Optimizing the Joint Replenishment and Delivery Scheduling Problem under Fuzzy Environment Using Inverse Weight Fuzzy Nonlinear Programming Method

and Applied Analysis 3


Introduction
As a multi-item inventory problem, the joint replenishment problem (JRP) has been widely applied to lots of sizing problems in manufacturing applications (Hsu [1]; Goyal [2]; Wang et al. [3,4]).Beside the possible quantity discount, gathering several items on a single order to reduce the total of these fixed ordering costs is pretty reasonable (Kaspi and Rosenblatt [5]).The studies of the JRPs can be divided into two categories: (a) heuristics for the classic JRP under constant demand and special applications (Olsen [6]; Lee and Yao [7]; Porras and Dekker [8]) and (b) the JRPs under dynamic and/or stochastic demand.The extensive literature review is available in Khouja and Goyal [9] and Narayanan et al. [10].
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. [11]).The relatively scarce literature on the JRDs can be classified into two categories: (a) JRDs with deterministic demand, Chan et al. [12] addressed scheduling issues for multibuyer joint replenishments and Cha et al. [13] studied a JRD model of the one-warehouse and -retailer system and developed a genetic algorithm (GA) and an improved heuristic named RAND to solve this problem and Moon et al. [14] provided joint replenishment and consolidated freight delivery policies with deterministic demand for a third party warehouse; (b) JRDs with stochastic demand, Wang et al. [15] proposed an effective and efficient differential evolution algorithm (DE) for the integrated stochastic JRD.Similar studies can be found in [16][17][18].A limitation common in all the JRDs mentioned above is that all the key factors are assumed to be certain.
In reality, it is more realistic to handle imprecise values using the fuzzy theory for the JRD.In fact, decision-makers are always in front of imprecise and vague operational conditions (Pishvaee and Torabi [19]; Zeng et al. [20]).Uncertainties have been tackled in a lot of ways and fuzzy set theory has a long history for managing inventory (Chiang et al. [21]; Kacprzyk and Stanieski [22]; Wang et al. [23,24]; Wee et al. [25]).Generally, the imprecision may originate from two aspects in the JRD modeling: (1) the imprecise specification of objectives, for example, a decision-maker may have to face vague goals such as "this total cost should be around $20000"; (2) the imprecise specification of related parameters, an important task involved in the fuzzy JRD model is to predict parameters' values such as the holding cost and major ordering cost.However, due to the nonavailability of sufficient and precise input data, the precise predicted values cannot be obtained easily; just an "approximate" value may be ascertained, while fuzzy numbers can efficiently model the imprecise values.
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. [26]).Handfield et al. [27] defuzzified a (Q, r) model based on fuzzy-set representations of various sources of uncertainly in the supply chain.(2) Fuzzy dependent-chance programming (DCP).Fuzzy DCP maximizes the credibility of event such that the total cost in the planning periods does not exceed a certain budget level (Liu [28]).Wang et al. [29] studied a novel JRP model based on DCP with fuzzy minor replenishment cost and fuzzy inventory holding cost.Similar paper can be found in Peng and Zhou [30].For the JRP decision, managers also hope that the total cost does not exceed the budget level, especially when enterprises have much cash flow pressure.In this situation, a fuzzy DCP model is always regarded as a good choice.
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 (M-JRD) model with deterministic demand and fuzzy cost firstly.Moreover, an inverse weight fuzzy nonlinear programming (IWFNLP) adopted by [25] is applied to the M-JRD to satisfy the decision-maker's desirable achievement level of each fuzzy component.At last, novel hybrid algorithms are provided using the fuzzy simulation approach (FSA) and DE for this model handled by IWFNLP method.Results of examples show solutions derived from the IWFNLP method satisfy the decisionmaker's desirable achievement level of the cost and credibility objective.It is an effective decision tool since it can really reflect the relative importance of each fuzzy component.
The rest of this paper is organized as follows.The multiobjective JRD model is given in Section 2. In Section 3, the fuzzy simulation for credibility calculation and DE for searching for an optimum solution are introduced.Section 4 is numerical examples and analysis.Section 5 contains conclusions and future research directions.

Assumptions and Formulation of Fuzzy JRD Model
(1) Assumptions and Notations.Lu and Posner [31] addressed the problem of supplying multiple retail outlets with constant and continuous demand from a single warehouse.Each retailer sells one product, and other retailers sell the same product.All demands must be met without shortages or backlogging.Orders placed by a retailer will result in demand generated at the warehouse.The objective is to minimize the average total cost.This one warehouse, n-retailer JRD problem has received considerable attention from researchers [13,14].In this study, we focus on the JRD model under fuzzy environment.The following assumptions are similar with Lu and Posner [31] and Cha et al. [13].(1) All parameters including demand rates and costs (except warehouse's major ordering cost) are known and constant; (2) replenishment is instantaneous; (3) replenishment lead time is constant.
As reported in Wang et al. [23], an interesting real problem is when human originated data like ordering cost are not precisely known but subjectively estimated or linguistically expressed because of the lack of the accurate history data.Suppose that the major ordering cost (S) is not precisely known.For example, let the linguistic estimates of S be as follows."The major ordering cost is about 40 dollars per order, but not less than 20 dollars per order nor more than 60 dollars per order."In reality, it is very hard to obtain the precise cost.So fuzzy variables are also utilized to handle the JRD problem under uncertainty.In this study, warehouse's major ordering cost is treated as a fuzzy number.Our work introduces fuzziness into the JRD which makes it become more practical.Due to the JRD's difficult mathematical properties, we just suppose the major ordering cost as a fuzzy number to make the study become more possible because two different objectives are considered simultaneously.
As presented in Cha et al. [13], the warehouse will deliver to an individual retailer after it replenishes goods jointly from the suppliers by taking into account the demand of each item at the retailer.The warehouse that has a lot of associated retailers can obtain significant cost savings (logistics cost and ordering cost) by replenishing jointly.The reduction effects will be higher if materials are imported from overseas with the high procurement cost.Referring to [13], the JRD model in this study can be described in Figure 1.Note the warehouse's major ordering cost is a fuzzy number.
In order to discuss the JRD problem, the following notations are defined: (2) Formulation with Fuzzy Major Ordering Cost.Similar to [13], we also discuss  different types of productions and assume that all productions can be stored at the warehouse.Further, we assume that item  is stored and sold only by retailer .The warehouse replenishes item  at every integer multiple (  ) of the basic cycle time () and delivers it to retailer .
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 [31]).Most scholars have concentrated on developing good heuristics for special policies.In this study, a popular stationary policy used in [13] is also adopted that a warehouse delivers item  to retailer  at the same time interval with a fixed order quantity.
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 (1)

Defuzzified Total Cost by Signed Distance
(1) Defuzzified TC.The signed distance method is simple and easy to handle, and this is why the extension principle and centroid method are not applied to this fuzzy JRD model.It is difficult for the extension principle and centroid to obtain the estimated TC in fuzzy sense.Moreover, Chiang et al. [21] found there was not a significant difference between signed distance method and the extension principle or centroid method.So the signed distance method is used to obtain the TC in fuzzy sense.
From (1), we can obtain the following equations: From ( 2), the defuzzified TC to be minimized using signed distance can be obtained as (3)

Credibility Object Based on DCP.
Generally speaking, DCP is related to maximizing some chance functions events in an uncertain environment (Liu [28]).As a widely used type of stochastic programming, DCP has been extended to the area of fuzzy programming (R. Wang and L. Wang [32]; Wang et al. [33]).In practice, the goal of JRD policy cannot be confirmed exactly due to inevitable uncertainty.Hence, a realistic approach for decision makers (DMs) may be to maximize the credibility of achieving the optimization goals.Sometimes, DMs are not concerned with minimizing the total cost but hope that TC does not exceed the budget level TC , especially under much cash flow pressure.In this situation, a natural idea is to maximize the credibility of the event such that the total cost is less than or equal toTC .So an objective function can be written as

Proposed Fuzzy Multiobjective JRD (M-JRD) Model.
Due to the uncertainty of the decision-making, 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 M-JRD model can be described as where the wavy bar "∼" denotes the fuzziness of the two objectives goal.
(1) The first one minimizes and the result is set as  0 ( C, 0) 0 .
Then, the membership function is used to describe the attainable degree of the two objectives.In this study, the linear membership functions  d0 ( C,0) ( 0 ( C, 0)),  C{⋅} ({⋅}) are used, where The pictorial representations of these membership functions are given in Figure 2.

Formulation Using Traditional Fuzzy Additive Goal
Programming.In order to specify imprecise aspiration levels of the goals under fuzzy environment, Narasimhan [34] had firstly developed fuzzy goal programming (FGP) using membership functions.Chen and Tsai [35] reformulated the fuzzy additive goal programming (FAGP) by incorporating different important and preemptive priorities of fuzzy goals.In contrast to other methods, the FAGP allows managers to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals.In order to better understand the following proposed M-JRD model using IWFNLP, we give the formulation using traditional FAGP as follows: Figure 2: Membership functions for fuzzy total cost and credibility objectives.
where  1 and  2 are weights for the defuzzification objective and credibility objective, respectively, and  1 +  2 = 1 ( 1 > 0;  2 > 0).These weights reflect the relative importance of each goal in the decision model.

Formulation Using IWFNLP Approach
(1) Rationales of Using IWFNLP.Although the FAGP can be applied to handle the fuzzy decision problem, Wee et al. [25] found that the ratio of the achievement levels ( 1 ,  2 ) obtained by this approach cannot really reflect the ratio of the weights.To overcome the shortages of the FAGP method, a novel IWFNLP method was adopted.This method embeds the idea of inverse weights into the Max-Min fuzzy model.

Results of a lot of experiments show IWFNLP method can
give solutions in which the ratio of the fuzzy components achievement levels is as close to the ratio of the assigned weights as possible.So the IWFNLP is also adapted to handle the M-JRD.The effectiveness of this method is further verified in Section 4.
(2) Formulation.For the weights   ( = 1, 2, . . ., ), one-toone transformation from the weight   to the inverse weight IW  is defined as In this study,  = 2.So the fuzzy model is formulated as where According to the definition of inverse weight, ( 12) is equivalent to

Outlet of the Proposed Algorithm
(1) The FSA Is Utilized to Solve the Fuzzy DCP.Usually, it is hard to obtain credibility value with an analytical method.Wang et al. [33] designed a FSA to calculate the credibility value for fixed decision variables.The advantage of the FSA is that it can calculate the credibility whether the membership functions of the fuzzy numbers are simple or not and whether several fuzzy numbers are involved or not.In fact, analytical methods usually can also be utilized for the same target.However, it can work only when member functions are simple, for example, linear functions and triangular functions.
Moreover, analytical methods do not work well when too many fuzzy numbers are involved.But the FSA does not subject to such a restraint.So the FSA is adopted in this study.
(2) DE Is Adopted to Find the Solution Quickly.When the objective functions to be optimized are multimodal or the search spaces are particularly irregular, intelligent algorithms should be designed to solve the fuzzy DCP models.Moreover, optimization algorithms need to be highly robust in order to avoid getting stuck at the local optimal solution.Among these algorithms, genetic algorithm (GA) has been proved to be effective for the DCP models (Ke and Liu [36]).However, the GA displays inherent difficulties in performing local search for some applications because of the difficulty of the selection of the suitable probability of crossover and mutation.So it is important to find a novel algorithm to handle the fuzzy DCP more effectively.DE is one of the best evolution algorithms (EAs) for solving nonlinear, nondifferentiable, and multimodal optimization problems (Storn and Price [37]).Due to its simple structure, easy implementation, quick convergence, and robustness, DE has been applied to a variety of fields [38][39][40][41][42][43].Wang et al. [4] found that the basic DE is a good candidate for the similar JRPs.But, the effectiveness of DEs for the fuzzy JRD should be studied further because of the difficult mathematical properties.
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 [42].A good trade-off between convergence and diversity should be designed.Zou et al. [44] developed an improved DE (IDE) with a modified mutation factor according to the objective function values in mutation steps and adjusted crossover factor in terms of iteration number in crossover step.The IDE can not only diversify candidate solutions, but also increase the convergence rate.So the IDE is utilized in case of DE having unsatisfactory performance.

Fuzzy Simulation Approach for Solving DCP-Based Model.
Usually, it is difficult to compute {⋅} with an analytic method.Instead, we use a FSA to obtain the value of {⋅} for a fixed (,    ,    ).According to [33], a classic fuzzy simulation approach can be adopted as follows.
Step 2. Uniformly generate a sequence  1 from Θ such that Pos { 1 } ≥  where  is a sufficiently small number.Thus, a real vector ( 1 ) can be obtained.
Step Step 1 (initialization and representation) Initialization.The initial population Ω is created by assigning random values which lies inside the feasible bound of the decision variable.Each individual is generated by where  is the number of individuals;   is the dimension of each individual;  ()  and  ()  represent the low and upper bound of the th decision parameter respectively; rand[0, 1] is a uniformly distributed random number in range [0,1].
Representation.Now, we will demonstrate how our chromosomes can be decoded to a feasible solution and how each chromosome of population is evaluated.For  = 6, Figure 3 shows what the chromosomes is composed of.
Assuming  ,min is the lower bound of th gene and  ,max is the upper bound of th gene, it is obviously that  ,min is 1 for    and    and  ,min is 0 for .It is difficult to give the maximum values for decision variables of the fuzzy M-JRD model because   and   influence each other.We consider that RAND algorithm mentioned in Section 3.4 can obtain the optimum solution for the crisp JRD model.So we set  ,max as values of three times the optimal upper bounds obtained by the RAND JRD for this fuzzy JRD model following the similar experience of Cha et al. [13]; Wang et al. [15].
Step 3 (crossover).The basic DE crossover operator implements a discrete recombination of the trial individual V ,+1 and the parent individual  , to produce offspring  ,+1 .The crossover is implemented as follows: where  , refers to the th element of the individual  , . ,+1 and V ,+1 are similarly defined.rand() is the th evaluation of a uniform random number generator between [0,1].() is a randomly chosen index from 1, 2, . . .,  which ensures that  ,+1 gets at least one parameter from V ,+1 .Otherwise, no new parent individual would be produced and the population would not alter.CR is the crossover or recombination rate between [0, 1] which has to be determined by the user.
Step 4 (selection).The selection in DE is deterministic and simple.The evaluation function of an offspring is one-to-one competition in the DE.It means 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 if ( ,+1 ) = ( , ) = 0, and then where  is obtained by (13).This is the same for all variants of the DE.Although the selection pressure is only one, the best individual of the next generation will be at least as fit as the best individual of the current generation.
Step 5 (stop and output results).When stopping criterion is met, output the optimal results; otherwise, repeat Step 2-Step 4. In this study, stopping criterion is met when maximum number of iteration ( max ) is reached.

The IDE.
The key difference between the IDE and basic DE is in the way of adjusting scale mutation factor  and crossover rate CR.The method modifies mutation factor  according to the objective function values of all candidate solutions in mutation step and adjusts crossover rate CR in terms of iteration number in crossover step.Both modified operators can not only diversify candidate solutions, but also increase the convergence of algorithm.In short, the IDE and DE are different in two aspects.
(1) For mutation  of DE, it is set to a fixed value for all candidate solutions over all iterations.That is to say, all candidate solutions have the same magnification factor of the differential variation   2 , −   3 , .Here, if the objective function is minimizing the total cost, an adaptive scale factor   for the th candidate solution is stated as follows: If the objective function is maximizing the credibility, an adaptive scale factor   for the th candidate solution is stated as follows: where rand  belongs to a uniform distribution in the ranges [0, 1];   represents the objective value of the th solution;  min and  max represent the minimal and maximal objective function values of all candidate solutions, respectively;  aver represents the average objective function value of all solutions.
(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 CR  is adopted for all candidate solutions at iteration , and it is stated at follows: where  and  max are current iteration number and maximal iteration number, respectively; CR min and CR max represent minimal crossover rate and maximal crossover, respectively.

Modified Algorithm for the Defuzzified JRD Model
(1) Optimal T. For a given set of    and   , the optimal basic cycle time  can be easily obtained as shown in (23) from the first order derivative of the defuzzified TC function since the TC function is convex in : (2) Optimal   and   .According to experiences of [13][14][15]17], for a given set of    and , the optimality condition of   can be derived from the following two conditions: ( C, 0,   ) ≤  ( C, 0,   + 1) ,  ( C, 0,   ) ≤ ( C, 0,   − 1) .
Therefore, the optimality condition of   is defined as follows: ≤   (  + 1) .
Similarly, the optimality condition of   is defined as follows: (3) RAND JRD: A Modified Algorithm for the Defuzzified JRD.Cha et al. [13] proposed a modified RAND algorithm for the crisp JRD model to minimize the TC.Making corresponding changes in  max , an algorithm named RAND JRD can be used to find the optimal solution.The procedures are as follows: Step 1.
Step 6.For a given value of   1 (−1) and a given set of   (), find the optimal values of   using (26).Put   () =   .

Numerical Examples
4.1.Experiment 1. Basic Experiment 1 is given to compare the DE and IDE.According to the recommendation of the inventor of DE [37] and the similar experiences of [4,15,17], the factors setting for DE used in all experiments are listed as follows: the population  = 50, the mutation factor  = 1.6, the crossover factor CR = 0.7, and the maximum generation is set to 500.Factors of IDE are set as follows: CR max = 0.9, CR min = 0.4.The data and results of this experiment are shown in Tables 1 and 2. Convergent curves of DE and IDE are given in Figure 5.
Then, DE and IDE are performed 50 times and the results are reported in Table 3.
From Table 2, the radios of the achievement levels gained by DE and IDE can be computed as follows: Results in Tables 2 and 3 show (1) the ratios of the achievement levels and the weights for the fuzzy objectives are nearly equivalent using IWFNLP and DE/IDE; (2) IDE outperforms DE with a rapid convergence speed.Given (T, k  i s, f  i s) (1) e 1 = 0, e 2 = 0 and z=1, simulation times N 1 Step (2) Generate  1z , calculate  S(S( 1z )) Calculate  by (13) Step   4. The results are listed in Table 5. Figure 6 shows the convergent processes.
From Table 5 and Figure 6, we can conclude that the IDE can also find satisfactory solutions with different values of major ordering cost.The ratios of the achievement levels and the weights for the fuzzy objectives are still nearly equivalent using IWFNLP method and IDE.

Conclusions and Future Research
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 [45].However, to our best knowledge, the JRD model under fuzzy costs that simultaneously minimizes the total cost and maximizes the credibility to assure the safety of cash flow is nonexistence.Our work provides a useful approach for the joint replenishment and delivery scheduling under uncertainty.
(2) The formulation of the proposed M-JRD 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 decision-maker'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 genetic-simulated annealing algorithm [46] and quantum evolution algorithm [47], also show good performances to solve complex optimization problems.In the future, we will design hybrid algorithms by taking the advantages of the above algorithms to handle more complex fuzzy JRD problems.Moreover, another future research direction is to optimize the fuzzy JRD problem under different operational risks because of the inevitable uncertainty under supply chain environment for enterprises in the network economic era [48][49][50].

Figure 5 :
Figure 5: Convergent curve of DE and IDE.
: integer number that decides the replenishment schedule of item  (decision variable), it means the replenishment cycle of item  is   .
3.3.1.Basic DE.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.
,  1 , and  2 as shown in Table 2.But the values of S are different as shown in Table

Table 2 :
The results for Experiment 1.

Table 3 :
Comparison of DE and IDE.

Table 5 :
Results for Experiment 2 using IDE.