Consider a newsboy problem with known demand and random replenishment. The distribution of the replenishment quantity is determined by a location-scale parameter, (μ, σ), which can be controlled. The usual expected cost function includes a separate term associated with the choice of σ. The asymptotic behavior of the optimal parameter values, as functions of the holding and shortage costs, is examined

Rose, John S.

English

University of Richmond

University of Richmond UR Scholarship Repository Robins School of Business White Paper Series, 1980-2011 Robins School of Business 1990 The Newsboy with Known Demand and Uncertain Replenishment: Applications to Quality Control and Container Fill John S. Rose University of Richmond Follow this and additional works at: https://scholarship.richmond.edu/robins-white-papers  Part of the Business Commons Recommended Citation Rose, John S. 1990. "The Newsboy with Known Demand and Uncertain Replenishment: Applications to Quality Control and Container Fill." E.C.R.S.B. 90-1. Robins School of Business White Paper Series. University of Richmond, Richmond, Virginia. This White Paper is brought to you for free and open access by the Robins School of Business at UR Scholarship Repository. It has been accepted for inclusion in Robins School of Business White Paper Series, 1980-2011 by an authorized administrator of UR Scholarship Repository. For more information, please contact scholarshiprepository@richmond.edu. THE NEWSBOY WITH KNOWN DEMAND AND UNCERTAIN REPLENISHMENT: APPLICATIONS TO QUALITY CONTROL AND CONTAINER FILL JOHNS. ROSE E. C. Robins School of Business University of Richmond Richmond, VA 23173 U.S.A. (804) 289-8579 Abstract Consider a newsboy problem with known demand and random replenishment. The distribution of the replenishment quantity is determined by a location-scale parameter, (µ,a), which can be controlled. The usual expected cost function includes a separate term associated with the choice of a. The asymptotic behavior of the optimal parameter values, as functions of the holding and shortage costs, is examined. KEYWORDS Inventory; Uncertainty; Sensitivity; Newsboy; Container Fill 1. Inventory Models with Uncertain Fill Inventory models with uncertain fill have been investigated by several researchers; among them are Silver [11], Shih [10], Moinzadeh and Lee [6], Posner and Berg [8], and Henig and Gerchak [4]. Explicit integration of quality control with inventory control has been considered by Peters et. al. [7] and Lee and Rosenblatt [5]. Investigating a container fill problem in which the fill is normally distributed, Golhar and Pollock [2] assume that the mean is controllable but that the variance is not. However, they can affect the quality of the outgoing containers by setting an upper control limit - fuller containers are scrapped and refilled. Recently, Golhar and Pollock [3] computed the cost saving expected from a reduction in the variance of the fill distribution. Gerchak and Parlar [l] consider an EOQ model in which the variance of the fill distribution is a 2 Q, where Q denotes the order quantity. The proportionality parameter, a 2 , can be controlled, and they minimize the long-run average cost with respect to Q and a 2 . 2. Problem Statement Consider a newsboy problem characterized by the following elements: A= 1 known (deterministic) demand; X = fill (replenishment quantity), a random variable with absolutely continuous distribution F; c = material cost; p = shortage cost, with p > c; h = holding cost; no initial inventory; and no fixed cost. The expected cost for the period is cµ + A pf(A-x)F(dx) -DF JBF + h (x-A)F(dx), A (1) whereµ= EX and the interval (-DF,BF) = supp(F), the support of F; DF~oo,BF~ 00 • The formulation given above is a nearly complete reversal of the usual newsboy problem; see, for example, Ross [9, pp. 169-171]. Indeed, (1) is also the expected cost for a newsboy with demand distribution F, reorder quantity A, holding cost p-c, and penalty cost h+c. The optimal order, A*, is known: F(A*) h/(p+h). (2) 2 The demand, A, is treated like any other parameter, and our objective henceforth is to control F, the distribution of the fill, X. We shall restrict our choices to a family of absolutely continuous distributions, {F }, determined by a location-µ,a scale parameter, (µ,a), a>O. Letting f denote the density of F , then F (x) µ,a µ,a µ,a = F((x-µ)/a) and f (x) = f((x-µ)/a)/a for some distribution F with density f. µ,a Without loss of generality, we assume that the mean of Fis zero: B Jxf(x)dx -D 0, (3) where supp(F) = (-D,B), D~oo, B~oo. shall not explicitly be alluded to. In what follows, (3) is invoked frequently and It follows that, if X - F , then EX=µ, µ,a VarX cr a2 , and supp(F ) = (-D ,B ) = (µ-aD,µ+aB). Common examples for the µ,a µ,a µ,a types of applications we envision are the normal, uniform, and triangular distributions. We shall frequently use another parameter, 0 = (A-µ)/a, the number of "deviation" units below demand at which we set the expected fill. Ideally, we should select the degenerate distribution, FA,O, thereby exactly satisfying demand with no deviation. Unfortunately, a=O may be prohibitively expensive to achieve, so we perforce settle for less "quality." Quality, in this context, refers to the precision of F. Denote the cost associated with the choice of a by v(a), which we add to (1) to obtain the expected cost function, L(µ,a) 0 B cµ + p(A-µ)F(O) + h(µ-A)[l-F(O)] - pafyf(y)dy + hafyf(y)dy + v(a). (4) -D 0 We want to choose µ~O and a>O to minimize L. 3 Assume that the quality cost function, v(a), a~O, is strictly convex, decreasing, and nonnegative. Indeed, we allow the possibility that v(0)= 00 • Finally, we shall impose an upper bound, S, on a, even if the support of F is infinite µ,a (D =oo or B =oo) - greater variability cannot be tolerated. Assume without µ,a µ,a loss of generality that v(S)=O. 3. Examples CONTAINER FILL A company sells its product in containers of A pounds, the so-called label weight. The container filling process is subject to some degree of randomness, a. More or less independently of a, the fill process is calibrated to provide a mean weight ofµ pounds of product per container. Ifµ is set nearly equal to A, then some containers will be underfilled, subjecting the company to penalty. Such penalty could include fines from a regulatory agency, the cost of handling customer returns, and lost goodwill. To compensate, the fill process may be recalibrated to yield higher mean fill. This action results in giving away free product, at unit cost c. Another possibility is to keepµ near A but improve the quality of the fill process by reducing a. Obviously, average outgoing quality could be improved simply through the application of more rigorous sampling rules at the end of the production line, but changes in management practices and/or technological innovations might be more cost effective. In any event, to improve quality is also expensive. Note that we have made no allusion to holding cost; it is a meaningless concept for this application. It should be mentioned that this application was the motivational stimulus for the present paper. QUALITY CONTROL A product's specifications call for a value, A, on some criterion, or dimension. For any unit of product, let X denote its value on this criterion, and suppose that costs incurred from producing an item with an excess or a deficit relative to A are locally approximately linear. If the process 5 which, surprisingly, does not explicitly depend on µ 0 • If the solution to (5) is negative, setµ* 0 -- the holding cost is so great relative top that we want the mean fill to be as far below A as possible. (Because the notion of negative fill has little meaning in a bonafide single-period inventory problem, the constraint µ*~O may be easily replaced by µ*~U, say, where U is chosen, in conjunction with S, such that Fu,s(O)=F(-U/S) is appropriately small.) Let R(h,p,c) denote the right hand side of (6). If there is no solution to (6), then either -v'(S) ~ R(h,p,c), or -v'(O) 5 R(h,p,c). In the former case, L2 (µ,a)<O, so let a*=S, with µ*=µ 0 , if µ 0 >0 in (5) with a 0 =S, and µ*=0 otherwise. If (9) holds, then L2 (µ,a)~O, so let a*=O and (8) (9) µ*=A. Then, (µ*,a*) minimizes L. Essentially, what (8) ((9)) says is that quality is dear (cheap) relative to the expected deviation costs, so we allow maximum variability (eliminate variability altogether) in the fill. Note the local behavior ofµ*: µ*>(<)A only if [1-F(O)]h < (>) pF(O)-c. (10) The inequality indicates which error is more serious. Also, note the implications of a linear quality cost function. If v(a)=a(S-a), for some a>O, then either (8) or (9) holds and we should select either the most variable fill or fill exactly at A. The remainder of the paper is devoted to the asymptotic behavior of the optimal solution. 5. Asymptotic Behavior of Optimal Solution 1) AS FUNCTION OF h, WITH B = ~ Because of the constraint, µ*~D, the behavior ofµ* and L(µ*,a*) ash increases requires a fair amount of work, especially if there is no upper bound on the support of F. By considering this case first, we should have an easier time with the subsequent analysis. First, IGNORE THE CONSTRAINT THATµ*~ 0 and consider the extreme point, (µ 0 ,a 0 ), as a function of h. To mitigate against rising holding costs, it seems reasonable to let a 0 • 0, unless perhaps the quality cost becomes prohibitive, in which case we might let µ 0 decrease, which in turn would yield larger shortage costs. It turns out that indeed a 0 • 0, but the limiting behavior of µ 0 depends on 6 the behavior of v near the origin. Consequently, we shall suppose v varies regularly at the origin -- definitions and the necessary results are presented in the Appendix. From (5), we have lim 9° = B = oo and 89°/8h = (p-c)/f(9°)(p+h) 2 > 0. Then, h• oo -9°[l-F(9°)] + bf(y)dy > 0, so R(•,p,c) is increasing. 90 l'Hospital's rule to obtain lim R(h,p,c) h• oo (p-c)B 00 Apply (11) Suppose -v'(O)<oo, which, of course, implies that v(O)<oo. Then, (9) holds for large h, so a*=O and µ*=A, a reasonable result. Quality is cheap compared to the holding cost, so X = A w.p.l; and the total expected cost is constant with respect to h. Hereafter, assume that -v'(O) = oo, which does not necessarily imply that v(O)=oo. Then, (6) is soluble and lim -v'(a 0 ) h• oo oo, from (11). Also, from (6), we have -v' '(a 0 )aa 0 /ah = R1 (h,p,c) > 0. From the convexity of v, a 0 is decreasing, and we have established a 0 i O as ht oo (12) Suppose now that v(O) < oo. Then, vis slowly varying at the origin, and we easily obtain some intuitively appealing results. From (5) and (6), -v'(a 0 ) = R(h,p,c) > (p+h)0°[l-F(0°)] = (p-c)0°, or -a 0 v'(a 0 ) > (p-c)(A-µ 0 ). From (A.l) and (12), -a 0 v'(a 0 ) • 0. It follows from (10) that From (7) we have lim µ 0 h • oo A. lim L(µ 0 ,a 0 ) h• oo cA + v(O). Obviously, µ*=µ 0 and a*=a 0 for large h, and the analysis is complete. (13) (14) Hereafter, we shall also assume that v(O)=oo. Although v(a 0 )too, we find that v(a 0 ) is small relative to h, i.e., v(a 0 )=o(h). Indeed, lim v(a 0 )/h h• oo lim v' (a 0 ) aa0 /ah=lim[ -v' (a 0 ) /v" (a 0 ) ]R1 (h, p, c). Referring to the paragraph leading h• oo h • oo to (11), we see that lim R1 (h,p,c) h • oo lim[Jy;(y)dy x(l-F(x))]=O, so it remains X• OO X to bound -v' (a 0 ) /v" (a 0 ). If - p is the exponent of v, then x- p lim v(xt) /v( t) t • O = lira xv'(xt)/v'(t), so v' is regular with exponent -(l+p). Write -v'(x) t • O x- cl+ P, M(x), where M is slow. Then, -v" (x) /v' (x) = x- 1 [ l+p-xM' (x) /M(x)] • oo as x + 0. Obviously, the inverse tends to zero, and we have established that lira v(a 0 )/h 0. (15) h• oo Next, it would be interesting to know what proportion of the expected cost is due to the improvement in quality. Indeed, from (7), (12), and (A.2), lim L(µ 0 ,a 0 )/v(a 0 ) h• oo 1 + P, (16) 7 where-pis the exponent of v. If vis slow, then p=0 and, in the limit, v accounts for 100% of the total cost. If p>0, then there is a fraction, p/(l+p), of the cost 8 due to the deviation from demand, A. We shall interpret this fraction after giving some consideration to the behavior of µ 0 • To this end, assume that lim f(x)/(1-F(x)] > 0. (17) If Fis nondecreasing failure rate (NDFR), at least on some right hand half-line, then (17) will be satisfied. Next, let K lim -xv'(x), x• O (18) where O~K~00 • The existence of the limit follows from the convexity of v. Examples are given in the Appendix. Applying (17) gives lim x[l-F(x)J/hf(y)dy=lim [1-[l-x F(x)]/xf(x)]=l. Hence, l=lim 8°[l-F(B 0 )J/hf(y)dy h• ro 90 so (p-c)(A-µ 0 ) • Kor lim µ 0 h• ro A - K/(p-c). lim (p-c)(A-µ 0 )/(-a 0 v'(o 0 )], h• ro (19) For the moment, think of Kasa measure of how steeply v grows at the origin. Then, in the limit, µ 0 is decreasing with K. It's a compromising result. We want o to vanish, in order to control holding costs. But, if a+ 0 too fast, then v too rapidly, so we letµ drift below demand, A, thereby saving somewhat on quality while still containing the excess. Such "saving" is not free, however; a (net) shortage cost, (p-c)(A-µ), is incurred. Indeed, we can now explain p>O in (16). From (A.2), p = lim -xv'(x)/v(x), so K= and µ 0 • -co x• O preceeding (19), we have lim (p-c)(A-µ 0 )/v(a 0 ) h• oo p. From the argument immediately (20) 

The Newsboy with Known Demand and Uncertain Replenishment: Applications to Quality Control and Container Fill

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The Newsboy with Known Demand and Uncertain Replenishment: Applications to Quality Control and Container Fill

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