Robust optimal policies of production and inventory with uncertain returns and demand

Int. J. Production Economics 134 (2011) 357–367

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Robust optimal policies of production and inventory with uncertain returns and demand Cansheng Wei a, Yongjian Li b,, Xiaoqiang Cai c a

College of Information Technical Science, Nankai University, Tianjin 300071, PR China Business School, Nankai University, Tianjin 300071, PR China c Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong b

a r t i c l e in fo

abstract

Available online 14 December 2009

We consider an inventory and production planning problem with uncertain demand and returns, in which the product return process is integrated into the manufacturing process over a finite planning horizon. We first propose an inventory control model for the return and remanufacturing processes with consideration of the uncertainty of the demand and returns. Then a robust optimization approach is applied to deal with the uncertainty of the problem through formulating a robust linear programming model. Moreover, properties on the robust optimization model are studied, and an equivalent robust optimization model based on duality theory is obtained which allows the solutions to be derived more efficiently. Finally, we provide a set of numerical examples to verify the effectiveness of the approach and analyze the effects of the key parameters on the solutions. & 2009 Elsevier B.V. All rights reserved.

Keywords: Robust optimization approach Product return Inventory control Production planning Reverse logistics

1. Introduction In the manufacturing industry, product recovery has been given an increasing attention with the aims to protect the environment and to save production costs in the supply chain. For example, some car manufacturers such as BMW have set up a recovery process for the reuse of end-of-life cars, and manufacturers of products such as one-off cameras(Kodak) and toner cartridges(Xerox) are also undertaking remanufacturing activities. In these examples, the remanufacturing process is integrated into the manufacturing process of the new products. The management of any company with remanufacturing of returned products often finds that they have to deal with two major uncertainties: the uncertainty with the market demand for the new product and the uncertainty with the quantity of the returned product. In particular, there is a great uncertainty involved in the collection process of the returned product, which may affect significantly the entire inventory and production planning decision. Generally, in a manufacturing/remanfuactring hybrid system, it is required to determine, in the presence of the abovementioned uncertainties, the quantities of new product to be manufactured, the quantities of returned product to be remanufactured or disposed off, and appropriate inventories of the sellable and returned products, over the planning horizon so that

 Corresponding author. Tel.: + 86 22 23505341; fax: +86 22 23501039.

E-mail addresses: [email protected], [email protected] (Y. Li). 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.11.008

the total production and inventory cost is minimized. Such a problem is hard to solve efficiently, especially in a multi-period environment. It is much complicated by the difficulty to forecast the returned products, because the used products are usually returned through different channels, from different customers in different circumstances. In many cases, information on the probabilistic distribution concerning the quantity of a returned product is not available (Trebilcock, 2002; Fleischmann et al., 1997), and what we can estimate is just the likely mean and its possible maximum/minimum values. Considering this inherent difficulty in remanufacturing with returned products, in this paper we will adopt the robust optimization approach to tackle a multi-period inventory and production planning problem with uncertain market demand and uncertain quantity of the returned product. As it is well known, robust optimization is particularly powerful in dealing with uncertain variables with only known intervals, which is different from stochastic optimization which needs distributional information on the random variables. Many studies have discussed production planning and inventory control problems with consideration of the product return process by using stochastic optimization approach. Two early models have been proposed by Heyman (1977) and Simpson (1978), respectively. Heymand considers a continuous time problem with the stochastic returns independent of the demand, while Simpson considers a periodic situation with mutually dependent stochastic demand and returns. Simpson also proposes an optimal structure of purchasing, repairing, and junking policies. Inderfurth (1997) investigates optimal procurement,

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remanufacturing, and disposal policies through a dynamic programming method. Erwin et al. (1999) study a hybrid system by comparing the ‘‘push’’ and ‘‘pull’’ control strategies in the manufacturing process with consideration of the remanufacturing process. Dobos and Richter (2004) examine a production/ recycling system with predetermined production-inventory policy and assume that there is no difference between newly produced and recycled items. Inderfurth (2004) shows how the manufacturing and remanufacturing decisions can be coordinated in order to maximize the total expected profit with new and remanufactured item substitution. Recently, Kim et al. (2006) propose a supply planning model for a remanufacturing system of reusable parts in reverse logistics. They develop a general model to maximize the total cost savings by optimizing the quality of the parts to be remanufactured and the quantity of parts purchased from outside supplier. Geyer et al. (2007) introduce a strategic model of a production system with remanufacturing, which is subjected to limited components durability and a finite life cycle. They demonstrate the necessity to carefully coordinate the production cost structure, collection rate, product life cycle, and component durability to maximize cost savings. Jaber and Saadany (2008) consider the optimal production, remanufacturing and disposal decisions when the lost-sales demand for the manufactured item is different from that for the remanufactured one. Mukhopadhyay and Ma (2008) give a model to derive the joint procurement and production decision in a hybrid system where both the returned and new parts serve as inputs of the system and the demand and quality are uncertain. Yang (2004) studies a production control problem with random raw material supply, which shares some similarity with the problem with uncertain product returns. Yang’s model involves, however, only one manufacturing process. The multi-echelon inventory system with remanufacturing is also the focus of many studies. DeCroix et al. (2005) study a series inventory system with stochastic demands and returns over an infinite horizon, and they prove that an optimal stationary echelon base-stock policy can be derived under the condition of nonnegative demand. An assembly system with product or component returns is studied by DeCroix and Zipkin (2005). They show that returns may disrupt the long-run balance in two main ways. Conditions on the item-recovery pattern and restrictions on the inventory policy are proposed to maintain the balance. DeCroix (2006) identifies the optimal policy structures of the remanufacturing/ordering/disposal activities for a multi-echelon inventory system, and decomposes the system into a sequence of single-stage systems. The echelon base-stock policy in each downstream stage and a three-parameter simple structure in most upstream stages are derived. Scarf (1958) appears to be the first attempting to apply a distribution-free approach to solve the newsboy problem. He considers the optimal policy corresponding to the worst distribution in a set of distribution functions with the same mean and variance. Gallego and Moon (1993), Moon and Silver (2000), Hesham and Hassan (2005), and others extend Scarf’s work. For the finite-period problem, Gallego et al. (2001) consider inventory models with discrete distributions that are incompletely specified. They propose an inventory policy that minimizes the maximum expected cost over the class of demand distributions. A robust optimization method that incorporates an uncertain data set is proposed by Soyster (1973), who deals with uncertain parameters by considering their worst cases. However, the result from Soyster’s model is very conservative because only the worst case parameters in the uncertain set are considered. Since then, optimization under uncertainty has received greater attention. Mulvey et al. (1995) propose that a solution is robust when it is close to the optimal value in all uncertain scenarios. They also

point out that a model is robust if it remains ‘‘almost’’ feasible for all scenarios, and present a robust optimization model. For the important generic convex optimization problem, Ben-Tal and Nemirovski (1998) first show that the corresponding robust convex program is either exact or approximate if the uncertain set is ellipsoidal. Ben-Tal and Nemirovski (2000) study a linear program with uncertain data using the method of Ben-Tal and Nemirovski (1998). Further, Ben-Tal et al. (2004) extend the method by considering a linear program with both nonadjustable and adjustable variables. Bertsimas and Sim (2004) flexibly adjust the level of conservation of the robust solutions in terms of the probabilistic bounds on constraint violation. An attractive aspect of their method is that the new robust formulation is also a linear optimization problem. Iyengar (2005) is the first to present robust dynamic programming, and argues that a robust dynamic program is equivalent to a stochastic zero-sum game with perfect information. Regarding applications of robust optimization, Escudero et al. (1993) consider production planning and outsourcing policies using scenarios to characterize the uncertainty in demand. Gutierrez and Kouvelis (1995) develop a robust optimization approach to international sourcing to ensure that the sourcing network is relatively robust concerning the potential exchange rate change. Ben-Tal and Nemirovski (2002) survey the main results of robust optimization that is applied to uncertain linear, conic quadratic, and semidefinite programming problems. They discuss some applications, especially, for problems of antenna design, truss topology design, and stability analysis/synthesis, in uncertain dynamic systems. Bertsimas and Thiele (2006) are the first to use the above robust methods to solve inventory problems in supply chain management. Ben-Tal et al. (2005) consider a twoechelon multi-period supply chain problem, and Adida and Perakis (2006) apply the robust method to dynamic pricing and inventory control problems based on flow models. Two critical issues have been given great attention in the existing studies of applying robust optimization to practical problems. The first is how to ensure the optimal solution meets the constraints of the problem under consideration with an acceptable probability; and the second one is how to reduce the sensitivity of the optimal solution to the support interval of the uncertain variable if its lower and upper bounds are hard to be estimated accurately. To the best of our knowledge, there does not seem to be any work reported in the previous literature on integrated manufacturing, remanufacturing, and disposal policies using the robust optimization approach. However, as we have discussed above, uncertainty in the product return process is very complicated, and is prevalent in remanufacturing activities. In this paper, we will apply a robust optimization approach to tackle an inventory control and production planning problem in which the product return process is integrated into the manufacturing process over a finite planning horizon, where it is assumed that the demand and returns are uncertain with only intervals being known. Our contributions include:

(1) A robust optimization model is developed, which treats both the quantity of returns and the market demand as uncertain variables, with only upper and lower bounds being known. Such a model captures the decision needs in situations with very uncertain information on the returns and the demand. Our model exhibits a linear programming structure, and can be solved efficiently by any powerful linear programming algorithms. We further derive an equivalent model based on duality theory, which allows the solutions to be derived more efficiently.

C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

(2) We prove that the optimal solution for our robust model can meet the constraints with a high probability. (3) We examine, by numerical study, the sensitivity of the optimal solution to the uncertainty intervals, which shows that the solution fluctuates within a very small range (at a magnitude of 3%) even though the uncertainty sets change significantly. This is a desirable effect attributed to the robust approach we have developed, in particular the use of ‘‘budget of uncertainty’’ regarding the uncertain information. The rest of this paper is organized as follows. In Section 2, we present a description of our problem. In Section 3, a nominal model is given. Our robust optimization approach is proposed in Section 4. Numerical examples are presented in Section 5, and some concluding remarks are given in Section 6.

2. Problem description Taking the case of Xerox Corp. as an example, we review the process of used product take-back and remanufacturing in Fig. 1 given by Azar et al. (1995). Xerox takes back high-value printer and copy cartridges directly from customers. Returned parts with high reuse value are remanufactured into serviceable parts, and low-value returns are disposed of by the company. Therefore, Xerox needs to coordinate the remanufacturing process and manufacturing process effectively to satisfy the demands and simultaneously maximize its profits. Motivated by Xerox’s remanufacturing process, we consider a multi-period inventory control and product planning problem with used product returns and remanufacturing. To analyze the problem in the form of a mathematic model, Fig. 1 is simplified into Fig. 2, in which we focus on inventory control rather than the production process and abstract the return inventory and serviceable inventory from the inventory in the return process and the inventory in the distribution process, respectively, in Fig. 1. Furthermore, to simplify the model, the ‘‘Material Recovery’’ process in Fig. 1 is treated as a disposal process in Fig. 2.

359

The problem is considered over a finite planning horizon with T periods. Define Im ðtÞ and Ir ðtÞ as the inventory levels of the returned and serviceable products at the beginning of period t, respectively. The procedures to run the system are as follows. (1) At the beginning of the planning horizon, we assume that Ir ð1Þ ¼ Im ð1Þ ¼ 0. (2) At the beginning of every period, the inventory levels Im ðtÞ and Ir ðtÞ can be observed before the manufacturing and remanufacturing processes. Returns rðtÞ and demand dðtÞ are realized at the end of period t, where t ¼ 1; 2; . . . ; T. (3) At the beginning of every period, we first decide how many products will be remanufactured, which is noted as kðtÞ. Then we decide whether and how many new products will be manufactured, noted as mðtÞ. At the end of the period, we decide whether and how many returns will be junked, noted as uðtÞ. No setup costs are considered for the manufacturing, remanufacturing, and disposal processes, and the finished product demand can be backlogged, but returns cannot. A shortage cost will occur if the demand is unsatisfied. (4) At the end of the planning horizon, no demand occurs. The manufacturing, remanufacturing, and disposal processes all stop at period T þ1. The salvage value of the stock is assumed to be zero. We consider the situations where there is limited information on the quantity of the returned product and the market demand for the new product. Specifically, we assume: Assumption 1. The quantity of the returned product and the demand for the new product fall in lower and upper bounds, respectively, which can be estimated in advance. 3. The nominal model In the nominal problem, the returns and demand are realized with the probability one. The notations are as follows. cm : ck : cu : hm hr : hs :

The The The The The The

unit unit unit unit unit unit

production cost of a new product. cost of remanufacturing a returned product. disposal cost for junking a returned product. holding cost of the serviceable product. holding cost of the returned product. shortage cost of unmet demands.

From the practice, we assume that hr r hm , ck r cm and cu ohr , which means that a returned product holds a lower value than a new product. Then, the inventory equations are formulated as follows: Ir ðt þ1Þ ¼ Ir ð1Þ þ

t X

ðrðiÞkðiÞuðiÞÞ;

ð1Þ

i¼1

Im ðt þ 1Þ ¼ Im ð1Þ þ Fig. 1. Xerox: equipment recycling process (Azar et al., 1995).

u (t) Disposal r (t)

Manufacture

Im (t)

Ir (t) Returns

m (t)

k (t) Remanufacture Fig. 2. System structure.

d (t) Demand

t X

ðkðiÞ þ mðiÞdðiÞÞ;

ð2Þ

i¼1

where Im ðt þ 1Þ 4 0 means a surplus of the serviceable products in period t and Im ðt þ 1Þ o 0 means that a shortage occurs in this period. The cost in period t is given as follows: the manufacturing cost is cm mðtÞ, the remanufacturing cost is ck kðtÞ, and the disposal cost is cu uðtÞ. The holding cost for returns is given as hr ðIr ðtÞ þrðtÞkðtÞuðtÞÞ, and the holding cost of serviceable products is given as hm ðIm ðtÞ þ kðtÞ þ mðtÞdðtÞÞ if Im ðtÞ þ kðtÞ þ mðtÞdðtÞ Z 0. Otherwise, Im ðtÞ þkðtÞ þ mðtÞdðtÞ o0 leads to a shortage cost hs ðIm ðtÞ þ kðtÞ þ mðtÞdðtÞÞ. With the

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objective to minimize the cost, the nominal model is then formulated as NM :

min

kðtÞ;mðtÞ;uðtÞ

T X

þ hr

Ht ¼ max hm

hs

t X

ðmðiÞ þ kðiÞdðiÞÞ;

ðrðiÞkðiÞuðiÞÞÞ

ð6Þ

ðrðiÞkðiÞuðiÞÞ

ð7Þ

i¼1

i¼1

Ir ðt þ 1Þ Z 0; ð3Þ

0 where Im and Ir0 are pre-specified constants. Without loss of 0 generality, we consider Im ¼ 0 and Ir0 ¼ 0 in the following sections for convenience of discussion. Let rðtÞ and dðtÞ be the mean values of the returns and demand, respectively. If the returns and demand take the values rðtÞ ¼ rðtÞ and dðtÞ ¼ dðtÞ, respectively, with the probability one, we call problem (3) a nominal problem. However, if rðtÞ and dðtÞ are uncertain, we call it an uncertain problem.

4. Robust optimization Following the method given by Bertsimas and Sim (2004), to formulate a robust model, we give the definitions of the uncertainty rðtÞ and dðtÞ as follows. The quantity rðtÞ of returns is uncertain in every period t by taking a value in a support interval ½r t r^ t ; r t þ r^ t  (Ben-Tal and Nemirovski, 2000). We define the scaled deviation from its nominal value as zrt ¼ ðrt r t Þ=r^ t . Thus, we have jzrt j r 1 and zr :¼ ðzr1 ; zr2 ; . . . ; zrT Þ. Then the cumulaP tively scaled deviation of the returns up to a period t is ti ¼ 1 jzri j for t ¼ 1; 2; . . . ; T, and we give a budget of uncertainty at each period t to restrict the cumulatively scaled deviation as Pt r r r i ¼ 1 jzi j r Gt , where the budget of uncertainty Gt takes values in an interval ð0; tÞ for t ¼ 1; 2; . . . ; T. Accordingly, we can cut out the worst deviation in the cumulative demand and returns, and obtain an acceptable result. The uncertainty budgets are characterized as follows: Grt increases with regard to period t, and its mean value is ErðtÞ, t ¼ 1; 2; . . . ; T. Further, we assume that 0 r Grt þ 1 Grt r1, which means that the increase in the uncertainty budgets is slower than the increase in the periods. Similarly, we present the definition of uncertain demand, dðtÞ, in a support interval ½d t d^ t ; d t þ d^ t . The scaled deviation from the nominal value of demand is defined as zdt ¼ ðdt d t Þ=d^ t with jzdt jr 1 and zd :¼ ðzd1 ; zd2 ; . . . ; zdT Þ. The cumulatively scaled deviation of the demand up to a period t is subject to the budgets of the Pt d d d demand uncertainty, i ¼ 1 jzi jr Gt , and Gt takes values in an interval ð0; tÞ for t ¼ 1; 2; . . . ; T. Gdt holds the same characteristics as Grt . According to the definitions of uncertain demand and returns, we have dðtÞ ¼ d t þ d^ t zdt and rðtÞ ¼ r t þ r^ t zrt , respectively, where n o P and zd A Z d ¼ zdt jjzdt jr 1; ti ¼ 1 jzdt j r Gdt n o Pt r r r r r r z A Z ¼ zt jjzt jr 1; i ¼ 1 jzt j r Gt , respectively. Then we formulate the robust counterpart of the uncertain problem as follows: ( T X RM : min max cm mðtÞ þ cu uðtÞ þ ck kðtÞ t¼1

ð5Þ

) ðmðiÞ þkðiÞdðiÞÞ

t X

Ir ðt þ 1Þ ¼

ðkðtÞ;mðtÞ;uðtÞ;Ht Þðzr A Z r ;zd A Z d Þ

t X

i¼1

i¼1

Ir ðt þ1Þ Z0 0 Ir ð1Þ ¼ Ir0 ; Im ð1Þ ¼ Im

ð4Þ

i¼1

uðtÞÞ þ maxfhm ðIm ðtÞ þ kðtÞ þ mðtÞdðtÞÞ; hs ðIm ðtÞ þ kðtÞ þ mðtÞdðtÞÞgg subject to Im ðt þ 1Þ ¼ Im ð1Þ t X ðmðiÞ þ kðiÞdðiÞÞ þ Ir ðt þ1Þ ¼ Ir ð1Þ þ

) r^ i zri þ Ht

i¼1

subject to

t¼1

t X

ðr i kðiÞuðiÞÞ þ hr

i¼1

(

fcm mðtÞ þ cu uðtÞ þ ck kðtÞ þhr ðIr ðtÞ þ rðtÞkðtÞ

t X

t X

ð8Þ

where dðtÞ ¼ d t þ d^ t zdt and rðtÞ ¼ r t þ r^ t zrt . In the robust optimization model (RM), we first maximize the objective function with respect to the deviations. Then, we minimize the maximization of the objective function with respect to the decision variables. Through the above two steps, the robustness of the model protects it against the worst realization of the uncertain variables in the uncertain set. Ht is introduced as P a new decision variable to replace maxf hm ti ¼ 1 ðmðiÞ þkðiÞdðiÞÞ; Pt hs i ¼ 1 ðmðiÞ þ kðiÞdðiÞÞg in the objective function and is added as constraint (6). For the above RM, according to Vanderbei (2001), we propose the following lemma. Lemma 1. If the feasible region is bounded in the robust optimization model (RM), the objective function reaches the optimality at the vertexes of the feasible region. A solution B  ffkðtÞ; mðtÞ; uðtÞ; Ht gjt ¼ 1; 2; . . . ; Tg is feasible in the RM problem if and only if, for any period t, every component of the set fkðtÞ; mðtÞ; uðtÞ; Ht g satisfies the constraints (6)–(8). Let B  ffkðtÞ; mðtÞ; uðtÞ; H t gjfkðtÞ; mðtÞ; uðtÞ; H t g A B; t ¼ 1; 2; . . . ; Tg be a feasible set of solutions for the RM problem. Accordingly, if fmðtÞ; kðtÞ; uðtÞ; HðtÞg A B is given, then the region of the feasible values Ht is defined as ( ( t X ðmðiÞ þkðiÞdðiÞÞ; H ¼ Ht jHt ¼ max hm i¼1

hs

t X

)

ðmðiÞ þkðiÞdðiÞÞ ; t ¼ 1; 2; . . . ; Tg:

i¼1

Then, we get the following result. Proposition 1. The optimal solutions of the RM problem are equivalent to those by substituting the following constraints: Ht Z hm

t X

ðmðiÞ þ kðiÞdðiÞÞ;

ð9Þ

i¼1

Ht Z hs

t X

ðmðiÞ þ kðiÞdðiÞÞ

ð10Þ

i¼1

for constraint (6). Proof. If constraint (6) is satisfied, then we can easily obtain P P Ht Z hm ti ¼ 1 ðmðiÞ þ kðiÞdðiÞÞ and Ht Zhs ti ¼ 1 ðmðiÞ þ kðiÞdðiÞÞ. If constraints (9) and (10) are given, then we present a new ~ ¼ofHt jHt Z feasible region of Ht for the RM problem as H n P P max hm ti ¼ 1 ðmðiÞ þ kðiÞdðiÞÞ; hs ti ¼ 1 ðmðiÞ þ kðiÞdðiÞÞ g. From Lemma 1, the optimal solutions can be found at the boundary ~ which involves the vertexes of the convex set. That is, of H, ~ holds the same optimal solutions as the feasible region H H. &

C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

Then we have the revised robust counterpart of the nominal problem (4)–(8) as follows: RRM :

max

min

ðkðtÞ;mðtÞ;uðtÞ;Ht Þðzr A Z r ;zd A Z d Þ

þ hr

t X

ðr i kðiÞuðiÞÞ þhr

i¼1

t X

t¼1

r^ i zri þ Ht

bt ;ftT

Ht Z hm t X

Ht Zhs

Ir ðt þ 1Þ ¼

t X

ðmðiÞ þ kðiÞdðiÞÞ

ð12Þ

ðmðiÞ þ kðiÞdðiÞÞ

ð13Þ

t X

ðrðiÞkðiÞuðiÞÞ

ð14Þ

Ir ðt þ 1Þ Z 0;

ð15Þ

where dðtÞ ¼ d t þ d^ t zdt and rðtÞ ¼ r t þ r^ t zrt . First, because of the influence of uncertainty on the problem constraints, we have to deal with these effects by reconstructing constraints (12) and (13). According to the robust optimization approach given by Bertsimas and Thiele (2006), we need to maximize the right-hand side of constraints (12) and (13) over the ~ to protect the feasibility of all the possible realizations of set H the uncertain returns and demand. When t is given, we then need to solve an auxiliary linear programming problem: t X

max r r

t X

subject to

t X

zdi r Gdt

i¼1

0 r zdt r1:

ð16Þ

P This auxiliary problem comes from minimizing ti ¼ 1 d^ i zdi in (12) Pt d d ^ and maximizing i ¼ 1 d i zi in (13) over Z . According to strong duality theory, we can get a linear programming problem by using duality while introducing deterministic parameters (Bertsimas and Sim, 2004). Then, the duality problem of the auxiliary linear problem (16) is given as follows: qt Gdt þ

t X

zti r Grt

i¼1

0 rzrt r1;

ð20Þ

and the duality problem is formulated as

ot Grt þ

min ot ;eit

t X

eit

i¼1

subject to ot þ eit Z r^ i ot Z 0; eit Z 0; 8i rt:

ð21Þ

Let k  ðkð1Þ; kð2Þ; . . . ; kðTÞÞ and m  ðmð1Þ; mð2Þ; . . . ; mðTÞÞ and and u  ðuð1Þ; uð2Þ; . . . ; uðTÞÞ and q  ðq1 ; q2 ; . . . ; qT Þ x  ðw1 ; w2 ; . . . ; wT Þ and H~  ðH~ 1 ; H~ 2 ; . . . ; H~ T Þ. Further, let n  ðx1t ; x2t ; . . . ; xtt Þ, e  ðe1t ; e2t ; . . . ; ett Þ, and /  ðf1T ; f2T ; . . . ; fTT Þ ~ n; e; /; b g, which is the for t ¼ 1; 2; . . . ; T. Let V  fk; m; u; q; x; H; T

set of the decision variables for the robust counterpart of the nominal problem. According to the duality relationships between (17), (19), and (21) and (16), (18), and (20), respectively, we formulate the robust counterpart model (RCM) of the RM as follows: ( T X RCM : min cm mðtÞ þck kðtÞ þ cu uðtÞ V

xit

þ hr

t X

t¼1

!

ðr i kðiÞuðiÞÞ þ ftT

~ t Zhm H

t X

ðTt þ1Þr^ t zrt

t¼1

zrt r GrT

ðmðiÞ þ kðiÞd i Þ þqt Gdt þ

i¼1

~ t Z hs  H

t X

ð22Þ

i¼1 t X

ðr i kðiÞuðiÞÞ Z ot Grt þ

t X

!

xit

ð23Þ

i¼1

ðmðiÞ þ kðiÞd i Þ þ qt Gdt þ

i¼1

subject to

þhr bT GrT

ð17Þ

Then, by observing the objective function (11), the uncertainty of the returns also has an influence on the feasible solutions and Pt PT ^ r as optimization. We rewrite t ¼ 1 hr i ¼ 1 r i zi PT r hr t ¼ 1 ðTt þ1Þr^ t zt , and then we give the following auxiliary problem to deal with the impact of uncertainty on the objective function.

T X

) ~t þH

i¼1

subject to

max

t X

subject to

i¼1

subject to qt þ xit r d^ i qt Z0; xit Z0; 8i r t; t ¼ 1; 2; . . . ; T:

zrt A Z r

r^ i zri

i¼1

d^ i zdi

i¼1

T X

ð19Þ

Furthermore, the uncertainty of returns also affects the P P constraint (15). By observing ti ¼ 1 ðr i kðiÞuðiÞÞ þ ti ¼ 1 r^ i zri , the r worst case is reached when zt ¼ 1, so we rewrite it as Pt Pt r ^ r i ¼ 1 ðr i kðiÞuðiÞÞ i ¼ 1 r i zi for every zi A ½0; 1. Then our objective is to seek a realization of zrt to minimize the left-hand side of P inequality (15), to minimize  ti ¼ 1 r^ i zri . The auxiliary problem is then formulated as

zt A Z

i¼1

qt ;xit

ftT

t¼1

ð11Þ

i¼1

min

T X

subject to bT þ ftT rðTt þ1Þr^ t bT Z 0; ftT Z0:

)

i¼1

max

bT GrT þ

i¼1

subject to

zdt A Z d

Thus, the duality problem is min

T  X cm mðtÞ þ cu uðtÞ þ ck kðtÞ

361

t X

!

xit

ð24Þ

i¼1 t X

eit þ kðt þ 1Þ

ð25Þ

i¼1

qt þ xit Z d^ i

ð26Þ

ot þ eit Z r^ i

ð27Þ

bT þ ftT rðTt þ 1Þr^ t

ð28Þ

t¼1

0 r zrt r 1:

ð18Þ

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kðT þ 1Þ ¼ mðT þ 1Þ ¼ uðT þ 1Þ ¼ 0

ð29Þ

qt Z 0; xit Z 0; ot Z0; eit Z0; ftT Z 0; bT Z 0; 8i rt

ð30Þ

t ¼ 1; 2; 3; . . . ; T:

ð31Þ

Let G ¼ minmðtÞ;kðtÞ;uðtÞ cost,

PT

i¼1

d

r

Γt m (t)

Γt mean of d (t) u (t) k (t)

mean of r (t)

26 24 22

gt , where gt is the single period total

20 18

gt ¼ cm mðtÞ þ cu uðtÞ þ ck kðtÞ þhr ðIr ðtÞ þ rðtÞkðtÞuðtÞÞ

16

þ maxfhm ðIm ðtÞ þ kðtÞ þ mðtÞdðtÞÞ; hs ðIm ðtÞ þ kðtÞ þmðtÞdðtÞÞg; ð32Þ

14 12 10

and we get the following result.

8

Theorem 1. The objective function for the nominal problem P G ¼ minmðtÞ;kðtÞ;uðtÞ Ti ¼ t gt is supermodular with respect to mðtÞ and kðtÞ, 8t ¼ 1; 2; 3; . . . ; T.

6

Proof. Consider the single-period cost function gt , and we get @2 gt =@mðtÞ2 40, @2 gt =@kðtÞ2 4 0, and @2 gt =@kðtÞ@mðtÞ 4 0. Hence, gt is supermodular in mðtÞ and kðtÞ. In addition, P G ¼ minmðtÞ;kðtÞ;uðtÞ Tt ¼ 1 gt has a linear relationship to uðtÞ; thus, the result is proved according to the conditions in Porteus (2002). & Because the objective function has a linear relationship to uðtÞ, Theorem 1 guarantees the existence of optimal solutions mðtÞ and kðtÞ for the robust problem with respect to the nominal problem and its robust counterpart. We can also make sure that our robust linear program guarantees the existence of optimal solutions. Hence, this theorem guarantees that the RCM can be solved with optimal solutions.  Let m and k be the optimal solutions from the robust optimization model. zd is the vector solved from problem (16). To check the upper bound on the probability that the two main constraints, (23) and (24), are violated, we get the following results. Theorem 2. The probability that constraints (23) and (24) are violated satisfies ! " !# t X Gdt 1 ^ ~  ðtÞ; k ðtÞ; d ðtÞÞo hm ffiffi Pr Hðm ðm ðiÞ þ k ðiÞd i zd r 1F p i diÞ t i¼1 " !# ! t X Gdt 1 ^ ~  ðtÞ; k ðtÞ; d ðtÞÞo hr ffiffi ; ðm ðiÞ þk ðiÞd i zd r 1F p Pr Hðm i diÞ t i¼1

pffiffiffiffiffiffi R x   where FðxÞ ¼ ð1= 2pÞ 1 exp y2 =2 dy and t ¼ 1; 2; . . . ; T. Proof. The proof is similar to that of Theorem 2, given by Bertsimas and Sim (2004). & These two probabilities guarantee that the solutions of mðtÞ, kðtÞ and uðtÞ from solving our robust optimization model satisfy constraint (6) with an acceptable probability. This means that the robust method can run well in practice, especially, in the application of uncertainty. The main work of this section is to formulate an RCM that is a robust optimization model of the nominal model (NM). The RCM is formulated as a linear program and we can solve it easily by using optimization software for linear programming such as LINGO, Matlab, GIPALS and so forth. It is unnecessary, therefore, to discuss the solution methods for our model because of the offthe-peg and effective methods for linear programming models. In our paper, we solve our model in LINGO (8.0) and the numerical example of the model solution is given in the next section.

4 2 0 -2 5

10 T

15

20

Fig. 3. Results when T ¼ 20.

5. Numerical studies Our numerical studies have been conduced with the objectives: (1) to validate the existence of the optimal solutions of the robust optimization model and discuss their computability when the planning horizon is very long; (2) to examine the sensitivity of the optimal solutions to the cost parameters; and (3) to study the sensitivity of the optimal solutions to the uncertain data support length. The parameters in our numerical examples were determined as follows. Since hr rhm , ck r cm and cu ohr according to the assumption in Section 3, the cost parameters were selected as hr ¼ 4, hm ¼ 5, hs ¼ 6, cm ¼ 7, ck ¼ 4, and cu ¼ 2. From the modelling formulation described in Eq. (3), the initial conditions were chosen as Ir ð1Þ ¼ Im ð1Þ ¼ 0. The uncertainty budgets are shown in Figs. 3 and 4, where d t and r t are also given as ‘‘mean of dðtÞ’’ and ‘‘mean of rðtÞ’’ . We let jd^ t j r 3 and jr^ t j r3. We will also show later the impact of the uncertainty budgets on the optimal result through changing the values of d^ t and r^ t . We ran our LINGO model in the following environment: Windows XP(Professional SP2), Pentium(R)4 CPU 2.80 GHz, 760 MB memory. The optimal policies obtained using our robust optimization model are given in Figs. 3 and 4 for the planning horizons of 20 and 50 periods, respectively. From Figs. 3 and 4, we can make the following observations: (1) mðtÞ is lower than kðtÞ for most periods except for the first one. This result is based on the assumption that the remanufacturing cost is lower than the manufacturing cost. As t increases, more and more products are returned into the system. The number of remanufactured products then increases while the number of manufactured products decreases. (2) Comparing the curves of dðtÞ, rðtÞ, and uðtÞ, we find that the disposal process often occurs at the end of a period when the mean of demand is lower than the mean of returns. The disposal quantity uðtÞ is not always zero, which conforms to the practical situation in which the disposal process is carried out after the returns are collected over other periods.

C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

Γ dt m (t)

50

mean of d (t) Γ dt u (t) k (t)

363

mean of r (t)

40

30

20

10

0

10

20

30

40

50

T Fig. 6. mðtÞ with different hm and hs .

Fig. 4. Results when T ¼ 50.

Fig. 5. Probability that the constraints are violated.

Table 1 Computational requirements. Period

Fig. 7. kðtÞ with different hm and hs .

CPU time (s) Iterations (steps) Optimal cost State

T ¼ 20 o1 T ¼ 50 1 T ¼ 80 5 T ¼ 100 9

1342 8940 21 486 33 316

7126.48 31 613.66 70 419.81 104 690.20

Global Global Global Global

optimization optimization optimization optimization

Fig. 5 illustrates the probability that constraints (23) and (24) are violated over the planning horizon. We find that the probability convexly decreases and approximately tends to zero after about 10 periods, which means that the solutions from the

robust optimization model are feasible with a high probability, especially when dealing with a long-period problem. Table 1 shows that the time requirements of our robust optimization approach, which suggests that the proposed approach is computationally efficient. To analyze the effects of some key parameters on the robust solutions and total cost, we also designed some experiments to test the holding cost and shortage cost parameters. First, we analyzed the following three situations based on the relationship between hm and hs when hm is given: hm o hs , hm ¼ hs , and hm 4 hs . Figs. 6–8 show the changes of mðtÞ, uðtÞ, and kðtÞ when T ¼ 20,

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C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

Fig. 10. mðtÞ with different hr and hs .

Fig. 8. uðtÞ with different hm and hs .

11000 10000 9000

Cost

8000 7000 6000 5000 4000 10 hm

5 0

0

2

6

4

8

10

hs

Fig. 9. Cost with different hm and hs .

hm ¼ 5, and hs ¼ 10, 5, or 1, respectively. We got the following results.

(1) Along with the increase in the shortage cost hs , we see from Figs. 6 and 7 that both mðtÞ and kðtÞ increase, where in every period t, the value of mðtÞ gets the maximum when hm hs o0, and gets the minimum when hm hs 4 0 among the three scenarios, so is the value of kðtÞ. It shows that the increase of the shortage cost encourages the manufacturer to produce more products to guarantee a serviceable inventory level against the demand shortage. (2) From Fig. 8, we see that the disposal quantity of the returns decreases with the increase in the shortage cost hs . That is, the higher the shortage cost, the more returns the system remanufactures, to reduce the total cost and avoid a demand shortage.

Fig. 11. kðtÞ with different hr and hs .

Fig. 9 shows the change of the minimum cost in the robust optimization model when hm and hs change from 1 to 10, respectively, over the planning horizon with the length of 20. Here we neglected the assumption about hm 4 hr to obtain the whole surface of the minimum cost when hr is fixed. We find that the surface of the minimum cost is increasingly concave concerning hm and hs , which means that the decrease in the holding cost of serviceable products will lead to cost savings.

C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

365

kðtÞ ¼ 0 occurs frequently in the last several periods, because when the shortage cost is low and the holding cost of returns is high, the manufacturer will decide to reduce the inventory of returns and the quantity of products to be remanufactured. (3) In Fig. 12, the quantity uðtÞ of the products to be disposed of increases on hr to save the holding cost. The quantity uðtÞ reaches the maximum in the last few periods of the planning horizon, because there is no salvage value after the planning horizon. Fig. 13 shows the change of total cost when hr and hs change from 1 to 10, respectively, in the planning horizon with 20 periods. Here we neglected the assumption of hr ohm to get the _ whole surface of the minimum cost using the robust optimization approach. We can find that the objective function of the robust optimization model is increasingly concave concerning hr and hs . So, the decrease in hr will lead to cost savings. From the definitions of Gdt ðGrt Þ and zdt ðzrt Þ, we can get Pt Pt d r i ¼ 1 jðdi d i Þ=d i j r Gt and i jðri r i Þ=r i j r Gt . According to Bertd r simas and Sim (2004), Gt and Gt can balance the robustness and optimality of the solution by changing their values to control the

Fig. 12. uðtÞ with different hr and hs .

10000

Cost

8000 6000 4000 2000 10 8

hr 5

4 0 0

2

10

6 hs

Fig. 13. Cost with different hr and hs .

(1) Along with the increase in the shortage cost hs , we can see from Fig. 10 that mðtÞ increases, where in every period t, the value of mðtÞ gets the maximum when hr hs o 0, and gets the minimum when hr hs 40 among the three scenarios. In addition, mðtÞ ¼ 0 occurs more frequently in the case of hr 4 hs than that of hr o hs when a higher shortage cost is incurred. This result is consistent with the practical situation. (2) In Fig. 11, the remanufacturing quantity kðtÞ is mostly influenced by hr and hs . For a given hr , a higher hs leads to a higher remanufacturing quantity. If hs is very low, then

3.4 3.2 3 2.8

Δcost(%)

Then, we analyzed three situations based on the relationship between hr and hs , that is, hr ohs , hr ¼ hs , and hr 4 hs . Figs. 10–12 show the changes of mðtÞ, uðtÞ, and kðtÞ in the above three situations, where T ¼ 20, hr ¼ 4, and hs ¼ 7; 4; or 1, respectively. Analysis of Figs. 10–12, yields the following results about the robust optimal solutions of mðtÞ, kðtÞ, and uðtÞ in the three different scenarios: hr o hs , hr ¼ hs , and hr 4 hs .

Fig. 14. Impact of DG on cost.

2.6 2.4 2.2 2 1.8 1.6 20 20

15

ΔrΓ(%)

15

10

10 5

5

Fig. 15. Impact of DdG and DrG on cost.

Δd (%) Γ

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C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

conservatism degree, which is also called the price of the robustness (Bertsimas and Sim, 2004). Generally, the values Gdt and Grt have much effect on the solution of the robust optimization model. 0 0 Let Gdt ðGrt Þ and Gdt ðGrt Þ be two different values of the budget of d uncertainty, and CðGt ; Grt Þ be the robust optimization cost of the model RCM with the given values for Gdt and Grt . Then 0 0 Dcost ¼ ðCðGdt ; Grt ÞCðGdt ; Grt ÞÞ=CðGdt ; Grt Þ  100% is defined as the 0 change rate of the cost when the values Gdt and Grt change to Gdt r0 and Gt , respectively. Similarly, we further define three change 0 0 rates of the budget of uncertainty: DG ¼ ðGdt Gdt Þ=Gdt ¼ ðGrt Gdr Þ= d d d r d0 d Gr when Gt and Gt change simultaneously, DG ¼ ðGt Gt Þ=Gdt and 0 DrG ¼ ðGrt Gdr Þ=Gdr when Gdt and Grt change independently. We have also conducted the following experiments. 0 0 First, for Gdt ¼ Gdt ð1 þ DG Þ and Grt ¼ Grt ð1 þ DG Þ (0 r DG r 1), and DG varying from 0 to 0.2 with the step size of 0.01, we computed

Fig. 18. uðtÞ with different DG . 0

0

Dcost for different values of Gdt and Grt . The result is shown in Fig. 14. From Fig. 14, we find that Dcost concavely increases in DG until DG ¼ 13%, and after this point, Dcost gets a same value 3.3364%. Although the value of DG changes from 0 to 20%, the

Fig. 16. mðtÞ with different DG .

Fig. 17. kðtÞ with different DG .

change rate of the robust optimization cost is not more than 3.3364%. It seems to suggest that when DG exceeds a threshold (e.g., DG ¼ 13% in our example), changing DG , that is, changing the 0 0 values of Gdt and Grt , does not influence the robust optimization cost. This observation seems to conform with the theoretic result given by Ben-Tal and Nemirovski (2000), which shows that the robust optimization approach effectively guarantees the reliability of the solution and protects the solution against uncertainty. 0 0 For Gdt ¼ Gdt ð1 þ DdG Þ and Grt ¼ Grt ð1þ DrG Þ (0 r DdG ; DrG r1), and d r DG ; DG varying from 0 to 0.2 with the step size of 0.01, 0 respectively, we computed Dcost for different values of Gdt and r0 Gt , and the result is illustrated in Fig. 15. We find that Dcost converges to a stable value after DdG or DdG increases up to 13% (which is similar to that of Fig. 14). Figs. 16–18 illustrate the changes of the robust optimization decisions mðtÞ, kðtÞ and uðtÞ when DG takes the values of 0 and 0.2, respectively. We find that these decisions do not change significantly although the budget of uncertainty increases up to 20%. This result shows that the solution is reliable and effectively immune against data uncertainty. Finally, some more general observations can be made based on the numerical results above. (1) The robust optimization approach appears to be an effective method to enhance the decisions making in the complicated hybrid manufacturing/remanufacturing system, especially in situations with a long planning horizon and inexact probability distribution information on the demand or returns. It can limit the probability of violating the constraints to an upper bound. (2) The holding cost and shortage cost have a great impact on the optimal solution. These cannot be avoided in practice in the presence of uncertain demand and product return. They may, however, be reduced by more information sharing among partners of the supply chain. Moreover, the manufacturer may possibly reduce their impact by proper coordination/trade-off

C. Wei et al. / Int. J. Production Economics 134 (2011) 357–367

with respect to manufacturing of new product and remanufacturing of returned product. This is actually what we aim to investigate in this paper. (3) In our model the manufacturer may adjust the conservation level of the solution according to his/her risk preference, based on his judgement regarding the uncertainty of demand and returns. However, the changing rate of the optimal solution seems to be limited in a relative small interval. That is, our approach may effectively guarantee the robustness of the solution.

6. Concluding remarks We have applied the robust method to a periodic inventory control and production planning problem with uncertain product returns and demand. We have developed a robust optimization model for the nominal problem, in which the constraints are dealt with through a linear programming formulation. The uncertain sets of the demand and returns are defined according to the robust approach, and the constraints are replaced by their duality forms. Then we solve the linear program with deterministic duality parameters to obtain solutions that are feasible for the nominal problem with a high probability. The reliability of our results is also discussed by deriving the probability bounds. We find that the solutions to the robust problem violate the constraints with a low probability. Since the robust counterpart model (RCM) is a linear program, we can solve it easily through a linear programming tool and software. Finally, numerical examples are provided by using the LINGO tools to validate the effectiveness of our model and to examine the sensitivity of some parameters. The robust optimization model is an effective method to deal with the situations when returns of the product are very uncertain, with only lower and upper bounds known. On the other hand, this approach ignores completely any available information, even though partially, on the distributions of unknown variables. An interesting work is to combine both approaches, robust optimization and stochastic programming, so that partial knowledge on the support intervals and distributions can all be utilized. Another interesting future work is to consider information update, for situations where new information can be observed and utilized dynamically.

Acknowledgments This work was partly supported by the National Natural Science Foundation of China under Grant nos. 70501014, 70971069 and Key Program no. 70932005, and the Research Grants Council of Hong Kong under General Research Fund no. 410208 and NSFC/RGC Joint Research Scheme no. N-CUHK442/05 (NSFC Ref: 70518002), and the fund for National Philosophy and Social Sciences Innovation Institute on ‘‘Business Administration and Institutional Innovation in China’’. The authors would like to thank the valuable comments and suggestions provided by the special issue guest editors and the referees. References Adida, E., Perakis, G., 2006. A robust optimization to dynamic pricing and inventory control with no backorders. Mathematical Programming 104 (1), 97–129. Azar, J., DeJong, E., Boateng, V., George, J., 1995. Agent of change: Xerox design-forenvironment program. Electronics and the Environment, 1995. ISEE, 51–61.

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