between methods when the costs of transport and inventory are considered separately. Here it can be observed that there are no significant differences between methods if only inventory costs are taken into account. This differs from the conclusions obtained in |
Esparcia-Alcázar et al., 2006a, , where the choice of VRPsolver influenced the inventory cost results. In that work, however, the algorithms employed were suboptimal compared to the ones used here. So, it could be concluded that the choice of VRPsolver does not have an influence on the inventory costs provided a “good ... |
The big difference lies in the transport costs, which is where ACO clearly shows its inferiority, especially in the multiobjective approach. |
Regarding the computational time, the results clearly favour CWLS over all other VRP solvers. In general, when employing CWLS the time for a whole run took approximately the same as that of a single generation in when using CWTS or ACO. This is a point in favour of CWLS when considering a potential commercial applicati... |
6.1 Pareto fronts In this section we study the Pareto fronts obtained in the multiobjective approach in two instances of the problem, namely B35 and A32. The former has been chosen because of its uncharacteristic behaviour (as we have seen, in this instance ACO performs better than the other VRP solvers), and the latte... |
. In Table we show the number of non-dominated solutions vs. the number of different individuals for each algorithm. Comparing the number of non-dominated solutions in the final generation for each instance, we observe that the Pareto front yielded by the best performing VRP solver (CWLS in A32 and ACO in B35) has alwa... |
On the other hand the ratio of repeated individuals over the population size is very high when using CWTS or CWLS, whilst in ACO there are no repeated individuals. It is possible that algorithm ACO has a slower rate of convergence than the other two in most of instances (except in cases such as B35), so it requires mor... |
At this point it would be of interest to measure the quality of the Pareto front using one of the metrics that have been proposed for this purpose (Coello Coello,, 2005 . However, many of them (e.g. the |
error ratio or the generational distance ) assume that knowledge exists on the actual Pareto front, which is not the case here. Other metrics measure the distribution of solutions on the Pareto front by evaluating the variance of neighboring solutions. An example of this is the spacing [MATH] , which measures the relat... |
[EQUATION] where [MATH] is the number of non-dominated solutions found, the distance [MATH] is given by [EQUATION] where [MATH] is the fitness of point [MATH] on objective [MATH] [MATH] is the generation number and [MATH] is the mean of all [MATH] |
The values obtained are given in Table . From these we can see that the best values of the metric (i.e. the lowest spacing) are obtained by CWTS; however, from previous analysis we know that it is the other two VRP solvers that perform better: ACO for B35 and CWLS for A32. We can hence conclude that this metric is not ... |
Conclusions and future work We have shown how, for the problem presented here, the multiobjective approach does not yield any advantage over the single objective one. This could be explained by the fact that inventory costs are well above the transport costs. Given that the multiobjective approach does not prefer one o... |
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