USING A ONSTANT RATE TO APPROXIMATE A LINEARLY HANGING RATE FOR THE EOQ AND EPQ WITH PARTIAL BAKORDERING David W. Pentico, Palumo-Donaue Scool of Business, Duquesne University, Pittsurg, PA 158-18, pentico@duq.edu, 41-396-65 arl Toews, Dept. of Matematics & omputer Science, University of Puget Sound, Tacoma, WA 98416,toewsc@gmail.com, 53-879-3839 Mattew J. Drake, Palumo-Donaue Scool of Business, Duquesne University, Pittsurg, PA 158-18, drake987@duq.edu, 41-396-1959 ABSTRAT In order to gain some insigt into ow well partial ackordering models for te EOQ and EPQ wit a constant ackordering rate can approximate te performance of te comparale models wit a ackordering rate tat increases linearly wit te time until te ackorder can e filled, we conducted an experiment tat compared te costs of using te two models over a reasonale set of parameter cominations. Our asic conclusion is tat te simpler model wit a constant ackordering rate can perform virtually as well as te more complicated model as long as te critical value for te ackordering rate is positive. Keywords: EOQ wit partial ackordering, EPQ wit partial ackordering INTRODUTION Te early models for te asic deterministic economic order quantity wit partial ackordering (EOQ-PBO) developed y Montgomery et al. (1973), Rosenerg (1979), Park (198,1983), and Wee (1989), one of te models included in San José et al. (5), and te recent model y Pentico and Drake (9) made all te usual assumptions of te asic deterministic EOQ model wit full ackordering except tat tey assumed tat a constant percentage β of te demand wen tere is no stock will e ackordered, wit te remaining percentage 1 β eing lost sales. Te first models for te asic deterministic economic production quantity model wit partial ackordering (EPQ-PBO), y Mak (1987), Sarma and Sadiwala (1997) and Zeng (1), and te recent model y Pentico et al. (9), made te same assumption of a constant ackordering rate, altoug Pentico et al. assumed tat rate applied during te entire stockout interval, wile Mak, Sarma and Sadiwala, and Zeng assumed tat tere was full ackordering once production started again. Descriptions of all of tese models, and many oters, may e found in a survey y Pentico and Drake (11). Te simplest models for te EOQ-PBO and te EPQ-PBO tat do not assume tat te ackordering rate is a constant allow β to cange once during an inventory cycle. San José et al. (5) included a step function for β for te EOQ-PBO in wic tere is no ackordering at all (β = ) if te time until replenisment exceeds a and ten steps up to β = 1. Pentico et al. (11) extended teir EPQ-PBO model wit a constant ackordering rate to allow β to increase wen production starts, wic includes te models y Pentico et al (9), Mak (1987), Sarma and Sadiwala (1997) and Zeng (1) as special cases, wit Pentico et al. (9) aving te new ackordering rate te same as te original one and te oter tree papers aving β increase to 1. 669486-1
Wee and Tang (1) extended te model in Pentico et al. (11) to allow for a single cange in β at a time oter tan wen production egins. Te next simplest model type for te EOQ-PBO or te EPQ-PBO tat does not assume tat β is a constant is one in wic te ackordering rate is given y a linear function β(τ), were τ is te time remaining until te ackorder can e filled, wic is riefly discussed in Montgomery et al. (1973). More complete developments of te linear model for β(τ) are given in San José et al. (7) for te EOQ-PBO and Toews et al. (11), wic also considered te EPQ-PBO. Oter tan te linear model for β(τ) in Montgomery et al. (1973), te first models to include a ackordering rate tat increases over time were developed y Aad (1996), wo assumed tat β(τ) is eiter an exponential function or a rational function of τ. Given teir forms, in ot of tese functions β(τ) increases as τ decreases, approacing its maximum value, wic is usually assumed to e 1., wen τ =. San José et al. (6) developed a solution procedure wen β(τ) is an exponential function tat is ased on te same asic concepts as te approac tey developed for oter forms for β(τ) in San José et al. (5). Descriptions of te exponential and rational forms of β(τ), along wit te linear form and oters proposed y oter autors, are in Pentico and Drake (11). A significant prolem wit asic EOQ-PBO and EPQ-PBO models in wic β(τ) as any form oter tan a constant or a linear function of τ is tat tey do not ave a closed form solution. Solving te models for any oter form for β(τ) involves some sort of searc process, usually eiter non-linear programming or some type of iterative process tat involves a searc procedure. Tere are at least two prolems wit using tese non-closed-form solution metods. First, tey are more time-consuming and arder to automate. Second, and te primary impetus for te researc reported ere, is tat tey are more difficult for many, if not most, managers to understand. Wy te difficulty of understanding ow a model and/or its solution procedure works is a relevant issue for managing inventory can e summarized very succinctly y te following quote from Woolsey and Swanson (1975): People would rater live wit a prolem tey cannot solve tan accept a solution tey cannot understand. Here we consider te accuracy of approximating te EOQ-PBO and te EPQ-PBO wit a ackordering rate β(τ) tat is a linear function of τ, te time remaining until te ackorder can e filled, y te EOQ-PBO or EPQ-PBO wit a constant β = (1 + β )/, were β is te value of te linearly canging β(τ) at te time te stockout egins. Tis is our first step in developing approximations for te more complicated situations in wic β(τ) is eiter an exponential or rational function of τ, scenarios tat are not easily solved. NOTATION Te notation for te parameters and variales to e used, wic is asically te same as te notation used in Toews et al. (11), is given in Tale 1. 669486-
Tale 1 Symols Used and Teir Meanings Parameters D = demand per year P = production rate per year if constantly producing s = te unit selling price o = te fixed cost of placing and receiving an order p = te variale 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 ackordered for a year g = te goodwill loss on a unit of unfilled demand l = (s p ) + g = te cost for a lost sale, including te lost profit on tat unit and any goodwill loss β = te fraction of stockouts tat will e ackordered in a constant ackordering rate model β = te initial fraction of stockouts tat will e ackordered in a linearly canging ackorder rate model β(τ) = te fraction of stockouts tat will e ackordered in a linearly canging ackorder rate model τ = te time until te ackorder will e filled Variales T = te lengt of an order cycle F = te fill rate or te percentage of demand tat will e filled from stock SUMMARIES OF THE PARTIAL BAKORDERING MODELS We egin wit rief summaries of te models we will e using for te EOQ and EPQ wit partial ackordering, wic are tose used in Pentico and Drake (9) for te EOQ-PBO wit a constant β, Pentico et al. (9) for te EPQ-PBO wit a constant β, and Toews et al. (11) for te EOQ-PBO and te EPQ-PBO wit a linear function for β(τ). In all of tese models, as defined in Tale 1, T is te lengt of an inventory cycle or te time etween orders and F is te fill rate or percentage of demand filled immediately from stock. Te EOQ-PBO wit a constant ackordering rate β Tis model makes all te assumptions of te asic EOQ wit full ackordering model except it assumes tat only a given fraction or percentage β of te demand during te time tat te system is out of stock is ackordered, wit te complementary fraction or percentage 1 β eing lost sales. Te average cost per period is: Γ(T,F) = o T + DTF (1- ) + DT F + l D(1 β)(1 F) (1) As sown in Pentico and Drake (9), te equations for te values of T and F tat minimize te average cost per period are: 669486-3
T = o [(1 ) l ] D F = F(T) = (1 ) l T T ( ) only if β satisfies te condition given y Eq. (4): () (3) β > β = 1 D / ( D l) (4) Te EPQ-PBO wit a constant ackordering rate β Tis model makes te same assumptions as te model for te asic EOQ-PBO wit a constant β except tat it makes te usual assumption of te asic EPQ tat te delivery of te order is at a constant rate P rater tan eing instantaneous. Te average cost per period is: Γ(T,F) = o DTF + β 1 + DT( -F) + l D(1 β)(1 F) (5) T were = (1 D/P) and = (1 βd/p ). As a result, as sown in Pentico et al. (9), te equations for te values of T and F tat minimize te average cost per period are: T = o β [(1 β)l ] D β β F = F(T) = (1 β ) l βt T( β ) only if β satisfies te condition given y Eq. (8): β > β = 1 D / ( D l) (8) wic is identical to te condition given in Equation (4) if replaces. Te EOQ-PBO wit a linear function for β(τ) Tis model makes te same assumptions as te model for te asic EOQ-PBO wit a constant β except tat it assumes tat β(τ), te ackordering rate τ periods efore te time at wic te ackorder will e filled, is a positive linear function of τ. As sown in Toews et al. (11), te average cost per period is: Γ(T,F) = o + DTF + DT(1 F) ( 1 ) + l D(1 F) (9) T were ( 1 ) = and β is te initial value of β(τ) wen te stockout egins. Te 3 equations for te values of T and F tat minimize te average cost per period are: 669486-4 (6) (7)
T = F = F(T) = o [ l (1 ) / D [( 1 ) / ] l T T ( ) ] (1) (11) only if β > = 1 D /( D ). (1) l Notice tat te sutracted term in te equation for in Equation (1) is doule te sutracted term in te equation for β in Equation (4) for te model for te EOQ-PBO wit a constant β, wic makes sense since te average value of β(τ) is twice te value of β. Te EPQ-PBO wit a linear function for β(τ) Tis model makes te same assumptions as te model for te asic EOQ-PBO wit a linear function for β(τ) except tat it makes te usual assumption of te asic EPQ tat te delivery of te order is at a constant rate P rater tan eing instantaneous. As sown in Toews et al. (11), te average cost per period is: Γ(T,F) = o + DTF + DT(1 F) ( 1 ) + l D(1 F) (13) T were = (1 D/P), (1 ) (1 ) D =, and β is te initial value of β(τ) 3 4P wen te stockout egins. Te equations for te values of T and F tat minimize te average cost per period are: T = F = F(T) = only if β o [ l (1 ) / D [( 1 ) / ] l T T ( ) = 1 ] / ( l) 669486-5 (14) (15) D D (16) wic is identical to te condition given in Equation (1) if THE STUDY replaces. Our purpose is to evaluate te accuracy of approximating te EOQ-PBO and te EPQ-PBO wit a ackordering rate β(τ) tat is a linear function of τ, te time remaining until te order can e filled, y a constant β. We do tis y examining te average and worst-case performance of te approximations on a set of test prolems ased on reasonale parameter values, wic will elp
identify te conditions under wic te approximations perform less well and te conditions under wic tey can e expected to perform very well. Researc Metodology Our performance measure is te ratio of te cost of using te solution from te constant-β model to te cost of using te optimal solution for te linearly canging β(τ) model for a set of test prolems ased on reasonale values for five (six for te EPQ-PBO) situational caracteristics. Four of te caracteristics are asic prolem parameters ( o,, l /, and D). Te fift is te value of β, te initial value of β(τ), relative to, te minimum value of β for wic partial ackordering is optimal. Te sixt caracteristic added for te EPQ-PBO is P/D. All of tese caracteristics affect te values of T and F. Te values cosen for te parameters were selected to give a range of values for since tis, as we sall see, as an effect on te average performance of te euristics. For all te parameters except te l / ratio, te values used ave order of magnitude differences. For l / te larger ratio is.5 times te lower ratio. To determine te average cost ratios for te EOQ-PBO, te cost of te approximation will e otained y sustituting T and F from Eqs. () and (3) into Eq. (9). For te EPQ-PBO, te cost of te approximation will e otained y sustituting T and F from Eqs. (6) and (7) into Eq. (13). Te Test Sets For ot te EOQ and te EPQ, te values used for te first five situational caracteristics were: 1. o =.5, 5., 5: Tis affects te value of T, wit o =.5 leading toward JIT results. It also affects te value of, wit a smaller value for o resulting in a larger, wic sould improve te performance of te approximations.. =.5, 5.: F sould e lower for =.5 (since ackordering is less expensive) and iger for = 5. (since ackordering is more expensive) for te same value of T. 3. l / =., 5.: l is important in determining te value of, wit a larger l resulting in a larger approximations., wic sould result in etter performance of te 4. D =, : A larger D leads to a larger and, terefore, a narrower range for β witin wic PBO is optimal. Tus a larger D sould improve te performance of te approximations for ot te EOQ and te EPQ. 5. β relative to : Since te different cominations of te first four factors will usually give different values for large it is relative to used: β is 5% of te way etween, β for an experimental comination will e ased on ow, rater tan using fixed values for β. Tree values of β will e and 1., 5% of te way, and 75% of te way. 669486-6
Because < may lead to negative values for β, we used max(,) rater tan wen determining β s value. In all cases = 1. Tis gives 3 3 = 7 cominations of situational caracteristics for testing te approximation for te EOQ. For te EPQ comparisons, te values of te sixt caracteristic were: 6. P/D =, : Tis was included ecause of its effect on and, ot of wic are important in determining T and F, wile is important in determining. Adding tis sixt factor increases te numer of test prolems for te EPQ approximation to 144. Results for te Approximation for te EOQ-PBO wen β(τ) anges Linearly Tale summarizes te test results for te approximation of te EOQ wit a linearly canging β(τ) y te EOQ wit a constant β = (1 + β )/. Eac row sows cost ratio information roken down y te value of β relative to and 1.. For example, EOQ(.5) means tat row refers to te EOQ approximation wit β eing 5% of te way from to 1.. Looking at te first set of columns, we see tat EOQ(.5) as an average cost ratio of 1.6, wit a maximum of 1.56 and a minimum of 1. for 4 cases. Examining te individual case results, we found tat te igest cost ratios all came from te four cases for wic is negative. Tese cases also resulted in te worst cost ratios for EOQ(.5) and EOQ(.75). Te next set of columns Tale Summary of Test Results for Approximations for te EOQ All ases, n = 4 β >, n = β >.5, n = 17 Group# Avg Max Min Avg Max Min Avg Max Min EOQ(.5) 1.6 1.56 1. 1.1 1.4 1. 1. 1.1 1. EOQ(.5) 1.3 1.18 1. 1.1 1.4 1. 1. 1.1 1. EOQ(.75) 1.1 1.4 1. 1. 1.1 1. 1. 1. 1. All 1.3 1.56 1. 1. 1.4 1. 1. 1.1 1. # Tis refers to te position of β relative to β and 1.. contains te summaries of te results for te cominations of o,, l, and D for wic >. For tose cases te average ratio, ignoring te relative position of β, was less tan 1.1, wit a worst case less tan 1.4. Referring to te final tree columns, if we limit our attention to te 17 parameter cominations and 51 total cases for wic >.5, we find tat te average ratio was less tan 1.5, wit a maximum of 1.1. Tus, as long as is positive, 669486-7
we can expect te approximation to perform well, and if is at least.5, it can e expected to perform essentially as well as te EOQ-PBO wit linearly canging β(τ). Looking at te individual case results summarized in Tale in more detail, we see two tings of interest tat are not at all surprising: For any given value for, te lowest ratios occur wen te value of β is closer to 1.. Looking at te results roken down y te value of β relative to, te cost ratio is closer to 1. te closer is to 1.. Neiter of tese results is a surprise ecause tey stem from te same asic idea: Te closer β is to 1., te less cange tere will e in te value of β(τ) as τ decreases. Tus, te less difference tere will e etween a linearly canging β(τ) and a constant β. Since te average and igest cost ratios for all 7 total cases are only 1.6 and 1.56 respectively, it is not surprising tat tere is very little difference in te approximation s performance wen different parameter values are considered. 1. Grouped y o : Since a lower value of o leads to a sorter inventory cycle, te average (maximum) ratios are, as expected, lowest for o =.5 at 1. (1.1), in te middle for o = 5. at 1.4 (1.6) and igest for o = 5 at 1.15 (1.56).. Grouped y : Since a lower (iger) value of makes ackordering more (less) attractive, it sould e expected tat te approximation s performance would e etter for =.5 tan for =.5, wic it is. 3. Grouped y te l / ratio: Since a iger cost for a lost sale makes ackordering less attractive, is sould e expected tat te approximation s performance would e etter for l / = tan it is for l / =5, wic it is. Te average (maximum) ratios are 1. (1.33) for l / = and 1.5 (1.56) for l / = 5. 4. Grouped y D: As noted aove, a larger D leads to a larger and, terefore, a narrower range for β witin wic PBO is optimal. Tus a larger D sould improve te performance of te approximation, wic it does. Te average (maximum) ratios are 1.5 (1.56) for D = and 1.1 (1.6) for D =. 5. Grouped y te ratio of β to : For all cominations of te first four parameters, te lowest cost ratio was found wen β was 75 percent of te way from to 1.. For all cominations in wic is less tan.6, te igest ratio was found wen β is 5 percent of te way from to 1.. If is more tan.6, te igest ratio is found wen β is 5 percent of te way from to 1.; in all of tese cominations te cost ratio for all tree values for β is less tan 1.4, wic means tat tere is no real difference etween te cost ratios for te tree starting values wen >.6. Results for te approximation for te EPQ-PBO wen β(τ) canges linearly Tale 3 summarizes te results of te tests for te EPQ-PBO. As in Tale, eac row sows cost ratio information roken down y te value of β relative to β and 1.. Altoug te cost 669486-8
ratios for te approximation are not as good for te EPQ as tey are for te EOQ, most of te same conclusions can e drawn. Tale 3 Summary of Test Results for Approximations for te EPQ All ases, n = 48 β >, n = 4 β >.5, n = 36 Group# Avg Max Min Avg Max Min Avg Max Min EOQ(.5) 1.6 1.53 1. 1.6 1.68 1. 1.1 1.9 1. EOQ(.5) 1.15 1.11 1. 1.5 1.53 1. 1. 1.8 1. EOQ(.75) 1.5 1.7 1. 1. 1.19 1. 1.1 1.3 1. All 1.15 1.53 1. 1.5 1.68 1. 1.1 1.9 1. # Tis refers to te position of β relative to β and 1.. Breaking down te results on te asis of te values of o,, l /, D and te size of β relative to for te EPQ sows asically te same ting as it did for te EOQ, altoug te differences for te EPQ are a little igger since te average and maximum ratios for te 144 cases are 1.15 and 1.53 respectively. 1. Grouped y o : Te average (maximum) ratios are, as expected, lowest for o =.5 at 1.1 (1.9), in te middle for o = 5. at 1.9 (1.14) and igest for o = 5 at 1.36 (1.53).. Grouped y : Since a lower (iger) value of makes ackordering more (less) attractive, it sould e expected tat te approximation s performance would e etter for =.5 tan for =.5, wic it is. For =.5 te average (maximum) is 1. (1.31) versus 1.9 (1.53) for =.5. 3. Grouped y te l / ratio: Since a iger cost for a lost sale makes ackordering less attractive, is sould e expected tat te approximation s performance would e etter for l / = tan it is for l / =5, wic it is. Te average (maximum) ratios are 1.8 (1.157) for l / = and 1.3 (1.53) for l / = 5. 4. Grouped y D: As noted aove, a larger D leads to a larger and, terefore, a narrower range for β witin wic PBO is optimal. Tus a larger D sould improve te performance of te approximation, wic it does. Te average (maximum) ratios are 1.5 (1.53) for D = and 1.6 (1.14) for D =. 5. Grouped y te value of β relative to : For all 48 cominations of te first four parameters and P/D, te lowest cost ratio occurs wen β is 75 percent of te way from to 1.. Te issue of wic value of β o gives te igest cost ratio is not quite as clear for te EPQ as it is for te EOQ. For P/D =, te result is te same as it is for te EOQ: for all cominations in wic is less tan.6, te igest ratio occurs wen β is 5 percent of te way from to 1.; if is more tan.6, te igest ratio occurs wen β is 5 percent 669486-9
of te way from to 1., ut in all of tese cominations wit a iger value for te cost ratio for all tree values for β is less tan 1.3, wic means tat tere is no real difference etween te cost ratios for te tree starting values wen >.6. For P/D =, te result is almost te same: for all cominations except one in wic is less tan or equal to.6, for wic =.3675, te igest ratio occurs wen β is 5 percent of te way from to 1.; if is more tan.6, te igest ratio is found wen β is 5 percent of te way from to 1., ut in all of tese cominations wit a iger value for, te cost ratio for all tree values for β is less tan 1.41, wic means tat tere is no real difference etween te cost ratios for te tree starting values wen >.6. 6. Grouped y P/D: In addition to te results just oserved in wic te P/D ratio makes a small difference in determining wic value of β gives te largest cost ratio, we oserve te following: For all 7 cominations of te values of te first five experimental factors, te cost ratio wit P/D = is lower tan it is for te same case wit P/D =. Tis is not surprising since te effect of P/D = is to reduce te effective olding cost per unit y alf, wile wit P/D = it is only reduced y five percent. Among oter tings, te lower value of leads to a iger value for performance for ot approximations., wic typically results in etter SUMMARY AND SUGGESTIONS FOR FUTURE RESEARH In order to gain some insigt into using a constant β to approximate a ackordering-rate function β(τ) tat is ased on τ, te time until te ackorder can e filled, we ave evaluated te accuracy of approximating te EOQ-PBO and te EPQ-PBO wit a ackordering rate β(τ) tat is a linear function of τ y te comparale model wit a constant β. We do tis y examining te average and worst-case performance of te approximations on a set of test prolems ased on reasonale parameter values, wic will elp identify te conditions under wic te approximations perform less well and te conditions under wic tey can e expected to perform very well. Our asic conclusion is tat, wile differences in te situational parameter values ave some impact on te performance of te approximations, in most cases te approximations are extremely good, costing a small fraction of a percent more tan using te model wit a timeased β(τ). At worst, te cost for te EOQ approximation was less tan.6 percent iger tan te cost from te model wit a linearly canging β(τ) and for te EPQ approximation it was no more tan aout.5 percent, and tese worst-case results occurred only if was negative. Te ig quality of te results in tis experiment otained y using te simpler model, wic as a closed-form solution, suggests tat tere is great potential for using eiter a constant-β EOQ- PBO or EPQ-PBO model or te comparale model wit a linearly canging ackorder-fraction β(τ), ot of wic also ave closed-form solution equations, to do well in approximating te results from models for te EOQ-PBO and EPQ-PBO wit β(τ) aving an exponential or rational 669486-1
form, ot of wic ave β(τ) increasing in a non-linear fasion as τ decreases and ot of wic require more complicated solution procedures. Wat will make conducting experiments to evaluate tese approximations more difficult tan wat we ave done ere is te following: an important caracteristic of te EOQ-PBO and EPQ- PBO wit a linear function for β(τ) is te assumption tat it is possile to determine β, te initial value for β(τ) wen te stockout egins. Tis makes it possile to determine an average value for β(τ) over te stockout interval. Te models using an exponential or rational form for β(τ) do not make tis assumption. Instead, tey just specify ow te value of β(τ) decreases as τ increases, in ot cases gradually approacing as τ gets aritrarily large. To use eiter te linearly canging form of β(τ) or a constant β, it will e necessary to develop a metod for estimating β or te average value of β over te stockout interval. REFERENES Aad, P. L. (1996). Optimal pricing and lot-sizing under conditions of perisaility and partial ackordering. Management Science, 4(8), 193-114. Aad, P. L. (8). Optimal price and order size under partial ackordering incorporating sortage, ackorder and lost sale costs. International Journal of Production Economics, 114(1), 179-186. Mak, K. L. (1987). Determining optimal production-inventory control policies for an inventory system wit partial acklogging. omputers & Operations Researc, 14(4), 99-34. Montgomery, D.., M. S. Bazaraa, & M. S., Keswani, A. K. (1973). Inventory models wit a mixture of ackorders and lost sales. Naval Researc Logistics Quarterly, (), 55-63. Park, K. S. (198). Inventory model wit partial ackorders. International Journal of Systems Science, 13(1), 1313-1317. Park, K. S. (1983). Anoter inventory model wit a mixture of ackorders and lost sales. Naval Researc Logistics Quarterly, 3(3), 397-4. Pentico, D. W., & Drake, M. J. (9). Te deterministic EOQ wit partial ackordering: a new approac. European Journal of Operational Researc, 194(1), 1-113. Pentico, D. W., & Drake, M. J. (11). A survey of deterministic models for te EOQ and EPQ wit partial ackordering. European Journal of Operational Researc, 14(), 179-198. Pentico, D. W., Drake, M. J., & Toews,. (9). Te deterministic EPQ wit partial ackordering: a new approac. Omega, 37(3), 64-636. Pentico, D. W., Drake, M. J., & Toews,. (11). Te EPQ wit partial ackordering and pase-dependent ackordering rate. Omega, 39(5), 574-577. 669486-11
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