A vendorpurchaser economic lot size problem with remanufacturing and deposit


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1 A endorpurchaser economic lot size problem with remanufacturing and deposit Imre Dobos Barbara Gobsch Nadezhda Pakhomoa Grigory Pishchulo Knut Richter European Uniersity Viadrina Frankfurt (Oder) Department of Business Administration and Economics Discussion Paper No. 304 August 011 ISSN
2 A endorpurchaser economic lot size problem with remanufacturing and deposit Imre Dobos 1, Barbara Gobsch, Nadezhda Pakhomoa 3, Grigory Pishchulo, Knut Richter August 1, Corinus Uniersity of Budapest, Institute of Business Economics, H1093 Budapest, Főám tér 8., Hungary, European Uniersity Viadrina Frankfurt (Oder), Chair of Industrial anagement, D1530 Frankfurt (Oder), Große Scharrnstr. 59, Germany, 3 St. Petersburg State Uniersity, Chair of Economic Theory, 6 Chaykoskogo str., St. Petersburg, Russia, Abstract. An economic lot size problem is studied in which a single endor supplies a single purchaser with a homogeneous product and takes a certain fraction of the used items back for remanufacturing, in exchange for a deposit transferred to the purchaser. For the gien demand, productiity, fixed ordering and setup costs, amount of the deposit, unit disposal, production and remanufacturing costs, and unit holding costs at the endor and the purchaser, the costminimal order/lot sizes and remanufacturing rates are determined for the purchaser, the endor, the whole system assuming partners cooperation, and for a bargaining scheme in which the endor offers an amount of the deposit and a remanufacturing rate, and the purchaser responds by setting an order size. Keywords: Joint economic lot size, Reerse logistics, Closed loop supply chain, Collection, Remanufacturing, EOQ 1. Introduction In the present paper, an economic lot size problem is studied in which a single endor supplies a single purchaser with a homogeneous product and takes a certain fraction of the used items back for remanufacturing, in exchange for a deposit transferred to the purchaser. For the gien demand, productiity, remanufacturing rate, fixed ordering and setup costs, amount of the deposit, unit disposal, production and remanufacturing costs, and unit holding costs at the purchaser and the endor, the costminimal order/lot sizes and remanufacturing rates are determined for the purchaser, the endor, the whole system assuming partners cooperation, and for a bargaining scheme in which the endor offers the amount of the deposit and the remanufacturing rate, and the purchaser reacts by setting the order size. The problem of coordinating a supply chain with a single buyer demanding the product at a constant rate and a single supplier operating in a lotforlot fashion has been addressed by onahan (1984) who considered quantity discounts as a coordination means. Subsequently Banerjee (1986a,b) has put forward a more general model that accounts for the inentory holding costs at the endor, and coined the term joint economic lot size. Their work has triggered a broad research interest in this and related supply chain coordination problems. In particular, omission of the lotforlot assumption identifies a major research direction which 1
3 addressed different forms of production and shipment policy (see, among others, Lee and Rosenblatt, 1986; Goyal, 1988; Hill, 1997, 1999; EbenChaime, 004; Hill and Omar, 006; Zhou and Wang, 007; Hoque, 009; see also Goyal, 1976, for an earlier model); we refer the reader to Sucky (005) and Leng and Parlar (005) for excellent reiews, as well. The work by Affisco et al. (00) and Liu and Çetinkaya (007) takes imperfect product quality into account and accommodates the opportunity of inesting in quality improement and setup cost reduction at the supplier. Kohli and Park (1989) hae addressed the question of benefit sharing between the supply chain members from a gametheoretical standpoint. Sucky (006) has further analysed the Banerjee s model in a setting with a priately informed buyer; similarly, Voigt and Inderfurth (011) study a problem with information asymmetry and opportunity of inesting in setup cost reduction. An approach to implement the joint optimization by the supply chain members without explicit disclosure of one s own sensitie data has been recently addressed by Pibernik et al. (011). At the same time, closedloop supply chain problems are receiing an increasing attention in the literature. In the early work by Richter (1994, 1997) and Richter and Dobos (1999), an EOQlike system with manufacturing, remanufacturing and disposal has been analyzed and the structure of optimal policies and cost functions established. ore recently, Chung et al. (008), Gou et al. (008), itra (009) and Yuan and Gao (010) address the inentory costs minimization in a closedloop supply chain context and offer an optimal manufacturing/remanufacturing strategy throughout the planning horizon. Bonney and Jaber (011) extend the classical reerse logistics inentory models to enironmentally related manufacturing factors in order to inole further metrics in the cost functions of inentory management other than just the costs of operating the system. GrimesCasey et al. (007), Lee et al. (010) and Subramoniam et al. (010) inestigate the practical application of reerse logistics models. In this context the models of GrimesCasey et al. (007) and Lee et al. (010) analyze the lifecycle aspects of the reuse systems. Ahn (009) and GrimesCasey et al. (007) suggest gametheoretical approaches for determining the best closedloop supply chain strategies. In the present paper we study a family of problems that represent, on the one hand, an extension of the basic model by Banerjee (1986a,b) to the case of reerse logistics flows which hae not been addressed in his work. On the other hand, they generalize the manufacturing and remanufacturing models of Richter (1994, 1997) and Richter and Dobos (1999) by accommodating a purchaser into the analysis (see Fig. 1).
4 Banerjee Richter, Dobos anufacturing, no remanufacturing anufacturing & remanufacturing anufacturer & purchaser anufacturer only, no purchaser anufacturer & purchaser, manufacturing & remanufacturing Fig. 1: Extending the models by Banerjee and Richter, Dobos Furthermore, we inestigate whether the reuse process helps to coordinate the endor purchaser supply chain. The paper is organized as follows. In the second section we present the models, classify them and construct the cost functions. In section 3 we determine the optimal lot sizes and minimum cost functions for the purchaser, the endor and the total system for the case that the manufacturing process starts the production cycle. In the next section an auxiliary function is introduced. The explicitly found minimum point of this function helps to determine the optimal remanufacturing rates for the partners of the supply chain. In section 5 the same problems will be soled for the case of starting the production process with remanufacturing. In the sixth section, a bargaining scheme is considered in which the endor offers a deposit amount and a remanufacturing rate, and the purchaser reacts by setting the order size. The final section concludes with a summary.. The model.1. General description The model deals with the setting in which a single endor (he) ships a homogeneous product to a single purchaser (her) according to her deterministic constant demand of D units per time unit. The endor and the purchaser will hae to agree on the regular shipment size q which has also to be the production lot size of the endor. This size can be determined by the endor, by the purchaser or jointly by both of them. The endor manufactures new items and can also produce asgoodasnew items by remanufacturing used ones. Both kinds of items which are shortly called sericeables sere the demand of the purchaser. The endofuse items (nonsericeables) can be collected by the purchaser and taken back to the endor for remanufacturing. By assumption, the same ehicle that deliers the sericeables to the purchaser, also picks up the nonsericeables accumulated by her and returns them back to the endor when backhauling (see Fig. ). The parameters and decision ariables employed in the model are presented in Fig. 3. 3
5 anufacturing Remanufacturing Vendor Shipment to the purchaser Stock of sericeables Stock of sericeables Purchaser Use process Stock of nonsericeables Shipment back to the endor Stock of nonsericeables End of use products Disposal Fig. : Operations, transports and inentories in the closed loop chain At both the endor and the purchaser, inentories of sericeables and nonsericeables are held due to the rhythmic deliery of q units and the take back actiities. The manufacturing and remanufacturing productiities P, P R at the endor s site, measured in units per time unit, are assumed to exceed the demand D, i. e., min(p, P R ) > D. The endor and the purchaser incur fixed costs s and s p per order, respectiely, and inentory holding costs h > u and h p > u p per unit of sericeables and nonsericeables per time unit, respsectiely. The unit manufacturing and remanufacturing costs are respectiely denoted by c, c R. The purchaser is assumed to collect βq units for the subsequent remanufacturing at the endor s site, while the other part (1 β)q is disposed at the cost c per unit. She receies d dollars per unit shipped back as a deposit returned by the endor. The length of a cycle, which is the time between two shipments, is denoted by T =q/d. Parameter β is called collection rate for the purchaser and remanufacturing rate for the endor. P productiity c unit man. cost P R productiity c R unit rem. cost u holding cost (nonsericeables) d deposit Vendor 1 β h holding cost (sericeables) s setup cost β 1. Offering d and β βq β remanufacturing rate q h p holding cost (sericeables) s p order cost. Setting q u p holding costs (nonsericeables) d deposit Purchaser D demand β β collection rate 1β c disposal cost Fig. 3: Parameters and decision ariables of the model 4
6 The following complex of problems will be subsequently studied (Fig. 4). The problems of the purchaser consist in determining her optimal order size and the corresponding minimum cost function for a gien collection rate (problem PL) as well as determining the optimal collection rate which minimizes the minimum cost function (problem PC). The corresponding problems studied for the endor hae to be distinguished with respect to two options. If the endor first manufactures new products and then remanufactures used ones (option br), his costs will be different compared to the costs due to the alternatie sequence Remanufacturing before manufacturing (Rb). Like in the purchaser s case, the optimal lot size and the optimal collection rate will be determined, yet separately for the first option (problems VLbR and VCbR) and the second option (problems VLRb and VCRb). In the case of studying the endor purchaser system as a whole, the total of endor s setup costs, his inentory holding costs for sericeables and nonsericeables, the purchaser s setup costs, her inentory holding costs for sericeables and nonsericeables, and her disposal costs, has to be minimized. Then the order of executing the manufacturing and remanufacturing processes is also significant, and therefore the respectie problems SLbR, SCbR, SL Rb and SCRb need to be considered. Finally the problems of determining the deposit and the collection (remanufacturing) rate which meet the purchaser s optimal order size VPC br and VPCRb hae to be studied. Vendor s problem Total system s problem Purchaser s problem anufacturing before remanufacturin g Remanufacturi ng before manufacturing anufacturing before remanufacturing Remanufacturing before manufacturing VLbR: Vendor s optimal lot size VCbR: Vendor s optimal collection rate VLRb: Vendor s optimal lot size VCRb: Vendor s optimal collection rate Fig. 4: Complex of problems studied in the paper SLbR: System s optimal lot size SCbR: System s optimal collection rate SLRb: System s optimal lot size SCRb: System s optimal collection rate VPCbR: Optimal deposit and collection rate under bargaining VPCRb: Optimal deposit and collection rate under bargaining PL: Purchaser s optimal lot size PC: Purchaser s optimal collection rate 5
7 .. Purchaser s cost function Before the models can be formulated analytically some further assumptions hae to be made. The purchaser is supposed to sort out the items which are suitable for remanufacturing continuously oer time and at a constant rate. Hence the aerage inentory leel of her sericeables per order cycle is ( q) q / I n p where ( q,β) β q / I s p = and the inentory leel of nonsericeables is =. Then the purchaser s total costs per time unit are expressed by TC p ( q,β)= C p ( q,β)+ R p β C p D q ( q β ) s + ( h + u β ) = p p p q ( ) (1), and ( β ) c ( c + d ) R p = ( β )D () represent the cost components which respectiely do and do not depend on the order size..3. Vendor s cost function for br The inentory leels of sericeables and nonsericeables at the endor depend obiously on the order in which manufacturing and remanufacturing are executed. Consider first the br option which assumes that in each order cycle, first (1β)q items are manufactured and then βq items are remanufactured. The Rb option will be studied subsequently in Section 5. Figures 5 and 6 illustrate the eolution of the inentory leel of sericeables and nonsericeables at the endor during an order cycle, where t and t 3 represent the duration of manufacturing and remanufacturing operations, respectiely, and t 1 = T t t 3 is the slack time. Note that t + t 3 < T holds due to the assumption P,P R > D made in Section.1. IT s, ( q, β) q (1β)q P R P t 1 t t 3 T t Fig. 5: The sericeables inentory leel at the endor during an order cycle 6
8 IT n, ( q, β) β q P R t 1 t t 3 T t Fig. 6: The nonsericeables inentory leel at the endor during an order cycle Let ( q, β), j s, n H j =, denote the holding costs per time unit for the endor s sericeables and nonsericeables, respectiely. It is straightforward to obtain that H s ( q,β)= q h (β ) β D P D + D, P R P H n q D ( q, β ) = u β β PR what in total gies the following expression of the endor s total holding costs per time unit: H q D D D, = h + + ( β ) β u β β P PR P ( q β ) D P R By rearranging the terms within the curly braces in the aboe cost expression, the following three terms can be identified: one that does not depend on the collection rate β, another one being linear in β, and the third one being quadratic in β, multiplied by the factors which we define respectiely as so that D D Ω and h D V =, = h u P P P R H q ( q, β ) = { Δ β Ω β V} + D D D = h u P P R PR Δ (3). (4) D Note that Ω = u and Δ = u if P = PR. Furthermore, the assumption D < P R made P R in Section.1 obiously implies that Ω < Δ < V. Also note that (4) is nonnegatie for all q 0 and 0 β 1 as H s (q,β) and H n (q,β) are both nonnegatie by construction. 7
9 The endor s ordering and holding costs per time unit express consequently as C D q ( q, β ) = s + { V + β Δ β Ω } q. (5) The endor incurs also costs which do not depend on the lot size, amounting per time unit to R ( β)= ( c + ( d + c R c )β)d. Accordingly, the endor s total costs per time unit are expressed by TC ( q,β)= C ( q,β)+ R ( β). (6).3. The total system costs Now the costs of the endor and the purchaser are considered jointly. The joint ordering and holding costs per time unit are then expressed by Let D q ( q, ) = ( s + s ) + { β Δ + β ( u Ω ) + V h } C T β p p + p. (7) q W + = V h p and U u p Ω =. (8) Then (7) can be rewritten as C D q ( q, β ) = ( s + s ) + { β Δ + βu W} T p + q and the systemwide total costs per time unit expressed by where TC T ( q,β)= C T ( q,β)+ R T β R T lot size. Note that ( ) (9) ( β)= ( c + c + β( c R c c ))D is the cost component that does not depend on the R T ( ) = R ( β ) + R ( ) d = 0 p β d = 0 β. (10) 3. Analysis of the optimal lot sizes and minimal cost functions Firstly, the cost functions of the purchaser, the endor and the whole system are analyzed to determine the optimal order and lot sizes, respectiely. 8
10 The purchaser s PL problem is merely a modification of the classical EOQ model. The optimal order size for a gien collection rate! is then expressed as q p ( β)= s p D h p + β u p (11) with the corresponding minimum total costs TC p ( β)= D s p ( h p + β u p )+ R p ( β). (1) In the same way, the endor s optimal lot size under br policy (i.e., the optimal solution to the VLbR problem) is the one that minimizes (6) and expresses accordingly as q ( β)= D s β Δ β Ω +V. (13) His resulting minimum total costs are then represented by the function ( β ) = D s ( β Δ β Ω + V ) R ( β ) TC +. (14) It is straightforward to see that the sign of this function s nd deriatie does not depend on β, therefore (14) is either conex or concae in β. Thus by analyzing its 1 st deriatie and the function alues at the borders, we can easily determine whether its minimum is attained at β = 0, β = 1, or some point in between. In the latter case, the optimal collection rate will be called nontriial. Section 4. below proides a detailed analysis of the respectie VCbR problem. Similarly, it follows from (7) (9) that the systemwide optimal lot size (i.e., the solution to the SLbR problem) is gien by q * (!) =! D!(s + s p ) W +!! " +!U (15) and implies the following minimum total costs for the system: ( β ) = D ( s + s ) ( W + β Δ + β U ) R ( β ) TC +. (16) P It can be seen that in all of the aboe three cases, the classical EOQ formula is employed with merely straightforward modifications that reflect the composition of the cost function coefficients in the purchaser s, the endor s, and the systemwide problems, respectiely. T 9
11 4. The optimal collection and remanufacturing rates Before we proceed with the analysis of the optimal collection and remanufacturing rates for the purchaser, the endor and the whole system, respectiely, it makes sense to utilize the fact that the respectie minimum cost functions (1), (14), (16) hae a similar form and to introduce the following omnibus function: ( β ) = G A + Bβ Cβ + E β F K + for which we further assume that A,G > 0 and that A + Bβ Cβ > 0 (18) (17) holds on the entire domain 0 β 1. It is straightforward to see that if and only if K(β) is strictly conex AB > C, (19) and otherwise concae. If strictly conex, the function has a minimum in the interior of its domain wheneer inequalities K ʹ (0) < 0 and K ʹ (1) > 0 what, in turn, holds if and only if the C B A + B C < E G < C A hold. In all other cases the minimum is found at one of the boundary points proides us with the following β = 0, (0) β =1. This Lemma 1. Assume that minimum at AB > C. If condition (0) is satisfied then K(β) has a global β * = C B E B and otherwise at AB C BG E (1) β * = 0 wheneer the righthand inequality in (0) does not hold, and at β * =1 wheneer the lefthand one does not. Proof: As discussed aboe, the assumption of the lemma implies that K(β) is strictly conex. Let (0) hold. Then the function is decreasing at β = 0 due to β = 1 due to K ʹ (0) < 0 and increasing at K ʹ (1) > 0. By the strict conexity, the global minimum is attained at the unique interior point β * (0,1) with K ʹ (β * ) = 0, which we accordingly determine by soling K ʹ ( β)= G Bβ C A + Bβ Cβ + E = 0 () 10
12 what implies: C BCβ + B β A + Bβ Cβ = E G and gies the following quadratic equation after rearranging the terms: B(BG E )β C(BG E )β + C G AE = 0. Note that otherwise C G = AE what contradicts (0). Hence the aboe equation can be rewritten as B > 0 holds due to the assumption of the lemma. Also note that β C B β + C G AE B(BG E ) = 0. Its roots are gien by BG E 0 since β 1, = C B ± C B C G AE B(BG E ) = C B ± ABE C E B (BG E ) = C B ± E B AB C BG E. (3) Substituting both roots for! in (), one can see that! * =! if E > 0, otherwise! * =! 1. Thus the following applies: β * = C B E B AB C BG E. Assume now that the righthand inequality in (0) does not hold what implies By the strict conexity, K ʹ ʹ > 0 and thus K ʹ (0) 0. K ʹ is growing at each point of the domain. Therefore K ʹ (β) > 0 at all 0 < β 1, what proes K(β) to grow on the right from β = 0. Then obiously β * = 0 is its global minimum. By a similar argument we obtain β * =1 when the lefthand inequality in (0) does not hold. Note that with iolated simultaneously. K ʹ ʹ > 0, both inequalities in (0) cannot be It follows from the aboe proof that realalued roots (3) which sole () exist under the assumption of Lemma 1 if and only if which K ʹ (β) = 0. This proides us with the following BG > E, otherwise there is no such interior point at Corollary 1. The condition of Lemma 1 can only be satisfied if (but not necessary) condition for that is condition of Lemma 1 hold. B > E holds. A sufficient G C > 0,E 0, proided that the assumption and the 11
13 Proof: Indeed, assume that C > 0 and E 0 holds. The condition of Lemma 1 requires E G < C E, what is then equialent to requiring A G < C, gien that A,G > 0 holds for K(β) A C by assumption. At the same time, the assumption of the Lemma implies < B. This gies A E < B, as required. G The following lemma treats the case of concae Lemma. If AB C then K(β). K(β) has a global minimum at β * = 0 wheneer the inequality A A + B C E G holds, and at β * =1 wheneer it holds in the opposite direction. The case of equality implies that both β = 0 and β =1 delier a global minimum to K(β). Proof: The condition AB C implies that the nd deriatie K ʹ ʹ is nonnegatie and constant eerywhere on the function s domain, hence K(β) is either strictly concae or linear. Hence if a local optimum is found in the domain s interior, it can only be a global maximum; otherwise the function is monotonic on the entire domain and, in either case, it suffices to compare its alues at the boundary points to figure out a global minimum. This immediately leads to the assertion of the lemma. We now proceed to the analysis of the respectie optimal collection rates for the purchaser, the endor and the whole system Purchaser s solution It is straightforward to see that the purchaser s minimum cost function (1) represents K(!) with A = h p, B = 0, C =!u p /, E =!(c + d )D, F = cd, G = Ds p and obiously complies with the assumptions made for K(!) at the beginning of Section 4. It is also obious to see that it satisfies the condition of Lemma, hence the purchaser is not interested in collecting any used items (! * = 0) if (c + d ) D < ( h p + u p! h p )" s p and will collect all of them (! * =1) in the opposite case. He is indifferent between the two policies if the aboe inequality be equality. 1
14 4.. Vendor s solution (VCbR) The endor s minimum cost function (14) represents K(!) with A = V, B =!, C =!, E = (d + c R! c )D, F = c D and G = Ds. There holds accordingly the following Lemma 3. Under conditions (18) (0), the endor s optimal remanufacturing rate is nontriial and gien by! * = " $ E # % V $ " # # # % Ds $ E. (4) Corollary. Equation (4) can be used to determine the endor s optimal remanufacturing rate wheneer the following conditions simultaneously hold:! > " V! > D" (d + c R # c ) s u! (D / P R "1) < (d + c R " c )! D < # (h " u )! D / P R + u Ds V (5) (6) (7) In particular, either of conditions (5) and (6) imply: P R P >1 + u h. (8) Hence P R must necessarily exceed P for! * being a nontriial remanufacturing rate. If any of conditions (5) (7) is iolated then the endor s optimal remanufacturing rate happens to be either! * = 0 or! * =1. Proof: It can be easily seen that conditions (19) and (0) immediately translate to (5) and (7), whereas (18) translates to:! " # $ " +V > 0. It is straightforward to see that the quadratic function on the left has no realalued roots wheneer! < " V what is, in turn, required by (5). Also by the irtue of (5),! > 0 and the quadratic function has its branches directed upwards. This guarantees the required positiity of the function for all feasible alues of "; thus, (5) implies (18) to hold. Further,! 13
15 condition (6) is an immediate consequence of Corollary 1. Finally, either of (5) and (6) imply! > 0. This together with the definition of! in (3) leads to (8). Example 1 (VCbR). Let D = 100, s = 1000, h = 100, u = 5, c = 35, c R = 0, d = 17, P = 00, P R = 50 (see Fig. 7). Then A = V = 50, B = Δ = 8, C = Ω = 5, E = 00, F = 3500, G! It is straightforward to see that conditions (5), (6), (7) respectiely hold:! = 8 > " V = 5 50 = 0.5! = 8 > D" (d + c # c R ) 100" ( # 35) = s 000 = 0. u! (D / P R "1) # "0.433 < (d + c " c )! D R # < $ (h " u )! D / P R + u Ds V # Hence by Corollary, the endor s optimal remanufacturing rate is nontriial and found to be! * " 0.37 with the associated endor s total costs TC * (! * ) " , see Fig. 7. Fig. 8 tracks! * and TC * (! * ) for a range of alues of deposit d. TC Β Β Β Fig. 7: The minimum cost cure TC ( β) Example 1 and the nontriial optimal remanufacturing rate of 14
16 Β d d TC Β d d Fig. 8: The optimal collection rate (left) and the endor s minimum total costs (right) as functions of the deposit in Example The total system solution (SCbR) The systemwide cost function (16) represents K(!) with A = W, B = Δ, U u p C = = Ω, E = (c R! c! c )D, F = ( c + c)d and G = D(s + s p ). Lemma 4. Under conditions (18) (0), the systemoptimal collection and remanufacturing rate is nontriial and gien by! T * = " # $ E # # % W $ (" $ u p / ) # % D(s + s p ) $ E $ u p #. (9) Corollary 3. Equation (9) can be used to determine the systemoptimal collection and remanufacturing rate wheneer the following conditions simultaneously hold:! > (" # u p / ) V + h p! > D" (c R # c # c ) (s + s p ) (30) (31) u! (D / P R "1) " u p / < (c R " c " c )! D < # " u p / (3) (h " u )! D / P R + u + h p + u p D(s + s p ) V + h p In particular, either of conditions (30) and (31) imply (8) and, hence, P R must necessarily exceed P for! * T being a nontriial collection and remanufacturing rate. If any of (30) (3) is iolated then the systemoptimal collection and remanufacturing rate is either! * T = 0 or! * T =1. 15
17 16 Proof: analogous to the proof of Corollary. Β T Β TC T Β Fig. 9: The minimum cost cure TC T * (!) and the nontriial systemoptimal collection rate of Example Note that equations (4) and (9) coincide if 0 = = p p u s and R R c c d c c c + = +. Example. Let the same data as in Example 1 hold with the following modification: c R = 45, and further let s p = 400, h p = 90, u p = 5, c = 10. Then the systemwide optimal collection and remanufacturing rate is! T * " with the associated total costs TC T * (! T * ) " , see Fig The case of remanufacturing before manufacturing (Rb) 5.1. Vendor s inentory costs Consider first the endor s problem. Due to the assumption that remanufacturing precedes manufacturing, the inentory leels of sericeables and nonsericeables during an order cycle eole as illustrated in Fig. 10 and 11, respectiely. The endor s total inentory holding costs per time unit express accordingly as follows: ( ) ( ) = = + R R n s P D β P D P D β u P D P D P D β h q q,β H q,β H + + = R R P D β P D β ) ( β u P D ) β ( P D β h q 1 1 (33)
18 IT s, ( q, β) q P βq P R t 1 t 3 t T t Fig. 10: Eolution of the sericeables inentory leel during an order cycle IT n, ( q, β) β q P R t 1 t 3 t T t Fig. 11: Eolution of the nonsericeables inentory leel during an order cycle By comparing endor s holding cost expressions (3) (4) and (33) in br and Rb settings, respectiely, it is straightforward to obtain the following Lemma 5. Let the same order size q and remanufacturing rate " apply in br and Rb settings. Then: 1. The endor s total holding costs in the br setting are strictly below those in the Rb! setting if and only if P R P >1 + u h! u (34). They are equal if and only if (34) holds as equality or! "{0,1}. Corollary 4. The br setting outperforms the Rb one if the remanufacturing productiity sufficiently exceeds the manufacturing one. Corollary 5. Note that (34) represents a stronger condition than (8) which is necessary for a nontriial optimal remanufacturing rate in the br setting, from the endor s perspectie. 17
19 5.. Vendor s solution (VCRb) As in Section, let us define #! R = (h " u ) D " D & D % ( + u $ P R P ' P # and! R = 1" D & % () u $ P. (35) ' Then the endor s holding costs (33) can be rewritten as H q ( q,β) { V + β Δ + βω } his total costs accordingly as = R R, (36) s D q TC R R + q ( q,β) = + { V + β Δ + βω } R ( β) and his minimum total costs for a gien remanufacturing rate as ( β) = D s ( V + β Δ + βω ) R ( β) TC R R +, (37), The latter expression represents function K(!) as defined in (17), with A = V, B = ΔR, C = Ω R, E = ( d + cr c )D, F = c D and G D s =. Lemma 6. Under conditions (18) (0), the endor s optimal remanufacturing rate in the Rb setting is nontriial and gien by *! Rb = " # R " E $ R % V "# R $ R $ R $ R % Ds " E. (38) Corollary 6. Equation (38) can be used to determine the endor s optimal remanufacturing rate wheneer the following conditions simultaneously hold:! R > " R V! R > D" (d + c R # c ) s (h! u )( D / P R! D / P ) + u > (c! d! c R )D > " R (h! u )D / P R + u Ds V (39) (40) (41) In particular, either of conditions (39) and (40) imply: P u >1!. (4) P R h! u 18
20 Hence gien a relatiely low holding cost rate u of nonsericeables, the manufacturing productiity P must necessarily be aboe a sufficiently large fraction of the remanufacturing productiity P R for! * being a nontriial remanufacturing rate. Furthermore, condition (41) implies: c > c R + d. If any of conditions (39) (41) is iolated then the endor s optimal remanufacturing rate happens to be either! * = 0 or! * =1. Proof: analogous to the proof of Corollary. Example 3. Let D = 000, s = 500, h = 00, u = 10, c = 40, c R = 0, d = 15, P = 3000, and P R = 500. Then P R u = 0.83 < 1 + ".5 P h! u and by Lemma 5, the Rb setting outperforms its br counterpart for eery fixed order size q and remanufacturing rate! "(0, 1). Further, V!133.33,! R " 90.67,! R = 40, E =!10000, and it is straightforward to see that conditions of Corollary 6 are satisfied, what * results in! Rb " 0.81 and the endor s minimum total costs TC * * (! Rb ) " Fig. 1 displays the endor s total costs for all! "[0,1] in both Rb and br settings, respectiely assuming the optimal choice of the order size. As one can see, conditions of Corollary are not satisfied for the br setting in the gien example, leading to the optimal remanufacturing * rate! br =1 in that setting. TC Β br Rb Β Β Rb Fig. 1: Vendor s cost functions in the Rb and br settings of Example 3 19
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