Application of fuzzy set to lot-sizing production planning with remanufacturing and heterogeneous demands

Abstract

Remanufacturing, which processes end-of-life products with disassembly, testing, reprocessing, and reassembly operations in such a way that their quality and performance is restored, has attracted substantial interest in recent years. In this article, we address a capacitated dynamic lot-sizing problem with remanufacturing, in which there are heterogeneous demand streams for newly manufactured products assembled wholly by brand-new components and remanufactured ones reassembled by reprocessed components. The demand for remanufactured products could also be satisfied by new ones, as remanufactured products do not meet self-demand, but not vice versa. For this planning problem, a fuzzy mixed integer linear programming model, which considers the uncertainties of market demands, the quantity of end-of-life products available, the remanufacturable rate, selling prices, unit cost values, and capacity constraints, is developed by triangular fuzzy numbers. After that, the fuzzy mixed integer linear programming model is transformed into a crisp equivalent by clarifying fuzzy constraints and objective. The solution approach is designed by genetic algorithm, in which self-adaptive formula is adopted to resolve difficulty in obtaining optimum value of crossover probability and mutation probability. Finally, a numerical example is suggested to demonstrate the applicability and effectiveness of the proposed model and solution approach.

Keywords

Lot-sizing production planning remanufacturing heterogeneous demands fuzzy mathematical programming genetic algorithm

Introduction

In recent years, due to increasing environmental deterioration, depletion of energy and natural resources, strict government legislation, and economic factors, more and more companies engage positively and voluntarily in the product recovery business, which entails activities to regain materials and realize value added from end-of-life (EOL) products for repairing, refurbishing, remanufacturing, and recycling.^1,2 Among these, remanufacturing is a very important field of product recovery, in which used products or components are restored to like-new condition which can be marketable again having the same characteristics with the new product in terms of appearance, quality, reliability, and performance.³ A complete remanufacturing process often involves collection, disassembly, cleaning, testing, component reprocessing, and reassembly operations.⁴ Therefore, the traditional forward supply chain of “resources-production-consumption-abandonment” is transformed into the closed-circuit feedback circulating process through integrating remanufacturing.

The traditional models of production planning usually do not take into account remanufacturing processes. Only in recent years, researches have been started to focus on the value of the EOL products by remanufacturing, which integrate into production planning systems as an effective approach to improve resource utilization efficiency, as well as get great benefits of energy saving and emission reduction.

However, note that all researches reviewed concerning lot-sizing production planning with remanufacturing imply that remanufactured products can be sold as new ones with the identical quality and performance, as well as selling price. In other words, remanufactured products are not distinguished from newly manufactured ones, and both of them jointly face the identical market demand. Actually, from customer’s viewpoint, even though remanufactured products are as-good-as-new in terms of both quality and performance, there are often different valuations for remanufactured products and new ones because remanufactured products derive from the recovery of used products or component, rather than processing of raw materials. Therefore, demands of remanufactured products and new ones are satisfied by themselves only, and remanufactured products should offer an inferior selling price than new ones.⁵ Piñeyro and Viera⁶ and Zhang et al.⁷ investigated a lot-sizing production planning based on the circumstances above. Meanwhile, Piñeyro and Viera⁶ assumed that a market strategy was to allow substitution of remanufactured products by new ones, maintaining the selling price of the remanufactured products in order to avoid losing potential customers. But their research simplified the processes of both remanufacturing and manufacturing, that is, used products were restored directly without disassembly; of course, there was also not the assembly or reassembly in their research framework. This assumption implied that a product has only a single remanufacturable component. Obviously, it limits the applicability of the developed model.

On the other hand, in a real lot-sizing production planning with remanufacturing, environmental coefficients and related parameters, including market demands, the quantity of EOL products available, the remanufacturable rate, selling prices, unit cost values, and capacities of every main operation, are normally imprecise because of some information being incomplete or unobtainable over the intermediate planning horizon. Few researches address the production planning with remanufacturing in an uncertain environment, all of which cannot be solved by the conventional deterministic solution techniques. Fuzzy set theory provides the appropriate framework to describe and treat uncertainty.⁸ Fuzzy mathematical programming is therefore viewed as an alternative to the stochastic problems, where the parameters or objectives or both are modeled as fuzzy sets. Some researchers dealt with production planning problems, which face just traditional forward supply chain without remanufacturing, in an uncertain environment through fuzzy set theory included those by Pai,⁹ Sahebjamnia and Torabi,¹⁰ and Figueroa-García et al.¹¹

Accordingly, the lot-sizing problem we study in this article can be characterized as follows: (1) the heterogeneous demands for remanufactured products and new ones as well as the one-way substitution between them are considered into lot-sizing production planning; (2) a used product could be disassembled to multiple remanufacturable components, and remanufacturing process is described as a complete process involving collection, disassembly, cleaning, testing, component reprocessing, and reassembly operations; (3) multiple products are considered in our article; (4) besides the limitation of inventories, the capacity constraints of every main operation are also taken into account; and (5) market demands, the quantity of EOL products available, the remanufacturable rate, selling prices, unit cost values, and capacities of every main operation are all regarded to be uncertain and are characterized by triangular fuzzy numbers. In summary, the main purpose of this article is to develop a multi-product, multi-component, multi-period lot-sizing production planning model considering the heterogeneous demands between remanufacturing products and new ones via fuzzy mixed integer programming.

The rest of the article is organized as follows. Section “Literature review” presents a brief literature review relevant to lot-sizing production planning with remanufacturing. Section “Model formulation” describes the problem, details the assumption, and proposes a new fuzzy mixed integer linear programming (FMILP) model for the lot-sizing production planning with remanufacturing and heterogeneous demands under uncertainty. Then, section “Crisp equivalent form for FMILP model” applies appropriate strategies for converting the FMILP model into a crisp equivalent model. In section “Solution approach for crisp equivalent model based on GA,” we go on to design the solution approach based on genetic algorithm (GA) to solve the crisp equivalent model. In section “Numerical experiments,” we provide the computational analysis of a numerical example. Finally, section “Conclusion” gives the conclusions with possible areas for further research.

Literature review

Richter and Sombrutzki¹² and Richter and Weber¹³ studied a set of the Wagner/Whitin dynamic production planning and inventory control models. In their research, the first model assumed that the quantity of return products is sufficient to satisfy all demands without delay, and only remanufacturing process was taken into account. The manufacturing process was added to the second model, and therefore market demand could be satisfied by either remanufactured products or newly manufactured products. The third model considered further variable manufacturing and remanufacturing costs, rather than minimized only setup costs and inventory costs. Golany et al.¹⁴ considered a similar but more general model and presented respective solution algorithms for the model when costs are linear, convex, and arbitrary.

Teunter et al.¹⁵ studied the lot-sizing production planning with remanufacturing of return products, without constraints on the quantity of return products. Two different setup cost schemes were considered. In the first model, there was a joint setup cost for manufacturing and remanufacturing aiming at single production line. And in the second model, there were separate setup costs for manufacturing and remanufacturing aiming at dedicated production lines. The objective function in two models also considered to minimize total costs consisting of inventory costs and setup costs. Schulz¹⁶ put forward a solution approach of the Silver-Meal-based heuristic aiming at the second model above. Li et al.¹⁷ discussed an optimal decision model including not only manufacturing decision and remanufacturing decision but also disposal decision of the returned product, and both the products demand and the quantity of the returned products are considered to be stochastic. Pan et al.¹⁸ considered to add capacity constraints for manufacturing and remanufacturing to the problem, in order to formulate a general model of the capacitated dynamic lot-sizing problem with manufacturing, remanufacturing, and disposal, as well as to analyze how capacity constraints affect the operation planning. Zhang et al.¹⁹ also addressed a dynamic capacitated production planning with remanufacturing. The difference in this research was manufacturing, remanufacturing, and inventory levels all have limits, and the proposed model was applied to steel enterprise.

Li et al.²⁰ investigated an unlimited-capacity multi-product production planning problem. Similarly, market demands of every product type could be satisfied by manufactured products as well as by remanufactured ones. And different product types corresponded to different levels of quality and performance. The demand for a product type with lower quality level could also be satisfied by a higher level type as lower level items did not meet self-demand. Then, Li et al.²¹ considered the capacity constraints in manufacturing and remanufacturing process for the same scenario.

Kim et al.²² focused on developing a general framework of remanufacturing system in reverse logistics environment and a corresponding mathematic model. In the general framework, the parts used for assembling finished products could be acquired from three channels, which were external suppliers offering new parts, remanufacturing process inside factories, and remanufacturing subcontractor. The function objective of the model was interested in maximizing total profits for overall system through minimizing total remanufacturing costs. Amin and Zhang²³ extended the general framework above. DePuy et al.²⁴ developed a remanufacturing planning approach, which was a probabilistic form of standard material requirement planning (MRP), to provide estimates of the expected number of remanufactured components to be completed in each future period, as well as a component purchase schedule to avoid shortages in periods. Xanthopoulos and Iakovou²⁵ presented a two-phased model of medium-range tactical production planning for the EOL electric and electronic products for a remanufacturing-driven reverse supply chain, with a special focus on the disassembly process. Doh and Lee²⁶ focused on production planning in remanufacturing systems over a set of remanufacturing processes including disassembly, disposal, reprocessing, and reassembly. The objective was to maximize the total profits.

Ferguson et al.²⁷ considered a tactical production planning problem for remanufacturing when return products have different quality levels. The research results showed that a grading system classing return products into different levels could increase profit over a wide range of systems whether in an unlimited-capacity environment or limited-capacity environment.

Subulan et al.²⁸ developed a fuzzy mixed integer programming model for medium-term planning in a closed-loop supply chain, which consisted of multiple manufacturing plants, multiple remanufacturing facilities, multiple wholesalers, multiple collection centers, and multiple retailers. Then, the fuzzy model was transformed into an equivalent model by different aggregation operators and fuzzy solution approaches, and the final crisp model was solved by the standard branch and bound technique.

In this article, the following main research contributions are made: (1) there are lots of researches related to the lot-sizing production planning with remanufacturing, which assume that remanufactured products are not distinguished from new ones. Thus, this research focus on modeling of lot-sizing production planning with the heterogeneous demands between remanufacturing products and new ones. (2) For our developed FMILP model, the appropriate strategies based on possibility measure are presented to transform into a crisp equivalent. (3) The solution approach based on GA is designed to solve the transformed crisp equivalent model, and solution results indicate that the solution approach is superior to the standard branch and bound technique used by Subulan et al.²⁸

Model formulation

Problem definition

In this article, we consider a lot-sizing production planning with remanufacturing problem, where the structure of the considered production system can be illustrated by Figure 1. The operational procedures of this system can be described as follows: (1) a factory collects EOL products from retailers or customers and deposits them in the inventory of return products; (2) return products must be disassembled to components, and there is a testing operation to know the extent of the damage of these components; (3) if these components are up to the remanufacturing standard, they can be deposited in the inventory of qualified components, or else would be disposed; (4) the factory can restore the quality and performance of qualified components through remanufacturing technology, and these reprocessed components are deposited in the inventory of remanufactured components; (5) meanwhile, the factory processes raw materials to produce new components, which deposited in the inventory of new components; (6) the remanufactured products are reassembled with remanufactured components and deposited in the inventory of remanufactured products, which are used to meet the demand for the secondary market; and (7) the new products are assembled with new components and deposited in the inventory of new products, which are used to meet the demand for the original market.

Figure 1.

The production system with remanufacturing and heterogeneous demands.

Meanwhile, there are two kinds of one-way substitution in this production system. One of them is for components. New components can be used to reassemble remanufactured products as a substitute for remanufactured components, when remanufactured components do not meet reassembly requirements, but not vice versa. The other is for products. The demand for the secondary market can also be satisfied by new products with maintaining the selling price of remanufactured products, as remanufactured products do not meet self-demand, but not vice versa.

Furthermore, the following assumptions are made in our model:

The demands whether for the original market or the secondary market in each period need not be satisfied absolutely; backordering is allowable, and penalty cost in the case of backordering should be taken into account.

There are capacity constraints for the main operations including disassembly and testing, reprocessing, processing, reassembly, and assembly, as well as inventory constraints.

Both the accumulative lead times of manufacturing and remanufacturing operations are assumed to be identical and equal to 1 period. That is to say, products manufactured or remanufactured in period $t - 1$ can be delivered to meet the demands in period $t$ .

Due to some information being incomplete or unobtainable over the planning horizon, market demands, the quantity of EOL products available, the remanufacturable rate, selling prices, unit cost values, and capacity constraints are regarded to be imprecise and uncertain, which are characterized by triangular fuzzy numbers.

Establishment of FMILP model

After reviewing literatures and considering practical situations, the original multi-product, multi-component, multi-period lot-sizing production planning with remanufacturing and heterogeneous demands model was designed in this article to determine the quantity for manufacturing/remanufacturing and inventory level for each type of product/component in each period, in order to achieve the maximum total profit. Particularly, some imprecise parameters, such as market demands, the quantity of EOL products available, the remanufacturable rate, selling prices, unit cost values, and capacities of every main operation, are represented by triangular fuzzy numbers as these parameters are sensitive to many unforeseen factors. This allows solutions, falling near the boundaries of corresponding restrains, be more realistic.

In the production system with remanufacturing, the total profit, which is denoted by $\tilde{F}$ in this article, is as follows:

\begin{matrix} \tilde{F} & = Sales revenue - Total manufacturing cost \\ - Total remanufacturing cost \\ - Total collection cost \\ - Total disassembly and testing cost \\ - Total disposal cost - Total penalty cost \\ - Total inventory cost \end{matrix}

(1)

where

\begin{matrix} Sales revenue \end{matrix} ≅ \sum_{t = 1}^{T} [\sum_{p = 1}^{P} SP \tilde{N} P_{pt} \cdot (D \tilde{O} M_{pt} - bo m_{pt}) + SP \tilde{R} P_{pt} \cdot (D \tilde{S} M_{pt} - bs m_{pt})]

(2)

Total manufacturing cost ≅ \sum_{t = 1}^{T} [\sum_{p = 1}^{P} (S A_{pt} \cdot η_{pt} + U \tilde{A} C_{pt} \cdot x_{pt} \cdot a b_{p}) + \sum_{c = 1}^{C} (S P_{ct} \cdot π_{ct} + U \tilde{P} C_{ct} \cdot v_{ct} \cdot p b_{c})]

(3)

Total remanufacturing cost ≅ \sum_{t = 1}^{T} [\sum_{p = 1}^{P} (SA R_{pt} \cdot δ_{pt} + UR \tilde{A} C_{pt} \cdot y_{pt} \cdot ra b_{p}) + \sum_{c = 1}^{C} (SR P_{ct} \cdot τ_{ct} + UR \tilde{P} C_{ct} \cdot z_{ct} \cdot rp b_{c})]

(4)

Total collection cost ≅ \sum_{t = 1}^{T} \sum_{p = 1}^{P} UC \tilde{C} P_{pt} \cdot γ_{pt}

(5)

Total disassembly and testing cost ≅ \sum_{t = 1}^{T} \sum_{p = 1}^{P} SD T_{pt} \cdot σ_{pt} + UD \tilde{T} C_{pt} \cdot {dt}_{pt} \cdot dt b_{p}

(6)

Total disposal cost ≅ \sum_{t = 1}^{T} \sum_{c = 1}^{C} U \tilde{D} C_{ct} \cdot d_{ct}

(7)

Total penalty cost ≅ \sum_{t = 1}^{T} \sum_{p = 1}^{P} (UB \tilde{C} O_{pt} \cdot bo m_{pt} + bs m_{pt} \cdot UB \tilde{C} S_{pt})

(8)

Total inventory cost ≅ \sum_{t = 1}^{T} [\sum_{p = 1}^{P} (ICR P_{pt} \cdot α_{pt} + ICN P_{pt} \cdot λ_{pt} + ICRM P_{pt} \cdot χ_{pt}) + \sum_{c = 1}^{C} (ICQ C_{ct} \cdot β_{ct} + ICN C_{ct} \cdot ζ_{ct} + ICR C_{ct} \cdot ξ_{ct})]

(9)

The symbol “ $≅$ ” is the fuzzified version of “ $=$ ” and means “essentially equal to some value.” Accordingly, the FMILP model of the considered production planning problem is presented in the following way. The objective function, describing the maximization of the total profit, comes in the form

Maximize \tilde{F} (maximize the total profit)

(10)

The inventory flow conservation equations for return products, qualified components, new components, remanufactured components, new products, and remanufactured products are as follows, respectively

α_{pt} = α_{p, t - 1} + γ_{pt} - d t_{pt} \cdot dt b_{p}, \forall p, t

(11)

\begin{matrix} β_{ct} = β_{c, t - 1} & + \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \\ - d_{ct} - z_{ct} \cdot rp b_{c}, \forall c, t \end{matrix}

(12)

\begin{matrix} ζ_{ct} & = ζ_{c, t - 1} + v_{ct} \cdot p b_{c} \\ - \sum_{p = 1}^{P} BO C_{pc} \cdot x_{pt} \cdot a b_{p} - sub c_{ct}, \forall c, t \end{matrix}

(13)

\begin{matrix} ξ_{ct} & = ξ_{c, t - 1} + z_{ct} \cdot rp b_{c} \\ - \sum_{p = 1}^{P} BO C_{pc} \cdot y_{pt} \cdot ra b_{p} + sub c_{ct}, \forall c, t \end{matrix}

(14)

λ_{pt} = λ_{p, t - 1} + x_{pt} \cdot a b_{p} - np d_{pt}, \forall p, t

(15)

χ_{pt} = χ_{p, t - 1} + y_{pt} \cdot ra b_{p} - rp d_{pt}, \forall p, t

(16)

The delivery equations for new products and remanufactured products, which display that all demands in each period may not be satisfied, and force to satisfy the remaining outstanding demands from previous period, are as follows, respectively

np d_{p, t - 1} ≅ D \tilde{O} M_{pt} + sub p_{pt} + bo m_{p, t - 1} - bo m_{pt}

(17)

rp d_{p, t - 1} ≅ D \tilde{S} M_{pt} - sub p_{pt} + bs m_{p, t - 1} - bs m_{pt}

(18)

The limits on inventory levels of return products, qualified components, new components, remanufactured components, new products, and remanufactured products are as follows, respectively

\sum_{p = 1}^{P} α_{pt} \leq \bar{α}, \forall t

(19)

\sum_{c = 1}^{C} β_{ct} \leq \bar{β}, \forall t

(20)

\sum_{c = 1}^{C} ζ_{ct} \leq \bar{ζ}, \forall t

(21)

\sum_{c = 1}^{C} ξ_{ct} \leq \bar{ξ}, \forall t

(22)

\sum_{p = 1}^{P} λ_{pt} \leq \bar{λ}, \forall t

(23)

\sum_{p = 1}^{P} χ_{pt} \leq \bar{χ}, \forall t

(24)

The capacity constraints of disassembly and testing, processing, reprocessing, assembly, and reassembly, where a setup is made whenever an operation is carried out in a period for a positive quantity, are as follows

M {\tilde{P}}_{c} \cdot π_{ct} \tilde{\geq} v_{ct} \cdot p b_{c}, \forall c, t

(25)

M \tilde{R} P_{c} \cdot τ_{ct} \tilde{\geq} z_{ct} \cdot rp b_{c}, \forall c, t

(26)

M {\tilde{A}}_{p} \cdot η_{pt} \tilde{\geq} x_{pt} \cdot a b_{p}, \forall p, t

(27)

M \tilde{R} A_{p} \cdot δ_{pt} \tilde{\geq} y_{pt} \cdot ra b_{p}, \forall p, t

(28)

M \tilde{D} T_{p} \cdot σ_{pt} \tilde{\geq} d t_{pt} \cdot dt b_{p}, \forall p, t

(29)

Here, the symbol “ $\tilde{\geq}$ ” is the fuzzified version of “ $\geq$ ” and means “essentially more than or similar to some value.”

The quantity constraint for disposed components, which requires that unqualified components have to be disposed, is as follows

d_{ct} ≅ \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \cdot (1 - {\tilde{θ}}_{c}), \forall c, t

(30)

The quantity constraint for collected EOL products, which requires that the quantity of collected EOL products cannot be more than the quantity of EOL products available in the market, “if possible,” is as follows

E \tilde{P} A_{pt} - γ_{pt} \tilde{\geq} 0, \forall p, t

(31)

The nonnegative integer constraint for variables is as follows ( $Z$ denotes integer set)

\begin{matrix} d t_{pt}, v_{ct}, z_{ct}, x_{pt}, y_{pt}, d_{pt}, γ_{pt}, sub p_{pt}, sub c_{pt}, \\ α_{pt}, β_{ct}, ζ_{ct}, ξ_{ct}, λ_{pt}, χ_{pt}, bo m_{pt}, bs m_{pt}, np d_{pt}, rp d_{pt} \\ \geq 0 \begin{matrix}  \end{matrix} and \begin{matrix}  \end{matrix} \in Z, \forall p, c, t \end{matrix}

(32)

The binary number constraint for variables is as follows

σ_{pt}, π_{ct}, τ_{ct}, η_{pt}, δ_{pt} \in (0, 1), \forall p, c, t

(33)

Crisp equivalent form for FMILP model

In this article, we assume that planners have already adopted the pattern of triangular distribution to represent all fuzzy parameters above. The primary advantages of the triangular fuzzy number are generally the simplicity and flexibility of the fuzzy arithmetic operations.^29,30 A typical triangular fuzzy number can be constructed by three prominent data, namely, the most pessimistic value, the most likely value, and the most optimistic value. For example, the demand of the original market $D \tilde{O} M_{pt}$ can be represented as a triangular fuzzy number $({DOM}_{pt}^{k}, {DOM}_{pt}^{l}, {DOM}_{pt}^{o})$ , in which ${DOM}_{pt}^{l}$ is the most likely value, ${DOM}_{pt}^{k}$ is the most pessimistic value, and ${DOM}_{pt}^{o}$ is the most optimistic value. Other fuzzy parameters are represented in a similar way. More contents concerning fuzzy set theory and operation for triangular fuzzy numbers can be acquired by referring to Liang and Wang,³¹ Zadeh³² and Kaufmann.³³

Since the FMILP model has a lot of fuzzy parameters, we need a fuzzy optimization approach that jointly considers the possible lack of information in data and existing fuzziness. There are two commonly used approaches to solve a fuzzy programming. One of them is the technology of fuzzy simulation, but it is quite complicated and time consuming to simulate uncertain functions. And the other, converting into crisp equivalent form, is comparatively simple and practical. For these reasons, based on possibility theory,^34,35 we define an approach to convert the FMILP model into a crisp equivalent model for the considered production planning problem under collection, process, and demand uncertainties.

In fuzzy set theory, $Pos {A}$ describes the possibility of event $A$ happening, and $Pos {A} \geq ε_{0}$ denotes the possibility for event $A$ happening should not be less than confidence level $ε_{0}$ . Assume that $\tilde{ϕ} = (a^{k}, a^{l}, a^{o})$ is a triangular fuzzy number and $ρ$ is a nonnegative real number, the following formulas are true³⁴

Pos {\tilde{ϕ} \geq ρ} = {\begin{matrix} 0, & ρ \geq a^{0} \\ \frac{a^{o} - ρ}{a^{o} - a^{l}}, & a^{l} \leq ρ \leq a^{0} \\ 1, & a^{l} \geq ρ \end{matrix}

(34)

Pos {\tilde{ϕ} \leq ρ} = {\begin{matrix} 0, & ρ \leq a^{k} \\ \frac{ρ - a^{k}}{a^{l} - a^{k}}, & a^{k} \leq ρ \leq a^{l} \\ 1, & a^{l} \leq ρ \end{matrix}

(35)

Strategy for converting the fuzzy constraints

Due to the uncertainties of some parameters in the production planning with remanufacturing, all constraints cannot necessarily be met strictly. A planner always expects only these constraints to be met to some extent in an uncertain environment. Therefore, we describe those constraints containing fuzzy parameters by fuzzy chance-constrained based on possibility measure as follows

\begin{matrix} Pos {np d_{p, t - 1} & = D \tilde{O} M_{pt} + sub p_{pt} + bo m_{p, t - 1} \\ - bo m_{pt}} \geq ε_{1}, \forall p, t \end{matrix}

(36)

\begin{matrix} Pos {rp d_{p, t - 1} & = D \tilde{S} M_{pt} - sub p_{pt} + bs m_{p, t - 1} \\ - bs m_{pt}} \geq ε_{2}, \forall p, t \end{matrix}

(37)

Pos {M {\tilde{P}}_{c} \cdot π_{ct} \geq v_{ct} \cdot p b_{c}} \geq ε_{3}, \forall c, t

(38)

Pos {M \tilde{R} P_{c} \cdot τ_{ct} \geq z_{ct} \cdot rp b_{c}} \geq ε_{4}, \forall c, t

(39)

Pos {M {\tilde{A}}_{p} \cdot η_{pt} \geq x_{pt} \cdot a b_{p}} \geq ε_{5}, \forall p, t

(40)

Pos {M \tilde{R} A_{p} \cdot δ_{pt} \geq y_{pt} \cdot ra b_{p}} \geq ε_{6}, \forall p, t

(41)

Pos {M \tilde{D} T_{p} \cdot σ_{pt} \geq d t_{pt} \cdot dt b_{p}} \geq ε_{7}, \forall p, t

(42)

Pos {d_{ct} = \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \cdot (1 - {\tilde{θ}}_{c})} \geq ε_{8}, \forall c, t

(43)

Pos {E \tilde{P} A_{pt} \geq γ_{pt}} \geq ε_{9}, \forall p, t

(44)

Here, $ε_{i} (i = 1, 2, \cdot \cdot \cdot, 9)$ denotes confidence level given beforehand by planners. According to Dubois and Prade,³⁵ there is a lemma to assist to convert constraints (36)–(44) into crisp equivalent form.

Lemma

Assume that $\tilde{ϕ} = (a^{k}, a^{l}, a^{o})$ is a triangular fuzzy number and $ε_{10}$ , $ε_{11}$ , $ε_{12}$ is confidence level, the following equivalent forms are true

Pos {\tilde{ϕ} \geq a} \geq ε_{10} \Leftrightarrow a \leq (1 - ε_{10}) \cdot a^{o} + ε_{10} \cdot a^{l}

(45)

Pos {\tilde{ϕ} \leq a} \geq ε_{11} \Leftrightarrow a \geq (1 - ε_{11}) \cdot a^{k} + ε_{11} \cdot a^{l}

(46)

\begin{matrix} Pos {\tilde{ϕ} = a} \geq ε_{12} \Leftrightarrow (1 - ε_{12}) \cdot a^{k} \\ + ε_{12} \cdot a^{l} \leq a \leq (1 - ε_{12}) \cdot a^{o} + ε_{12} \cdot a^{l} \end{matrix}

(47)

The proof for Lemma refers to Liu.³⁶ Accordingly, constraints (36)–(44) can be converted further into crisp equivalent form as follows

\begin{matrix} (1 - ε_{1}) \cdot {DOM}_{pt}^{k} + ε_{1} \cdot {DOM}_{pt}^{l} \leq np d_{p, t - 1} - sub p_{pt} \\ - bo m_{p, t - 1} + bo m_{pt} \leq (1 - ε_{1}) \cdot {DOM}_{pt}^{o} \\ + ε_{1} \cdot {DOM}_{pt}^{l}, \forall p, t \end{matrix}

(48)

\begin{matrix} (1 - ε_{2}) \cdot {DSM}_{pt}^{k} + ε_{2} \cdot {DSM}_{pt}^{l} \leq rp d_{p, t - 1} + sub p_{pt} \\ - bs m_{p, t - 1} + bs m_{pt} \leq (1 - ε_{2}) \cdot {DSM}_{pt}^{o} \\ + ε_{2} \cdot {DSM}_{pt}^{l}, \forall p, t \end{matrix}

(49)

[(1 - ε_{3}) \cdot {MP}_{c}^{o} + ε_{3} \cdot {MP}_{c}^{l}] \cdot π_{ct} \geq v_{ct} \cdot p b_{c}, \forall c, t

(50)

[(1 - ε_{4}) \cdot {MRP}_{c}^{o} + ε_{4} \cdot {MRP}_{c}^{l}] \cdot τ_{ct} \geq z_{ct} \cdot rp b_{c}, \forall c, t

(51)

[(1 - ε_{5}) \cdot {MA}_{p}^{o} + ε_{5} \cdot {MA}_{p}^{l}] \cdot η_{pt} \geq x_{pt} \cdot a b_{p}, \forall p, t

(52)

[(1 - ε_{6}) \cdot {MRA}_{p}^{o} + ε_{6} \cdot {MRA}_{p}^{l}] \cdot δ_{pt} \geq y_{pt} \cdot ra b_{p}, \begin{matrix}  \end{matrix} \forall p, t

(53)

[(1 - ε_{7}) \cdot {MDT}_{p}^{o} + ε_{7} \cdot {MDT}_{p}^{l}] \cdot σ_{pt} \geq d t_{pt} \cdot dt b_{p}, \forall p, t

(54)

\begin{matrix} \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \cdot [(1 - ε_{8}) \cdot θ_{c}^{k} + ε_{8} \cdot θ_{c}^{l}] \\ \leq \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} - d_{ct}, \forall c, t \end{matrix}

(55)

\begin{matrix} \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} - d_{ct} \leq \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \\ \cdot [(1 - ε_{8}) \cdot θ_{c}^{o} + ε_{8} \cdot θ_{c}^{l}], \forall c, t \end{matrix}

(56)

(1 - ε_{9}) \cdot {EPA}_{pt}^{o} + ε_{9} \cdot {EPA}_{pt}^{l} \geq γ_{pt}, \forall p, t

(57)

Strategy for converting the fuzzy objective

Similarly, there are the uncertainties of some parameters in the considered production system, which will affect the achievement of the optimal operation objectives for the system. A planner always expects only to maximize the possibility that the realized operation objectives outstrip the intended targets. Therefore, we describe the objective function containing fuzzy parameters by dependent chance based on possibility measure as follows

Maximize PF = Pos {\tilde{F} \geq G_{0}}

(58)

where $G_{0}$ denotes the intended target profit. And the objective function (58) can be converted further into the following crisp equivalent form

Maximize PF = Pos {\tilde{F} \geq G_{0}} = {\begin{matrix} 0, & G_{0} \geq F^{o} \\ \frac{F^{o} - G_{0}}{F^{o} - F^{l}}, & F^{l} \leq G_{0} \leq F^{o} \\ 1, & F^{l} \geq G_{0} \end{matrix}

(59)

where

\begin{array}{l} F^{o} = R^{o} - U, F^{l} = R^{l} - U, R^{l} = \sum_{t = 1}^{T} {\sum_{p = 1}^{P} [S P N P_{p t}^{l} \cdot (D O M_{p t}^{l} - b o m_{p t}) + S P R P_{p t}^{l} \cdot (D S M_{p t}^{l} - b s m_{p t}) - (U A C_{p t}^{l} \cdot x_{p t} \cdot a b_{p} \\ + U R A C_{p t}^{l} \cdot y_{p t} \cdot r a b_{p} + U D T C_{p t}^{l} \cdot d t_{p t} \cdot d t b_{p} + U B C O_{p t}^{l} \cdot b o m_{p t} + U B C S_{p t}^{l} \cdot b s m_{p t} + U C C P_{p t}^{l} \cdot γ_{p t})] - \sum_{c = 1}^{C} (U P C_{c t}^{l} \cdot v_{c t} \cdot p b_{c} + U R P C_{c t}^{l} \\ \cdot z_{c t} \cdot r p b_{c} + U D C_{c t}^{l} \cdot d_{c t})}, \\ R^{o} = \sum_{t = 1}^{T} {\sum_{p = 1}^{P} [S P N P_{p t}^{o} \cdot (D O M_{p t}^{o} - b o m_{p t}) + S P R P_{p t}^{o} \cdot (D S M_{p t}^{o} - b s m_{p t}) - (U A C_{p t}^{o} \cdot x_{p t} \cdot a b_{p} + U R A C_{p t}^{o} \\ \cdot y_{p t} \cdot r a b_{p} + U D T C_{p t}^{o} \cdot d t_{p t} \cdot d t b_{p} + U B C O_{p t}^{o} \cdot b o m_{p t} + U B C S_{p t}^{o} \cdot b s m_{p t} + U C C P_{p t}^{o} \cdot γ_{p t})] - \sum_{c = 1}^{C} (U P C_{c t}^{o} \cdot v_{c t} \cdot p b_{c} + U R P C_{c t}^{o} \cdot z_{c t} \cdot r p b_{c} \\ + U D C_{c t}^{o} \cdot d_{c t})}, \\ U = \sum_{t = 1}^{T} [\sum_{p = 1}^{P} (S A_{p t} \cdot η_{p t} + S A R_{p t} \cdot δ_{p t} + S D T_{p t} \cdot σ_{p t} + I C R P_{p t} \cdot α_{p t} + I C N P_{p t} \cdot λ_{p t} + I C R M P_{p t} \cdot χ_{p t}) + \sum_{c = 1}^{C} (S P_{c t} \cdot π_{c t} \\ + S R P_{c t} \cdot τ_{c t} + I C Q C_{c t} \cdot β_{c t} + I C N C_{c t} \cdot ζ_{c t} + I C R C_{c t} \cdot ξ_{c t})] \end{array}

Proof

According to the fuzzy operation rules, $\tilde{R}$ is a triangular fuzzy number, which can be represented as $R (R^{k}, R^{l}, R^{o})$ . In addition, $U$ is a real number, so that $\tilde{F} = \tilde{R} - U$ is also a triangular fuzzy number, which can be represented as $(F^{k}, F^{l}, F^{o}) = (R^{k} - U, R^{l} - U, R^{o} - U)$ . Referring to formula (34), we can get the crisp equivalent form (59) for the objective function containing fuzzy parameters.

By above analysis, the complete crisp equivalent model for the lot-sizing production planning with remanufacturing and heterogeneous demands can be formulated as follows

The objective function (59);

The converted constraints (48)–(57);

The nonfuzzy constraints (11)–(16), (19)–(24), (32), and (33) are also included in the model in a similar way.

Solution approach for crisp equivalent model based on GA

The majority of the lot-sizing problems are regarded as nondeterministic polynomial time (NP)-hard questions. And the above crisp equivalent model is a nonlinear programming model, for which it is difficult to search the optimal solution by commonly used exact algorithms. GA is a kind of highly parallel, stochastic, global probability search algorithm based on the evolutionism such as natural selection, genetic crossover, and gene mutation, which is applied a lot in planning optimization. Therefore, in this article, GA is adopted to solve the crisp equivalent model. Meanwhile, self-adaptive formula is introduced to resolve difficulty in obtaining optimum value of crossover probability and mutation probability, thus to enhance searching efficiency. Key steps of designed GA are as follows.

Encoding and decoding

According to the characteristic of the model, $d t_{pt}$ , $v_{ct}$ , $z_{ct}$ , $x_{pt}$ , $y_{pt}$ , $d_{ct}$ , $γ_{pt}$ , $np d_{pt}$ , $rp d_{pt}$ , $sub p_{pt}$ , $sub c_{ct}$ , $bo m_{pt}$ , and $bs m_{pt}$ are regarded as the decision variables, while $σ_{pt}$ , $π_{ct}$ , $τ_{ct}$ , $η_{pt}$ , $δ_{pt}$ , $α_{pt}$ , $β_{ct}$ , $ζ_{ct}$ , $ξ_{ct}$ , $λ_{pt}$ , and $χ_{pt}$ are the state variables. Every decision variable is regarded as a sub-chromosome, which initially generates chromosomes $Q_{m = 1}^{n} = {({dt}_{pt}, v_{ct}, z_{ct}, x_{pt}, y_{pt}, d_{ct}, γ_{pt}, np d_{pt}, rp d_{pt}, {subp}_{pt}, {subc}_{ct}, {bom}_{pt}, {bsm}_{pt})}_{m = 1}^{n}$ , in which $m = {1, 2, \cdot \cdot \cdot, M}$ is the index set of evolutional generation, $n = {1, 2, \cdot \cdot \cdot, N}$ is the index set of population size, $M$ is the maximum evolutional generation, and $N$ is the maximum population size.

Nonnegative integer encoding is adopted for all decision variables. Take $v_{ct}^{n}$ , for example, a gene fragment is $(v_{1}, v_{2}, \cdot \cdot \cdot, v_{T})$ ( $\forall c, m, n$ ), in which gene point $v_{t}$ indicates the quantity of batches of new components to be processed in the $t$ period. When the population is initialized, $d t_{pt}$ is first defined to generate randomly in the interval of $[0, ({MDT}_{p}^{o} - ε_{7} \cdot {MDT}_{p}^{o} + ε_{7} \cdot {MDT}_{p}^{l}) / dt b_{p}]$ , and $v_{ct}$ , $z_{ct}$ , $x_{pt}$ , $y_{pt}$ , $γ_{pt}$ , $sub p_{pt}$ are encoded and generated in a similar way; then, $d_{ct}$ , $sub c_{ct}$ , $np d_{pt}$ , $rp d_{pt}$ are encoded and generated on the basis that above decision variables have been generated. Finally, $bo m_{pt}$ and $bs m_{pt}$ are generated. Each generation interval of decision variables is shown in Table 1.

Table 1.

The generation intervals of decision variables in encoding.

Variables	Generation interval	Variables	Generation interval
$d t_{pt}$	$[0, ({MDT}_{p}^{o} - ε_{7} \cdot {MDT}_{p}^{o} + ε_{7} \cdot {MDT}_{p}^{l}) / dt b_{p}]$	$d_{ct}$	$\begin{matrix} [\sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \cdot (1 + ε_{8} \cdot θ_{c}^{o} - θ_{c}^{o} - ε_{8} \cdot θ_{c}^{l}), \\ \sum_{p = 1}^{P} BO C_{pc} \cdot d t_{pt} \cdot dt b_{p} \cdot (1 + ε_{8} \cdot θ_{c}^{k} - θ_{c}^{k} - ε_{8} \cdot θ_{c}^{l})] \end{matrix}$
$v_{ct}$	$[0, ({MP}_{p}^{o} - ε_{3} \cdot {MP}_{p}^{o} + ε_{3} \cdot {MP}_{p}^{l}) / p b_{c}]$	$sub c_{ct}$	$[0, v_{ct} \cdot p b_{c} - \sum_{p = 1}^{P} BO C_{pc} \cdot x_{pt} \cdot a b_{p} + \bar{ζ}]$
$z_{ct}$	$[0, ({MRP}_{c}^{o} - ε_{4} \cdot {MRP}_{c}^{o} + ε_{4} \cdot {MRP}_{c}^{l}) / rp b_{c}]$	$np d_{pt}$	$[0, x_{pt} \cdot a b_{p} + \bar{λ}]$
$x_{pt}$	$[0, ({MA}_{p}^{o} - ε_{5} \cdot {MA}_{p}^{o} + ε_{5} \cdot {MA}_{p}^{l}) / a b_{p}]$	$rp d_{pt}$	$[0, y_{pt} \cdot ra b_{p} + \bar{χ}]$
$y_{pt}$	$[0, ({MRA}_{p}^{o} - ε_{6} \cdot {MRA}_{p}^{o} + ε_{6} \cdot {MRA}_{p}^{l}) / ra b_{p}]$	$bo m_{pt}$	$\begin{matrix} [(1 - ε_{1}) \cdot {DOM}_{pt}^{k} + ε_{1} \cdot {DOM}_{pt}^{l} - np d_{p, t - 1} + sub p_{pt} + bo m_{p, t - 1}, \\ (1 - ε_{1}) \cdot {DOM}_{pt}^{o} + ε_{1} \cdot {DOM}_{pt}^{l} - np d_{p, t - 1} + sub p_{pt} + bo m_{p, t - 1}] \end{matrix}$
$γ_{pt}$	$[0, {EPA}_{pt}^{o} - ε_{9} \cdot {EPA}_{pt}^{o} + ε_{9} \cdot {EPA}_{pt}^{l}]$	$bs m_{pt}$	$\begin{matrix} [(1 - ε_{2}) \cdot {DSM}_{pt}^{k} + ε_{2} \cdot {DSM}_{pt}^{l} - rp d_{p, t - 1} - sub p_{pt} + bs m_{p, t - 1}, \\ (1 - ε_{2}) \cdot {DSM}_{pt}^{o} + ε_{2} \cdot {DSM}_{pt}^{l} - rp d_{p, t - 1} - sub p_{pt} + bs m_{p, t - 1}] \end{matrix}$
$sub p_{pt}$	$[0, {DSM}_{pt}^{o}]$

The decoding for the chromosome is mainly to obtain corresponding value of the state variable. After one chromosome generates, by using equations (11)–(16) and (50)–(54), all corresponding state variables can be decoded and thus one complete solution set, called as complete individual, $I_{m}^{n}$ can be obtained.

Structuring the fitness function

Obtained state variables $α_{pt}$ , $β_{ct}$ , $ζ_{ct}$ , $ξ_{ct}$ , $λ_{pt}$ , and $χ_{pt}$ in the above decoding process are not necessarily to meet the constraints (19)–(24) and nonnegative integer constraints, and the decision variables $bo m_{pt}$ and $bs m_{pt}$ are also not necessarily to meet the nonnegative integer constraints. Therefore, we use the objective function and the constraint penalty function to structure the fitness function $H$ with the complete individual $I_{m}^{n}$ , namely, $H (I_{j}^{n}) = PF (I_{m}^{n}) - ω (m) \cdot W (I_{m}^{n})$ , in which $ω (m)$ is the penalty factor and $W (I_{m}^{n})$ is the penalty function corresponding to the constraints (19)–(24) and (32).

In this article, self-adaptive penalty function method is adopted to guide adjustment of the penalty factor by searching the process information of evolution. In other words, the penalty factor $ω (m + 1)$ has the following three update modes with the change in evolutional generation $m$

ω (m + 1) = {\begin{matrix} ω (m) / ϖ_{1}, & case 1 \\ ϖ_{2} ω (m), & case 2 \\ ω (m), & others \end{matrix} .

(60)

where $ϖ_{1} > ϖ_{2} > 1$ . In general, $ϖ_{1} = 4$ and $ϖ_{2} = 2$ . Case 1 indicates the best individual is feasible solution in the past $m$ evolutional generation, and Case 2 indicates found the best individual is infeasible solution in the past $m$ evolutional generation. Through this self-adaptive regulation, when the found best individual in the past is feasible solution, showing penalty factor has been large enough, then the penalty pressure for the infeasible solution can be appropriately reduced; when the found best individual in the past is infeasible solution, showing penalty factor is too small, then the penalty for the infeasible solution is needed to be strengthened.

Genetic operation

Selection operation: Roulette wheel method is adopted for selection strategy. The complete individuals are copied to the next-generation population according to the replication probability $P_{s}$ by using the formula

P_{s} = \frac{H (I_{m}^{n})}{\sum_{n = 1}^{N} H (I_{m}^{n})}

(61)

Crossover operation: in order to make the individual still meet the initial constraints of the decision variable after crossover, two-point crossover is adopted in this article. Meanwhile, the self-adaptive crossover probability $P_{cr}$ of the chromosomes is calculated. The calculation formula is as follows

P_{cr} = {\begin{matrix} P_{cr 1} - \frac{(P_{cr 1} - P_{cr 2}) (H (I_{m}^{n}) - H_{avg})}{H_{max} - H_{avg}}, & H (I_{m}^{n}) \geq H_{avg} \\ P_{cr 1}, & H (I_{m}^{n}) < H_{avg} \end{matrix}

(62)

where $H_{avg}$ is the average fitness value in the current population and $H_{max}$ is the maximum fitness value in the current population. In general, $P_{cr 1} = 0.9$ and $P_{cr 2} = 0.6$ .

Mutation operation: in this article, inverted operation is adopted to implement mutation for the chromosomes selected with probability. At the same time, the self-adaptive mutation probability $P_{mu}$ of the chromosomes is calculated according to formula (63)

P_{mu} = {\begin{matrix} P_{mu 1} - \frac{(P_{mu 1} - P_{mu 2}) (H (I_{m}^{n}) - H_{avg})}{H_{max} - H_{avg}}, & H (I_{m}^{n}) \geq H_{avg} \\ P_{mu 1}, & H (I_{m}^{n}) < H_{avg} \end{matrix}

(63)

In general, $P_{mu 1} = 0.1$ and $P_{mu 2} = 0.001$ .

Numerical experiments

Numerical example description

In this section, a numerical example abstracted from a production enterprise of automobile engine is considered to illustrate the application and efficiency of the proposed model and solution approach above. We make a production plan for a series of electronic fuel injection (EFI) engines including three different types of engines ( $P = 3$ ), which are denoted by type I, type II, and type III. And in this production enterprise, main remanufacturing components are cylinder blocks, cylinder heads, connecting rods, and crankshafts. The displacements of the three types of engines are all 1.6 L, cylinder diameters are all 81 mm, and piston strokes are all 77.4 mm. The cylinder blocks and crankshaft of the three engines are all identical in terms of size and material. All remanufacturing components can be interchanged between type II and type III engines because the main differences of the two types of engines are only electronic control and intake manifold assembly. But the cylinder heads and connecting rods of type I engine are distinct from type II and type III engines, so they cannot be interchanged. Therefore, there are six kinds of component ( $C = 6$ ) segmented in detail, namely, cylinder block, cylinder head (a), cylinder head (b), connecting rod (a), connecting rod (b), and crankshafts. Table 2 shows the bill of components to each product ( $BO C_{pc}$ ). The planning horizon is decided to span 12 base planning periods ( $T = 12$ ). According to historical data, the predicted values of demand data for the original market ( $D \tilde{O} M_{pt}$ ) and the secondary market ( $D \tilde{S} M_{pt}$ ) are shown in Tables 3 and 4, respectively; the predicted values of supplies of EOL products available are shown in Table 5. Other fuzzy parameters are shown in Table 6. The main crisp parameters about products are shown in Table 7. The main crisp parameters about components are shown in Table 8. And other parameters are as follows: the quantity of every main operation per batch $dt b_{p} = 25$ , $p b_{c} = 25$ , $rp b_{c} = 25$ , $a b_{p} = 25$ , $ra b_{p} = 25$ ; maximum capacities of all inventories are $\bar{α} = 200$ , $\bar{β} = 300$ , $\bar{ζ} = 300$ , $\bar{ξ} = 300$ , $\bar{λ} = 200$ , $\bar{χ} = 200$ ; the initial quantity of backordered product $bo m_{p 0} = 0$ , $bs m_{p 0} = 0$ ; the initial quantity of finished products to be delivered $np d_{p 0} = 610$ , $rp d_{p 0} = 265$ ; and all initial inventories are considered to be 0.

Table 2.

The bill of each kind of core component to each type of product.

$c$ $p$	Cylinder block	Cylinder head (a)	Cylinder head (b)	Connecting rod (a)	Connecting rod (b)	Crankshafts
I	1	1	0	4	0	1
II	1	0	1	0	4	1
III	1	0	1	0	4	1

Table 3.

The demand data for the original market.

$p =$	1			2			3
$D \tilde{O} M_{pt} =$	${DOM}_{pt}^{k}$	${DOM}_{pt}^{l}$	${DOM}_{pt}^{o}$	${DOM}_{pt}^{k}$	${DOM}_{pt}^{l}$	${DOM}_{pt}^{o}$	${DOM}_{pt}^{k}$	${DOM}_{pt}^{l}$	${DOM}_{pt}^{o}$
$t = 1$	576	604	632	592	620	648	670	698	726
$t = 2$	618	646	674	632	660	688	690	718	746
$t = 3$	561	589	617	590	618	646	604	632	660
$t = 4$	587	615	643	651	679	707	691	719	747
$t = 5$	629	657	685	634	662	690	675	703	731
$t = 6$	614	642	670	611	639	667	655	683	711
$t = 7$	571	599	627	624	652	680	649	677	705
$t = 8$	553	581	609	639	667	695	599	627	655
$t = 9$	613	641	669	678	706	734	664	692	720
$t = 10$	580	608	636	583	611	639	686	714	742
$t = 11$	625	653	681	594	622	650	633	661	689
$t = 12$	645	673	701	659	687	715	676	704	732

Table 4.

The demand data for the secondary market.

$p =$	1			2			3
$D \tilde{S} M_{pt} =$	${DSM}_{pt}^{k}$	${DSM}_{pt}^{l}$	${DSM}_{pt}^{o}$	${DSM}_{pt}^{k}$	${DSM}_{pt}^{l}$	${DSM}_{pt}^{o}$	${DSM}_{pt}^{k}$	${DSM}_{pt}^{l}$	${DSM}_{pt}^{o}$
$t = 1$	252	268	284	264	280	296	285	301	317
$t = 2$	266	282	298	245	261	277	288	304	320
$t = 3$	246	262	278	255	271	287	284	300	316
$t = 4$	275	291	307	273	289	305	275	291	307
$t = 5$	252	268	284	283	299	315	285	301	317
$t = 6$	235	251	267	267	283	299	274	290	306
$t = 7$	243	259	275	254	270	286	281	297	313
$t = 8$	251	267	283	257	273	289	276	292	308
$t = 9$	254	270	286	271	287	303	284	300	316
$t = 10$	264	280	296	286	302	318	266	282	298
$t = 11$	277	293	309	278	294	310	291	307	323
$t = 12$	252	268	284	260	276	292	270	286	302

Table 5.

The quantity of EOL products available.

$p =$	1			2			3
$E \tilde{P} A_{pt} =$	${EPA}_{pt}^{k}$	${EPA}_{pt}^{l}$	${EPA}_{pt}^{o}$	${EPA}_{pt}^{k}$	${EPA}_{pt}^{l}$	${EPA}_{pt}^{o}$	${EPA}_{pt}^{k}$	${EPA}_{pt}^{l}$	${EPA}_{pt}^{o}$
$t = 1$	266	284	302	322	340	358	324	342	360
$t = 2$	273	291	309	271	289	307	309	327	345
$t = 3$	261	279	297	275	293	311	300	318	336
$t = 4$	291	309	327	285	303	321	306	324	342
$t = 5$	320	338	356	276	294	312	323	341	359
$t = 6$	271	289	307	322	340	358	311	329	347
$t = 7$	289	307	325	258	276	294	286	304	322
$t = 8$	267	285	303	310	328	346	318	336	354
$t = 9$	261	279	297	288	306	324	315	333	351
$t = 10$	282	300	318	293	311	329	321	339	357
$t = 11$	299	317	335	290	308	326	298	316	334
$t = 12$	313	331	349	257	275	293	294	312	330

Table 6.

The fuzzy parameters.

Parameters	Triangular fuzzy distribution
$UD \tilde{T} C_{pt}$	$UDT C_{pt} (190, 200, 210)$
$U \tilde{P} C_{ct}$	$UP C_{1 t} (3900, 4000, 4100)$ , $UP C_{2 t} (2400, 2500, 2600)$ , $UP C_{3 t} (2700, 2800, 2900)$ , $UP C_{4 t} (70, 75, 80)$ , $UP C_{5 t} (95, 100, 105)$ , $UP C_{6 t} (950, 1000, 1050)$
$UR \tilde{P} C_{ct}$	$URP C_{1 t} (950, 1050, 1100)$ , $URP C_{2 t} (550, 600, 650)$ , $URP C_{3 t} (650, 700, 750)$ , $URP C_{4 t} (22, 25, 28)$ , $URP C_{5 t} (28, 30, 32)$ , $URP C_{6 t} (325, 350, 375)$
$U \tilde{A} C_{pt}$	$UA C_{1 t} (5900, 6000, 6100)$ , $UA C_{2 t} (7400, 7500, 7600)$ , $UA C_{3 t} (7900, 8000, 8100)$
$UR \tilde{A} C_{pt}$	$URA C_{1 t} (5900, 6000, 6100)$ , $URA C_{2 t} (7400, 7500, 7600)$ , $URA C_{3 t} (7900, 8000, 8100)$
$U \tilde{D} C_{ct}$	$UD C_{1 t} (32, 35, 37)$ , $UD C_{2 t} (28, 30, 32)$ , $UD C_{3 t} (28, 30, 32)$ , $UD C_{4 t} (23, 25, 27)$ , $UD C_{5 t} (23, 25, 27)$ , $UD C_{6 t} (28, 30, 32)$
$UC \tilde{C} P_{pt}$	$UCC P_{1 t} (950, 1000, 1050)$ , $UCC P_{2 t} (1150, 1200, 1250)$ , $UCC P_{3 t} (1150, 1200, 1250)$
$UB \tilde{C} O_{pt}$	$UBC O_{1 t} (3150, 3200, 3250)$ , $UBC O_{2 t} (3750, 3800, 3850)$ , $UBC O_{3 t} (3950, 4000, 4050)$
$UB \tilde{C} S_{pt}$	$UBC S_{1 t} (2250, 2300, 2350)$ , $UBC S_{2 t} (2650, 2700, 2750)$ , $UBC S_{3 t} (2850, 2900, 2950)$
$SP \tilde{N} P_{pt}$	$SPN P_{1 t} (16500, 16700, 16900)$ , $SPN P_{2 t} (18800, 19000, 19200)$ , $SPN P_{3 t} (19800, 20000, 20200)$
$SP \tilde{R} P_{pt}$	$SPR P_{1 t} (11400, 11600, 11800)$ , $SPR P_{2 t} (13500, 13700, 13900)$ , $SPR P_{3 t} (14300, 14500, 14700)$
$M \tilde{D} T_{p}$	$MD T_{p} (350, 375, 400)$
$M {\tilde{P}}_{c}$	$M P_{1} (2300, 2400, 2500)$ , $M P_{2} (750, 800, 850)$ , $M P_{3} (1500, 1600, 1700)$ , $M P_{4} (3100, 3200, 3300)$ , $M P_{5} (6200, 6400, 6600)$ , $M P_{6} (2300, 2400, 2500)$
$M \tilde{R} P_{c}$	$MR P_{1} (1000, 1050, 1100)$ , $MR P_{2} (325, 350, 375)$ , $MR P_{3} (650, 700, 750)$ , $MR P_{4} (1300, 1400, 1500)$ , $MR P_{5} (2700, 2800, 2900)$ , $MR P_{6} (1000, 1050, 1100)$
$M {\tilde{A}}_{p}$	$M A_{p} (750, 800, 850)$
$M \tilde{R} A_{p}$	$MR A_{p} (325, 350, 375)$
${\tilde{θ}}_{c}$	$θ_{1} (0.93, 0.95, 0.97)$ , $θ_{2} (0.89, 0.92, 0.94)$ , $θ_{3} (0.90, 0.92, 0.94)$ , $θ_{4} (0.87, 0.90, 0.92)$ , $θ_{5} (0.88, 0.90, 0.92)$ , $θ_{6} (0.83, 0.85, 0.87)$

Table 7.

The product-related parameters.

$p =$	1	2	3		1	2	3		1	2	3
$SD T_{pt}$	1000	1200	1200	$SA R_{pt}$	25,000	30,000	35,000	$ICN P_{pt}$	20	20	20
$S A_{pt}$	25,000	30,000	35,000	$ICR P_{pt}$	10	10	10	$ICRM P_{pt}$	20	20	20

Table 8.

The component-related parameters.

$c =$	1	2	3	4	5	6		1	2	3	4	5	6
$S P_{ct}$	25,000	20,000	24,000	15,000	18,000	18,000	$ICN C_{ct}$	15	15	15	10	10	10
$SR P_{ct}$	10,000	6000	7000	5000	6000	5000	$ICR C_{ct}$	15	15	15	10	10	10
$ICQ C_{ct}$	10	10	10	5	5	5

Model results

We set all confidence levels ( $ε_{i}, i = 1, 2, \cdot \cdot \cdot, 10$ ) to 90% and make the intended target profit $G_{0} = 1.4 \times 10^{8}$ . The lot-sizing production planning for the numerical example can be solved according to the procedure set out above. Using these parameters, we can formulate directly the crisp equivalent form for the FMILP model. And by the solution approach based on GA for the crisp equivalent model, we can obtain decision results for the lot-sizing problem.

The solution approach is coded with MATLAB, and the computational experiences for the numerical example are conducted on such a computer with Windows XP, Core 2 Duo 2.20 GHz CPU, and 1.99 GB RAM. And in genetic parameters, the penalty factor $ω (m)$ , the crossover probability $P_{cr}$ , and the mutation probability $P_{mu}$ are all self-adaptive, so they do not have a significant impact on solution efficiency and results. But solution results will be changed by selecting different values of maximum evolutional generation $M$ and maximum population size $N$ . The change in solution results with $M$ and $N$ is shown in Figure 2, in which the total profit is represented by the most likely value only.

Figure 2.

The change in solution results with the values of $M$ and $N$ .

As can be seen from Figure 2, when the values of $N$ are too small, satisfactory solution results are difficult to obtain regardless of the value of $M$ , even GA cannot converge; when $N \geq 150$ , we can obtain satisfactory solution results, and value of $M$ can be adjusted gradually to a lower level with increasing values of $N$ . The satisfactory solution results are given in Table 9 in detail. According to the results, the most likely value of the total profit is $F^{l} = 1.338 \times 10^{8}$ , the most pessimistic value of the total profit is $F^{k} = 1.124 \times 10^{8}$ , and the most optimistic value of the total profit is $F^{o} = 1.564 \times 10^{8}$ . The possibility that the realized total profit outstrips the intended profit is $PF = 0.73$ . In addition, the standard branch and bound techniques used by Subulan et al.²⁸ are compared with our approach. The branch and bound techniques are implemented in Lingo 12.0, and the corresponding results are $F_{BAB}^{l} = 1.326 \times 10^{8}$ , $F_{BAB}^{k} = 1.112 \times 10^{8}$ , $F_{BAB}^{o} = 1.551 \times 10^{8}$ , and $P F_{BAB} = 0.67$ . Obviously, the performance of our solution approach based on GA is superior to the standard branch and bound techniques.

Table 9.

The computational variable results for the lot-sizing problem.

Variable	Results
$d t_{pt}$	$d t_{11} = 11$ , $d t_{12} = 12$ , $d t_{13} = 10$ , $d t_{14} = 13$ , $d t_{15} = 11$ , $d t_{16} = 13$ , $d t_{17} = 11$ , $d t_{18} = 11$ , $d t_{19} = 10$ , $d t_{1, 10} = 13$ , $d t_{1, 11} = 12$ , $d t_{1, 12} = 0$ , $d t_{21} = 13$ , $d t_{22} = 11$ , $d t_{23} = 11$ , $d t_{24} = 13$ , $d t_{25} = 12$ , $d t_{26} = 10$ , $d t_{27} = 11$ , $d t_{28} = 13$ , $d t_{29} = 12$ , $d t_{2, 10} = 12$ , $d t_{2, 11} = 12$ , $d t_{2, 12} = 0$ , $d t_{31} = 12$ , $d t_{32} = 14$ , $d t_{33} = 13$ , $d t_{34} = 13$ , $d t_{35} = 14$ , $d t_{36} = 12$ , $d t_{37} = 13$ , $d t_{38} = 13$ , $d t_{39} = 13$ , $d t_{3, 10} = 13$ , $d t_{3, 11} = 12$ , $d t_{3, 12} = 0$
$v_{ct}$	$v_{11} = 87$ , $v_{12} = 75$ , $v_{13} = 80$ , $v_{14} = 83$ , $v_{15} = 78$ , $v_{16} = 77$ , $v_{17} = 75$ , $v_{18} = 82$ , $v_{19} = 75$ , $v_{1, 10} = 76$ , $v_{1, 11} = 82$ , $v_{1, 12} = 0$ , $v_{21} = 27$ , $v_{22} = 26$ , $v_{23} = 24$ , $v_{24} = 28$ , $v_{25} = 26$ , $v_{26} = 24$ , $v_{27} = 23$ , $v_{28} = 26$ , $v_{29} = 23$ , $v_{2, 10} = 26$ , $v_{2, 11} = 27$ , $v_{2, 12} = 0$ , $v_{31} = 61$ , $v_{32} = 52$ , $v_{33} = 57$ , $v_{34} = 54$ , $v_{35} = 52$ , $v_{36} = 56$ , $v_{37} = 52$ , $v_{38} = 56$ , $v_{39} = 52$ , $v_{3, 10} = 50$ , $v_{3, 11} = 55$ , $v_{3, 12} = 0$ , $v_{41} = 114$ , $v_{42} = 96$ , $v_{43} = 100$ , $v_{44} = 109$ , $v_{45} = 104$ , $v_{46} = 96$ , $v_{47} = 92$ , $v_{48} = 104$ , $v_{49} = 98$ , $v_{4, 10} = 105$ , $v_{4, 11} = 109$ , $v_{4, 12} = 0$ , $v_{51} = 246$ , $v_{52} = 213$ , $v_{53} = 224$ , $v_{54} = 224$ , $v_{55} = 208$ , $v_{56} = 224$ , $v_{57} = 211$ , $v_{58} = 227$ , $v_{59} = 208$ , $v_{5, 10} = 201$ , $v_{5, 11} = 222$ , $v_{5, 12} = 0$ , $v_{61} = 96$ , $v_{62} = 76$ , $v_{63} = 80$ , $v_{64} = 86$ , $v_{65} = 82$ , $v_{66} = 79$ , $v_{67} = 77$ , $v_{68} = 86$ , $v_{69} = 80$ , $v_{6, 10} = 81$ , $v_{6, 11} = 81$ , $v_{6, 12} = 0$
$z_{ct}$	$z_{11} = 34$ , $z_{12} = 35$ , $z_{13} = 32$ , $z_{14} = 37$ , $z_{15} = 35$ , $z_{16} = 34$ , $z_{17} = 30$ , $z_{18} = 35$ , $z_{19} = 33$ , $z_{1, 10} = 36$ , $z_{1, 11} = 33$ , $z_{1, 12} = 0$ , $z_{21} = 10$ , $z_{22} = 10$ , $z_{23} = 10$ , $z_{24} = 11$ , $z_{25} = 10$ , $z_{26} = 10$ , $z_{27} = 10$ , $z_{28} = 11$ , $z_{29} = 11$ , $z_{2, 10} = 12$ , $z_{2, 11} = 11$ , $z_{2, 12} = 0$ , $z_{31} = 23$ , $z_{32} = 23$ , $z_{33} = 22$ , $z_{34} = 24$ , $z_{35} = 24$ , $z_{36} = 20$ , $z_{37} = 22$ , $z_{38} = 24$ , $z_{39} = 22$ , $z_{3, 10} = 24$ , $z_{3, 11} = 22$ , $z_{3, 12} = 0$ , $z_{41} = 34$ , $z_{42} = 48$ , $z_{43} = 36$ , $z_{44} = 47$ , $z_{45} = 40$ , $z_{46} = 40$ , $z_{47} = 42$ , $z_{48} = 42$ , $z_{49} = 38$ , $z_{4, 10} = 47$ , $z_{4, 11} = 43$ , $z_{4, 12} = 0$ , $z_{51} = 90$ , $z_{52} = 83$ , $z_{53} = 88$ , $z_{54} = 97$ , $z_{55} = 96$ , $z_{56} = 79$ , $z_{57} = 87$ , $z_{58} = 91$ , $z_{59} = 88$ , $z_{5, 10} = 95$ , $z_{5, 11} = 86$ , $z_{5, 12} = 0$ , $z_{61} = 25$ , $z_{62} = 34$ , $z_{63} = 32$ , $z_{64} = 33$ , $z_{65} = 30$ , $z_{66} = 31$ , $z_{67} = 30$ , $z_{68} = 31$ , $z_{69} = 29$ , $z_{6, 10} = 33$ , $z_{6, 11} = 31$ , $z_{6, 12} = 0$
$x_{pt}$	$x_{11} = 26$ , $x_{12} = 24$ , $x_{13} = 24$ , $x_{14} = 27$ , $x_{15} = 26$ , $x_{16} = 24$ , $x_{17} = 23$ , $x_{18} = 26$ , $x_{19} = 23$ , $x_{1, 10} = 26$ , $x_{1, 11} = 27$ , $x_{1, 12} = 0$ , $x_{21} = 27$ , $x_{22} = 25$ , $x_{23} = 27$ , $x_{24} = 27$ , $x_{25} = 25$ , $x_{26} = 26$ , $x_{27} = 27$ , $x_{28} = 28$ , $x_{29} = 24$ , $x_{2, 10} = 25$ , $x_{2, 11} = 27$ , $x_{2, 12} = 0$ , $x_{31} = 32$ , $x_{32} = 26$ , $x_{33} = 29$ , $x_{34} = 29$ , $x_{35} = 27$ , $x_{36} = 27$ , $x_{37} = 25$ , $x_{38} = 28$ , $x_{39} = 28$ , $x_{3, 10} = 25$ , $x_{3, 11} = 28$ , $x_{3, 12} = 0$
$y_{pt}$	$y_{11} = 11$ , $y_{12} = 12$ , $y_{13} = 10$ , $y_{14} = 12$ , $y_{15} = 10$ , $y_{16} = 10$ , $y_{17} = 10$ , $y_{18} = 11$ , $y_{19} = 11$ , $y_{1, 10} = 12$ , $y_{1, 11} = 11$ , $y_{1, 12} = 0$ , $y_{21} = 11$ , $y_{22} = 11$ , $y_{23} = 11$ , $y_{24} = 12$ , $y_{25} = 12$ , $y_{26} = 11$ , $y_{27} = 10$ , $y_{28} = 12$ , $y_{29} = 11$ , $y_{2, 10} = 12$ , $y_{2, 11} = 11$ , $y_{2, 12} = 0$ , $y_{31} = 14$ , $y_{32} = 12$ , $y_{33} = 11$ , $y_{34} = 12$ , $y_{35} = 12$ , $y_{36} = 12$ , $y_{37} = 12$ , $y_{38} = 12$ , $y_{39} = 11$ , $y_{3, 10} = 12$ , $y_{3, 11} = 11$ , $y_{3, 12} = 0$
$d_{ct}$	$d_{11} = 46$ , $d_{12} = 48$ , $d_{13} = 44$ , $d_{14} = 50$ , $d_{15} = 45$ , $d_{16} = 42$ , $d_{17} = 45$ , $d_{18} = 48$ , $d_{19} = 45$ , $d_{1, 10} = 49$ , $d_{1, 11} = 44$ , $d_{1, 12} = 0$ , $d_{21} = 22$ , $d_{22} = 24$ , $d_{23} = 20$ , $d_{24} = 26$ , $d_{25} = 22$ , $d_{26} = 26$ , $d_{27} = 22$ , $d_{28} = 22$ , $d_{29} = 20$ , $d_{2, 10} = 26$ , $d_{2, 11} = 24$ , $d_{2, 12} = 0$ , $d_{31} = 50$ , $d_{32} = 50$ , $d_{33} = 48$ , $d_{34} = 51$ , $d_{35} = 51$ , $d_{36} = 45$ , $d_{37} = 49$ , $d_{38} = 53$ , $d_{39} = 51$ , $d_{3, 10} = 51$ , $d_{3, 11} = 47$ , $d_{3, 12} = 0$ , $d_{41} = 113$ , $d_{42} = 123$ , $d_{43} = 103$ , $d_{44} = 128$ , $d_{45} = 108$ , $d_{46} = 131$ , $d_{47} = 108$ , $d_{48} = 108$ , $d_{49} = 100$ , $d_{4, 10} = 128$ , $d_{4, 11} = 118$ , $d_{4, 12} = 0$ , $d_{51} = 250$ , $d_{52} = 252$ , $d_{53} = 236$ , $d_{54} = 255$ , $d_{55} = 255$ , $d_{56} = 216$ , $d_{57} = 236$ , $d_{58} = 260$ , $d_{59} = 245$ , $d_{5, 10} = 245$ , $d_{5, 11} = 236$ , $d_{5, 12} = 0$ , $d_{61} = 136$ , $d_{62} = 138$ , $d_{63} = 126$ , $d_{64} = 148$ , $d_{65} = 137$ , $d_{66} = 132$ , $d_{67} = 133$ , $d_{68} = 140$ , $d_{69} = 130$ , $d_{6, 10} = 141$ , $d_{6, 11} = 134$ , $d_{6, 12} = 0$
$γ_{pt}$	$γ_{11} = 283$ , $γ_{12} = 292$ , $γ_{13} = 265$ , $γ_{14} = 310$ , $γ_{15} = 310$ , $γ_{16} = 290$ , $γ_{17} = 275$ , $γ_{18} = 275$ , $γ_{19} = 274$ , $γ_{1, 10} = 301$ , $γ_{1, 11} = 300$ , $γ_{1, 12} = 0$ , $γ_{21} = 325$ , $γ_{22} = 282$ , $γ_{23} = 294$ , $γ_{24} = 304$ , $γ_{25} = 295$ , $γ_{26} = 250$ , $γ_{27} = 275$ , $γ_{28} = 325$ , $γ_{29} = 300$ , $γ_{2, 10} = 300$ , $γ_{2, 11} = 300$ , $γ_{2, 12} = 0$ , $γ_{31} = 336$ , $γ_{32} = 328$ , $γ_{33} = 319$ , $γ_{34} = 325$ , $γ_{35} = 342$ , $γ_{36} = 320$ , $γ_{37} = 305$ , $γ_{38} = 325$ , $γ_{39} = 325$ , $γ_{3, 10} = 325$ , $γ_{3, 11} = 300$ , $γ_{3, 12} = 0$
$np d_{pt}$	$np d_{11} = 650$ , $np d_{12} = 587$ , $np d_{13} = 613$ , $np d_{14} = 659$ , $np d_{15} = 644$ , $np d_{16} = 597$ , $np d_{17} = 579$ , $np d_{18} = 639$ , $np d_{19} = 606$ , $np d_{1, 10} = 651$ , $np d_{1, 11} = 675$ , $np d_{1, 12} = 0$ , $np d_{21} = 666$ , $np d_{22} = 616$ , $np d_{23} = 684$ , $np d_{24} = 660$ , $np d_{25} = 637$ , $np d_{26} = 650$ , $np d_{27} = 669$ , $np d_{28} = 704$ , $np d_{29} = 609$ , $np d_{2, 10} = 620$ , $np d_{2, 11} = 685$ , $np d_{2, 12} = 0$ , $np d_{31} = 800$ , $np d_{32} = 634$ , $np d_{33} = 721$ , $np d_{34} = 701$ , $np d_{35} = 681$ , $np d_{36} = 675$ , $np d_{37} = 625$ , $np d_{38} = 690$ , $np d_{39} = 712$ , $np d_{3, 10} = 659$ , $np d_{3, 11} = 702$ , $np d_{3, 12} = 0$
$rp d_{pt}$	$rp d_{11} = 275$ , $rp d_{12} = 261$ , $rp d_{13} = 289$ , $rp d_{14} = 270$ , $rp d_{15} = 252$ , $rp d_{16} = 260$ , $rp d_{17} = 268$ , $rp d_{18} = 271$ , $rp d_{19} = 279$ , $rp d_{1, 10} = 294$ , $rp d_{1, 11} = 269$ , $rp d_{1, 12} = 0$ , $rp d_{21} = 274$ , $rp d_{22} = 270$ , $rp d_{23} = 281$ , $rp d_{24} = 298$ , $rp d_{25} = 282$ , $rp d_{26} = 269$ , $rp d_{27} = 268$ , $rp d_{28} = 286$ , $rp d_{29} = 297$ , $rp d_{2, 10} = 299$ , $rp d_{2, 11} = 276$ , $rp d_{2, 12} = 0$ , $rp d_{31} = 340$ , $rp d_{32} = 299$ , $rp d_{33} = 286$ , $rp d_{34} = 300$ , $rp d_{35} = 291$ , $rp d_{36} = 297$ , $rp d_{37} = 291$ , $rp d_{38} = 299$ , $rp d_{39} = 281$ , $rp d_{3, 10} = 306$ , $rp d_{3, 11} = 285$ , $rp d_{3, 12} = 0$
$sub p_{pt}$	$sub p_{11} = 4$ , $sub p_{12} = 6$ , $sub p_{24} = 7$ , $sub p_{28} = 4$ , $sub p_{34} = 4$ , other $sub p_{pt} = 0$
$sub c_{ct}$	$sub c_{11} = 50$ , $sub c_{22} = 50$ , $sub c_{24} = 25$ , $sub c_{31} = 50$ , $sub c_{36} = 75$ , $sub c_{41} = 25$ , $sub c_{43} = 100$ , $sub c_{44} = 25$ , $sub c_{49} = 150$ , $sub c_{4, 10} = 25$ , $sub c_{4, 11} = 25$ , $sub c_{51} = 25$ , $sub c_{52} = 125$ , $sub c_{56} = 100$ , $sub c_{57} = 25$ , $sub c_{58} = 125$ , $sub c_{5, 10} = 25$ , $sub c_{5, 11} = 50$ , $sub c_{61} = 75$ , $sub c_{62} = 25$ , $sub c_{64} = 75$ , $sub c_{65} = 100$ , $sub c_{66} = 50$ , $sub c_{67} = 50$ , $sub c_{59} = 100$ , $sub c_{5, 10} = 50$ , other $sub c_{ct} = 0$
$bo m_{pt}$	$bo m_{21} = 8$ , $bo m_{31} = 86$ , $bo m_{32} = 3$ , other $bo m_{pt} = 0$
$bs m_{pt}$	$bs m_{14} = 1$ , $bs m_{21} = 14$ , $bs m_{2, 10} = 4$ , $bs m_{31} = 35$ , other $bs m_{pt} = 0$
$σ_{pt}$	$σ_{1, 12} = 0$ , $σ_{2, 12} = 0$ , $σ_{3, 12} = 0$ , all other $σ_{pt} = 1$
$π_{ct}$	$π_{1, 12} = 0$ , $π_{2, 12} = 0$ , $π_{3, 12} = 0$ , $π_{4, 12} = 0$ , $π_{5, 12} = 0$ , $π_{6, 12} = 0$ , all other $π_{ct} = 1$
$τ_{ct}$	$τ_{1, 12} = 0$ , $τ_{2, 12} = 0$ , $τ_{3, 12} = 0$ , $τ_{4, 12} = 0$ , $τ_{5, 12} = 0$ , $τ_{6, 12} = 0$ , all other $τ_{ct} = 1$
$η_{pt}$	$η_{1, 12} = 0$ , $η_{2, 12} = 0$ , $η_{3, 12} = 0$ , all other $η_{pt} = 1$
$δ_{pt}$	$δ_{1, 12} = 0$ , $δ_{2, 12} = 0$ , $δ_{3, 12} = 0$ , all other $δ_{pt} = 1$
$α_{pt}$	$α_{11} = 8$ , $α_{13} = 15$ , $α_{15} = 35$ , $α_{19} = 24$ , $α_{22} = 7$ , $α_{23} = 26$ , $α_{24} = 5$ , $α_{31} = 36$ , $α_{32} = 14$ , $α_{33} = 8$ , $α_{34} = 8$ , $α_{36} = 20$ , other $α_{pt} = 0$
$β_{ct}$	$β_{11} = 4$ , $β_{12} = 6$ , $β_{13} = 12$ , $β_{14} = 12$ , $β_{15} = 17$ , $β_{17} = 80$ , $β_{18} = 82$ , $β_{19} = 87$ , $β_{1, 10} = 88$ , $β_{1, 11} = 19$ , $β_{1, 12} = 19$ , $β_{21} = 3$ , $β_{22} = 29$ , $β_{23} = 9$ , $β_{24} = 33$ , $β_{25} = 36$ , $β_{26} = 85$ , $β_{27} = 88$ , $β_{28} = 66$ , $β_{29} = 21$ , $β_{2, 10} = 20$ , $β_{2, 11} = 21$ , $β_{2, 12} = 21$ , $β_{33} = 2$ , $β_{34} = 1$ , $β_{36} = 5$ , $β_{37} = 6$ , $β_{38} = 3$ , $β_{39} = 27$ , $β_{3, 10} = 1$ , $β_{3, 11} = 4$ , $β_{3, 12} = 4$ , $β_{41} = 137$ , $β_{42} = 14$ , $β_{43} = 11$ , $β_{44} = 8$ , $β_{46} = 69$ , $β_{47} = 11$ , $β_{48} = 53$ , $β_{49} = 3$ , $β_{4, 11} = 7$ , $β_{4, 12} = 7$ , $β_{52} = 173$ , $β_{53} = 137$ , $β_{54} = 57$ , $β_{55} = 2$ , $β_{56} = 11$ , $β_{58} = 65$ , $β_{59} = 12$ , $β_{5, 11} = 14$ , $β_{5, 12} = 14$ , $β_{61} = 139$ , $β_{62} = 76$ , $β_{64} = 2$ , $β_{65} = 40$ , $β_{66} = 8$ , $β_{68} = 10$ , $β_{69} = 30$ , $β_{6, 10} = 14$ , $β_{6, 11} = 5$ , $β_{6, 12} = 5$ , other $β_{ct} = 0$
$ζ_{ct}$	$ζ_{32} = 25$ , $ζ_{33} = 50$ , $ζ_{57} = 50$ , $ζ_{69} = 25$ , $ζ_{6, 10} = 75$ , other $ζ_{ct} = 0$
$ξ_{ct}$	$ξ_{14} = 25$ , $ξ_{15} = 50$ , $ξ_{16} = 75$ , $ξ_{17} = 25$ , $ξ_{18} = 25$ , $ξ_{19} = 25$ , $ξ_{1, 10} = 25$ , $ξ_{1, 11} = 25$ , $ξ_{1, 12} = 25$ , $ξ_{47} = 50$ , $ξ_{54} = 25$ , $ξ_{55} = 25$ , other $ξ_{ct} = 0$
$λ_{pt}$	$λ_{12} = 13$ , $λ_{14} = 16$ , $λ_{15} = 22$ , $λ_{16} = 25$ , $λ_{17} = 21$ , $λ_{18} = 32$ , $λ_{19} = 1$ , $λ_{21} = 9$ , $λ_{22} = 18$ , $λ_{23} = 9$ , $λ_{24} = 24$ , $λ_{25} = 12$ , $λ_{26} = 12$ , $λ_{27} = 18$ , $λ_{28} = 14$ , $λ_{29} = 5$ , $λ_{2, 10} = 10$ , $λ_{32} = 16$ , $λ_{33} = 20$ , $λ_{34} = 44$ , $λ_{35} = 38$ , $λ_{36} = 38$ , $λ_{37} = 38$ , $λ_{38} = 48$ , $λ_{39} = 36$ , $λ_{2, 10} = 2$ , $λ_{26} = 12$ , other $λ_{pt} = 0$
$χ_{pt}$	$χ_{12} = 39$ , $χ_{14} = 30$ , $χ_{15} = 28$ , $χ_{16} = 18$ , $χ_{18} = 4$ , $χ_{1, 10} = 6$ , $χ_{1, 11} = 12$ , $χ_{21} = 1$ , $χ_{22} = 6$ , $χ_{24} = 2$ , $χ_{25} = 20$ , $χ_{26} = 26$ , $χ_{27} = 8$ , $χ_{28} = 22$ , $χ_{2, 10} = 1$ , $χ_{31} = 10$ , $χ_{32} = 11$ , $χ_{35} = 9$ , $χ_{36} = 12$ , $χ_{37} = 21$ , $χ_{38} = 22$ , $χ_{39} = 16$ , $χ_{3, 10} = 10$ , other $χ_{pt} = 0$

Sensitivity analysis

For the process that the FMILP model converts into the crisp equivalent form, an important step is to define the values of all confidence levels and the intended target profit. However, since generally planners cannot determine correctly the exact values of these parameters, it is important to know the influence they have on the results when some changes occur in their values. More clearly, the robustness of the results must be demonstrated. Therefore, we must perform a sensitivity analysis on all confidence levels and the intended target profit.

First, we need to demonstrate the influence of different confidence levels on the results. Single-factor sensitivity analysis is adopted. And the values of every confidence level range from 0.5 to 1, in which minimum interval is 0.05. Meanwhile, confidence levels $ε_{i} (i = 3, 4, 5, 6, 7)$ for capacity constraints are regarded as identical confidence levels $ε_{3 - 7}$ . The analysis results are shown in Figures 3 and 4, in which the total profit is represented by the most likely value only.

Figure 3.

Change trend of profit with increasing values of $ε_{1}$ , $ε_{2}$ , and $ε_{3 - 7}$ .

Figure 4.

Change trend of profit with increasing values of $ε_{8}$ and $ε_{9}$ .

As can be seen from Figures 3 and 4, the total profit will be decreased with increasing values of most confidence levels, and the rest of them have too little influence. The confidence level $ε_{1}$ has the most important influence on the total profit, closely followed by confidence level $ε_{2}$ . And overall, the decrease extent of the total profit induced by confidence level $ε_{9}$ is greater than the one induced by confidence level $ε_{8}$ , but change sensitivity induced by confidence level $ε_{8}$ is greater than the one induced by confidence level $ε_{9}$ . Confidence levels $ε_{3 - 7}$ have too little influence on the total profit. However, it must be pointed out that confidence levels $ε_{3 - 7}$ do not always have effect on the results. On the contrary, confidence levels $ε_{3 - 7}$ should bring about obvious changes in profit, as emergent demands occur, which is not considered in our numerical example.

Then, the influence of the intended target profit on the results is explored. The values of the intended profit range from $1.22 \times 10^{8}$ to $1.61 \times 10^{8}$ , in which minimum interval is $0.3 \times 10^{7}$ . The analysis results are presented in Figure 5.

Figure 5.

Change trend of profit and possibility value with increasing intended target.

As can be seen from Figure 5, when $G_{0} \geq 1.58 \times 10^{8}$ or $G_{0} \leq 1.34 \times 10^{8}$ , the possibility values are constantly 0 or 1, respectively; when $1.34 \times 10^{8} \leq G_{0} \leq 1.58 \times 10^{8}$ , the possibility values decrease gradually with the increase in $G_{0}$ . On the other hand, the most likely values of the total profit keep stable at an optimal level when $G_{0}$ exceeds $1.34 \times 10^{8}$ , for otherwise the most likely values are irregular. Therefore, so long as $G_{0}$ is set to a relatively large value, the solution results are not affected.

Conclusion

In a real lot-sizing production planning with remanufacturing, environmental coefficients and related parameters are often imprecise due to incomplete and/or unavailable information over the intermediate planning horizon. In this article, we discussed a kind of production system with remanufacturing and heterogeneous demands. For this system, we developed a FMILP model to solve the lot-sizing problem with considering market demands, the quantity of EOL products available, the remanufacturable rate, selling prices, unit cost values, and capacity constraints as imprecise and uncertain parameters, which are characterized by triangular fuzzy numbers. And based on possibility measure, we presented appropriate strategies to converting the FMILP model into a crisp equivalent model. Then, according to the structure and characteristic of the crisp equivalent model, we designed the solution approach based on GA in order to obtain near-optimal solution. Finally, a numerical example abstracted from a real business situation was presented to demonstrate the applicability and effectiveness of the model and solution approach proposed, and the sensitivity analyses for all confidence levels and the intended target profit were performed.

Meanwhile, based on our review, there are several opportunities in the future research directions for this important area. First, the analysis of heterogeneous demands remains a wide open area for research because there are different market structures and quality levels of remanufactured products in real world. Second, other fuzzy mathematical programming-based approaches can be applied and compared with one another. Finally, it is essential to design an expert system to solve the problem in which planners, according to their aspirations, experiences, and business, could have that index for ranking fuzzy numbers that better is adapted to their requirements.

Footnotes

Appendix 1 Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by the Fundamental Research Funds for Central Universities of China (No. CDJXS12110001 and No. CDJZR12118801).

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