The stochastic joint replenishment and delivery scheduling (JRD) problem is a key issue in supply chain management and is a major concern for companies. So far, all of the work on stochastic JRDs is under explicit environment. However, the decision makers often have to face vague operational conditions. We develop a practical JRD model with stochastic demand under fuzzy backlogging cost, fuzzy minor ordering cost, and fuzzy inventory holding cost. The problem is to determine procedures for inventory management and vehicle routing simultaneously so that the warehouse may satisfy demand at a minimum long-run average cost. Subsequently, the fuzzy total cost is defuzzified by the graded mean integration representation and centroid approaches to rank fuzzy numbers. To find optimal coordinated decisions, a modified adaptive differential evolution algorithm (MADE) is utilized to find the minimum long-run average total cost. Results of numerical examples indicate that the proposed JRD model can be used to simulate fuzzy environment efficiently, and the MADE outperforms genetic algorithm with a lower total cost and higher convergence rate. The proposed methods can be applied to many industries and can help obtaining optimal decisions under uncertain environment.
The joint replenishment problem (JRP) has been heavily researched since the early work of Shu [
Supply chain management has received much attention and global sourcing has been widespread in recent years. Since cost savings can be achieved by using joint replenishment policy, managers have realized that jointly considering JRP and delivery scheduling (JRD) can obtain a scale effect of replenishment and transportation simultaneously, thus further reducing total cost (Qu et al. [
In reality, managers often have to make decisions under imprecise operational conditions (Zeng et al. [
Moreover, it is very difficult to find optimal solutions for the defuzzified model effectively. The JRP has been proven to be an NP-hard problem (Arkin et al., [
On the other hand, metaheuristics have grown quickly in order to attain better solutions. Among these algorithms, the evolutionary algorithms (EAs) and especially genetic algorithm (GA) have been proved to be effective algorithms for the JRD. For example, Cha et al. [
The aim of this paper is to model and optimize the practical JRD policy with stochastic demands under uncertainty. This topic is interesting because of the widely adoption of the JRD policy in many industries and the operability and rationality of the method to handle uncertain costs. This study makes the work of Qu et al. [
The rest of this paper is organized as follows. In Section
We consider the similar situation of a central warehouse (where all stocks are kept) and several geographically dispersed suppliers asserted by Qu et al. [ TC: the total annual cost;
Sometimes, the stock-out factor
The total cost consists of the following parts: (1) the inventory cost, which includes ordering, holding, and backlog cost; (2) the transportation cost, which includes dispatching and stopover and routing costs. The details of different costs will be discussed as follows.
We suppose the demand of each item follows the normal distribution and the lead time is
A stockout occurs following an order placed at time
For each replenishment period
As reported in Vujošević et al. [
These imprecise definements in linguistics are always described by fuzzy numbers. So, fuzzy variables are also utilized to handle the JRD problem under uncertainty. Two popular kinds of fuzzy numbers characterized by triangular and trapezoidal membership functions (MFs) are presented in the appendix. They were widely used to solve uncertainty problem because of their intuitive appeal and their perceived computational efficacy (Pramanik and Biswas [
According to Definition
Take each TFN into (
Hence,
Defuzzification has been a favorite approach in many inventory studies for its simplicity. Defuzzification can easily transfer fuzziness to be explicit without complex analysis. In this study, two approaches will be utilized to defuzzify the fuzzy total cost.
So, the simplified objective is
Hence, the simplified objective is expressed as
Differential evolution algorithm (DE) was proposed by Storn and Price [
However, the mutation factor
The model aforementioned in Section
Representation and initialization. Let
Because
According to the related experience of Khouja and Goyal [
By combining
Then, create the initial population randomly.
If the maximum number of iterations is reached, the algorithm will be stopped and output the optimal solution; otherwise, go to the next step.
For each target individual
With randomly chosen integer indexes
The crossover operator implements a discrete recombination of the trial individual
The evaluation function of an offspring is one-to-one competition in the MADE. It means that the resulting trial individual will only replace the original if it has a lower objective function value. Otherwise, the parent will remain in the next generation. The rule is as follows:
Then, the procedure will return to Step
Output the optimal total cost TC* and the corresponding replenishment cycle
The steps can be described by the flow chart as shown in Figure
Flow chart of the MADE.
In this section, a simple example of four-product and three-supplier problem is presented in order to better understand the implementation of MADE.
2
Sequences of 3
Sequences of 4
Sequences of 2
The mutant vector 4.2
0.25
And rnb
Since “4.2” is not an integer, we round it to the nearest integer; that is, it will be changed to “4.” Meanwhile, “3.22” exceeds the range of variety; it is substituted by a number randomly generated from
In Section
The basic data under deterministic situation.
Item 1 | Item 2 | Item 3 | Item 4 | |
---|---|---|---|---|
|
25 | 14 | 20 | 30 |
|
600 | 900 | 1200 | 1000 |
|
800 | 600 | 700 | 500 |
|
0.02 | 0.02 | 0.02 | 0.02 |
|
5.6 | 21 | 42 | 15 |
|
28 | 35 | 40 | 30 |
The items supplied by suppliers.
Supplier 1 | Supplier 2 | Supplier 3 | |
---|---|---|---|
Item 1 | 1 | 0 | 0 |
Item 2 | 0 | 1 | 0 |
Item 3 | 0 | 0 | 1 |
Item 4 | 0 | 0 | 1 |
|
40 | 50 | 60 |
The distance between the warehouse and suppliers.
Warehouse | Supplier 1 | Supplier 2 | Supplier 3 | |
---|---|---|---|---|
Warehouse | 0 | 11 | 9 | 7 |
Supplier 1 | 11 | 0 | 5 | 8 |
Supplier 2 | 9 | 5 | 0 | 10 |
Supplier 3 | 7 | 8 | 10 | 0 |
Results comparison of three approaches.
|
|
|
TC, min | Convergence rate | CPU time (seconds) | |
---|---|---|---|---|---|---|
Approach of [ |
3, 1, 1, 1 | 1.7035, 1.7125, 1.4060, 1.7907 | 0.0759 | 9021.2 | / | / |
MADE | 2, 1, 1, 1 | 1.8537, 1.6576, 1.3765, 1.7441 | 0.0811 | 9005.9 | 100% | 4.3734 |
GA | 2, 1, 1, 1 | 2.2442, 2.1071, 1.9193, 2.2863 | 0.0775 | 9213.2 | 66% | 4.4655 |
Both GA and MADE are intelligence EAs. For a perspicuous and direct understanding, we give two convergence curves as shown in Figure
Convergence curve of MADE and GA.
Results listed in Table
Obviously, MADE is better than GA and the algorithm of Qu et al. [
In this section, we will design three different scenarios to compare the results of JRD and fuzzy JRD.
We set the range of the fuzzy set to [
The basic data after extension.
Item |
|
|
|
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1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
The major ordering cost
Results comparison under different situations (scenarios 1).
|
Situation |
|
|
|
TC, min | CPU time (seconds) |
---|---|---|---|---|---|---|
100 | Certainty | (2, 1, 1, 1) | (1.8459, 1.6578, 1.3710, 1.7442) | 0.0811 | 9005.90 | 4.3686 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.8399, 1.6514, 1.3637, 1.7379) | 0.0822 | 8805.18 | 4.4798 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.8349, 1.6459, 1.3575, 1.7327) | 0.0832 | 8643.61 | 20.1834 | |
| ||||||
300 | Certainty | (2, 1, 1, 1) | (1.7319, 1.5343, 1.2301, 1.6252) | 0.1041 | 11165.68 | 4.3143 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.7249, 1.5268, 1.2214, 1.6179) | 0.1057 | 10934.50 | 4.4220 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.7191, 1.5205, 1.2142, 1.6118) | 0.1070 | 10747.98 | 20.258 | |
| ||||||
500 | Certainty | (2, 1, 1, 1) | (1.6525, 1.4478, 1.1302, 1.5421) | 0.1231 | 12926.66 | 4.2566 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.6449, 1.4396, 1.1206, 1.5342) | 0.1250 | 12668.79 | 4.3772 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.6387, 1.4328, 1.1127, 1.5277) | 0.1266 | 12460.48 | 19.8587 | |
| ||||||
700 | Certainty | (1, 1, 1, 1) | (1.8962, 1.3601, 1.0276, 1.4581) | 0.1448 | 14415.60 | 4.1407 |
Uncertainty, GMIR | (1, 1, 1, 1) | (1.8893, 1.3514, 1.0174, 1.4497) | 0.1471 | 14132.48 | 4.2276 | |
Uncertainty, centroid | (1, 1, 1, 1) | (1.8836, 1.3442, 1.0089, 1.4428) | 0.1491 | 13903.65 | 19.9156 |
Table
Set the range of the fuzzy set [
The basic data under uncertainty.
Item |
|
|
|
---|---|---|---|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
When
Results comparison under different situations (scenarios 2).
|
Situation |
|
|
|
TC, min | CPU time (seconds) |
---|---|---|---|---|---|---|
100 | Certainty | (2, 1, 1, 1) | (1.8459, 1.6578, 1.3710, 1.7442) | 0.0811 | 9005.90 | 4.4341 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.8459, 1.6578, 1.3710, 1.7442) | 0.0811 | 9005.90 | 4.6048 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.8459, 1.6578, 1.3710, 1.7442) | 0.0811 | 9005.90 | 21.0726 | |
| ||||||
300 | Certainty | (2, 1, 1, 1) | (1.7319, 1.5343, 1.2301, 1.6252) | 0.1041 | 11165.68 | 4.3173 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.7319, 1.5343, 1.2301, 1.6252) | 0.1041 | 11165.68 | 4.4136 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.7319, 1.5343, 1.2301, 1.6252) | 0.1041 | 11165.68 | 20.3023 | |
| ||||||
500 | Certainty | (2, 1, 1, 1) | (1.6525, 1.4478, 1.1302, 1.5421) | 0.1231 | 12926.66 | 4.2955 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.6525, 1.4478, 1.1302, 1.5421) | 0.1231 | 12926.66 | 4.37988 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.6525, 1.4478, 1.1302, 1.5421) | 0.1231 | 12926.66 | 19.8634 | |
| ||||||
700 | Certainty | (2, 1, 1, 1) | (1.8962, 1.3601, 1.0276, 1.4581) | 0.1448 | 14415.60 | 4.1261 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.8962, 1.3601, 1.0276, 1.4581) | 0.1448 | 14415.60 | 4.2493 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.8962, 1.3601, 1.0276, 1.4581) | 0.1448 | 14415.60 | 19.8194 |
Results shown in Table
We extend the range of the fuzzy set to [
The basic data after extension.
item |
|
|
|
---|---|---|---|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
We set the major ordering cost
Results comparison under different situations after extension (scenarios 3).
|
Situation |
|
|
|
TC, min | CPU time (seconds) |
---|---|---|---|---|---|---|
100 | Certainty | (2, 1, 1, 1) | (1.8459, 1.6578, 1.3710, 1.7442) | 0.0811 | 9005.90 | 4.3782 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.8516, 1.664, 1.378, 1.7501) | 0.0801 | 9205.31 | 4.5183 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.856, 1.6687, 1.3833, 1.7547) | 0.0793 | 9363.95 | 20.6277 | |
| ||||||
300 | Certainty | (2, 1, 1, 1) | (1.7319, 1.5343, 1.2301, 1.6252) | 0.1041 | 11165.68 | 4.2908 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.7386, 1.5416, 1.2385, 1.6322) | 0.1026 | 11394.79 | 4.4135 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.7437, 1.5472, 1.2449, 1.6376) | 0.1015 | 11576.66 | 20.0418 | |
| ||||||
500 | Certainty | (2, 1, 1, 1) | (1.6525, 1.4478, 1.1302, 1.5421) | 0.1231 | 12926.66 | 4.2934 |
Uncertainty, GMIR | (2, 1, 1, 1) | (1.6597, 1.4558, 1.1393, 1.5497) | 0.1212 | 13181.85 | 4.3672 | |
Uncertainty, centroid | (2, 1, 1, 1) | (1.665, 1.4616, 1.1461, 1.5553) | 0.1199 | 13384.18 | 20.1400 | |
| ||||||
700 | Certainty | (1, 1, 1, 1) | (1.8962, 1.3601, 1.0276, 1.4581) | 0.1448 | 14415.60 | 4.1279 |
Uncertainty, GMIR | (1, 1, 1, 1) | (1.9029, 1.3685, 1.0374, 1.466) | 0.1426 | 14695.63 | 4.2845 | |
Uncertainty, centroid | (1, 1, 1, 1) | (1.908, 1.3749, 1.045, 1.4722) | 0.1410 | 14917.54 | 19.8705 |
Table
From Tables the ranges of fuzzy parameters will inevitably influence the decision of the JRD. So, it is important to utilize every useful information and to correct judgment to confirm the ranges of fuzzy numbers scientifically and reasonably; MADE is quite effective for this fuzzy JRD problem because it can find the best minimum long-run average total cost with 100% convergence rate. The CPU time is also accepted for decision makers.
This paper is an interdisciplinary research of the fuzzy inventory model and intelligent optimization algorithm. A fuzzy JRD model for the one-warehouse, Due to the nonavailability of sufficient and precise input data, accurate predicted values of the minor ordering cost, inventory holding cost, and backlogging cost cannot be obtained easily; while fuzzy numbers can efficiently model the imprecise values. Obviously, it is more reasonable to handle imprecise values using the fuzzy theory in the JRD model. It is the first time to introduce fuzziness into the stochastic JRD, which will widen the application field of fuzzy theory and will make the JRD become more practical. Since the stochastic JRD policy is widely used in many industries, such as manufacturing, wholesale, maintenance, repair, and operating (MRO) supplies, the proposed fuzzy JRD models with two defuzzification methods can also be applied in these industries. The proposed simple and effective MADE-based approach can find the optimal cycle time and safety factor of each item for the defuzzified JRD effectively. Moreover, the convergence rate of MADE outperforms another popular GA-based approach. Therefore, managers can always know the corresponding minimum total cost under fuzzy environment, just like the case of precise data. This study expands the application field of the DE.
In the future, a dependent-chance programming model or a chance-constrained programming model can be designed for the JRD model of the one-warehouse, n-retailer system in fuzzy environment. The DE still can provide good solutions to these problems.
In order to solve the fuzzy JRD model, we need to use the following definitions and properties.
A fuzzy set
Let the family of all fuzzy points be
A fuzzy set
A fuzzy set
When
A fuzzy set
This research is partially supported by National Natural Science Foundation of China (70801030, 71371080, 71131004, 70871050, and 71171093), Humanities and Social Sciences Foundation of Chinese Ministry of Education (no. 11YJC630275), and Fundamental Research Funds for the Central Universities (HUST: 2012TS065, and CXY12Q043). The authors are very grateful for their contribution of data handling of Chen Xiao-xi and Dun Cai-xia.