Scott E. Grasman 1, Zaki Sari 2 and Tewfik Sari 3

Transcription

1 RAIRO Operations Research RAIRO Oper. Res. 41 (27) DOI: 1.151/ro:2731 NEWSVENDOR SOLUTIONS WITH GENERAL RANDOM YIELD DISTRIBUTIONS Scott E. Grasman 1, Zaki Sari 2 an Tewfik Sari 3 Abstract. Most systems are characterize by uncertainties that cause throughput to be highly variable, for example, many moern prouction processes an services are substantially affecte by ranom yiels. When yiel is ranom, not only is the usable quantity uncertain, but the ranom yiel reuces usable capacity an throughput in the system. For these reasons, strategies are neee that incorporate ranom yiel. This paper presents the analysis of the newsvenor moel with a general ranom yiel istribution, incluing the erivation of the optimal orer quantity. Results are shown to converge to the basic newsvenor moel for the case of perfect yiel, an are further emonstrate using the case of general multiplicative ranom yiel. Results have significant impact on both manufacturing an service sectors since the newsvenor moel applies to many real-worl situations. Keywors. Newsvenor moel, operations management, planning an control, ranom yiel. Mathematics Subject Classification. 9B. 1. Introuction an literature review Many moern prouction processes an services are substantially affecte by ranom yiels. Semiconuctor manufacturers, for instance, often have yiels below 5% an plastics proucers often run at aroun 75% (see, for example, Gilbert Receive October 9, 25. Accepte May 25, EMSE, University of Missouri - Rolla, Rolla, Missouri , USA; grasmans@umr.eu 2 Laboratoire Automatique e Tlemcen, Faculté es Sciences e l Ingénieur, Université Abou Bekr Belkaï e Tlemcen, Algeria; z sari@univ-tlemcen.eu 3 Mathematics, University of Mulhouse, France; t.sari@uha.fr c EDP Sciences, ROADEF, SMAI 27 Article publishe by EDP Sciences an available at or

2 456 S.E. GRASMAN ET AL. an Emmons [4], Bohn an Terwiesch [1], an Grasman et al. [6]). Many of these processes are new an prouction yiels are well below the ieal. As more is learne, yiel increases, but will never reach 1 percent or even come close in some cases. Rapily changing technologies may become obsolete before the process is well unerstoo, or it may not be financially justifiable to correct the yiel problem; therefore, in these situations, methos are neee to cope with uncertainty. Yano an Lee [12] points out that while the ultimate goal is to make yiels perfect, ranom yiel moels are valuable for many reasons. First, results can be use in the short term to help an operation run more efficiently so that efforts can be focuse on yiel improvement. Secon, process improvement an supplier ecisions can be assesse more accurately an effectively if system effects of these ecisions are moele accurately an optimize. Thir, moels can assist in capacity planning ecisions when ranom yiel is expecte to be a long run concern. Ranom yiels are not confine to manufacturing inustries. Service inustries, for example, hotels, airlines, an healthcare facilities, incur large costs ue to the ranom nature of confirme no-shows not meeting appointments. Consier a hotel, which has a given number of rooms available on a specific ate, an airline, which has a given number of seats available for a specific flight, or a healthcare facility, which has a given number of available appointments. Some customers will not show up an also not cancel their reservations/appointments. The question is how many reservations to take - on one han there is a penalty associate with unerbooking, an, on the other han, there is a penalty associate with overbooking. These types of problems may be consiere newsvenor problems with ranom yiel. The subject of process yiels has receive consierable attention, which is not surprising consiering its significance across manufacturing an service inustries. Numerous papers have been written regaring prouction/inventory moels with ranom yiel; however, few eal with the generalize system presente here. Yano an Lee [12] provies a comprehensive review of the literature on quantitativelyoriente approaches for etermining lot sizes when prouction or procurement yiels are ranom. Issues iscusse relate to the moeling of costs, yiel uncertainty, an performance. Grosfel-Nir an Gerchak [7] provies a review of recent avances in moels, analytic results, an insights pertaining to ranom yiel prouction environments. Khouja [1] buils a taxonomy of newsvenor-relate literature incluing extensions to ranom yiels. Although many classical moels, incluing newsvenor moels, provie optimal results for a wie variety of problems, they cannot be irectly applie to systems with ranom yiel. Existing classical prouction approaches may be moifie to consier average yiel rates by simply ajusting the prouction rates an normalizing the prouction costs per unit to account for average yiel losses, but this approach is insufficient. Such approaches o not capture the fact that, in the presence of ranom yiel, if the inventory level falls below a critical level, the amount prouce epens on yiel variability (see, Henig an Gerchak [8]); thus these approximations yiel optimal results only if the yiel is eterministic. If the

3 NEWSVENDOR MODELS WITH GENERAL RANDOM YIELD DISTRIBUTIONS 457 quantity starte is etermine simply by allowing for average yiel rate, eman will only be met a fraction of the time an large shortage costs will generally be incurre. Bollapragaa an Morton [2] iscuss these approaches at length. In the authors opinion, the most closely relate paper is Gerchak, Vickson an Parlar [3], which presents a ynamic version of the problem an proves certain structural results. The main ifference is that yiel is represente using a multiplicative factor, Y Q = QU, whereu is a nonnegative ranom variable. This assumption is a special case of the results presente in this paper an is highligheinsectionfour. AlthoughGerchak,Vickson an Parlar exten their moel to allow for the initial inventory to be positive in a multiperio moel, they again o not consier general ranom yiel. It shoul be note that allowing for initial inventory oes not allow the conitions for myopic solutions to be satisfie; therefore, a challenging extension to this paper woul be to aress this issue in the context of general ranom yiel. Thus, this paper presents the analysis of the newsvenor moel with a general ranom yiel istribution, an is organize as follows. Section Two provies an overview of ranom yiel istributions, Section 3 provies the formulation of the newsvenor moel with general ranom yiel an emonstrates the convergence to the basic newvenor solution for the case of perfect yiel. Section Four further emonstrates the moel for general multiplicative yiel. Finally, Section Five provies conclusions an extensions. 2. Overview of ranom yiel istributions The most important aspect of ranom yiel prouction moels is the istribution of the yiel. Yiel istributions may or may not be inepenent of the prouction quantity. It is important to note that the yiel rate may be inepenent of the prouction quantity, however, the number of efects prouce (or the actual yiel) is almost certainly epenent on the prouction quantity. Various ranom yiel istributions are now etaile. Deterministic: A known fraction of the prouction quantity is efective. Heyman an Sobel [9](p. 79) have shown that problems with eterministic yiel may be moele as perfect yiel systems. Multiplicative: A ranom multiple, m, of the prouction quantity, Q, is efective. Thus, Y = Qm where, m follows a given ranom istribution. Also calle stochastically proportional common-cause yiel, the istribution of the fraction of goo items prouce is specifie (an multiplie by Q to get the number of goo units). This approach allows for specification of both the mean an variance of the fraction of goo items, but forces the yiel rate to be inepenent of the number prouce. This moeling approach applies when large batches are prouce. Thus, for a uniform multiplier where m Uniform[,1], Y = Qm, E[Y ]= 1 2 Q an

4 458 S.E. GRASMAN ET AL. E[Y 2 ]= 1 3 Q2. Note that the most common multiplier is istribute uniformly, but shoul not be confuse with uniformly istribute yiel, where the number of goo items is equally istribute. Uniform: The number of goo items is equally istribute between an Q. Thus, P (Y Q = j) = 1 for j =, 1, 2,..., Q. Q +1 Binomial: Each item prouce has a Bernoulli istribute probability of being efective, i.e., occurrences of efects uring a prouction run are inepenent an ientically istribute. The creation of goo units is a Bernoulli process, hence the number of goo units in a batch of size Q is binomially istribute with parameters Q an p, wherep is the probability of proucing a goo unit. One avantage of this moel is that only the value of p nees to be specifie. One isavantage is that is oes not allow for specification of variance. The expecte value of the fraction of goo units prouce remains constant (p), but the variance of the fraction of goo units, p(1 p) Q, ecreases with Q. This moeling approach is appropriate for systems that are in control for long perios of time since each unit prouce may be consiere inepenent of the previous unit. Thus, P (Y Q = j) = ( ) Q p j (1 p) Q j for j =1, 2,..., Q. j A significant contrast between binomial an multiplicative yiel is inepenence of the iniviual yiels exhibite by the binomial moel, versus perfect correlation as implie by the multiplicative moel. Accoringly, in the binomial moel, the variance of the yiel rate (fraction nonefective) iminishes as batch size increases, while in the proportional moel it is not affecte by the batch size. Lot size epenent: The most common form of lot size epenent yiel is when the yiel ecreases ue to eteriorating prouction processes. If no efects are prouce when the process is in control, the shift to out of control is immeiately signale by the presence of a efect. The istribution of the fraction of goo items prouce is epenent on the number of items prouce, an requires the istribution of the time until the process goes out of control (usually exponential) or the probability that the process will go out of control after each unit has been prouce (leaing to a geometric istribution). Let β enote the probability that the process remains in control after the completion of each unit. If the time until the process goes out of control is exponential, then β =e λ µ where µ is the prouction rate an 1 λ is the mean time that the process remains in control. Then the number of units prouce until the next failure is istribute accoring to the geometric istribution. The istribution of the fraction of goo items may also increase with the number of units prouce. This may occur

5 NEWSVENDOR MODELS WITH GENERAL RANDOM YIELD DISTRIBUTIONS 459 ue to setup, start up an learning, an may be moele in the opposite manner to a eteriorating process, by assuming that the process is out of control until the first (or first n) goo units are prouce. 3. Moel formulation This section presents the moel formulation for the newsvenor moel with generalize ranom yiel. Notation use throughout the paper is presente first, followe by erivation of the expecte total cost an optimal orer quantity. The section conclues by emonstrating that the results simplify to the stanar newsvenor moel for the case of perfect yiel Notation The following notation will be use throughout the paper. X = Ranom Deman F (x) = CDF of X f(x) = x F (x) = PDF of X Q = Orer quantity Y = Ranom Yiel variable g Q (y) = PDF of Y P = Ranom Percentage of Defects π Q (p) = PDF of P C o = Penalty Cost for Overage C s = Penalty Cost for Shortage C(Q) = Expecte Total Cost 3.2. Total cost an optimal orer quantity In orer to efine the function for expecte total cost, efine sets O ={(x, y) : x y} an U ={(x, y) :x y}. Figure 1 graphically isplays the outcome space for realize yiel an eman. The expecte overage an unerage can be calculate by integrating over the sets O an U, respectively, thus C(Q) = = = O Q y= Q y= C o (y x)f(x)xg Q (y)y + [C o y g Q (y) (y x)f(x)xg Q (y)+c s y [C o U C s (x y)f(x)xg Q (y)y ] (x y)f(x)xg Q (y) y (y x)f(x)x + C s (x y)f(x)x ] y. (1)

6 46 S.E. GRASMAN ET AL. y y=x Overage O={(x,y): y > x} Ranom Yiel, Y Unerage U={(x,y): y < x} Ranom Deman, X Figure 1. Realization of yiel an eman. Taking the erivative of Equation (1) with respect to Q, we have Q Q C(Q) = y= Q g Q(y)L(y)y + g Q (Q)L(Q), where (2) y L(y) = C o (y x)f(x)x + C s (x y)f(x)x. Finally, since Q y= g Q(y)y =1 Q y= Q g Q(y)y + g Q (Q) =, Q Q C(Q) = y= Q g Q(y)[L(y) L(Q)] y. (3) C(Q) is a convex function, an an expression for the optimal orer quantity may be obtaine by setting Equation (3) equal to zero an solving for Q ;thusthe equation for Q is Q y y= Q g Q [C (y) o (y x)f(x)x + C s (x y)f(x)x Q ] C o (Q x)f(x)x C s (x Q )f(x)x y =. x=q

7 NEWSVENDOR MODELS WITH GENERAL RANDOM YIELD DISTRIBUTIONS 461 Rearranging, the equation for Q becomes = Q (C o + C s ) y= Q g Q (y) Q y= (Q y) Q g Q (y)y Q (x y)f(x)xy ( (C o + C s ) Q ) f(x)x C s. (4) 3.3. Case of perfect yiel For the case of perfect yiel, the PDF g Q (y) fory is the elta function δ Q. Thus Q Q y= Q g Q (y) (x y)f(x)xy =, an Q (Q y) y= Q g Q(y)y =1. Therefore, Equation (4) reuces to Q (C o + C s ) f(x)x C s =, which correspons to the well-known optimal orer for the basic newsvenor moel. 4. Newsvenor solutions for multiplicative yiel The previous section presents results for general ranom yiel istributions. To emonstrate the applicability, the analysis is continue in this section for the case of multiplicative ranom yiel, which is commonly foun in literature an practice (incluing Gerchak et al. [3]). While this sections assumes multiplicative ranom yiel, no assumption is mae regaring the inepenence of the multiplicative factor, p, fromthequantity, Q. Let y =(1 p)q, then [ 1 (1 p)q C(Q) = Qg Q ((1 p)q) C o [(1 p)q x]f(x)x p= + C s x=(1 p)q ] [x (1 p)q]f(x)x p. (5) Notice that Qg Q ((1 p)q) =π Q (p). Inee, since Y =(1 P )Q, wehave P(Y y) =P(P 1 y/q) = 1 1 y/q π Q (α)α = y 1 1 Q π Q(1 β/q)β,

8 462 S.E. GRASMAN ET AL. thus, g Q (y) = 1 Q π Q ( 1 y ) Qg Q ((1 p)q) =π Q (p). Q Finally, the expecte total cost may be given by [ 1 (1 p)q C(Q) = π Q (p) C o [(1 p)q x]f(x)x p= + C s x=(1 p)q 4.1. Optimal orer quantity expression ] [x (1 p)q]f(x)x p. (6) Since C(Q) is a convex function, we may fin an expression for the optimal orer quantity by setting C(Q) =. From (6) we obtain x=(1 p)q Q C(Q) = R(Q)+S(Q), where (7) Q [ 1 (1 p)q R(Q) = p= Q π Q(p) C o [(1 p)q x]f(x)x ] + C s [x (1 p)q]f(x)x p, an S(Q) = 1 p= [ (1 p)q π Q (p) C o (1 p)f(x)x C s Notice that S(Q) can be rewritten as 1 S(Q) = (C o + C s ) π Q (p) p= (1 p)q x=(1 p)q (1 p)f(x)xp C s (1 ] (1 p)f(x)x p. 1 ) pπ Q (p)p. Setting Q C(Q ) = leas to the following equation from which the optimal orer quantity Q can be etermine. 1 R(Q )+(C o + C s ) π Q (p) p= Special cases (1 p)q (1 p)f(x)xp C s (1 1 ) pπ Q (p)p = (8) Special cases of (1) inepenent yiel an (2) perfect yiel are now presente. Since perfect yiel is a special case of inepenent yiel, it is presente last.

9 NEWSVENDOR MODELS WITH GENERAL RANDOM YIELD DISTRIBUTIONS 463 Case 1. π Q (p) =π(p) If the PDF π Q (p) =π(p) ofp is inepenent of the orer quantity Q we have R(Q) = an Equation (8) reuces to 1 p= π(p) (1 p)q (1 p)f(x)xp = ( 1 1 ) C s pπ(p)p. C o + C s Integrating first with respect to p, then with respect to x we have Q f(x) 1 x/q p= (1 p)π(p)px = ( 1 1 ) C s pπ(p)p (9) C o + C s This result correspons to previous results (Shih [11], Eq. (19)). Case 2. Y = Q The case of perfect yiel can be obtaine as a particular case of Equation (9). In this case, π(p) is a Dirac Function, an 1 x/q p= Thus, Equation (9) reuces to Q (1 p)π(p)p = 1, an 1 pπ(p)p =. C s f(x)x = (1) C o + C s ( ) Q = F 1 Cs. C o + C s Again, it can be seen that Equation (1) correspons to the well-known optimal orer quantity for the basic newsvenor moel. 5. Conclusions This paper presents the analysis of the newsvenor moel with a general ranom yiel istribution, incluing the erivation of the optimal orer quantity. Results are shown to converge to the basic newsvenor moel for the case of perfect yiel, an are further emonstrate using the case of general multiplicative ranom yiel. Results have significant impact on both manufacturing an service sectors since the newsvenor moel applies to many real-worl situations, thus improving the ecision making capabilities of complex system environments. Aitionally, the integration an implementation of this work in the form of common elements which may be applie to a variety of applications with consistent managerial implications will lea to cost reuctions, increase profits, an improve

10 464 S.E. GRASMAN ET AL. quality. The results will have application in the areas of manufacturing, as well as service inustries, such as hotels, airlines, an healthcare. Extensions of the moel inclue constraine multiple proucts (see, Grasman [5]), multi-echelon systems, an others. As previously mentione, a challenging extension relates to the inclusion of initial inventory. Acknowlegements. The lea author acknowleges the Missouri Research Boar for support of research on ranom yiel systems. Aitionally, the authors acknowlege the eucational partnership supporte by the Unites States Department of State (Grant S-ECAAS-2-GR-281 PS) that mae this collaborative effort possible. References [1] R.E. Bohn an C. Terwiesch, The Economics of Yiel Driven Processes. J. Oper. Manage. 19 (1999) [2] S. Bollapragaa, an T.E. Morton, Myopic Heuristics For the Ranom Yiel Problem. Oper. Res. 47 (1999) [3] Y. Gerchak, R.G. Vickson an M. Parlar, Perioic Review Prouction Moels with Variable Yiel an Uncertain Deman. IIE Trans. 2 (1988) [4] S. Gilbert an H. Emmons, Managing a Deteriorating Process in a Batch-Prouction Environment. IIE Trans. 27 (1993) [5] S.E. Grasman, Capacitate Multiple Prouct Systems with Ranom Yiel. Working Paper. [6] S.E. Grasman, T.L. Olsen an J.R. Birge, Setting Basestock Levels in Multiprouct Systems with Setups an Ranom Yiel. IIE Trans. (to appear). [7] A. Grosfel-Nir an Y. Gerchak, Multiple Lotsizing in Prouction to Orer with Ranom Yiels: A Review of Recent Avances. Ann. Oper. Res. 26 (24) [8] M. Henig an Y. Gerchak, The Structure of Perioic Review Policies in the Presence of Ranom Yiel. Oper. Res. 38 (199) [9] D.P. Heyman an M.J. Sobel, Stochastic Moels in Operations Research, Vol. II: Stochastic Optimization. McGraw-Hill (1984). [1] M. Khouja, The Single-Perio (Newsvenor) Problem: Literature Review an Suggestions for Future Research. OMEGA Int. J. Manag. Sci. 27 (1999) [11] W. Shih, Optimal Inventory Policies When Stockouts Result From Defective Proucts. Int. J. Pro. Res. 18 (198) [12] C.A. Yano an H.L. Lee, Lot Sizing with Ranom Yiels: A Review. Oper. Res. 43 (1995)