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Applied Mathematics and Computation 256 (2015) 742753Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amcRemanufacturing production planning with compensationfunction approximation methodhttp://dx.doi.org/10.1016/j.amc.2015.01.0700096-3003/Crown Copyright 2015 Published by Elsevier Inc. All rights reserved.

Corresponding authors at: School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei 230009, China.E-mail addresses: [email protected] (H. Wen), [email protected] (M. Liu).Haijun Wen a,b,, Mingzhou Liu a,, Changyi Liu a, Conghu Liu aa School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei 230009, Chinab School of Mechanical and Power Engineering, North University of China, Taiyuan 030051, China

a r t i c l e i n f o a b s t r a c tKeywords:RemanufacturingTwo-stage uncertain programmingProduction planningApproximation methodHybrid intelligent algorithmRemanufacturing is becoming a strategic emerging industry in China. However, there aremany uncertain factors such as remanufacturing rate of recycling products, reprocessingcosts, quantity of recycling products during a remanufacturing process. Hence, it is difficultto make an accurate production planning. This paper aims at studying a new remanufac-turing production planning model in view of some possible uncertain factors in a reman-ufacturing enterprise according to the features and characteristics of remanufacturing.Considering the production capacity constraint of recycling, reprocessing and reassemblyunder the condition of uncertain reprocessing amount, unpredictable reprocessing cost,unknown purchase volume of new parts, and uncertain customer demand, this paperdevelops a two-stage, multi-period hybrid programming model with compensation func-tion based on uncertainty theory to minimize the total remanufacturing cost. A hybridintelligent algorithm is designed combined with compensation function approximation,neural network training, and virus particle algorithm to optimize this two-stage uncertainremanufacturing production planning. By use of compensation function approximationmethod, it is to convert an infinite optimization problem in this algorithm into that of afinite one. Finally, one remanufacturing simulation case is studied to validate the efficiencyand rationality of the proposed approach.

Crown Copyright 2015 Published by Elsevier Inc. All rights reserved.1. Introduction

Remanufacturing is a recycling process to manufacture a recycling product as good as a new one [1]. Chinas 12th Five-Year Plan for Circular Economy put forward a strategy deployment to build a circular industrial system clearly. As a strategicemerging industry, remanufacturing is also becoming an effective way to develop circular economy and promote social sus-tainable development [2]. Just taking automobile as example, it is estimated that the total number of Chinese civilian vehi-cles will be 200 million in 2015 and the discarded automobiles more than 10 million for the first time. With the emergencyof discarded industrial products in a large scale and booming of remanufacturing industries, remanufacturing productionplanning and scheduling as one of key links in the production management, the study on it is of important theoreticaland practical value for improving production management within a remanufacturing enterprise.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.amc.2015.01.070&domain=pdfhttp://dx.doi.org/10.1016/j.amc.2015.01.070mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.amc.2015.01.070http://www.sciencedirect.com/science/journal/00963003http://www.elsevier.com/locate/amc

H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753 743Great stride has been made in the study of remanufacturing production planning optimization. Teunter et al. [3] create amanufacturing/remanufacturing batch production model when either of the two new and remanufacturing product canmeet the production demand, which aims to minimize the total remanufacturing cost if demand and supply are given. Liet al. [4] develop a non-constraint, multi-period production planning model considering the possibility of substitution forremanufacturing products and find solution using a dynamic programming and heuristic algorithm. Li et al. [5] build a plan-ning model for both manufacturing and remanufacturing hybrid production scheme including emergency procurement. Toachieve the minimum cost, this model is solved by adopting the combination of genetic algorithm and heuristic dynamicprogramming method. Doh and Lee [6] establish a remanufacturing production planning model including mixed integer pro-gramming to maximize profit based on considering the influence and constraint of some links such as recycling disposal,disassembly, inspection, remanufacturing, and reassemble. However, a large number of uncertain factors exist in the reman-ufacturing production process [7], including: Uncertainty of recycling factors, such as unknown recycling time, variable recy-cling quantity, and unknown quality; Uncertainty of disassembly rate of recycling product and its component, it means thatit is unknown whether a product or a component can be disassembled or not; Uncertainty of remanufacturing rate of recy-cling products due to out-of-tolerance, residual stress, internal cracks, surface deformation, and so forth; Uncertainty of cus-tomer demand; and so forth. Due to the existence of these uncertain factors, traditional production planning and controlmethods cannot adapt to remanufacturing production system, and it is more complicated to make a remanufacturing pro-duction planning than a traditional production planning. Therefore, Guide [8] study the influence on remanufacturing pro-duction planning and control because of a random uncertainty in a remanufacturing production system, the imbalancebetween recycling supply and demand, and other unknown factors of recycling products. For one-off product and the impactof a random machine failure, Kenne et al. [9] study a strategy scheme for a mixed production of both manufacturing andremanufacturing under a closed-loop reverse logistics network in order to reduce the inventory costing. Jing et al. [10,11]establish a production planning model for a hybrid manufacturing and remanufacturing system under an uncertainty envi-ronment, and develop a diploid genetic algorithm to find solution against the uncertainty of customer demand, recyclingquantity, remanufacturing cost and remanufacturing rate. Wang and Wang [12] study a closed-loop optimal supply chaindecision-making approach from the view of the supply chain when it happens variation in market demand, remanufacturingcost, as well as recycling cost. Mukhopadhyay [13] studies a remanufacturing production planning system when customerdemand and recycling supply follow random distribution. Su et al. [14,15] create a hybrid uncertain programming modelconsidering the stochastic and fuzzy parameters in the remanufacturing process taking profit maximization in multi-periodand cost minimization in single period as goal respectively. Huang and Chen [16] construct a hybrid integer programmingmodel for a fuzzy remanufacturing system and present a crisp equivalent form for this model to minimize the total costbased on the capacity constraints of recycling, disassembly, reprocessing, and reassembly.

During these studies mentioned above, some remanufacturing production planning models are only suitable for thoseremanufacturing production system whose factors related are known, and some could not adjust and compensate theproduction plan when the production plan must be changed because of disturbance events although some uncertain fac-tors and disturbance events are taken into account. In this paper, a two-stage remanufacturing production planningmodel is established based on uncertainty theory in order to minimize the total cost. This model contains the influenceof uncertain factors on remanufacturing production planning and regards some uncertain parameters including customerdemand, remanufacturing quantity, purchase volume of new parts, and remanufacturing costs as uncertain variable withknown distribution. It is divided into two stages: The first stage includes recycling and disassembly, and the second stagecontains inspection, reprocessing, purchasing of new parts, and reassembly. Some decisions have to be made before someuncertain parameters are determined in the first stage, so it is allowable that the decision in the first stage is infeasible,then, some correspond measurements must be taken to find optimal scheme for the production plan in the second stage.Following, the compensation function approximation is studied and a hybrid intelligent algorithm combining compensa-tion function approximation, neural network training, and virus particle algorithm is developed to solve this model. Thepurpose of this paper is to provide a new way to make a remanufacturing production planning under uncertainconditions.

The paper is organized as follows: Section 2 gives a mathematical description of a two-stage uncertain programming.Then a two-stage uncertain remanufacturing production planning model is established in Section 3. Following, a solutionmethod of this proposed model is developed in Section 4. Section 5 gives a case study to validate the model and algorithmproposed in Sections 3 and 4. Finally, Section 6 draws conclusions and suggests the next study direction of this theme.

2. Two-stage uncertain programming definition and its mathematic description

Some decision parameters in the remanufacturing production plan are uncertain, and they are connected with each other[17,18].

At first, several definitions are identified as follows:

Definition 1. Let C be a nonempty set, L be one of algebra r of C, and each element K in algebra r is called an event. Iffunction M meets the following three principles when L 2 0;1:

744 H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753(1) MfCg 1.(2) For any event K, MfKg MfKcg 1.(3) For any sequences with countable events fKig, Mf[1i1Kig 6

P1i1MfKig

then M is called an uncertain measure, triple C;L;M is called an uncertainty space.Definition 2. An uncertain variable is a function n from an uncertainty space C;L;M to the set of real numbers such thatfn 2 Bg is an event for any Borel set B, it is:fn 2 Bg fc 2 Cjnc 2 Bg 1Definition 3. Suppose n be uncertain variable, and at least one of integral from the right side of formula (2) is limited (inorder to avoid the appearance of 11), thenEn Z 10

Mfn P rgdr Z 01

Mfn 6 rgdr 2is called an expected value of uncertain variable n.Then a programming model with uncertain parameters is constructed as follows:min cTx qTny;s:t: Ax b;

TnxWny hn;x P 0; y P 0;

8>>>>>: 3

where qn, Tn, Wn and hn are uncertain variables defined in uncertainty space C;L;M. Due to the influence of uncer-tain parameters, the model is difficult to solve by conventional methods. So a new approach is: to divide the whole processinto two stages. At first, to determine the decision variables x and observe the realization of the uncertain variable n, anddeclare the decision-making y at the end of the first phase. Following, to compensate the insufficiency caused by the firstphase because of uncertain factors during the second phase. The compensation function in the second phase is then formed:min qTny;s:t: Wny hn Tnx;

y P 0:

8>: 4

Let Qx; n minfqTnyjWny hn Tnx; y P 0g, Qx; n is called the optimal function of the second stage. To avoid

infeasible solutions, Qx; n is restricted to 1 and 1 1 1. Let K fxjx 2 Rn1 ;CrfQx; n < 1g 1g, x 2 K iscalled induced constraint, which can ensure to find a feasible solution for each realization in n to solve the compensationproblem. Compensation function QEx is denoted as QEx EnQx; n, where En is operator of an expected value for uncer-tain variable n.

Combined with the compensation function, the two-stage uncertain programming problem can be shown by thefollowing:min cTx QEx;s:t: Ax b;

x P 0;x 2 K;

8>>>>>: 5

Qx; n min qTny;s:t: Wny hn Tnx;

y P 0:

8>: 6

The aforementioned model contains uncertain variables with unlimited support and could not be solved easily because of

infinite dimensional optimization problem, so it is necessary to converted it into finite dimensional optimization problem byuse of the approximation method. Suppose uncertain variable n n1; n2; . . . ; nrT has unlimited support N

Qri1ai; bi,

where ai; bi is the support of ni, then the possibility distribution of uncertain variable n can be approximated by that ofuncertain variable sequence ffmg, where fm fm;1; fm;2; . . . ; fm;rT . 8i 2 f1;2; . . . ; rg, fm;i gm;ini can be defined as follows:gm;ini ai; ni 2 ai; ai 1m;

sup kimki 2 Z; s:t: kim 6 nin o; ni 2 ai 1m ; bi;

8

H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753 745rim

n ri mai; ri ki mai 1; . . . ;Kio; 8

where r is the greatest integer no more than r. If mbi is an integer, then Ki mbi 1, otherwise Ki mbi. The possibilityvm;i of uncertain variable fm;i is denoted by vm;i rim

Pos cj rim 6 nic : 103. Design of a two-stage uncertain remanufacturing production planning model

3.1. Project statement

This paper studies only remanufacturing production planning-making for one kind of product/component with recyclingproducts. This product may consist of N parts, its production plan involves T identical cycles, and uncertain variables in eachcycle are mutual independent. Remanufacturing work flow is shown in Fig. 1, and its operation principle is as follows:

(1) Recycle discarded products by a third party recycling company, then a remanufacturing enterprise purchases the recy-cling parts according to the contract from this third party company and forms a recycling product inventory.

(2) Disassemble the recycling products then divide all parts into groups according to material attribution. Finally, aftercleaning and inspection, each part is identified as use directly, remanufacturable, or scrap according to the technicalrequirements.

(3) Renew all remanufacturable parts through remanufacturing technology.(4) Assemble the finished products with remanufacturing parts. If necessary, buy some new parts from suppliers to make

up for the shortage. Finally, store them in the warehouse.(5) Deliver the finished products to customer.

There are a lot of uncertain factors during the remanufacturing production, even some will be found during the produc-tion process gradually. For example, the accurate reprocessing cost and recycling quantity of some constituent parts could beknown only after disassembly and inspection because of their different utilization time, operating environment and damagedegree. However, some decisions must be made before uncertain parameters are identified, so the remanufacturing processis divided into two phases to find optimal production planning scheme: An expected value model including uncertainparameters is built in the first phase, and it is allowable to make some infeasible decision; In the following second phase,it will revise the mistakes of the first phase by use of compensation method after the uncertain parameters are identified.Meanwhile, this two-stage production planning model still needs to meet the following assumption:third party recycling manufacturers

inventory of recycled product

wash and test remanufacturingdisassemble

scrap new parts

recycling

use directly

assemble

Product recycling

inventory of remanufactured products

demand for remanufactu

red products

Fig. 1. Work flow of a remanufacturing production system.

746 H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753(1) The total remanufacturing cost contains recycling cost, disassembly costs, cleaning costs, inspection costs, remanufac-turing costs, purchasing costs of new parts, assembly costs, and inventory costs for recycling products and remanu-facturing products.

(2) Disassembly, reprocessing and reassembly are done within the constraints of production capacity.(3) New parts can be free supplied.(4) Customer demand, amount of new parts, amount of remanufacturable parts and reprocessing cost are uncertain vari-

ables in each cycles.(5) The recycling products must be inspected roughly in the first phase and it is impossible to discard a product in whole

in the second phase.

3.2. Model study

In order to establish a two-stage uncertain remanufacturing production planning expected value model, it is firstly toclarify that the indices, parameters and decision variables in it are defined as follows:

(1) Subscripti, i 1;2; . . . ;N: indicate the type of a part or a component.t, t 1;2; . . . ; T: denote the period number of production plan.(2) NotationsCR: recycling Cost per unit product.x1R;t: amount of recycling products in the first phase at the period t.

x2R;t: amount of recycling products in the second phase at the period t.CI: inventory cost per recycling product.x1I;t: inventory of recycling products in the first phase at the period t.

x2I;t: inventory of recycling products in the second phase at the period t.CD: disassembly cost per recycling product.x1D;t: disassembly amount of recycling products in the first phase at the period t.

x2D;t: disassembly amount of recycling products in the second phase at the period t.cnit: purchasing cost of part i at the period t.CA: assembly cost per remanufacturing product.yA;t: assembly amount of remanufactured products at the period tCPI: inventory cost of remanufactured products.yP;t: inventory volume of remanufactured products at the period t.ED;t: maximum production capacity of disassembly process.EM;it: maximum production capacity of remanufacturing process for part i.EA;t: maximum production capacity of reassembly process.HD;t: disassembly time per recycling product.HM;it: remanufacturing time for part i.HA;t: reassembling time per final product.ni: amount of part i in a product.max IR;t: the maximum inventory capacity for recycling products.maxPR;t: the maximum quantity for recycling product.max IP;t: the maximum inventory capacity for remanufacturing finished products.(3) Uncertain variablesnmit c: reprocessing amount of part i at the period t.nnitc: purchase volume of part i at the period t.ndt c: customer demand at the period t.ncitc: reprocessing cost of part i at the period t.

Following, it is to describe how to establish this model.It is difficult to predict damage degree of each part before a recycling product is dismantled, thus some uncertain variables

such as purchase volume and remanufacturing costs are all unknown, and they can be identified only after disassembly andinspection. However, some decisions must be made before that, so the remanufacturing production process is divided intotwo phases to optimize such kind of production planning. x1R;t ; x1I;t ; x1D;t is called the decision vectors in the first phase, i.e.x1R;t ; x1I;t ; x1D;t must be determined before the realization of uncertain variable n

mit c; n

nitc; n

ct c; n

dt c and x1R;t ; x1I;t ; x1D;t is also

allowed to be infeasible. In the second phase, an enterprise can continue purchasing the recycling products from the third partyrecycling company if remanufacturing finished products cannot meet customer demand and compensate the insufficient esti-mation in the first stage. For convenience, we use x1 and n to denote decision vectors x1R;t; x1I;t; x1D;t and uncertain variables

H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753 747nmit c; nnitc; n

ct c; n

dt c respectively. To minimize the total cost, the planning model in the first phase is constructed as

follows:min TC XTt1

CRx1R;t CIx1I;t CDx1D;t

EnQx; n;

s:t: x1I;t 6 max IR;t;x1R;t 6 maxPR;t;x1D;tHd;t 6 ED;t ;x1R;t ; x

1I;t ; x

1D;t P 0;

8>>>>>>>>>>>>>>>>>>>:11The total cost in the first phase involves recycling costs, inventory costs of recycling products, disassembly costs andremanufacturing costs expected value of the second phase.

To give the decision vector x1 and uncertain variables n of the first stage a realization value nc, the programming modelof the second phase can be expressed as the following form:Qx; nc minXTt1

CRx2R;t CIx2I;t CDx2D;t

XTt1

CAyA;t CPIyP;t XTt1

XNi1

nmit cncitc n

nitccnit

; 12The total cost in the second phase mainly includes three sections: the compensation value for the first stageCRx2R;t CIx2I;t CDx2D;t , sum of assembly costs and inventory costs for finished products, and sum of reprocessing costs andnew parts purchasing costs, where reprocessing cost, amount of reprocessing part, and purchase volume of new part areall uncertain variables.s:t: x1D;t x2D;t

HD;t 6 ED;t t 1;2; . . . ; T; 13

nmit cHM;it 6 EM;it ; i 1;2; . . . ;N; t 1;2; . . . ; T; 14

yA;tHA;t 6 EA;t ; t 1;2; . . . T; 15

x1I;t x1I;t1

x2I;t x2I;t1

x1D;t x2D;t x1R;t x2R;t 6 maxPR;t; t 1;2; . . . ; T; 16

yA;t yP;t1 yP;t P ndt c; t 1;2; . . . ; T; 17

x1D;t x2D;t

ni P yA;tni P nmit c n

nitc; i 1;2; . . . ;N; t 1;2; . . . ; T; 18

x1D;t x2D;t P yA;t ; t 1;2; . . . ; T; 19

x1I;t x2I;t 6 max IR;t ; t 1;2; . . . ; T; 20

yP;t 6 max IP;t ; t 1;2; . . . ; T; 21

x1R;t ; x1I;t ; x

1D;t; x

2R;t ; x

2I;t ; x

2D;t; yA;t; yP;t P 0; 22Formulas (13)(15) denote the production capacity constraints of disassembly process, remanufacture machining processand remanufacture assembly process respectively; formula (16) shows that amount of recycling products is less than themaximum recycling capacity at the period t; formula (17) denotes that remanufacturing product quantity is more than cus-tomer demand; formula (18) shows that amount of part i after disassembly should not be less than the reassembly amountand the reassembly quantity should not be less than the sum of new-buying parts and the qualified remanufacturing parts;formula (19) means that disassembly amount should not be less than that of reassembly; formulas (20) and (21) show theinventory capacity constraints of recycling products and remanufacturing products respectively; formula (22) expresses thenon-negative constraints of variables.

Suppose S be a set of collection of x that meets restraints of formula (11), for some x 2 S, the model solution of the secondphase may be nonexistent. Here, it is to add induced constraints K fxjCrfcjQx; nc < 1g 1g on decision vector of thefirst phase x 2 S, the feasible region of model is K \ S. Corresponding to the remanufacturing production planning model, thedecision vector is feasible if the followed condition is met:niXTt1

max PR;t P niXTt1

max ndt c XTt1

min nnitc: 23The maximal quantity of recycling products of a part obtained by remanufacturing production should not be less than thedifference between the maximal total demand and the minimal purchasing quantity.

748 H. Wen et al. / Applied Mathematics and Computation 256 (2015) 7427534. Solution method

4.1. Approximation algorithm of expectation function

Since n is a continuous uncertain variable with infinite support in this two-stage uncertain production planning model, itis impossible to evaluate the precise value of EnQx; n at point x. In this paper, it is to convert infinite dimensional optimi-zation problem into a finite one by taking an approximation strategy.

For a continuous uncertain variable n nmit c; nnitc; n

ct c; n

dt c, let fm gmn gm;1n

mit c; gm;2n

nitc; gm;3n

ct c;

gm;4ndt c be a discrete uncertain variable. f x EnQx; n is approximated to f mx Efm Qx; fm, for 8m,bfqm bfqm;1;bfqm;2;bfqm;3;bfqm;4 q 1;2; . . . ; L, where L L1L2L3L4. Let vq vm;1 bfqm;1 ^ vm;2 bfqm;2 ^ vm;3 bfqm;3 ^ vm;4 bfqm;4 , vm;iis the possibility distribution of uncertain variable fm;i. The optimum value of an uncertain remanufacturing productionplanning of the second stage is expressed by the following formula:gqx Q x;bfqmc min

XTt1

CRx2R;t CIx2I;t CDx2D;t

XTt1

CAyA;t CPIyP;t XTt1

XNi1

bfqm;1cbfqm;3c bfqm;2ccnit : 24

Without loss of generality, the subscript of vq and gqx are rearranged, and meeting g1x 6 g2x 6 gLx, wq is

denoted as below:wq 12

maxq

p1vp max

q1

p0vp

12

maxL

pqvp max

L1

pq1vp

; 25where v0 vL1 0. When the decision vector is x, f mx can be calculated as follows:f mx XLq1

wqgqx; 26When m is large enough, expectation function f x of a remanufacture production planning in the second stage can beapproximated by use of formula (26). Then the uncertain production planning model can be represented through the follow-ing approximation algorithm:min TC XTt1

CRx1R;t CIx1I;t CDx1D;t

f mx: 274.2. Neural network training

In view of the complexity of the model mentioned above, this paper selects a three layer radial basis function neural net-work (RBFNN) to replace formula (26) in order to accelerate the solution tempo. RBFNN belongs to the feedforward neuralnetwork and has a simple structure, strong approximation ability and rapid learning convergency, so it is applicable to prob-lems such as pattern recognition, function approximation and adaptive filtering. When used in function approximation,RBFNN adopts nonlinear activation function as basis function of hidden layer nodes and linear function for output nodes.RBF is radial symmetry, so for neurons, the farther the input value from the center, the lower the degree of activation, i.e.local feature.

4.3. A virus particle swarm algorithm

Uncertain optimization with compensation is a nonconvex optimization problem and the model is complex and difficultto solve. The application of particle swarm optimization (PSO) helps to find the satisfied solution in the feasible time. But alarge number of engineering practice shows that the optimization efficiency is low and the global optimization ability is poorfor the basic particle swarm algorithm. This paper introduces the virus evolution principles based on PSO algorithm tostrengthen the information exchange between the same generation particles, enhance the population diversity and improvethe precision and convergence speed against these weaknesses. Virus particle swarm optimization algorithm [19] (VPSO) isadopted to solve the uncertain production planning problem. Two groups are generated in the process of evolution: maingroups and viruses groups. The main groups correspond to the solution space of problem and each particle represents a fea-sible solution. Main groups are responsible for communication longitudinally between generations to make the populationglobal optimization.

Part of the chromosome gene of virus individual is changed to change their genetic information by transcribing gene seg-ment of some individual and transcribing reversely to other individual. Virus transfers evolutionary information crosswiseamong individuals of same generation and conducts the local search of particle swarm.

H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753 7494.3.1. Algorithm parameter settingsFitness function of main groups: according to the principle of the minimum total production cost, formula (27) is taken as

the fitness function of main groups.Infection depth: The infection depth of virus individual to main groups is measured by the fitness function of virus

individual.fitv irusj XSk1

FithostPrej;k FithostAftj;k; 28where fitvirusj is the fitness of virus j, S is the set of main individual infected by virus j, FithostPrej;k and FithostAftj;k representthe fitness of kth individual before and after infected by virus j respectively.

Virus vitality: The vitality of virus j in t + 1 generation is defined as:LFj;t1 z LFj;t fitv irusj; 29where z is the rate of life attenuation.Virus activity: The probability that a virus is selected as infection subject during evolutionPCj 1

1 aefitvirusj 11 bem

; 30wherem is the virus number that has the same gene encoding with virus j. a and b are weighting coefficients used to balancethe activity of virus individual and group. Generally a > b is met to increase influence of the activity of virus individual on theselection probability. Bigger probability indicates the stronger virus individual vitality and the greater individual fitnessvalue. Yet the excessive probability will reduce population diversity and converge until local optimal. Too small probabilitywill reduce the convergence speed.

Virus update mechanism: whether the virus individual updates is measured by the similarity of individual chromosome.The similarity between individual k and j is represented as:Hjk 1

1 djk; djk 2 0;1; 31where djk is called the chromosome euclidean distance between individual k and j, djk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

l1Bjl Bkl2

q, n is the chromo-

some genes quantity, Bjl and Bkl are the lth chromosome gene of individual j and k respectively. When Hjk reaches the givenstandard H, new virus individual will be generated in the next generation to ensure the diversity of population.

Encoding scheme: The papulation is made up by pop size particles, for a two-stage remanufacturing production plan-ning model, the search space of decision variables is 3T dimension, so using X xR;1; . . . ; xR;T ; xI;1; . . . ; xI;T ; xD;1; . . . ; xD;Tdenotes the decision vectors in the first stage. Virus particles in the group are taken as the code substring of the main indi-vidual, so it is to select a variable of a random period from the decision about recycling quantity, inventory amount and dis-assembly quantity of recycling products, xR;u; xI;v ; xD;w, where u;v ;w 2 1; T.

4.3.2. Calculation procedure

Step 1. Parameter initialization. Recycling quantity, inventory and disassembly quantity of each period should not exceedthe maximal number of recyclable products because most of remanufacturing production are addressed with a cus-tomer order. Therefore, it can generate a main group particle and corresponding virus individual random from0;maxPR;t , then validate the feasibility of decision vector X. If it meets formula (23), it is feasible and can be takenas a particle of initial population. Repeating above-mentioned process until to obtain pop size feasible particles ofmain group X1;X2; . . . ;Xpopsize.

Step 2. Use formula (26) again to obtain the approximate compensation value of every particle Xj, then set the particlesposition and the target as their own optimal position and optimal value, denoted by pbest. Find out the optimal par-ticle in the population and set its optimal position and target, denoted by gbest. Set V be the upper bound of particlevelocity and value V the difference of maximum quantity and minimum quantity of recycled products.

Step 3. Each particle updates its speed and position in the solution space according to the formula of a basic particle swarmoptimization.v ti wv t1i c1rand1pi Xt1i c2rand2pg X

t1i ; 32

Xti Xt1i v ti ; 33

where i 1;2; . . . ; pop size, w is called inertia coefficient, c1 and c2 are learning factors, rand1 and rand2 are randomnumbers distributed evenly in the range of [0, 1]. The value of w can be determined by the followed formula:

w wmax wmax wzmin

itermax iter; 34

750 H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753where wmax, wmin are the maximum and minimum of w respectively; iter and itermax are the current iteration numberand maximum number of iterations respectively. The feasibility of every particle Xti can be verified by use of constraintformula (23). If feasible, if is treated as a new particle; otherwise, it is to regenerate through (32) and (33).Step 4. Virus infection. Virus individual j infects the main group particles xk randomly to generate another main individual xlthrough selection probability PCj.

Step 5. Recalculate the fitness value of each main group particle and update pbest and gbest. The fitness value, vitality andactivity of virus individual will be calculated according to the fitness before and after being infected.

Step 6. t t 1.Step 7. Termination condition of algorithm: The algorithm will be terminated when the preset maximum number of iter-

ations is reached or some acceptable solutions come true. Otherwise, turn to step 3 to repeat this calculation.

5. Case study

In order to validate this model and algorithm, this paper takes a following case as example. A remanufacturing productionplanning for 4102QB diesel engine for 3 months is made using the proposed approach. Considering the complex structureand numerous parts of engine, this paper only takes engine camshaft assembly. The camshaft assembly parts is shown inFig. 2 and their parameters in Table 1.

Suppose that the recycling product inventory and the finished products inventory are zero at the beginning of the period.The other parameters are as follows: CR 60, CI 2, CD 5, CA 17, CPI 4; max IR;t 210, max IP;t 280;ED;t 29000 min, EA;t 35000 min; HD;t 90 min, HA;t 110 min; For the sake of simplicity, the change of the time dimen-sion of these parameters is not taken into account in this example. Remanufacturing processing capability and operationtime are as shown in Table 2.

Furthermore, remanufacturing amount, purchase volume, market demand and remanufacturing cost per month are allnormal distribution Ne;r. The concrete parameters are as shown in Table 3.

Using the approximation method above mentioned, it generates 3000 samples bfqm, q 1; . . . ;3000. Put each sample intoformula (24) and calculate the optimal value in the second stage Qx;bfqmc. Repeat the above-mentioned process and find aset of input and output data for goal function. A RBF neural network is trained to approximate the goal function through thespecified training data (the neuron number of input layer is 5T 2TN = 45, the hidden layer is 50, output layer is 1), thenembed the trained neural network into VPSO algorithm to search the optimal solution. Parameter Settings of VPSO algorithmare as follows: Inertia coefficient linear decreases from 0.8 to 0.3; Learning factors c1 and c2 is 2; pop size 100; Virus lifeattenuation rate z 0:8; Weighted coefficient of virus individual activity a 0:6; Weighted coefficient of virus group activityb 0:4; Number of iterations iter = 3000. The algorithm is programmed in MatlabR2009 under the computer systemrequired as follows: Intel i3CPU, 2.3 GHZ processor, 4 GB RAM. The algorithm is run independently replicated 10 times.The optimal solution of remanufacturing production planning model in the first stage is (310, 0, 310, 296, 15, 281, 234,27, 249), the optimal solution in the second stage is (0, 0, 0, 290, 10, 36, 0, 36, 276, 21, 0, 0, 0, 293, 10), the correspondingoptimal value of the total remanufacturing cost is 175664.2.

To illustrate the optimization performance of VPSO algorithm, this paper adopts PSO and VPSO to solve the two-stageuncertain remanufacturing production planning model under the same initial conditions. The results is shown in Table 4,where BF, AF, and MI represent the iterative optimal solution, average optimal solution after 10 cycles, and the largest num-ber of iterations respectively.

The results show that the lower total cost value can be got through VPSO than by PSO algorithm. Because VPSO importsthe virus evolution infection function, the population diversity during evolution process especially at its late period isensured by virus individual infecting main groups. And VPSO also avoids the prematurity of the population and improvesthe solution precision of the algorithm. Iterative times of VPSO algorithm is less than that of PSO algorithm, although the1

2

345

6

78

9

10

12

13

110

Fig. 2. Graphical representation of camshaft assembly.

Table 1Camshaft assembly parts.

No. Parts code Parts name Remanufacturable* Number New product price ()

1 4100QB-01-014-II Camshaft bush II N 1 10.34100QB-01-014 Camshaft bush III N 1 6.4

2 4102QB-02-001 Camshaft Y 1 197.53 GB1096-79 Flat key A8 22 N 1 2.54 GB93-87 Washer 8 N 2 0.85 GB21-76 Bolt M8 25 N 2 1.56 4100QB-01.02-001 Camshaft idle gear Y 1 62.57 4100QB-01.02-002 Idle Gear N 1 1.48 GB93-87 Washer 12 N 1 0.89 GB21-76 Finished hexagon head bolt M12 32 N 1 1.410 4100QB-02-004 Camshaft thrust plate Y 1 14.511 4100QB-01-013A Camshaft bush I N 1 7.612 4100QB-2-02-006 Valve pushrod Y 8 12.213 495QA-02-005 Valve tappet Y 8 13.4

* For security guarantee, all fasteners and wearing parts should be forced to scrap whatever damaged or not.

Table 2Remanufacturing processing capacity constraints (min).

Parts Camshaft Camshaft idle gear Camshaft thrust plate Valve pushrod Valve tappet

EM;it 16,000 15,000 5000 50,000 75,000HM;it 45 35 15 5 10

Table 3Parameters of remanufacturing plan model.

Parts name Uncertain variable 1 2 3

Camshaft nmit c (230,20) (275,15) (249,10)nnitc (20,3) (70,20) (38,6)ncitc (50,10) (25,10) (25,5)

Camshaft idle gear nmit c (256,11) (250,15) (170,15)nnitc (43,13) (50,15) (134,11)ncitc (34,14) (25,5) (20,8)

Camshaft thrust plate nmit c (132,12) (148,5) (165,9)nnitc (25,3) (26,5) (25,4)ncitc (5,3) (6,3) (7,3)

Valve pushrod nmit c (632,62) (445,31) (525,58)nnitc (342,30) (530,80) (480,20)ncitc (1.5,0.4) (2.1,0.5) (2.5,0.6)

Valve tappet nmit c (914,50) (580,30) (450,45)nnitc (255,21) (368,24) (240,25)ncitc (3.1,0.4) (1.9,0.3) (2.4,0.3)

Camshaft assembly ndt c (280,20) (300,10) (230,30)

Table 4Comparison of optimized results by PSO and VPSO.

Optimization algorithm BF AF MI

PSO 180368.6 182547.4 3180VPSO 175664.2 176734.5 2650

H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753 751former is more complex than the latter. The reason is that VPSO algorithm is not the simple superposition of PSO and virusmechanism, but the optimized combination of both. The speed of the genetic mutations caused by virus infection is dozensof time orders of magnitude faster than that on their own and its influence is dozens of orders of magnitude larger. The algo-rithm can avoid population trapping in local optimum and improve the convergence speed of VPSO at the same time.

For any given feasible decisions, approximate expectation function f mx is a sequence famg about number m of sample.When m tends to infinity, the sequence famg converges to a0. The limitation a0 is the value of expectation function corre-

0 500 1000 1500 2000 2500 3000110300

110400

110500

110600

110700

110800

110900

sample size

appr

oxim

ate

valu

e of

com

pens

atio

n fu

nctio

n

Fig. 3. The convergence of approximate expectation function f mx.

752 H. Wen et al. / Applied Mathematics and Computation 256 (2015) 742753sponding to decision x, so f mx converges to expectation function f x when m tends to infinity. This situation can be seenfrom Fig. 3. To select an optimal solution of decision variable x = (310, 0, 310, 296, 15, 281, 234, 27, 249), then the value off mx is 110876.11. Observing Fig. 3, the limiting value of f mx located in [110,800,110,900]. So the relative deviation of f mxand a0 are small. The results show that this hybrid algorithm to solve this two-stage uncertain remanufacturing productionplanning problem is effective.

6. Conclusion

A production planning is one of important job of a remanufacturing production system. However, it is difficult for dis-patcher and scheduler to make efficient production planning because of the influence of large amount of uncertain factors,and such production planning leads to the chaos of production and low economic benefit generally. Hence, it studies aremanufacturing production planning scheme under uncertain conditions in this paper. Its main contents are as follows:

(1) To establish a new two-stage remanufacturing production planning expected value model based on uncertainty the-ory and two-stage uncertain optimization method in view of the uncertainty of remanufacturing production. Someparameters must be given expected value although they are unknown in the first stage, and some measurements mustbe taken to compensate the difference between the estimation in the first stage and the actual demand in the secondstage, which aims at getting close to the actual production requirement.

(2) To study the approximation method of expectation function and to develop a hybrid intelligent algorithm combinedapproximation method, neural network training, and virus particle algorithm in order to find solution for this two-stage remanufacturing production planning model. This algorithm can convert an infinite optimization problem intothat of a finite one efficiently, and it can ensure the accuracy of solution through virus particle algorithm and accel-erate the solution tempo.

(3) To study a case in order to verify the feasibility and effectiveness of the proposed planning model and algorithm. Theresult shows that this model and algorithm can optimize remanufacturing production planning-making under someunknown factors.

Despite some uncertain factors in the remanufacturing production is considered, it is insufficient to mirror the wholeactual remanufacturing production environment. Hence, the next study will add the impact on remanufacturing productionplanning caused by recycling amount, recycling product quality, and so on. In addition, the study on a remanufacturing pro-duction planning for diversified products will be the direction in the future.

Acknowledgements

This work was supported by Major Project of National Basic Research Program of China (No. 2011CB013406).

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Remanufacturing production planning with compensation function approximation method1 Introduction2 Two-stage uncertain programming definition and its mathematic description3 Design of a two-stage uncertain remanufacturing production planning model3.1 Project statement3.2 Model study

4 Solution method4.1 Approximation algorithm of expectation function4.2 Neural network training4.3 A virus particle swarm algorithm4.3.1 Algorithm parameter settings4.3.2 Calculation procedure

5 Case study6 ConclusionAcknowledgementsReferences