The integration with different decisions in the supply chain is a trend, since it can avoid the suboptimal decisions. In this paper, we provide an effective intelligent algorithm for a modified joint replenishment and location-inventory problem (JR-LIP). The problem of the JR-LIP is to determine the reasonable number and location of distribution centers (DCs), the assignment policy of customers, and the replenishment policy of DCs such that the overall cost is minimized. However, due to the JR-LIP’s difficult mathematical properties, simple and effective solutions for this NP-hard problem have eluded researchers. To find an effective approach for the JR-LIP, a hybrid self-adapting differential evolution algorithm (HSDE) is designed. To verify the effectiveness of the HSDE, two intelligent algorithms that have been proven to be effective algorithms for the similar problems named genetic algorithm (GA) and hybrid DE (HDE) are chosen to compare with it. Comparative results of benchmark functions and randomly generated JR-LIPs show that HSDE outperforms GA and HDE. Moreover, a sensitive analysis of cost parameters reveals the useful managerial insight. All comparative results show that HSDE is more stable and robust in handling this complex problem especially for the large-scale problem.
As a multiitem inventory replenishment policy, the joint replenishment (JR) which can save the total costs by grouping multiitems in the same order had received numerous attentions [
Silva and Gao [
Except Berman et al. [
The periodic-review
JRPs and uncapacitated facility location problems had been proven to be the NP-hard problems and they were rather hard to find effective algorithms [
The aim of this study is to propose a new and effective approach to handle the modified JR-LIP model based on the study of Silva and Gao [
The rest of this paper is organized as follows. Section
A three-level supply chain consisting of multidistribution centers (DCs), an outside supplier, and multicustomers is considered. The item is ordered and collected by the DCs and then is distributed to customers. The objective is to decide the following policy: (1) how many DCs should be opened, and where to locate them; (2) how the customers are assigned to appropriate DCs; (3) how many and when to order, to minimize the total cost. The JR-LIP model is studied based on the following assumptions. Demand rates and costs are known and constant. Shortages are not allowed. Replenishment lead time is constant. Each customer is only assigned to one DC, while other DCs cannot serve it. There is no limitation for the capacity of storage and shipment.
Only one item is considered in our model, DCs replenish their demand jointly. DC
The three-level supply chain of proposed JR-LIP.
Notions used in the model are as follows:
The total cost is composed of the fixed location costs of DCs, assignment costs of DCs, and replenishment costs of DCs. For a given problem, we assume that the maximum number (
In (
The objective of JR-LIP model is
The goal of this problem is to find the optimal
Since all involved costs in the model are associated with the location of the DC, the thinking of solving methodology is converting (
For a given set
Due to its easy implementation, quick convergence, and robustness, DE has turned to be one of the best evolutionary algorithms in a variety of fields [
Neri and Tirronen [
The DE-based algorithms referred in Neri and Tirronen [
Based on the above analysis, we propose a DE-based algorithm named HSDE by integrating the GA and self-adapting parameters of DE.
The difference between HSDE and original DE is the structure of individual and selection operation. Table
HSDE’s notations.
Notation | Explanation |
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Population: the number of individuals |
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The dimension of the specific problem |
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The maximum generation for evolution |
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The target vector of individuals |
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The mutated vector of individuals |
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The trial vector of individuals |
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Mutation factor of individuals |
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Crossover factor of individuals |
Figure
Flow chart of HSDE for the JR-LIP.
In the stage of initialization, we should determine the parameters of HSDE, the representation of chromosome, and the encoding and decoding scheme for the JR-LIP.
A decoded chromosome (
The first part of the decoded individual represents that customers 1, 2, 4, 6, and 9 are assigned to DC1; customers 3, 5, 8, 11, and 12 are assigned to DC2; customers 7, 10 are assigned to DC3. Thus, the decision variables
The target of decoding is converting the initial chromosome to practical solution. Denote
The practical values of the last two parts of chromosome are between 0 and 1, since the value of gene can directly map into practical value.
From (
To verify the performance of the proposed HSDE, three numerical examples were designed. Another two intelligent algorithms, GAs and HDE which had been proven to be effective approaches for solving JRPs and JRDs [
The decoding scheme of GA and HDE is the same with HSDE. Besides the number of populations (
In order to verify the performance of HSDE, seven well-known benchmark functions are used in the following experiments. To assure a relatively fair comparison, the functions were selected according to their different properties [
Benchmark functions.
Test functions |
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Global optimum ( |
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10, 30 |
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10, 30 |
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10, 30 |
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10, 30 |
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10, 30 |
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−12569.5 |
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10, 30 |
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10, 30 |
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0 |
Three algorithms including HSDE, HDE, and GA are compared with two different dimensions, that is, the speed of convergence and the ability to obtain optimal solution. In all cases, the number of populations (
Results of seven functions with three algorithms (
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Results of seven functions with three algorithms (
Function | HSDE | HDE | GA |
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Convergence results of
Convergence results of
Convergence results of
Convergence results of
Results in Tables the ability to obtain optimal solutions of HSDE is better than HDE and GA, especially for functions with the large dimension (when although HSDE and HDE can both obtain the global optimum for
In this section, three different scales of the JR-LIPs are used to test the performance of three intelligent algorithms. The part of basic data listed in Table
Parameters for test problems.
Parameters | Value | |
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Common parameters | ||
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Annual demand rate |
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Major ordering cost | 45 |
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Minor ordering cost |
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Annual inventory holding cost |
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Fixed location cost |
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Changed parameters | ||
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Maximum number of opened DCs | 5, 10, 20 |
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Number of customers (potential sites of DCs) | 30, 50, 100 |
We denote
Comparison of the results for different scales.
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Algorithm | Avg. CPU times | Optimal TC | Avg. |
Ratio of finding the optimal TC |
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HSDE | 17.6 | 1397.4 | 1397.4 | 100% |
HDE | 17.3 | 1397.4 | 1397.4 | 100% | |
GA | 19.2 | 1397.4 | 1397.4 | 100% | |
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HSDE | 99.0 | 2511.2 | 2511.2 | 100% |
HDE | 105.1 | 2511.2 | 2511.2 | 100% | |
GA | 114.1 | 3064.8 | 3126.0 | 0% | |
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HSDE | 613.2 | 4444.7 | 4709.2 | 45% |
HDE | 689.9 | 7318.3 | 7846.4 | 0% | |
GA | 742.3 | 8539.4 | 9497.5 | 0% |
Convergence trend (
Convergence trend (
Results in Table
Figures
Section
The size of problem in this section is
Comparative results using three algorithms under different
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Algorithms | Location sites (DCs) | CPU time |
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−40 | HSDE |
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85.6 |
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2174.7 | 298.7 | 2473.4 |
HDE |
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81.0 |
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2174.7 | 298.7 | 2473.4 | |
−20 | HSDE |
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89.5 |
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2177.0 | 342.3 | 2519.3 |
HDE |
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88.3 |
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2177.0 | 342.3 | 2519.3 | |
20 | HSDE |
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106.4 |
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2177.0 | 419.2 | 2596.2 |
HDE |
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113.6 |
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2177.0 | 419.2 | 2596.2 |
Comparative results using three algorithms under different
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Algorithms | Location sites (DCs) | CPU time |
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−40 | HSDE |
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105.4 |
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2888.9 | 394.5 | 3283.4 |
HDE |
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109.9 |
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2881.4 | 403.0 | 3284.4 | |
−20 | HSDE |
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96.7 |
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2951.2 | 358.2 | 3309.4 |
HDE |
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106.8 |
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2924.2 | 401.2 | 3325.4 | |
20 | HSDE |
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107.9 |
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2966.1 | 429.2 | 3395.3 |
HDE |
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111.5 |
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2966.1 | 429.2 | 3395.3 |
The comparative results of Tables The robustness of HSDE is better than HDE. When inventory holding cost and major ordering cost vary from −40% to 20%, the optimal number of DCs is always three. Moreover, the location sites of DCs have little change ( DCs have more motivation to order item jointly when the major ordering cost S increases (
In this paper, we proposed an effective intelligent algorithm for a modified joint replenishment and location-inventory (JR-LIP) model. The objective of the JR-LIP is to determine the number and locations of DCs, the assignment decision, and replenishment policy to minimize the total system cost. To handle this NP-hard problem effectively, an intelligent algorithm named HSDE is designed to solve the proposed model. To verify the effectiveness of HSDE, GA and HDE were chosen to be compared with it by benchmark functions tests and numerical examples. We can easily come to useful conclusions and managerial insight as follows. Results of benchmark functions tests show the good ability of HSDE in handling the large-scale problems. When the dimension of test function is 10, HSDE and HDE have the same precision, while the dimension is 30, HSDE has the higher precision and faster speed than HDE. The similar conclusion can be obtained from example 2. The rate of convergence obtained by HSDE and HDE is both 100% when Example 3 illustrates the impacts of cost parameters on the optimal decision and reveals that when the major ordering cost is bigger, DCs have more incentive to replenish jointly for sharing related costs.
All numerical examples verify that the HSDE is an easy and effective algorithm to handle the JR-LIP. To our best knowledge, this is the first time to use the improved DE-based algorithm to solve this NP-hard problem.
In our model, the demand is constant which generated from a uniform distribution, and there is no capacity limitation. These assumptions are not required. The future direction about this problem includes relaxing the assumption to match real-world scenario and looking for more quickly and efficiently solving method.
The authors are very grateful for the constructive comments of editors and referees. This research is partially supported by the National Natural Science Foundation of China (70801030; 71131004; 71371080; 71373093), Humanities and Social Sciences Foundation of Chinese Ministry of Education (No. 11YJC630275), and Fundamental Research Funds for the Central Universities (HUST: 2012TS065).