In reality, decisionmakers are always in front of imprecise and vague operational conditions. We propose a practical multiobjective joint replenishment and delivery scheduling (JRD) model with deterministic demand and fuzzy cost. This model minimizes the total cost defuzzified by the signed distance method and maximizes the credibility that the total cost does not exceed the budget level. Then, an inverse weight fuzzy nonlinear programming (IWFNLP) method is adopted to formulate the proposed model. This method embeds the idea of inverse weights into the MaxMin fuzzy model. Thirdly, the fuzzy simulation approach and differential evolution algorithm (DE) are utilized to solve this problem. Results show that solutions derived from the IWFNLP method satisfy the decisionmaker’s desirable achievement level of the cost objective and credibility objective. It is an effective decision tool since it can really reflect the relative importance of each fuzzy component. Our study also shows that the improved DE outperforms DE with a faster convergence speed.
As a multiitem inventory problem, the joint replenishment problem (JRP) has been widely applied to lots of sizing problems in manufacturing applications (Hsu [
Typically speaking, if the warehouse is assumed as the central of a supply chain for all the new JRPs, two extensions of JRP should be noted; one extension is in the supplying end and the other extension is in the selling end. For both of two extensions, the delivery considerations should be considered. Here we call them the joint replenishment and delivery scheduling (JRD) problems. In fact, most corporations with global purchasing have realized that considerable cost savings can be achieved by a JRD policy (Sindhuchao et al. [
In reality, it is more realistic to handle imprecise values using the fuzzy theory for the JRD. In fact, decisionmakers are always in front of imprecise and vague operational conditions (Pishvaee and Torabi [
At present, there are two common approaches to handle fuzzy parameters. (1) Defuzzification. As a favorite approach in many inventory modeling for its simplicity, defuzzification can easily transfer fuzziness to be explicit without complex analysis (Roy et al. [
However, no study has simultaneously considered total cost and credibility of the total cost does not exceed the budget level as performance criterions for the JRD with fuzzy cost. This unexplored area is important and interesting since it integrates into a single model two main decision criterions: total cost and credibility. The aim of this paper is to develop a practical multiobjective JRD (MJRD) model with deterministic demand and fuzzy cost firstly. Moreover, an inverse weight fuzzy nonlinear programming (IWFNLP) adopted by [
The rest of this paper is organized as follows. The multiobjective JRD model is given in Section
As reported in Wang et al. [
As presented in Cha et al. [
The JRD model (source: Cha et al. [
In order to discuss the JRD problem, the following notations are defined:
Procedures to find optimal policies are very difficult. There are no known good approaches for solving this problem in time polynomial in the number of retailers (Lu and Posner [
The total cost (TC) is composed of the sum of the major ordering cost, minor ordering cost, inventory holding cost, and outbound transportation cost of a warehouse as well as the total of the inventory holding costs of retailers. According to the above definitions, the total relevant fuzzy cost per unit time to be minimized is given by
From (
From (
Generally speaking, DCP is related to maximizing some chance functions events in an uncertain environment (Liu [
Due to the uncertainty of the decisionmaking, it is quite natural to assume two main goals: (1) nonrigid total cost goal; (2) credibility goal to assure the safety of cash flow. These goals represent different attitudes of managers for handling the inevitable uncertainty. In reality, it is not surprising that managers have their own opinions on the goal under uncertainty. For the fuzzy objectives, the MJRD model can be described as
To describe the fuzzy objectives in (
Alternatively, the following four cases can be solved to confirm the values of
The first one minimizes
The second one minimizes
The third one maximizes
The forth one maximizes
Then, the membership function is used to describe the attainable degree of the two objectives. In this study, the linear membership functions
The pictorial representations of these membership functions are given in Figure
Membership functions for fuzzy total cost and credibility objectives.
In order to specify imprecise aspiration levels of the goals under fuzzy environment, Narasimhan [
In this study,
According to the definition of inverse weight, (
DE is one of the best evolution algorithms (EAs) for solving nonlinear, nondifferentiable, and multimodal optimization problems (Storn and Price [
DE has a good performance in convergence speed, but the faster convergence may cause the diversity of population to descend quickly during the solution process [
Usually, it is difficult to compute
When the objective functions to be optimized are multimodal or the search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at local optimal solution. The advantage of DE is just to obtain the global optimal solution fairly. In the following, we discuss two DEs.
Basic DE consists of three evolution operators: mutation, crossover, and selection. In mutation operator, DE uses the differences between randomly selected individuals to generate a trial individual. Then crossover operator is used to produce one offspring which is only accepted if it improves on the fitness of parent individual. The process of choosing individuals is called selection. A brief description of the DE algorithm is as follows.
Decoding chromosome.
Assuming
The key difference between the IDE and basic DE is in the way of adjusting scale mutation factor
(1) For mutation
If the objective function is maximizing the credibility, an adaptive scale factor
(2) For crossover rate CR of DE, it is set to a fixed value for any dimension of any candidate solution over all iterations. In other words, any dimension of any candidate solution has the same crossover rate, which does not change with the evolution process. Here, a dynamic crossover
Therefore, the optimality condition of
Similarly, the optimality condition of
Otherwise, stop and select
Main flowchart of the proposed algorithm is shown in Figure
Main flowchart of the proposed algorithm.
Basic Experiment 1 is given to compare the DE and IDE. According to the recommendation of the inventor of DE [
Data for Experiment 1.
Item 
1  2  3  4  5  6 


10,000  5,000  3,000  1,000  600  200 

45  46  47  44  45  47 

1  1  1  1  1  1 

5  5  5  5  5  5 

1.5  1.5  1.5  1.5  1.5  1.5 




TC  5000  



0.44 

0.56 
The results for Experiment 1.
Algorithm 







DE  0.195 


0.341  4831.7  0.8417 
IDE  0.195 


0.341  4831.7  0.8417 
Convergent curve of DE and IDE.
Convergent curve of DE
Convergent curve of IDE
We can calculate
Then, DE and IDE are performed 50 times and the results are reported in Table
Comparison of DE and IDE.
Algorithm  Convergence times  Average iteration times  Maximum iteration times  Minimum iteration times  Average 
Minimum 

DE  50  234.88  283  180  0.3704  0.3704 
IDE  50  85.66  104  68  0.3704  0.3704 
From Table
Results in Tables
Khouja and Goyal [
Data for Experiments 2a, 2b, and 2c.
Experiment 2a 

(4, 10, 16)  TC  3639 
Experiment 2b 

(38, 45, 52)  TC  3912 
Experiment 2c 

(1000, 1060, 1280)  TC  8267 
Results for Experiment 2 using IDE.
Experiment number 









2a  0.1066  (1, 1, 2, 3, 4, 6)  (2, 2, 3, 2, 2, 2)  0.2745  3625.1  0.6240 


2b  0.1410  (1, 1, 1, 2, 3, 5)  (3, 2, 2, 2, 2, 2)  0.2781  3899.0  0.6320 


2c  0.3683  (1, 1, 1, 1, 1, 2)  (8, 6, 5, 3, 2, 2)  0.3534  8013.0  0.8033 


Convergent curve of IDE for Experiment 2.
Experiment 2a
Experiment 2b
Experiment 2c
From Table
This paper is an interdisciplinary research of the fuzzy inventory model and intelligent optimization algorithm. Due to the inevitable uncertainty, it is quite natural for decisionmaker to assume two main goals: (1) nonrigid total cost goal; (2) credibility goal to assure the safety of cash flow. We developed a practical JRD model under uncertainty and provided an effective algorithm for this model. The main contributions are as follows.
(1) Actually, there are lots of papers discussed inventory and risk management issues [
(2) The formulation of the proposed MJRD model is handled by the IWFNLP which can make the ratios of the achievement levels of objectives and the weights for the fuzzy objectives are nearly equivalent. The IWFNLP method gives the solution that satisfies the decisionmaker’s desirable achievement level of the total cost objective and credibility objective. It is an effective decision tool to ensure a decisionmaker’s expectation is achieved.
(3) Hybrid intelligent algorithms are designed to solve the proposed JRD handled by the IWFNLP method using the FSA and DE/IDE. Results of numerical examples show the IDE can find satisfactory solutions faster than DE.
Other intelligent algorithms, such as geneticsimulated annealing algorithm [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is partially supported by National Natural Science Foundation of China (nos. 71371080 and 71131004), Humanities and Social Sciences Foundation of Chinese Ministry of Education (no. 11YJC630275), and Fundamental Research Funds for the Central Universities (HUST: 2014QN201).