A New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem

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1 This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: lgtclli@polyu.edu.hk SHORT COMMUNICATION November 2007 Revised April 2008

2 Abstract We cosider the fiite-horizo discrete-time ecoomic order quatity problem. Kovalev ad Ng (2008) have developed a solutio approach for solvig this problem. Their approach requires a search for the optimal umber of orders, which takes O(log ) time. I this ote, we preset a modified solutio method which ca determie the optimal solutio without the eed of such a search. Keywords: Combiatorial optimizatio; EOQ; ivetory maagemet

3 1 Itroductio We cosider the followig discrete versio of the fiite-horizo Ecoomic Order Quatity (EOQ) problem: There is a fiite plaig horizo with discrete time periods 1, 2,...,. Demad occurs at a costat rate of λ > 0 uits per period. Orders ca be placed at the begiig of ay period. Whe a order is placed, repleishmet takes place istataeously before the demad of that period is filled. A fixed orderig cost c > 0 is icurred whe a order is placed, ad a ivetory holdig cost h > 0 is icurred if a uit of the item is carried from oe period to the ext. The iitial ivetory at the begiig of period 1 is zero. We would like to determie the order quatity i each time period so that the total orderig ad holdig cost is miimized. We deote this problem as P. Problem P differs from the classical EOQ problem i that the plaig horizo is fiite ad demad occurs i discrete time periods. Problem P is also a special case of the classical dyamic lot size problem, which has time-depedet demad ad cost parameters. Kovalev ad Ng [1] have developed a algorithm for problem P. Their algorithm requires a search for the optimal umber of orders i the plaig horizo which takes O(log) time. I this ote, we preset a modified solutio method which does ot require such a search. Note that problem P is defied slightly differetly from that of Kovalev ad Ng [1], who have assumed that a holdig cost h is icurred if a demad uit is satisfied by a order received i the same time period. However, this differece is very mior, because the objective fuctio of Kovalev ad Ng s model differs from that of problem P by oly a costat amout of hλ. By the zero-ivetory property of the classical dyamic lot size problem (see, e.g., [4], page 82), it suffices to cosider the case i which orders are placed ad received oly whe the ivetory level reaches zero. Thus, the size of each order must be a multiple of λ. Let k deote the umber of orders placed durig the plaig horizo. Let s i = λx i deote the size of the ith order, where x i is a positive iteger (i = 1, 2,..., k). The total quatity ordered, k i=1 s i, is equal to λ, which implies that k i=1 x i =. Sice the ith order is used to satisfy the demad of the curret period as well as the demad of the ext x i 1 periods, the holdig cost of the items i this order equals 1

4 hλ [ (x i 1) + (x i 2) ] = hλ 2 x i(x i 1). Hece, i problem P, the total orderig cost is ck, ad the total ivetory holdig cost is hλ k 2 i=1 x i(x i 1). Sice hλ k 2 i=1 x i = hλ 2, which is a costat, problem P reduces to the followig discrete optimizatio problem: miimize ck + hλ 2 subject to k i=1 k x i = i=1 x 2 i k = 1, 2,..., x i = 1, 2,... for i = 1, 2,..., k, where k, x 1, x 2,..., x k are decisio variables. We deote this problem as P. (Note: P is the same as problem D i [1].) Kovalev ad Ng s [1] solutio method ca be described as follows: First, they have show that if k is give, the the optimal decisio is to place r orders of size λ( k 1) ad k r orders of size λ k, where r = k k. (The sequece of these k orders has o impact o the total cost.) This implies that the objective fuctio of problem P ca be rewritte as f(k) = ck + hλ 2 { ( ) 2 } 2 r 1 + (k r), k k or equivaletly, f(k) = ck + hλ 2 { 2 } (2 + k) k. (1) k k They have also show that fuctio f is covex over the iteger domai {1, 2,..., }. They the propose a three-sectio search procedure for determiig k {1, 2,..., } which miimizes f. 2 Determiig the Number of Orders To preset our ew solutio method, we first defie 2 g(y) = (2 + y) y y y 2

5 for y (0, + ). Cosider ay oegative iteger α. For simplicity, we deote α = + if α = 0. Whe α+1 y < α, we have α < y α + 1, which implies that g(y) = (2 + y)(α + 1) y(α + 1) 2, or equivaletly, g(y) = α(α + 1)y + 2(α + 1). (2) Hece, fuctio g is liear withi the iterval [ α+1, α). I additio, for α = 1, 2,..., lim (2 g(y) = + ) (α + 1) ( y (/α) α α (α + 1)2 = 2 + ) α ( α α α2 = g. α) Therefore, g is a cotiuous piecewise liear fuctio. Kovalev ad Ng [1] have show that fuctio g is covex whe the domai of the fuctio is restricted to the iteger set {1, 2,..., }. We ca exted this covexity property to the domai (0, + ). Note that equatio (2) implies that the slope of g(y) is α(α + 1) whe α+1 y < α for α = 0, 1, 2,.... Hece, the slope of g(y) decreases as α icreases, which implies that the slope of g(y) is odecreasig as y icreases. This i tur implies that g is covex over the iterval (0, + ). Recall that the objective fuctio of problem P is f(k) as defied i (1). We ow exted the defiitio of fuctio f to cover the domai (0, + ). Thus, f(y) = cy + hλ 2 { 2 } (2 + y) y = cy + hλ y y 2 [ g(y) ] for y (0, + ). Because g is a cotiuous piecewise liear covex fuctio, f is also a cotiuous piecewise liear covex fuctio. The slope of f(y) equals c hλ 2 α(α + 1) whe α+1 y < α for α = 0, 1, 2,... Fuctio f attais its miimum whe its slope chages from egative to oegative as α decreases. This occurs whe α chages from α +1 to α, where α satisfies the coditio c hλ 2 α (α + 1) = 0, or equivaletly, α = c hλ 1 2. (3) 3

6 Hece, f(y) attais its miimum whe y =. By covexity of f, the optimal umber of orders α +1 must be either y or y. Therefore, we set the umber of orders to y if f( y ) f( y ), ad y otherwise. We demostrate our solutio method usig the followig example: = 10, c = 75, h = 0.2, ad λ = 100. I this example, f(y) = 75y + 10 [ g(y) 10 ] = 75y + 10(20 + y) 10y 100. y y Fuctio f is depicted i Figure 1. We have α = Thus, f attais its miimum whe α chages from 3 to 2 (i.e., whe the slope of the curve chages from c hλ 2 (3)(3 + 1) = 45 to c hλ 2 (2)(2 + 1) = 15). Hece, y = α +1 = Note that f( y ) = f(3) = 565 ad f( y ) = f(4) = 560. Therefore, the optimal umber of orders is k = 4. Note also that r = k k = 2. This implies that the optimal decisio is to place two orders of size 200 ad two orders of size 300 i the plaig horizo. 3 Cocludig Remarks We have preseted a ew method for determiig the optimal umber of orders for the fiitehorizo discrete-time EOQ model. This provides us with a simple way to determie the optimal orderig policy for the problem without usig three-sectio search or ay biary search method. However, our method ca oly geerate a approximate solutio uless we ca obtai a exact value of α i the square root calculatio i (3). I fact, we ca geerate a exact solutio to problem P without the eed of a exact value of α. To do so, we modify our solutio method as follows: We first compare the values of c hλ ad ( ) 2. If c hλ ( + 1 2) 2, the α. I such a case, we have y < 1, which implies that the optimal umber of orders is equal to 1. If c hλ < ( + 1 2) 2, the we obtai a approximate value ˆα of α with ˆα α 1, ad the optimal umber of orders must be equal to oe of the followig values: ˆα, ˆα +1, ˆα +2, ˆα, ˆα +1, ad ˆα +2. Hece, we choose the best 4

7 amog these six values. (If ˆα = 0, the we choose the best oe amog ˆα +1 ad ˆα +2.), ˆα +2, ˆα +1, Note that this method requires oly simple arithmetic operatios, the floor ad ceilig operatios (i.e., ad ), ad the determiatio of ˆα. Kovalev ad Ng s [1] method has a ruig time of O(log ), assumig that the floor ad ceilig operatios require costat time (i.e., assumig that either the floor or the ceilig operatio is i the repertoire). Here, we make the same assumptio o the computatioal requiremet o the floor ad ceilig operatios. The, the computatioal complexity of our modified solutio method is domiated by the determiatio of ˆα. Note that B ca be computed to accuracy ɛ i O(loglog(B/ɛ)) time for ay B > 1 (see [2, 3]). This implies that ˆα ca be determied i O ( log log( c hλ )) O ( log log( 2 ) ) = O(loglog ) time. Therefore, our modified solutio method ca geerate a exact solutio to problem P i O(loglog ) time. Ackowledgmets The author would like to thak three aoymous referees for their helpful commets ad suggestios. This research was supported i part by The Hog Kog Polytechic Uiversity uder grat umber 1-BBZL. Refereces [1] Kovalev, A., Ng, C.T., A discrete EOQ problem is solvable i O(log) time, Europea Joural of Operatioal Research 189 (2008) [2] Reegar, J., O the worst-case arithmetic complexity of approximatig zeros of polyomials, Joural of Complexity 3 (1987) [3] Ye, Y., Combiig biary search ad Newto s method to compute real roots for a class of real fuctios, Joural of Complexity 10 (1994) [4] Zipki, P.H., Foudatios of Ivetory Maagemet, McGraw-Hill, Bosto,

8 f (y) α = 3 slope = -45 α = 2 slope = y y * = 3.33 Figure 1. Fuctio f ad its miimum

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