We solve a novel inventorylocation model with a stochastic capacity constraint based on a periodic inventory control (ILMPR) policy. The ILMPR policy implies several changes with regard to other previous models proposed in the literature, which consider continuous review as their inventory policy. One of these changes is the inclusion of the undershoot concept, which has not been considered in previous ILM models in the literature. Based on our model, we are able to design a distribution network for a twolevel supply chain, addressing both warehouse location and customer assignment decisions, whilst taking into consideration several aspects of inventory planning, in particular, evaluating the impact of the inventory control review period on the network configuration and system costs. Because the model is a very hardto solve combinatorial nonlinear optimisation problem, we implemented two heuristics to solve it, namely, Tabu Search and Particle Swarm Optimisation. These approaches were tested over small instances in which they were able to find the optimal solution in just a few seconds. Because the model is a new one, a set of mediumsize instances is provided that can be useful as a benchmark in future research. The heuristics showed a good convergence rate when applied to those instances. The results confirm that decision making over the inventory control policy has effects on the distribution network design.
Distribution network design (DND) is one of the most important problems for companies that distribute products to their customers. The problem consists of selecting specific sites to install plants, warehouses, and distribution centres, assigning customers to serving and interconnecting facilities by flow assignment decisions. DND is typically solved as part of a sequential approach that simplifies associated tactical and operational issues. Hence, the decisions that have been omitted are tackled only after the DND problem has already been solved. This means that decisions related to the location of warehouses and allocation of customers are made without taking into account operational aspects such as transportation and inventory. This situation leads to suboptimal designs because operational decisions are restricted to the current network design.
This paper considers a twolevel supply chain, in which a single plant serves a set of warehouses, which in turn serve a set of end customers or retailers. Unlike traditional approaches and in accordance with the recent inventorylocation literature, in this paper, we incorporate the inventory control policy as a relevant factor that directly affects DND. A distinctive feature of our model is that it is based on a periodic inventory control policy
The remainder of the paper is organised as follows. First, a literature review of the main ILM is presented in Section
Many authors have been focused on the DND problem and on ILM in particular during the last 15 years. For instance SimchiLevi and Zhao [
More recently, Kumar and Tiwari [
The authors have presented different techniques to solve these models. For instance, Bard and Nananukul [
As we stated previously, in this paper, we have considered a twolevel supply chain with a single plant, a set of warehouses, and a set of end customers or retailers. Because we incorporate the inventory control policy as a relevant factor that directly affects DND in this paper, based on a periodic inventory control policy
The model presented in [
It may be noted that a similar and more conservative capacity constraint is proposed in [
In terms of inventory capacity constraints, peak inventory levels are not controlled at any time, but only at specific times for each review period. The peak inventory level is reached only when orders arrive at the warehouse LT time units after the last order and naturally only if an order was submitted to the central warehouse or plant. Consequently, when an order arrives at a warehouse, the inventory level is
This expression is not surprising, as when an order is submitted to the plant, it is necessary that the total inventory position (on hand plus on order inventory) reaches level
Finally, the reorder point
Finally, substituting (
According to the previous inventory control assumption, we describe the proposed InventoryLocation Model with Stochastic Constraints on Inventory Capacity under Periodic Review (ILMSCCPR) as a stochastic nonlinear mixedinteger programming model (SNLMIP). Based on a periodic
Additionally, inventory and ordering costs related to order size or inventory cycle are evaluated in terms of the minimum order size
Number of available sites to install warehouses
Number of customers that must be served by the installed warehouses
Fixed location cost of a warehouse on site
Transportation unit cost from the warehouse on site
Fixed ordering cost at site
Holding cost per time unit at site
Mean demand of customer
Standard deviation of the demand of customer
Capacity at warehouse
Maximum order size at warehouse
Inventory check period at warehouse
Average leadtime at warehouse
Value of the Standard Normal Distribution, which accumulates a probability of
Value of the Standard Normal Distribution, which accumulates a probability of
Binary variable. It is equal to 1 if a warehouse is installed on site
Binary variable. It is equal to 1 if warehouse i serves customer
Order size at warehouse
Total mean demand at warehouse
Total variance of the demand at warehouse
Min
s.t.:
Equation (
Heuristic methods are a common approach to solve hard combinatorial optimisation problems. Despite the fact that heuristics do not guarantee optimality, the solutions provided by them can be considered as good suboptimal ones. In contrast exact methods guarantee optimality; however, they usually fail when dealing with medium and largesized problems. In this paper, two heuristics have been separately considered to solve our DND problem. The first one corresponds to a wellknown local search called Tabu Search. The second one is an evolutionary algorithm called Particle Swarm Optimisation. In the next subsections, an overview of these two heuristics is provided.
We can describe the TS approach as a local search technique guided by the use of adaptive or flexible memory structures. However, such a general definition fails to show the specificity of TS and could be confused with other types of Greedy Random Adaptive Search Procedure (GRASP) algorithms. The variety of the tools and search principles introduced and described in [
The TS algorithm starts with a random solution in which each customer is allocated to a randomly selected DC. When all customers are assigned, TS validates the model constraints. If the constraints are satisfied, the initial solution (
The PSO algorithm was proposed and developed by Kennedy and Eberhart [
Unlike other evolutionary algorithms, PSO does not use the “survival” concept. This is because all particles are kept “alive” throughout the algorithm execution time and at no time is their survival threatened. The algorithm starts when a set of particles
find best (
iteration without improvements = 0;
iteration without improvements + 1;
update (tabuList);
check (diversificationCriterion);
The general frame for our PSO algorithm is presented below.
The most important step in Algorithm
Initialise set
Calculate
Particles
As with local search algorithms, the PSO approach can also get trapped in local optima if the global best and local best positions are equal to the position of the particle over a number of iterations [
It is clear that in this paper, we do not attempt to develop an original strategy to solve our problem. Instead, we are proposing very simple procedures to obtain solutions for our new model that can be used as a baseline in future research. Moreover, validating some of the main model assumptions is also one of the goals of this work.
In this section, we present the experiments we have carried out, and we draft some conclusions about our numerical results. As we have noted before in this papaer, our goal is not to state whether the TS or PSO approach is the best way to solve this problem. In fact, we firmly believe that other approaches might be more effective than our approach, and, therefore, applying different strategies to solve our model appears to be a fruitful area for future research into artificial intelligence field. In spite of this, due to the simplicity that both TS and PSO offer, we present these results as a baseline for future studies. We have selected these two algorithms as samples of local search and evolutionary strategies, respectively. Therefore, two of the most important approaches used to solve large and complex combinatorial optimisation problems are covered with these two algorithms.
Moreover, in this section, we present a set of instances that can be used as a benchmark in future studies. The benchmark set consists of two base instances, namely,
Customer and warehouse distributions. (a) Shows that customers (blue squares) are uniformly distributed. (b) Shows that customers are distributed over two clusters (blue squares and green triangles). In both cases, warehouses (orange diamonds are uniformly distributed).
From these two main configurations, we generated a set of 22 instances based on them (11 for each one). To do that, we modified values of FC, HC, and OC in ±25%, and additionally we have varied parameter
The computational experiments were performed on an Intel Core Duo processor CPU T2700 2.33 GHz with 2 GB of RAM, using the Ubuntu 12.04 operating system. Both the TS and PSO algorithms were implemented in the JAVA programming language using NetBeans IDE. To validate the algorithms and measure their convergence, we used a set of small instances from [
We fixed both the nonstockout probability and the nonviolate capacity constraint probability at 95%. Hence, the values for
Summary results. Comparison between the results obtained by movements
Instance 






Gap (%) 


151541.81  1.57%  305.93  142772.05  0.69%  94.62  5.79% 

176910.48  0.76%  175.37  172204.30  0.50%  46.83  2.66% 

155167.37  7.69%  97.86  125920.71  9.85%  86.94  18.85% 

173094.71  7.54%  289.57  164272.51  9.61%  104.37  5.10% 

144888.67  8.96%  223.16  134974.87  11.98%  79.43  6.84% 

185303.83  6.05%  166.17  173147.78  7.57%  88.09  6.56% 

164188.72  1.48%  155.14  155644.07  0.67%  64.94  5.20% 

163959.14  1.39%  60.80  157637.04  0.30%  67.82  3.86% 

164081.02  7.75%  203.51  152860.21  16.25%  87.88  6.84% 

226197.19  5.71%  239.07  228169.26  11.13%  249.08 


300915.27  3.25%  217.26  301631.38  6.89%  198.21  −0.24% 

142217.21  0.43%  235.52  123147.56  2.31%  114.04  13.41% 

168438.53  0.33%  121.61  142471.58  1.52%  90.45  15.42% 

146509.00  4.64%  218.86  124372.47  14.23%  199.12  15.11% 

161718.90  6.10%  294.69  136471.36  15.16%  67.86  15.61% 

137153.56  7.16%  240.17  114363.85  17.62%  82.98  16.62% 

172214.43  4.57%  94.59  148850.52  11.69%  65.94  13.57% 

155132.44  1.61%  181.48  133414.40  0.83%  53.26  14.00% 

156367.93  1.39%  62.21  133270.49  0.73%  68.75  14.77% 

154938.46  9.34%  216.25  132817.52  10.40%  69.38  14.28% 

197953.21  7.59%  327.43  198309.61  7.06%  164.28  −0.18% 

267045.25  4.54%  254.97  266821.85  4.58%  259.85  0.08% 
Column
Table
(a) Shows the DND obtained when parameter
These two situations are produced for the same reason: when the value of
We now compare the best solutions obtained by the TS algorithm with the ones obtained by the PSO method. Table
Comparison between results obtained by the local search (TS) and the evolutionary algorithm (PSO).
Instance  TS  PSO  Dif (%) 


142,772.05  142,849.31  0.054% 

172,204.30  166,232.44  −3.468% 

125,920.71  147,587.20  17.206% 

164,272.51  164,450.20  0.108% 

134,974.87  141,538.00  4.862% 

173,147.78  178,852.42  3.295% 

155,644.07  157,435.47  1.151% 

157,637.04  157,176.59 


152,860.21  156,823.91  2.593% 

226,197.19  226,450.57  0.112% 

300,915.27  301,078.22  0.054% 

123,147.56  119,074.55  −3.307% 

142,471.58  145,809.30  2.343% 

124,372.47  122,065.28 


136,471.36  135,172.27  −0.952% 

114,363.85  108,741.32  −4.916% 

148,850.52  142,216.14 


133,414.40  127,997.33  −4.060% 

133,270.49  133,320.84  0.038% 

132,817.52  119,972.02  −9.672% 

198,309.61  197,929.53  −0.192% 

266,821.85  262,059.09  −1.785% 
According to the results the PSO algorithm seems to perform slightly better than the TS. That is more evident when we look at
Because we do not have any bound for our instances, we cannot say anything about the quality of our solutions. Another interesting research line would be to find lower bounds for this model to determine how good the solutions provided by the heuristic methods are.
Moreover, when we examine the components of our objective function, we can see how they behave. In Table
Variation of cost distribution at each instance as a percentage of base instances
Instance  FC (%)  TC (%)  Inv (%)  SS (%) 


13.75%  −3.59%  −8.11%  −7.35% 

−24.85%  15.95%  10.29%  10.88% 

8.15%  16.00% 

−10.61% 

−4.25%  −11.52%  6.07%  6.62% 

−16.76%  −11.70%  13.72%  14.86% 

9.79%  12.35%  −9.87%  −10.42% 

−5.81%  6.78%  1.70%  1.41% 

−7.10%  7.48%  1.29%  2.53% 






31.66%  31.96%  −28.61%  −31.50% 

48.95%  47.11%  −44.14%  −47.62% 

19.18%  −7.42%  −7.93% 


−14.05%  2.19%  6.82%  6.87% 

2.38%  16.64%  −7.00%  −6.63% 

5.64%  −17.16%  1.94%  2.95% 

−17.46%  −8.89%  11.43%  12.80% 

13.25%  12.61%  −11.09%  −11.40% 

−2.13%  0.07%  2.11%  0.62% 

0.97%  −1.34%  −1.04%  0.41% 






37.02%  36.05%  −30.89%  −32.30% 

52.70%  53.47%  −44.24%  −46.91% 
Another interesting finding is that individual subcosts (FC, TC, Inventory, and SS) do not increase to the same extent that parameters do. In other words, when, for example, FC is increased (decreased) by 25%, the portion of the total cost corresponding to FC does not increase by 25% when compared with its corresponding base instance. This can be associated with the ability of the model to seek other location/allocation alternatives to keep the total cost as low as possible.
Regarding our algorithms, they showed good performance in terms of time and convergence. As we mentioned before, we cannot compare our results with other works because this is a very new model. Figure
Current solution costs. The peaks correspond to the values of the current solution when the restart method is invoked.
Despite the fact that restart method proved to be effective in providing some diversification to our algorithm (and consequently allowing us to move away from local optima), other diversification strategies can be implemented to exploit information from previous iterations and, consequently, provide the algorithm a higher exploration level. With regard to the
In this paper, we solve a novel inventorylocation model that integrates inventory decisions at the strategic level. As a first contribution, our model considers a periodic inventory review policy
Moreover, some
The authors are indebted to anonymous referee comments and the editor for their valuable comments through the review process. The paper remarkably improved through their recommendations; yet, the authors are responsible for any remaining errors. Fernando Paredes is supported by FONDECYTChile Grant 1130455.