An Optimal Supply Policy for Multi-product Multi-retailer Using Simulated Annealing Method

We consider a multi-product multi-retailer supply and inventory system. The total costs which include inventory cost and transport cost belong to the supplier. The transport cost consists of a major carrier cost and a minor delivery cost for each product. The total costs saving can be achieved by coordinating the supply of a group of products. Applying for each individual product a cyclic supply and inventory policy, where the delivery cycle time is a multiple of basic cycle time, can reduce the total costs. A modified simulated annealing method (MSA) to determine the optimal supply policy is proposed in this paper. A numerical example presents that the MSA is feasible. Experiment results show that the MSA algorithm significantly outperforms other solution method in the quality of the solution as well as in the running time.


Introduction
In the real world, products are usually jointed to ship to multiple retailers by suppliers in order to reduce costs such as on construction material, steel piping material and steel structure products distribution.The supplier has to consider the holding cost of products and the transport cost to minimize the total cost.In the classical EOQ model without considering joint supply, optimal ordering size and basic cycle time is computed based on the demand of retailers.But when delivery products are considered to be jointed to replenish, the problem is termed as the joint replenishment problem (JRP).
The JRP has been extensively studied during the past two decades.Many iterative solution techniques of JPR have been presented in the literature.Kaspi and Rosenblatt (1991) proposed a RAND approach for solving the economic ordering quantity for jointly replenished products.The authors conducted an extensive simulation study and concluded that the RAND solution procedure outperforms previously known algorithm (Goyal 1974) for solving this problem.They claimed that using RAND 10 compared to optimal (full enumeration) solution, the maximum error value found is less than 0.2%.Goyal et al. (1993) suggested a modified procedure which used a better estimate for the lower bound of the basic cycle time.Wildeman (1997) presented an efficient optimal solution method for JRP by applying Lipschitz optimization to obtain a solution with an arbitrarily small deviation from the optimal solution.Frenk et al. (1999) proposed an efficient algorithm with 'Improved-feasiblity procedure' to determine the optimal policy of the multi-product and one-buyer inventory problem.Khouja et al. (2000) used 1200 randomly generated problems to make a comparison between genetic algorithms and the RAND method for solving the JRP.The GA found the solutions with the same total cost as the RAND for 761 problems, outperformed the RAND for six problems and under-performed the RAND for 433 problems.The maximum percentage savings in total cost provided by the RAND was 0.078%.Olsen (2005) developed an evolutionary algorithm (EA) that used a direct grouping method to solve JRP.The authors showed that EA for joint replenishment policy incurs a lower total cost than the best available algorithm for some parameters of factor.Overall the EA improved over the RAND for 5.1% of the 72,900 randomly generated problems.The maximum percentage savings in total cost provided by the RAND was 0.69% from the 72,900 problems.
to be applicable to this problem and another is to develop a genetic algorithm for the JPR with resource restriction.Hoque (2006) developed a generalized global optimal solution algorithm of the JPR extended model which includes some practical issues Most of the literature has considered the JRP as a problem of coordinating the supply of multi-product for a single retailer.But a common practice for a supplier is to have multi-retailers who order multi-product from the supplier.In this case, the JRP becomes a multi-product multi-retailer joint replenishment problem (MJRP).The MJRP has received little attention in the literature until recently.Chan et al. (2003) proposed a new modified genetic algorithm for solving the MJRP.The performance test was compared to the algorithm of Goyal (1974) and got a better result.Li (2004) proposed a modified RAND method to solve the MJRP, but without a performance test.Chan et al. (2006) addressed the delivery scheduling issue for MJRP, once the optimal replenishment cycles are determined, by formulating four problems according to four different objectives in cost and resource minimization.The authors used the solution of the optimal replenishment cycles which had been solved to do the post research for optimal scheduling.
In previous reviews, MJRP has received little attention, and although many heuristics and exact methods have been proposed to solve JRP or MJRP, very few papers so far adopts the simulated annealing (SA) method to solve these problems.When adapted efficiently to optimization problem, SA is often fast convergence and ease of implementation.These characteristics motivate the applying of SA for MJRP problem.In this paper we proposed a modified simulated annealing method (MSA) to determine the optimal supply policy for MJRP.A numerical example is employed to present the feasibility of MSA.Simulation experiments are conducted to confirm the performance of MSA algorithm comparing with other solution methods.

Problem Formulation
A single supplier supplies multi-product products to multiple retailers.The total costs of the supply system include the holding cost and transport costs.The holding cost of all consignment inventories belongs to the supplier.The transport costs consist of two components, one is major carrier cost and another is minor delivery cost.The major carrier cost charged is fixed and the minor delivery cost of each product for each retailer would continue to be added per delivery.Each retailer has a deterministic demand for each product.A supply network of joint supply distribution system is shown in Figure 1.
The basic definitions in this problem include deterministic demand, holding cost is a fraction of the product cost, major carrier cost is fixed, and no shortages allowed.The following notation is defined: The ratio of ij t to T , Total costs consist of holding cost, major carrier cost and minor delivery cost.
The goal of the supply policy is to arrange the optimal delivery scheduling, and to minimize the total costs.The annual . The annual major carriers cost is / .The total costs are To reduce total costs, products joint supply policy is considered.Thus ij t should be the integer multiple to T , i.e. , .The objective function can be obtained from equation ( 1) and the constraint is shown as In above model, all the values of ij K must be positive integers.It is a nonlinear integer problem.

Bounds on Basic Cycle Time
Before giving the solution method, bounds on basic cycle time T need to be discussed.The optimal basic cycle time can be derived from ( 2 T can be decided when If the product i for retailers j is delivered independently, the optimal cycle time of each product i for retailer j can be calculated from classical EOQ model. When the product is jointly delivered with other products, the common delivery cost 0 S , the optimal cycle time for each product i to retailer j is Kaspi and Rosenblatt (1991)  The formula ( 9) is obtained from the first iteration, which Kasp & Rosenblatt (1983) applied in an iterative way until the value of i K is converged in the computing procedure of JRP.
The upper bound max T is easily obtained from equation ( 5).The lower bound of * T can be obtained from equation (8) which is modified from one Kaspi and Rosenblatt (1991) proposed for solving JRP.In this paper we used another lower bound of * T from equation ( 9) which is modified from an equation Goyal at al (1993) suggested and tested by simulating a better estimate for min T to solve JRP.

The Genetic Algorithm Method
The JRP is a NP-hard problem proven by Arkin et al. (1989).Khouja et al. (2000) applied the genetic algorithm (GA) to solve JRP and Chan et al. (2003) applied the GA to solve MJRP.They used the ratio of product delivery cycle time to the basic cycle time ( ij K ) as the genes in a chromosome.The chromosomes represent the integer multipliers of basic cycle time ( T ) for each product delivered to each retailer.Khouja et al. (2000) used the upper bound and lower bound of T from equations ( 5) and ( 8). can be encoded and corresponded to the binary sequence bits ij u .The genetic representation for possible solutions is created by using the genes ij K The integer number from the range is the number between the lower bound and upper bound of ij K .Each individual chromosome in the population represents a possible solution to the problem.The binary chromosome can be converted into a decision variables representation of , ij K after the GA operation.The evaluation function is responsible for rating these possible solutions when each decision variable is assigned.Khouja et al. (2000) used an evaluation function which can be modified for MJRP.
Equation ( 12) is a function of the values ij K , the number of decision variable ij K is J I . Using the genes ij K , the total length of the binary chromosomes is bits.The disadvantage of using the genes ij K to form chromosome is that the genes need to be encoded and decoded during GA operation.However, encoding and decoding the chromosomes will increase computing time.In this paper we propose to directly use the decision variable vector of ij K as the presentation without encoding and decoding in simulated annealing method (SA).Kirkpatrick et al. (1983) proposed the optimization approach by simulated annealing method (SA).A SA algorithm repeats an iterative neighbor generation procedure and follows search direction that improves the objective function.While searching optimal solution, the SA method offers the possibility to accept worse neighbor solution in order to avoid getting a local optimal solution.A cooling scheme specifies how it should be progressively reduced to make the procedure more selective as the search progresses to neighborhood of good solutions.

Simulated Annealing
Using SA to an optimization problem should include following components: a solution representation of decision variables, a method for objective function value evaluation, a neighbor generation mechanism for possible solution exploration and a cooling scheme and stopping criteria.The general procedure of SA algorithm is as follows: 1. Choose a random vector i X , select the initial system temperature, and specify the cooling schedule.

Evaluate
using a simulation model.

Perturb i X to obtain a neighboring design vector
using a simulation model.

Modified Simulated Annealing
The MJRP model is a MINLP problem.The decision variables of optimizing total costs include the ij K and T , therefore the number of decision variables is 1 J I , the problem size depends on the products I and retailers J .It is not easy to directly obtain the optimal solution for a large size problem.We propose modified simulated annealing method to solve these problems.
Base on components of using SA, in this paper we directly use the decision variable vector of ij K as the presentation without encoding and decoding in SA.The presentation of design vector ] , [ , where we compute the low bound of ij K with the equation can be derived from equations ( 9) and ( 10).The objective function value can be evaluated by ( 4) and ( 2).The neighbor generation mechanism for possible solution exploration is created: The cooling scheme is set , where 1 0 , and if stop , then stop the algorithm.The algorithm procedure of MSA is as follows.
Step 1. Read problem parameters: Step 2. Randomly generate a design vector , and compute objective function value init T C from equation ( 4) and ( 2) Step 3. Save  4) and ( 2) In order to test the performance, a numerical example is shown in next section and simulation experiments are designed in section 5.

Numerical Example
A numerical example of five products and four retailers is employed to demonstrate the MSA algorithm.The supplier who supplies the piping insulation materials has four retailers.The insulation products have five items including 2", 3", 4", 6" and 8".The problem data are as follows. .
The convergence diagram of MSA operation is illustrated in figure 2, and the computing results are summarized in Table 1.In Table 1, the values of ij K is obtained, the optimal delivery scheduling is shown by transport frequency with twelve times.The optimal delivery basic cycle time * T is 0.07841 annual.The minimum total costs of annual inventory and transport cost are 165,790 dollars.Comparing the policy with independent supply which minimum total costs are 299,420 dollars, the total costs can be reduced 44.628%.

Simulation Experiments and Results
The simulation experiments are designed to confirm the performance of the MSA algorithm, and then are compared with the GA algorithm (Khouja et al. 2000) which has been tested and shown to have good performance.For comparison, the parameter ranges of test problems and the values for parameters of GA are the same with Khouja et al. 2000 For each combination of J I and S , 100 problems were generated and solved by using the GA and MSA for a total 1600 problems.The simulation results are summarized in Tables 2. In Table 2 better' is 72% for S =5, 83% for S =10, 85% for S =15 and 81% for S =20.For J I =50, ' MSA C better' is 91% for S =5, 94% for S =10, 96% for S =15 and 97% for S =20.And comparing the CPU running time, the average CPU time for each problem with MSA is 0.700 seconds and with GA it is 12.913 seconds.The average CPU time with GA can be improved 92.44% by the proposed MSA.For the large size problem ( J I =30 and 50), the average CPU time with GA can be improved 94.52%~96.58%.Eventually, MSA is superior to GA not only in the quality of solutions but also CPU running time.

Conclusions
In this paper, we consider a multi-product multi-retailer supply and inventory system and propose MSA to directly use the decision variable vector of ij K as the presentation to replace the genes ij K for genetic algorithm to solve these problems.The major advantages of MSA are easy to implement and the representation need not to be encoded or decoded.This MSA method shortens the CPU running time which is important from a practical point of view.The numerical example presents that the MSA is feasible, the optimal solution is found and the system with joint supply can save the total costs comparing the policy with independent supply.From simulation and results show that the MSA outperformed GA not only reducing CPU running time but also improving the quality of the solutions.Especially for solving the large size ( J I =30 and 50) problems, the MSA method is more efficient than GA method.The MSA method can aids supplier to plan the delivery scheduling and save the total costs.Finally, we would like to remark that the multi-product multi-retailer supply and inventory system model may be further researched to handle constrained problem.

T
is a function of ij K and ij K are the decision variables of * T .The lower bound of 1 ij K .The upper bound of * ij D = [100000 50000 4000 1800 6000; 4000 3000 4000 5000 3000; 4000 3000 1000 2000 1500; 2000 1800 1200 800 200], S =2000, ij s =500, ij h =0.1, ij c = [50 80 100 120 200; 50 80 100 120 200; 50 80 100 120 200; 50 80 100 120 200].The values of parameter for the MSA operation are set: . The values of ij D , ij c , ij h and ij s for the test problem are randomly generated from uniform distribution on the ranges [100, 100 000], [1, 3], [0.5, 5.0] and [0.2, 3.0] respectively.Four different values of J I (10, 20, 30and 50) and four values of S (5, 10, 15 and 20) are considered.The values of parameters for the GA operation are set: probability of crossover 0.6, the probability of mutation denoted m P , m P =1/ (string length of chromosome), population size 10 for J I =10 and 20, and size 20 for J I =30 and 50, elite=1.The above values of the parameters were selected by the best performance from testing the problems of the GA algorithm.For fair comparison, the termination condition of GA is to stop computing after the number of G generations, where G=MSA solutions/GA Population, or when no improved solution is obtained in 50 generations.The values of parameter for the MSA operation are set as follows:

FigureFigure 2 .
Figure 1.A network of jointed supply distribution system

of product i for retailer j , ij h Inventory holding cost of product i for retailer j , per unit per unit time, ij c Unit cost of product i for retailer j ,
ij D Demand ij t Delivery cycle time of product i for retailer j , ij K The lower bounds on the values of ij K are 1 ij u , the label of ' MSA C better' means that the solution of objective function T C minimized by MSA is better than by GA, and the label of ' MSA C better or equal' means that the solution of objective function T C minimized by MSA is better than or is equal to GA.The percentage of all test problems of ' MSA C better' is 55% and ' MSA C better or equal' is 84%.The maximum saving cost in the problem of Test 7 reaches 1.71 % of the total cost.The average of maximum saving cost is 0.60 %.For the large size problem ( J I =30 and 50), the performance of solutions quality with the MSA is much better than with the GA.As shown in Table 2, for

Table 1 .
Optimal solution of ij K and supply scheduling j i

Table 2 .
Comparison between MSA and GA solution for total costs and CPU time 44 § 'C MSA better(or equal)' means that comparing the effectiveness of the solution , the results of percentage by MSA algorithm is better (or equal) than GA method.