The deterministic EPQ with partial backordering: A new approach

Transcription

1 Omega 37 (009) Te deterministic EPQ wit partial backordering: A new approac David W. Pentico a, Mattew J. Drake a,, Carl Toews b a Scool of Business Administration, Duquesne University, 95 Rockwell Hall, Pittsburg, PA 158, USA b Department of Matematics and Computer Science, Duquesne University, Pittsburg, PA 158, USA Received 15 October 007; accepted 5 Marc 008 Available online 18 Marc 008 Abstract Several autors ave developed models for te EOQ wen only a percentage of stockouts will be backordered. Most of tese models are complicated, wit equations unlike tose for te EOQ wit full backordering. In tis paper we extend work by Pentico and Drake [Te deterministic EOQ wit partial backordering: a new approac. European Journal of Operational Researc 008; in press] tat developed equations for te EOQ wit partial backordering tat are more like tose for te EOQ wit full backordering to develop a comparable model for te EPQ wit partial backordering. 008 Elsevier Ltd. All rigts reserved. Keywords: Inventory control; Production lot sizing; Partial backordering 1. Introduction Te two basic questions any inventory control system must answer are wen and ow muc to order. Over te years, undreds of papers and books ave been publised presenting models for doing tis under various conditions and assumptions. Te best known of tese models is Harris s [] classic square root economic order quantity (EOQ) model tat appears in every basic textbook covering inventory management. Wile tis model as been criticized for te unreasonableness of its assumptions (see, e.g., [3]), surveys ave sown tat it is widely used. Furter, it forms te basis for many oter models tat relax one or more of its assumptions. See Jagieta and Micenzi [4] and Kouja and Park [5] for examples of extensions to te classic Tis manuscript was processed by Associate Editor Ruud Teunter. Corresponding autor. Tel.: address: drake987@duq.edu (M.J. Drake). EOQ model tat relax one or more of its traditional assumptions. An early extension of te basic EOQ model, generally referred to as te economic production quantity (EPQ) or economic manufacturing quantity (EMQ) model, replaced te assumption of instantaneous replenisment by te assumption tat te replenisment order is received at a constant finite rate over time. A key assumption of bot te basic EOQ and EPQ models is tat stockouts are not permitted. Assuming tat te lead time and demand are known and constant, tis means tat an order will be placed wen te inventory available is exactly sufficient to cover te demand during tat lead time. Under conditions of demand certainty, owever, it is possible to prove tat, assuming customers are always willing, altoug not necessarily appy, to wait for delivery, planned backorders can make economic sense, even if tey incur some actual or implied cost. Relaxing te basic EOQ and EPQ models assumption tat stockouts are not permitted led to te development of bot EOQ and EPQ models for te two /$ - see front matter 008 Elsevier Ltd. All rigts reserved. doi: /j.omega

2 David W. Pentico et al. / Omega 37 (009) basic stockout cases: backorders and lost sales. Wat took longer to develop was a model tat recognized tat, wile some customers are willing to wait for delivery, oters are not. Eiter tese customers will cancel teir orders or te supplier will ave to fill tem witin te normal delivery time by using more expensive supply metods. Wile tere ave been a number of models developed for te EOQ and EPQ wit partial backordering, most of tem incorporate considerably more complicated assumption sets tan te classic EOQ and EPQ models do. After reviewing te models for te pure stockout cases of backorders and lost sales, we will briefly discuss five models for te basic EOQ wit partial backordering and will summarize te only paper we are aware of tat develops a model for te basic EPQ problem wit partial backordering. Following tat, we will present an alternative approac to modeling te latter problem and determining expressions for wen and ow muc to order.. Notation and terminology A significant difficulty in reading te literature in inventory modeling is tat tere is no standard set of notation. We will use notation tat, in our opinion, makes it somewat easier to remember wat te different symbols represent. Parameters D P s C o C p C C b C g C l (s C p ) + C g β Variables Q T demand per year production rate per year if constantly producing te unit selling price te fixed cost of placing and receiving an order te variable cost of a purcasing or producing a unit te cost to old a unit in inventory for a year te cost to keep a unit backordered for a year te goodwill loss on a unit of unfilled demand te cost for a lost sale, including te lost profit on tat unit and any goodwill loss te fraction of stockouts tat will be backordered te order quantity te lengt of an order cycle I S B F te maximum inventory level, wit I being te average inventory level over te year te maximum stockout level, including bot backorders and lost sales te maximum backorder position, wit B being te average backorder level over te year (B βs) te fill rate or te percentage of demand tat will be filled from stock 3. Te pure stockout cases: backorders and lost sales Models for te EOQ and EPQ wit backorders appear in many basic texts. Wile te analysis for lost sales appears less frequently, Zipkin [6] and Pentico and Drake [1] give derivations for te basic EOQ wit lost sales tat are easily generalized to te EPQ case Te pure backorder models Since many basic texts derive models for te EOQ and EPQ wit full backordering, we will not go troug tose derivations. Tere ave also been many extensions of te full backordering EOQ to scenarios tat relax oter basic EOQ assumptions as well; see Wee et al. [7] for an example of an EOQ model wit defective units and full backordering. Using te notation in te previous section, te relevant equations for te optimal order quantity (Q ), maximum backorder quantity (B ), and time between orders (T ) are as follows: EOQ wit full backordering Q C o D C b + C, C C b ( ) B Q C and C b + C T C o C b + C. (1) DC C b EPQ wit full backordering Fig. 1 is a grap of te net inventory level for te EPQ wit full backordering. Te equations for te optimal order quantity (Q ) and time between orders (T ) for te EPQ wit full backordering problem are almost

3 66 David W. Pentico et al. / Omega 37 (009) Q DT Net Inventory Level B S T Fig. 1. Grap of pure backorder case for EPQ. identical to tose for te EOQ wit full backordering, te difference being te multiplication of C in te first square root of bot Q and T by (1 D/P), reflecting te relative sizes of te production and usage rates: Q C o D C b + C and T C (1 D/P) C o DC (1 D/P) C b I C b + C C b. () Te equation for B for te EPQ differs from te comparable equation for te EOQ in a different way, but is still involves adjusting for te relative sizes of te production and usage rates: B C o D C b C C b + C 3.. Te pure lost sales case 1 D P. (3) Zipkin [6] and Pentico and Drake [1] sow, for te EOQ problem, tat if demands during a stockout period are lost sales rater tan resulting in backorders, te optimal policy is to ave eiter no stockouts or all stockouts, te coice being te alternative wit te lower average cost per period. Zipkin s proof is actually for te situation in wic stockouts are backordered at a fixed cost per unit rater tan incurring a cost proportional to te amount of time tat te unit is backordered. He ten develops te cost equation for te lost sales case and, sowing tat it as te same form as te backorder incidence case, concludes tat te same result olds. Pentico and Drake prove te same result directly, using te notation and decision variables used in tis paper. It is straigtforward to sow tat te same result olds for te EPQ wit lost sales. 4. Previous researc on models wit partial backordering Since, for te EOQ model, it is optimal to allow some stockouts if all customers will wait (β1) and to eiter allow no stockouts or lose all sales if no customers will wait (β0), it is logical to assume tere will be a value of β below wic one sould use te optimal ordering policy for te lost sales case and above wic one sould allow stockouts, some of wic will be backordered. Tis is, in fact, wat appens. Determining an optimal policy for te partial backordering deterministic EOQ problem starts wit identifying te minimum value of β for wic stockouts sould be allowed and, if β is greater tan tis minimum value, determining te optimal order quantity. We begin tis brief survey wit a discussion of models tat are identical to te basic EOQ model except for allowing for partial backordering. Models for te partial backordering EOQ problem were developed by Montgomery et al. [8], Rosenberg [9], Park [10], San José et al. [11 14], and Pentico and Drake [1]. Tese papers took somewat different approaces to modeling te problem, differing in some cases in teir cost assumptions, but primarily in wic decision variables tey used. Bot Montgomery et al. [8] and Rosenberg [9] include a fixed backorder carge, wic is te same as te goodwill loss for a lost sale. Tey solve for te optimal ordering quantity troug a two-step optimization procedure since te overall cost function is not necessarily convex; te difference in teir models is due to te decision variables used. Montgomery et al. use te order quantity, Q, and te maximum backorder level, S, but transform tem into two new variables: U Q+(1 β)s, wic is te total amount of demand during an order cycle, or TD in our notation, and V Q βs, wic, in our notation, is I, te on-and inventory at te beginning of te cycle. Rosenberg also begins wit Q and S as is decision variables but replaces tem wit two oter variables: T, te lengt of te inventory review cycle, given by T [Q + (1 β)s]/d, and a fictitious demand rate X, defined as X (Q βs)/t. Park [10] does not include a fixed unit cost per unit backordered. His coice of variables S, te maximum

4 David W. Pentico et al. / Omega 37 (009) size of te stockout during an inventory cycle, and R, wic is te same as U in Montgomery et al. enables im to sow tat is cost function is convex, so e can develop a solution by simultaneously solving te two equations developed by setting te partial derivatives equal to 0. Determining te solution involves evaluating a complicated expression for S as te solution to a quadratic equation. From te equation for S e develops a statement of te range of values for β for wic S would ave to be 0, in wic case eiter all sales are lost or no stockouts are allowed and Q is determined wit te basic EOQ model. Te cost structure used by San José et al. [11 14] also includes a fixed unit cost of backordering, but it is not necessarily te same as te goodwill loss for a lost sale. More significantly, in all of teir models except te one presented in [11], tey assume tat te percentage of unmet demand backordered is not a constant. Along wit analyzing te constant-β case, tey consider a variety of different customer impatience functions in wic te percentage backordered increases as te replenisment date approaces. San José et al. s decision variables are T, te lengt of time in an inventory cycle during wic inventory is positive (FT in our notation), and Ψ, te complementary time during wic tere are backorders ((1 F)T in our notation). Tey replace tese two variables by Ψ and α T + Ψ, wic is te same as our T. As in Montgomery et al. [8] and Rosenberg [9], teir solution procedure is executed in two stages, first finding an optimal value for Ψ and ten substituting tat into an equation tat relates te value of α to te value of Ψ. Pentico and Drake s [1] objective was to develop a set of equations tat are bot simpler to use and ave a more understandable and intuitive form tat more closely resemble te comparable equations for te basic EOQ and EOQ wit full backordering. Because teir approac forms te basis for wat we will do ere, we will cover it in greater detail tan tose of te previous autors. As in Park [10], Pentico and Drake did not include a fixed unit cost to backorder, altoug tey did consider tat case in an Appendix in [1]. Teir decision variables are T, te lengt of an order cycle, wic is te same as San José et al. s α, and F, te fraction of demand to be filled from stock. Setting te derivatives of teir objective function wit respect to T and F equal to 0 gives an expression for T as a function of F C o T(F) D[C F + βc b (1 F) ] (4) and an equation for F as a function of T F(T) (1 β)c l + βc b T. (5) T(C + βc b ) Substituting te expression for F in (5) into Eq. (4) leads to: [ ] T C o C + βc b [(1 β)c l]. (6) DC βc b βc C b Recognizing tat T for partial backordering must be at least as large as T for te basic EOQ ( C o /(DC )) if partial backordering is optimal leads to te following bound on β: β β Co C 1. (7) DC l Tis is te same condition derived by Park [10]. An alternative way of stating tis condition on β, wic makes it comparable to te condition given in Rosenberg [9], wic is equivalent to te one in Montgomery et al. [8], is C o (1 β)c l. (8) DC C In addition to models based on te straigtforward partial-backordering generalization of te EOQ model discussed above, tere ave been a variety of additional studies tat model te partial backordering beavior in more complicated decision environments. Some of tese papers consider deteriorating inventory, items suc as fres produce and semiconductor cips tat lose teir value quickly from damage, obsolescence, or pilferage. Some include time-varying demand and/or pricing decisions. Abad [15] models te joint dynamic pricing and ordering decisions wit perisable inventory and timevarying backorder percentages defined by customer impatience functions similar to some of tose used by San José et al. [11,13]. He investigates te computational properties of tis model wen te unit selling price is fixed witin te inventory cycle in [16]. Neiter of tese models incorporates expressions for stockout or backordering costs, since te autor claims tat tese parameters are difficult to estimate in practice. Dye [17] extends te model in [16] by including te stockout and backorder costs. Wee [18] considers te same pricing and ordering decisions for perisable items under linear demand wit quantity discounting and a constant backordering percentage. Cang and Dye [19] develop an optimal ordering model for time-varying demand and

5 68 David W. Pentico et al. / Omega 37 (009) Abad s [15] backordering impatience function. An additional model utilizing partial backordering is Yang et al. [0], wic considers optimal lot sizes for integrated supply cains under bot perfect and monopolistic competition. In addition to tese papers, and oters, tat consider partial backordering in te context of batc ordering, tere are a number of papers tat consider te combination of partial backordering and a finite production rate, wic is te problem tat we are addressing in tis researc. As is te case wit most of te abovereferenced papers, most of tese autors also generally looked at more complex decision environments. Abad [1] extended is previous work based on te EOQ to consider te combination of perisable goods and a finite production rate. Unlike is approac in [15,16], in [1] Abad included costs for bot backordering and lost sales and used a constant backordering percentage. Goyal and Giri [] allow te demand, production, and deterioration rates to vary over time. Giri et al. [3] develop a model for te case in wic demand is increasing, te production rate is finite and can be adjusted for eac production cycle, and sortages will be partially backordered. In addition to a time-varying rate of inventory deterioration, Lo et al. [4] include inflation, an imperfect production process, and multiple deliveries in teir situational scenario. Jolai et al. [5] included perisable inventory, a constant backordering percentage, demand tat decreases linearly wit te decreasing inventory level, a finite production rate, inflation, and, unlike te oter papers referenced ere, a finite planning orizon. Tey assume tat eac production/demand cycle will be of te same lengt and solve for m, te integer number cycles during te finite planning orizon, and T, te lengt of time from wen inventory reaces 0 until te next production run begins. 5. Modeling net inventory during te stockout period for te EPQ wit backordering Te only paper we ave found tat develops a model for te basic EPQ wit partial backordering under basically te same assumptions as we consider in tis paper is by Mak [6]. Lietal.[7] discuss an EPQ model for multiple products wit planned backorders, but tey assume tat all of te demand not originally satisfied from stock is willing to wait for te units to be delivered upon production. Cakraborty et al. [8] develop a model for te EPQ tat simultaneously considers te effect of production defects and macine breakdowns; in teir model, owever, all sortages result in lost sales. Since Mak s approac to te treatment of demands tat Q Net Inventory Level B S T Fig.. Grap of partial backorder case for EPQ wit LIFO backorder filling (adapted from [6]). occur wile tere is no stock but te production run as started differs from our approac, we first discuss tat issue. From te time te system runs out of stock until te time te next order is received (EOQ wit backordering) or te next production run begins (EPQ wit backordering), a fraction β of incoming demand will be backordered until te maximum backorder level B βs is reaced. In te EOQ wit full or partial backordering, te entire order quantity Q is received simultaneously, so all te backorders can be filled at once, wit te inventory rising immediately to I Q B. In te EPQ wit full backordering, te order quantity Q is received in a constant stream at a rate of P. Since all demands tat occur during te time it takes to fill all te backorders are also backordered if tey are not filled immediately, it makes no difference weter te incoming orders are filled before te backorders (a LIFO approac) or te backorders are filled before te incoming orders (a FIFO approac). A grap of te net inventory level for te EPQ wit full backordering is sown in Fig. 1. For te EPQ wit partial backordering; owever, weter LIFO or FIFO is used to determine te order in wic new demands and backorders are filled after te production run begins can make a difference in te net inventory level. Te impact depends on te answer to an additional question: Wat appens to te demands tat occur wen tere is no stock on and but te production run as been started? If we assume I

6 David W. Pentico et al. / Omega 37 (009) Q Net Inventory Level B S T Fig. 3. Grap of partial backorder case for EPQ wit FIFO backorder filling. tat incoming demands will be filled from production before existing backorders are filled (LIFO) and tat none of te existing backorders will convert to lost sales, ten te net inventory level for te EPQ wit partial backordering will be as sown in Fig.. If, owever, we assume tat te existing backorders will be filled before any new demands are filled (FIFO) and assume tat only a fraction β of tese new orders tat cannot immediately be filled will be backordered, wit te rest being lost sales, ten te net inventory level for te EPQ wit partial backordering will be as sown in Fig. 3. (If all incoming orders will wait once te production run as started, it makes no difference weter LIFO or FIFO is used and Fig. applies.) 6. Mak s [6] model for te EPQ wit partial backordering Mak s assumptions are te usual ones for te basic EPQ wit full backordering except tat, as in all te basic EOQ wit partial backordering models summarized above except te ones by San José et al. [11 14], only a constant fraction β of te stockouts will be backordered, wit te rest being lost sales. Relative to te discussion immediately above about weter LIFO or FIFO is used to fill te backorders once te production run starts, Mak s approac is as sown in Fig.. He assumes tat tere will be no increase in eiter backorders or lost sales once te production pase begins so te backorders are filled at a rate of P D. A model I based on Mak s LIFO assumption tat uses our decision variables is developed in Appendix C. Mak s decision variables are T, te lengt of an inventory cycle, and t, te lengt of time from wen te inventory level reaces 0 until te next production run begins. Te cost function e develops is convex, and tus te optimal solution can be found by setting te two partial derivatives equal to 0 and solving te resulting equations simultaneously. He does tis by developing an equation for T as a function of t and ten, by using tis to eliminate T from te oter equation, finds an expression for t as a function of te parameters and, using tis, finds an expression for T. Bot of tese equations are quite complicated. Mak also develops a statement of te condition tat β must satisfy for te partial backordering EPQ equations to apply tat is comparable to te ones developed for te partial backordering EOQ models and te one to be developed ere. 7. An alternative approac to te EPQ wit partial backordering We use te same assumptions about costs and demand as used in te basic EOQ wit full backordering model and by Mak [6]. However, we assume tat a FIFO policy is used to fill te backorders once te production run starts, so tat Fig. 3 sows te level of net inventory over te course of an inventory cycle. As in Pentico and Drake [1], weuset, te lengt of an inventory cycle, and F, te fill rate, as te decision variables. Our objective, as in Pentico and Drake, is to develop a set of equations tat are simpler to use and ave a more understandable form tan tose in Mak Time intervals and te maximum inventory and backorder levels As sown in Fig. 4, an inventory cycle, wic as lengt T, can be divided into four sub-intervals. As developed in Appendix A, te lengts of tese intervals are t 1 (1 F)T(1 βd/p), t (1 F)T(βD/P), t 3 FTD/P, t 4 FT(1 D/P). Using tese values we develop expressions for I, te maximum inventory level, and B, te maximum backorder level. As in te EPQ wit full backordering model, during interval 3 te inventory level is increasing at a rate of

7 630 David W. Pentico et al. / Omega 37 (009) Net Inventory Level t 1 t B T t 3 t 4 Fig. 4. Grap sowing time intervals for EPQ wit partial backordering and FIFO backorder filling. P D,soI (P D)t 3. Substituting te expression for t 3 above, we get I FTD(1 D/P). During interval 1, no production is taking place and no demands are filled, so te backorder grows from 0 to its maximum value, B βdt 1. Substituting te expression for t 1 above, we get B βd(1 F)T(1 βd/p). 7.. Te profit and cost functions based on T and F Te average profit per year to be maximized is te revenue from filling demands, eiter from stock or as backorders, minus te cost of placing orders, te cost of te units used or sold, te cost of carrying inventory, te cost of te backorders, and te cost of lost sales. Tus, Π(T, F ) (s C p )D[F + β(1 F)] [C o /T + C I + C b B + C g D(1 β)(1 F)] (s C p )D Γ(T, F ), (9) were Γ(T, F ) C o /T + C I + C b B + C l D(1 β)(1 F). (10) Since (s C p )D is a constant, Π(T, F ) is maximized by te pair (T, F ) tat minimizes Γ(T, F ). I From Fig. 4, we see tat te average inventory is given by te average value of a triangle wit eigt I and base t 3 +t 4 FT, multiplied by te fraction of time for wic tere is inventory, F. Since I FTD(1 D/P), te average inventory is I DTF ( 1 D P ). (11) Also from Fig. 4, we see tat te average backorder level is given by te average value of a triangle wit eigt B and base t 1 + t (1 F)T, multiplied by te fraction of time for wic tere are backorders, 1 F. Since B βd(1 F)T(1 βd/p), te average backorder level is B βdt (1 F) ( 1 βd P ). (1) Substituting te expressions for I in (11) and B in (1) into (10) gives Γ(T, F ) C o T + C DTF ( 1 D ) P + βc bdt (1 F) ( 1 βd ) P + C l D(1 β)(1 F). (13) To simplify te notation, we define C C (1 D/P) and C b C b(1 βd/p), wic makes te average cost per year to be minimized: Γ(T, F ) C o T + C DTF + βc bdt (1 F) + C l D(1 β)(1 F). (14) 7.3. Determining te optimal values for T and F Taking te partial derivative of Γ(T, F ) in (14) wit respect to T and setting it equal to 0 gives Γ T C o T + C DF + βc bd(1 F) 0. (15) Tis gives, after some algebra: C o T D[C F + βc b (1 F) ]. (16) Note tat tis equation for T as te same general form as te equation for te optimal T for te EPQ wit full backordering model given in () and, in fact, reduces to tat equation if β 1. Furter, tis equation for T also reduces exactly to te equation for te basic

8 David W. Pentico et al. / Omega 37 (009) EPQ model if F 1, wic means tat tere will be no stockouts. Taking te partial derivative of Γ(T, F ) wit respect to F and setting it equal to 0 gives Γ F C DTF βc bdt (1 F) (1 β)c l D 0. (17) After some algebra, tis results in F(T) (1 β)c l + βc b T T(C + βc b ). (18) Substituting tis expression for F into Eq. (16), we get, after some algebra: T C o DC [ C + βc b ] [(1 β)c l]. (19) βc b βc C b Using te value of T from (19) in (18) gives F(T ) and completes te solution. Substituting te formulas for T in (19) and F(T ) in (18) into te equation for Γ(T, F ) in (14) gives, after a considerable amount of algebra, a simple expression for te optimal cost for te EPQ wit partial backordering: Γ Γ(T,F ) C DT F. (0) It is interesting to note tat Γ for tis model, as given in (0), as exactly te same form as it does for te comparable models for te basic EOQ, te EOQ wit full backordering, and te EOQ wit partial backordering and constant β, for all of wic Γ C FT F, and for te models for te basic EPQ and te EPQ wit full backordering, for bot of wic Γ C DT F, were C (1 D/P). Recognizing tat T for te partial backordering model must be at least as large as T for te basic EPQ ( C o /(DC )) if backordering is optimal gives te bound [ C o C + βc b ] [(1 β)c l] DC βc b βc C b C o DC. After some algebra, tis leads us to te following conclusion: For te equations for T and F to give an optimal solution, we must ave β β C o C 1. (1) DC l Wit a little algebra, an alternate form of tis condition on β is C o DC (1 β)c l C. () In Appendix B we prove tat te equations in (19) and (18) give an optimal solution for te EPQ wit partial backordering and constant β if te condition on β in (1) or () is met and a FIFO policy on filling backorders is followed. It is encouraging to note tat te form of te equation for T in (19) is very similar to te equation for T for te full backordering case given in (): Determining T begins wit multiplying te value of T for te basic EPQ model by a term tat reflects te relative sizes of te unit inventory cost per year and te unit backorder cost per year, altoug te backordering cost component as been multiplied by β to reflect te fact tat only a percentage of te stockouts will be backordered. Tis initial value of T is ten reduced by a term tat reflects te relative cost of aving a unit of demand result in a lost sale to te cost of aving tat unit of demand eventually satisfied, eiter from inventory or by being backordered. Similarly, te equation for F in (18) is logical in tat it reflects te relative sizes of te cost of not filling a unit of demand from stock and te cost of filling a unit of demand, weter immediately from stock or eventually by being backordered. It is also interesting to note tat, wit te replacement of C for C and C b for C b, te equations for T and F for te EPQ wit partial backordering given in (19) and (18) and te condition for determining te minimum value of β for wic backordering is optimal given in (1) and () are identical to te equations for T and F for te EOQ wit partial backordering in Pentico and Drake [1], reproduced ere in (6) and (5) and te condition on β reproduced in (7) and (8). Te procedure for determining te optimal values for T, F, Q, I, S, and B is, ten: 1. Determine β, te critical value for β, from (1).. (a) If β β, determine T from te basic EPQ model (T C o /(DC (1 D/P))) and determine te optimal cost of allowing no stockouts (Γ C o C D(1 D/P)). Compare tis wit te cost of losing all demand, C l D, to determine weter it is optimal to allow no stockouts or all stockouts. (b) If β > β, use (19) to determine te value of T and substitute it into (18) to determine te value of F.

9 63 David W. Pentico et al. / Omega 37 (009) Determine te values of te oter variables and cost as follows: Total demand during a cycle DT. Maximum inventory I F DT (1 D/P). Maximum stockout S Dt 1 (1 F )T D(1 βd/p). Maximum backorder B βs. Order quantity Q DT [F + β(1 F )]. Average total cost per period C DT F. 8. Numerical example To illustrate te procedure, we will use te numerical example from Mak [6]. However, te value of β tat Mak uses (β 0.75), wile acceptable wit a LIFO policy, does not meet te criterion for te optimality of partial backordering under a FIFO inventory policy. As a result, we use a different value of β and, tus, obtain different results. Te remainder of te parameter values for Mak s example are D 1100 units per year, P 900 per year, C o $75 per setup, [ T C o C + βc ] b DC βc b [(1 β)c l] βc C b For β 0.50:. Since β < β , compare te cost of te basic EPQ model wit te cost of not stocking at all. Using te EPQ model: T C o /(DC (1 D/P)) (75)/((1100)(.00)(1 1100/900)) Γ C o C D(1 D/P) (75)(.00)(1100)(1 1100/900) Since te cost of not stocking at all as an annual cost of C l D(4.00)(1100)4400, te optimal strategy for β 0.50 is to use te basic EPQ model wit T and Q DT (1100)(0.5387) 586. For β 0.90:. Since β > β , proceed to Step First, determine C b 3.0( β) 3.(1 ( )(0.90)) 3.0( ) 3.0(0.894) Using (19), [ ] (75) (0.90)(.8557) (1100)(1.761) (0.90)(.8557) ((0.8393)(1.685) ) [(1 0.90)(4.00)] (0.90)(1.761)(.8557) C $.00 per unit per year, C b $3.0 per unit per year, C l $4.00 per unit. Te first step is to determine te values of C and C b : C C (1 D/P).00(1 1100/900).00( ) C b C b(1 βd/p) 3.0( β), wic cannot be evaluated furter until we ave a value for β. 1. From (0) determine β 1 1 C o C DC l (75)(1.761) (1100)(16.00) Using (18), F (1 β)c l + βc b T T (C + βc b ) (1 0.90)(4.00) +[(0.90)(.8557)(0.6657)] (0.6657)( (0.90)(.8557)) Te values of te oter decision variables are: Total demand during a cycle DT (1100)(0.6657) 73.7 Maximum inventory I F DT (1 D/P) (0.73)(1100)(0.6657)(1 1100/900) Maximum stockout S (1 F )DT (1 βd/p) (1 0.73)(1100)(0.6657)(0.894) Maximum backorder B βs (0.90)(175.13)

10 David W. Pentico et al. / Omega 37 (009) Order quantity Q DT [F + β (1 F )] 73.7[ (0.90)(1 0.73)] 73.7[ ] (73.7)(0.973) Te average cost per period C DT F (1.761)(1100)(0.6657)(0.73) Conclusions and future work As noted by previous autors in te context of te EOQ model, determining te optimal ordering and stockout quantities wen demands tat cannot be filled from stock are partially backordered is muc more complicated tan for te cases in wic all stockouts are eiter backordered or result in lost sales. As sown ere, te same is true for te EPQ model. However, by canging te decision variables from Q, te order quantity, and S, te stockout level, to T, te time between orders, and F, te fill rate, we ave developed a model wit equations tat are more like tose for te basic EPQ model and its full backordering extension and are muc easier to solve tan te equations developed by Mak [6]. Tere are several possible extensions to our EPQ model. Te most obvious one is te relaxation of te assumption of a constant backordering rate. In many instances it is likely tat a larger percentage of customers would be willing to wait for a backordered product as te start of te next production run gets closer in time. Tis can be captured by incorporating a time-dependent backordering rate. Several EOQ models, suc as tose of San Jose et al. [11 14], ave been developed wit tis time-varying rate, but to our knowledge tere are no EPQ models for tis situation. Anoter possible extension is to include a fixed backordering carge in addition to or instead of our existing cost tat depends on te lengt of te backorder. Appendix A. Derivation of te subinterval lengts in Fig. 4 Interval 1, wit lengt t 1, extends from wen te net inventory level first becomes 0 until it reaces te maximum backorder level, B. During tis interval te backorder is accumulating at a rate of βd per year, so B βdt 1. Production starts at te beginning of interval, wic as lengt t. During tis time te backorder level is being reduced. However, since demands continue to come in and cannot be filled until all te backorder as been satisfied, te net inventory increases (and te backorder decreases) at a rate of P βd. Interval 3, wic as lengt t 3, begins wen te backorders ave finally been eliminated and continues until te full order quantity, Q, as been produced and te inventory reaces its maximum level, I. As in te EPQ wit full backordering model, te inventory level is increasing at a rate of P D during tis interval, so I (P D)t 3. Interval 4, wic as lengt t 4, begins wen te inventory level reaces its maximum and production stops. During tis interval te inventory level is decreasing at a rate of D. Wit tese considerations in mind, we can determine te values of t 1, t, t 3, and t 4 in terms of te parameters and decision variables. From te beginning of interval 3 to te end of interval 4, all demand is filled from stock, so t 3 + t 4 FT and te total amount demanded and produced during tis time is FTD, all of wic is produced during interval 3. Tus, t 3 FTD/P. At te end of interval 3 te inventory level is I, wic is te amount produced during interval 3 minus te amount demanded during interval 3. Tus I FTD t 3 D D(FT FTD/P ) FTD(1 D/P). Since t 4 FT t 3, t 4 FT FTD/P FT(1 D/P). Intervals 1 and complement intervals 3 and 4, so t 1 +t (1 F)T. During interval 1, no production is taking place and no demands are filled, so, as noted above, te backorder grows from 0 to its maximum value, B βdt 1. During interval, tis backorder will be eliminated. Assuming a FIFO policy on filling demand, te time required to eliminate te initial backorder is βdt 1 /P. During tis time, additional demands will be coming in, a fraction β of wic will be backordered, and te time to eliminate tis second amount of backorder is βd(βdt 1 /P )/P (βd/p) t 1. Similarly, additional demands will be coming in during tis time, a fraction β of wic will be backordered, and te time to eliminate tis tird amount of backorder is βd((βd/p) t 1 )/P (βd/p) 3 t 1. Continuing in tis fasion, we find tat [ ( βd βd t t 1 P + P t 1 [ βd/p 1 βd/p ) ( ) βd 3 ( ) βd ] P P ].

11 634 David W. Pentico et al. / Omega 37 (009) Tus, [ t 1 + t t βd/p ] (1 F)T, 1 βd/p wic gives t 1 (1 F)T(1 βd/p) and t (1 F)T(βD/P). Appendix B. Proof of te optimality of te solution in (19) and (18) if (1) and () old Altoug te cost function in (14) is not convex, we can prove tat te solution given by simultaneously solving (19) and (18) is a global optimum if β meets te condition in (1) or () by examining te caracteristics of te partial derivatives and te boundary conditions. Te cost function in (14), Γ(T, F ) C o T + C DTF + βc bdt (1 F) + C l D(1 β)(1 F), can be rewritten as Γ(T, F ) G 0 T + T(G 1F G F + G ) G 3 F + G 3, were G 0 C o, G 1 D(C + βc b )/, G DβC b /, G 3 C l D(1 β). Note tat all te G i s are positive and G 1 >G. For ease of notation, we will rewrite (B1) as Γ(T, F ) G 0 T were r(f) G 1 F G F + G and q(f) G 3 F + G 3. (B1) + Tr(F ) + q(f), (B) (B3) (B4) Our objective is to establis te conditions under wic Eq. (B) as a unique interior minimizer. Differentiating (B) wit respect to T yields Γ T G o T + r(f), wic equals zero if and only if T satisfies T T G 0 (F ) r(f). (B5) Note tat tis is te same result, wit appropriate canges of notation, given in (16). Since te discriminant of r(f) is negative, r(f) as no roots. Tus, r(f) is eiter all positive or all negative on its entire domain. Since r(0)g > 0, r(f)is strictly positive in [0,1]. Tus, (B5) gives, for eac F, a unique T T (F ) tat minimizes te cost function given by (B). Substituting te expression for T (F ) in (B5) into Γ(T, F ) given by (B) gives ˆΓ(F ) : Γ(T (F ), F ) G 0 r(f) + q(f), (B6) wic represents tat minimal possible cost for eac value of F. Note tat ˆΓ(F ) is continuous, so on te compact interval [0,1] it as one or more local minima, te smallest of wic will be te global minimum of te cost function. To find tese minima, take te first and second derivatives of ˆΓ(F ) wit respect to F, yielding ˆΓ (F ) G 0 r (F ) r(f) 1/ + q (F ), ˆΓ (F ) (B7) G0 [r (F )r(f ) (r (F )) ] r(f) 3/, (B8) respectively. Note tat ˆΓ (F ), wic is, wit te cange in notation, te same as Γ(T, F )/ F as given in (17), is continuous and satisfies ˆΓ (0)<0 (since r (0) G < 0, r(0)g > 0, and q (0) G 3 < 0). Moreover: ˆΓ (1) r (1) G 0 r(1) 1/ + q (1) (G 1 G ) G 0 G 3 G1 G G 0 (G 1 G ) G 3 C 0 DC C ld(1 β), wic means tat ˆΓ (1)>0 if and only if C o /DC >C l(1 β)/c. (B9) Note tat tis is te strict inequality part of te condition on β given in (). Finally, te second derivative ˆΓ (F )

12 David W. Pentico et al. / Omega 37 (009) given in (B8) factors into ˆΓ (F ) G G0 r(f) 3/ (G 1 G ), wic is positive for all F. It now follows from elementary calculus tat, if (B9) olds, ten ˆΓ(F ) as a unique minimizer in te open interval (0,1); wile if (B9) does not old, te minimizer will lie on te boundary point F 1. Note furter tat if ˆΓ (1) 0, wic is te equality part of te condition on β given in (), ten te solution given by (19) and (18) is identical to te solution at te boundary point F 1, wic is te basic EPQ solution. Tus, if te condition on β given by (1) or () olds, te solution given by (19) and (18) minimizes te cost function given by (14). Appendix C. Te optimal policy for LIFO filling of backorders Te basic difference between te model developed in tis paper and te one in Mak [6] is te assumption about ow backorders are filled during Interval in te order cycle, wen te backorders ave reaced teir maximum level B and production starts. As discussed and sown in Fig. 3, our model assumes a FIFO policy on filling te backorders. Tat is, te backorders are filled before te new demands. As sown in Fig., Mak assumes (1) a LIFO policy in wic incoming orders are filled first and te backorders are only filled from te excess production, and () tat none of tose backorders will convert into lost sales. In tis Appendix we sow ow tis affects te values of t 1, t, t 3, t 4, B and I and, because of tis, canges te total cost function Γ(T, F ) to be minimized. We will ten sow ow tis cange in te cost function canges te equations for T and F and te condition on β tat must be satisfied for partial backordering to be optimal. In order to eliminate any possible confusion about te meaning of F,we redefine it as te fraction of te cycle during wic tere is inventory. Tis means tat, as sown in Fig. 4, FT t 3 + t 4. C.1. Te Values of t 1, t, t 3, t 4, B and I Referring to Figs. and 4 and te reasoning in Appendix A, it is obvious tat te equations for t 3, t 4, and I are exactly te same for LIFO and for FIFO: t 3 FTD ( P, t 4 FT 1 D ), P ( I DTF 1 D ). (C1) P Using te same reasoning, B is still given by B βdt 1 but, due to te LIFO policy, t B/(P D) βdt 1 /(P D). Since t 1 + t (1 F)T,weave (1 F)T(P D) t 1, P D(1 β) β(1 F)TD t P D(1 β), B β(1 F) TD(P D). (C) (P D(1 β)) C.. Te revised cost function Recognizing tat lost sales can only occur during Interval 1 of te cycle, te cost function is Γ(T, F ) C o T + C I + C b B + C l D(1 β) t 1 T. (C3) Replacing t 1, I, and B by teir expressions from (C1) and (C), we get Γ(T, F ) C o T + C DTF ( 1 D ) P + βc bdt (1 F) (P D) (P D(1 β)) + C ld(1 F)(1 β)(p D). (C4) P D(1 β) Taking te partial derivatives of Γ(T, F ) wit respect to T and F, setting tem equal to 0, solving for T as a function of F and F as a function of T, and ten replacing F in T(F) by te expression for F(T),as was done in te development of te FIFO model in te paper, we get T Co [ C ] + βc b DC [(1 β)c l] βc b βc C, b F (1 β)c l + βc b T T (C + βc, (C5) b) were Co C o(1 D(1 β)/p ) and C C (1 D(1 β)/p ). Wit te cange from C o, C, and C b in te FIFO model to Co, C, and C b ere, tese are exactly te same as te expressions in Eqs. (19) and (18). Again following te same reasoning as in te development of te FIFO model, we can develop a condition on β similar to tat in inequality (1) for te FIFO model. For optimal backordering to be optimal wen a LIFO policy is used for filling backorders, β must

13 636 David W. Pentico et al. / Omega 37 (009) satisfy te following condition, wic is equivalent to te one in Mak [6]: C β β 1 o C. (C6) DC l Wile (C6) can be used to test weter any specific value of β is large enoug for te equations in (C5) to give an optimal solution, using it to determine te value of β requires a searc process since β appears in te formula on te rigt side of te inequality as part of Co and C. After going troug some algebra, we can transform (C6) into an inequality tat can be used directly to determine te value of β : β T EPQ PC 1 PC l + T EPQ DC PC l T EPQ C (P D), (C7) PC l + T EPQ DC were T EPQ is te optimal cycle time for te basic EPQ model wit no backordering. A proof tat te values of T and F given by (C5) are optimal if a LIFO policy is followed and β is at least as large as β given by (C7) follows te same outline as in Appendix B if a FIFO policy is used. References [1] Pentico DW, Drake MJ. Te deterministic EOQ wit partial backordering: a new approac. 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