Pricing and Inventory Control Strategy for a Periodic-Review Energy Buy-Back System

J Syst Sci Complex (2016) 29: 10181033Pricing and Inventory Control Strategy for aPeriodic-Review Energy Buy-Back SystemZHANG Jihong CHEN Hongqiao DING Xiaosong LI XianDOI: 10.1007/s11424-016-4101-0Received: 23 April 2014 / Revised: 20 March 2015c The Editorial Oce of JSSC DECIDE ResearchGroup, Department of Computer and Systems Sciences, Stockholm University, Forum 100, SE-164 40 Kista,Sweden. Email: .LI XianInternational Business School, Beijing Foreign Studies University, Beijing 100089, China.The work was partially supported by Young Faculty Research Fund of Beijing Foreign Studies University(2015JT005), YETP (YETP0851), the National Natural Science Foundation of China (71371032), Key Projectof Beijing Foreign Studies University Research Programs (2011XG003), the Humanities and Social ScienceResearch Project of Ministry of Education (13YJA630125), and the Fundamental Research Funds for the CentralUniversities.This paper was recommended for publication by Editor ZHANG Xun.PRICING AND INVENTORY CONTROL STRATEGY 10191 IntroductionIn the past fteen years, the world energy shortage has become an increasingly serious issuealong with the rapid economic development and enhancement of industrialization. Meanwhile,the power demand has been keeping rising. The frequent mismatch created between demandand supply in electricity provides energy buy-back programs with great opportunities; see, forexample, 1, 2. Such incentive-based programs, when activated, oer certain amount of nancialcompensations to participants for reducing their energy use. They aim at encouraging partici-pants to shift their electricity usage from peak to non-peak time, thereby releasing the demandpressure during peak time and smoothing the overall electricity consumption. Many high en-ergy consumption manufacturers including steel companies actively participate in miscellaneousenergy buy-back programs. Nonetheless, such participation will inuence a manufacturers pric-ing and productioin/inventory strategy since he needs to handle the balance between receivingnancial compensations from participation by reducing production and increasing sales fromsatisfying customers demands.Our work is motivated by the energy buy-back programs provided in the U.S. energy marketaccompanying the deregulation of the electricity industry. Nowadays, many energy buy-backprograms have been widely utilized as options for balancing supply and demand. For instance,Wisconsin Electric created an energy buy-back program referred to as power market incentivesthat pays large commercial and industrial customers for voluntarily reducing electric load whenthe wholesale spot market spikes; Commonwealth Edison (ComEd) Company created a loadresponse program named ComEd smart returns that pays business customers nancial incen-tives for reducing the electricity usage during times of high wholesale electricity prices; PugetSound Energy created similar programs including custom retrot grants, commercial rebatesand enhanced lighting incentives for industrial and manufacturing businesses. There are manyother similar incentive-based programs emerging under dierent names, interested readers arereferred to the ocial website of U.S. department of energy for more detailed case studies.In the literature, there exist a few papers concerning production-inventory or joint pricingand inventory control problems under an energy buy-back program; see 35. Chen, et al.4rst investigated a periodic-review production-inventory problem under an energy buy-backprogram over nite planning horizons. By assuming that the ordering cost is proportional tothe order quantity and there exists no xed setup cost, the authors showed that partly a base-stock policy is optimal for the normal market condition; whereas a state dependent (s, S) policyis optimal for a peak state. With a similar background but under a continuous time setting,Chao and Chen3 considered a production-inventory problem under an incentive program, inwhich two cases are both formulated as Markovian decision processes. It is shown that for theexponential peak period duration, the production and shutdown strategy is determined by asingle threshold level; whereas when peak duration becomes known at the beginning of a peakperiod, the strategy is determined by a sequence of threshold levels depending on the remainingtime prior to the end of the current peak period. Ding, et al.5 studied a periodic-review jointpricing and inventory model under an energy buy-back program over nite planning horizons.a non-trivial extension since the critical assumption in 5, Lt1 Lt+1 = i=1 p(t+1)iL(t+1)i,1020 ZHANG JIHONG, et al.With the concept of symmetric k-concavity proposed in 6 and by incorporating a xed setupcost, they showed that for the general random demand function, a state dependent (s, S, A, P )policy is optimal for a peak state.The other stream of related research is the joint pricing and inventory control models withoutthe energy buy-back programs. Research in this topic can be dated back to Whitin7, in whicha newsvendor problem with price dependent demand is considered for the rst time. Thomas8further extended the model with the incorporation of a xed ordering cost and shows thatthe optimal policy is of an (s, S, P ) type. Federgruen and Heching9 developed a combinedpricing and inventory control model over nite planning horizons. Under the assumption thatthe ordering cost is proportional to the order quantity, it is shown that a base-stock list pricepolicy is optimal. Chen and Simchi-Levi6 further extended the model proposed in 9 with theinclusion of a xed setup cost. They show that the prot-to-go function is k-concave when thedemand function is in the additive form, and thereby an (s, S, P ) policy is optimal; whereas it issymmetric k-concave when the demand function is in the general form, and thus an (s, S, A, P )policy is optimal. Following 6, Chen and Simchi-Levi10 showed that the optimal pricing andinventory policy for the corresponding model over an innite planning horizon is of an (s, S, P )type. Other results in joint pricing and inventory control problems can be categorized intocontinuous review models11, 12, the model with a Markovian demand13, the model with arandom supply capacity14, lost-sales models15, 16, the model with return and expediting17,and so on18, 19.This paper investigates a periodic-review joint pricing and inventory control system underan energy buy-back program with the incorporation of the compensation levels, setup cost andadditive random demand function. The primary objective aims at identifying the manufac-turers optimal pricing and inventory control policy in order to maximize his expected totalprot over nite planning horizons. Under mild assumptions and by utilizing Veinotts con-ditions, we can show that the manufacturers optimal decision is a state dependent (s, S, P )policy under a peak market condition, or partly an (s, S, A, P ) policy under the normal marketcondition.In the aforementioned papers, the work in 46 is most relevant to ours. This paper diersfrom the previous work in several aspects. Firstly, our paper diers from Chen and Simchi-Levi6 on joint pricing and inventory policy, in that we introduce an energy buy-back programinto their model so that it changes into a multi-state environment response model. Secondly,our paper diers from Chen, et al.4 in that, under the energy buy-back program, we inves-tigate a joint pricing and inventory system with the inclusion of variable costs as well as axed setup cost from the revenue maximization perspective; while in 4, the authors study aproduction-inventory system only with the inclusion of variable costs from the cost minimiza-tion perspective. Consequently, the manufacturers optimal policy derived herein is also quitedierent from the one in 4. Finally, we extend the work by Ding, et al.5 with the introduc-tion of the normal market condition (i.e., a state with no compensations), which is a practicalconsideration of the energy buy-back program, and thereby should not be neglected. This isMPRICING AND INVENTORY CONTROL STRATEGY 1021imposed on the sorted compensation levels for period t, t = 1, 2, , N 1, fails to hold. Theassumption in 5 requires that the lowest compensation level, Lt1, is at least as great as theexpected compensation level in the next period, and can be regarded as a variation of the con-dition K ti Kt+1 in order to carry out the backward induction by using k-convexity; see, forexample, 20, 21. Nevertheless, when we introduce the normal market condition for which thecompensation level is zero, such an assumption can never be satised since the lowest compen-Msation level becomes Lt0 = 0, which cannot exceed Lt+1 = i=0 p(t+1)iL(t+1)i 0. This makesit impossible to carry out the traditional backward induction by using k-convexity, and thuscauses a change in the methodology for the derivation of the manufacturers optimal policy. Inthis paper, we make use of Veinotts conditions in 22 instead of k-convexity, and the resultsderived herein are also dierent from those in 5.The rest of this paper is organized as follows. Under mild assumptions, Section 2 sets upa joint pricing and inventory model under an energy buy-back program for a manufactureraiming at maximizing his expected total prot over nite planning horizons. By using Veinottsconditions, Section 3 shows that the manufacturers optimal decision is either a state dependent(s, S, P ) policy or partly an (s, S, A, P ) policy. Section 4 concludes this paper.2 Assumptions and ModelSuppose that in period t, a manufacturer facing uncertain demand needs to make decisionon pricing and inventory simultaneously in order to maximize his expected total prot over Nconsecutive periods. Without loss of generality, we start from t = 1. In each period, thereexist M + 1 market states: Normal type i = 0, and peak type i, i = 1, 2, , M . For period t,denote by Lti the nancial compensation level of an energy buy-back program corresponding topeak type i, while for normal type, there is no compensation, that is, Lt0 = 0, and we can getthese Ltis sorted with 0 = Lt0 0, (yt xt)(Lti + K) Lti + c(yt xt), where (x) = 0, if x = 0.Due to backlogging, xt may be positive or negative. A cost h(x), which represents the inventoryholding cost if x 0 or the shortage cost if x 0, dene the function J(x) :=supy x Z(y) (y x)C. Then x2 x1, J(x1) J(x2) C.Proof First note thatJ(x) = sup Z(y) (y x)C = max Z(x), sup Z(y) C .y x yxConsider the following two cases.Case (a) If J(x2) = Z(x2), thenJ(x1) = max Z(x1), sup Z(y) C max Z(x1), Z(x2) Cyx 1= max Z(x1), J(x2) C J(x2) C.Case (b) If J(x2) = supyx 2Z(y) C, thenJ(x1) = max Z(x1), sup Z(y) Cyx 1 max Z(x1), sup Z(y) Cyx 2= max Z(x1), J(x2) J(x2) J(x2) C.The proof is completed.Lemma 3.2 For period t = 1, 2, , N , gt(y, d) is jointly continuous in (y, d), and thusthere exists a dt(y) that maximizes gt(y, d) for any given y, such that ydt(y) is a nondecreasingfunction of y.Proof See Lemma 2 in 6.With Lemmas 3.1 and 3.2, we can show that Veinotts conditions hold for Ft(x), t =1, 2, , N , which are presented in the following theorem.Theorem 3.3 For period t = 1, 2, , N , Veinotts conditions hold, that is, there existsan SN such that Ft(y2) Ft(y1) K + Lt+1, Ft(y2) Ft(y1) 0,y2 y1,y1 si .1024 ZHANG JIHONG, et al.Proof By the denition of Ft(x) in (5) and fN+1(x, i) 0 for all market states, we haveFN (y2) = FN (y1) = 0, and thus Veinotts conditions hold for period N . Next, we will use math-ematical induction to show that Veinotts conditions also hold for for period t = 1, 2, , N 1,since fN (x, i) is dierent from ft(x, i) in other periods.1) We rst show that Veinotts conditions hold for period N 1.By (1) and (6), for period N , gN (y, dN (y) = R(dN (y) cy GN (y, dN (y), andgN (y1 + (1 )y2, dN (y1 + (1 )y2)= R(dN (y1 + (1 )y2) cy1 + (1 )y2E h(y1 + (1 )y2 dN (y1 + (1 )y2) N ) R(dN (y1) + (1 )dN (y2) cy1 + (1 )y2E h(y1 + (1 )y2 dN (y1) (1 )dN (y2) N (1 ) N ) R(dN (y1) + (1 )R(dN (y2) cy1 (1 )cy2 E h(y1 dN (y1) N ) (1 )E h(y2 dN (y2) N )= gN (y1, dN (y1) + (1 )gN (y2, dN (y2). (9)In (9), the rst inequality holds since dN (y) is the maximizer of gN (y, dN (y) given a xed y,and the second inequality holds since both R(dt) and h(x) are concave. Hence, gN (y, dN (y) isconcave in y and there exists a maximi