A Proof of the EOQ Formula Using Quasi-Variational. Inequalities. March 19, Abstract

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1 A Proof of the EOQ Formula Using Quasi-Variational Inequalities Dir Beyer y and Suresh P. Sethi z March, 8 Abstract In this paper, we use quasi-variational inequalities to provide a rigorous proof of the familiar square root formula for the optimal economic order quantity (EOQ) in the classical deterministic average cost inventory model. Introduction The familiar square root formula for the optimal economic order quantity (EOQ) in simple inventory models is one of the most fundamental results in management science and operations research. The formula dates bac to Harris (5); see Erlenotter (8) for its history. The development of the formula and its subsequent derivations have been largely informal. Every OR textboo contains a calculus derivation of the EOQ formula. But such a derivation does not establish formally that the EOQ formula provides a production policy that minimizes the long-run average cost over the class of all admissible policies. Certainly there are many possible ways to prove the formula. But there does not seem to be a publication devoted directly to providing a mathematically rigorous proof of the famous formula. It is also possible that there are a number of papers dealing with complex inventory models, of which the simple lotsize model is a special case. One such paper is This research was supported in part by NSERC grant A6 and a grant from The University of Texas at Dallas. y Hewlett-Pacard Laboratories, Palo Alto, CA z School of Management, The University of Texas at Dallas, Richardson, TX

2 that of Chand, Sethi and Proth (), which deals with the issue of the existence of forecast horizons in the more general dynamic lotsize model. In this paper, we use quasi-variational inequalities (QVI) to provide a rigorous proof of the EOQ formula. Our purpose in selecting this approach over other possible ones is its applicability to more general inventory models that would be subsequently studied in our future wor. The lotsize model provides us with a simple example with which to illustrate the QVI methodology in average-cost models. Moreover, the paper would be a convenient reference to direct interested readers to a rigorous proof of the EOQ formula. In the next section, we state precisely a slightly more general lotsize problem, and formulate an appropriate system of QVIs for the problem. In Section, we characterize a suitably small class of policies containing an optimal policy. We prove a verication theorem and obtain an optimal policy. In Section we derive the EOQ formula for the simple lotsize model. Section 5 concludes the paper. Average-Cost Optimal QVIs In this section we provide a precise formulation of the simple economic lotsize problem. It is a deterministic stationary one-product lotsizing problem of a manufacturer (retailer) who faces continuous demand at a constant rate. To meet the demand she holds an inventory of the product. At any point in time, she has the opportunity to replenish the inventory, i.e., to produce a lot (or to place an order). Production (resp. delivery) is assumed to be instantaneous. Her goal is to minimize the long run average-cost for ordering and holding over the innite horizon while meeting all demand. We use the following notation: { the constant demand rate, x(t) { inventory level at time t, x(t), { instant of -th order,, v { amount of -th order, v, f(x) { holding cost rate at inventory level x, c(v) { cost of an order of amount v, { the discount rate.

3 We assume the function f() to be continuous, nonnegative and nondecreasing with lim x! f(x). The function c() is assumed to be continuous, nonnegative and nondecreasing for positive arguments. Additionally, we assume that c() and c(+) >, where the "+" denotes the right hand limit of the function (see ()). Denition. Let ( ) be a strictly increasing sequence with lim! and. Let (v ) be a sequence of positive real numbers with v > for >. Then we call U ( ; v ; ; v ; : : :) an admissible strategy. The set of all admissible strategies is denoted by U. Following a strategy U U and given an initial inventory level x() x, the inventory process is described by x(t) x? t + N (t) X v with N(t) maxf : < tg: The function x() is left-continuous; thus x(t) lim h# x(t + h). Moreover, we dene The total discounted cost is given by J (x; U) x(t+) lim h# x(t + h): () f(x(t))e?t dt + X The average cost is dened as J(x; U) lim sup n! n+ X c(v )e? C f(x( +)? )e? d + c(v )e? A : nx C f(x( +)? )d + c(v ) A : Under more restrictive assumptions on the cost functions, it is shown in Bensoussan et al. (8) that the value function v (x) : inf UU J (x; U) is the maximum solution of the following system of QVIs: R v (x) t f(x? )e? d + v (x? t)e?t ; 8t x ; v (x) c(v) + v (x + v); 8v ; v (x) v (x) > >; ()

4 for some a priori bound v (x) on the value function, which is nite for each x. Moreover, a bound v (x) is easy to establish. For our analysis of the average-cost problem, we dene the dierential discounted value function (nown also as the potential function) w (x) v (x)? v (): The rst two inequalities in () can now be written as R w (x) t f(x? )e? d + w (x? t)e?t? (? e?t )v (); 8t x w (x) c(v) + w (x + v); 8v : If there is a sequence ( ) with! such that w (x)! w(x) 8x and v ()! g; > >; () then we can pass to the limit in () and obtain R w(x) t f(x? )d + w(x? t)? gt; 8t x ; w(x) c(v) + w(x + v) 8v : > >; () In what follows, we show that indeed the QVI system () has a solution, and that this solution provides an average optimal ordering policy. Analysis of the Average-Cost Optimal QVIs We begin with a well-nown result, which we shall prove for the sae of completeness. Lemma. Let U ( ; v ; ; v ; : : :) be an admissible strategy and x() the inventory process associated with this strategy. If there exists an integer such that x( ) > and v >, then there is an alternate strategy ~ U ( ~ ; ~v ; ~ ; ~v ; : : :) with the associated inventory process ~x(t) which satises x( ) for all. Furthermore, the average cost of the alternate strategy does not exceed that of U. Proof. Let i minf : x( ) > g and assume i <. Let t i + x( i ) be the rst instant after i, at which the inventory level would be zero provided that no order is placed at i. We distinguish two cases:

5 Case. t < i+. Dene the strategy ~ U according to ~ ; ~v v for < i; ~ i t ; ~v i x(t +) v i ; ~ ; ~v v for > i: The two strategies U and ~ U are the same except that ~ U places the i-th order later. Clearly, the strategy ~ U is admissible and J(x; U) J(x; ~ U). Case. t i+. Dene the strategy ~ U according to ~ ; ~v v for < i; ~ i i + ; ~v i v i + v i+ ; ~ ; ~v v for > i; The two strategies U and ~ U are the same except that ~ U places the i-th and i + -st order at i+. Clearly, the strategy ~ U is admissible and J(x; U) J(x; ~ U). When the approach described above is repeated "as we go along", we obtain a strategy that does not order if the inventory level is not zero. Lemma. Let U ( ; v ; ; v ; : : :) be an admissible strategy and x() be the inventory process associated with it. Then there exists a constant M < such that if there is a for which v > M, then there is an alternate strategy ~ U ( ~ ; ~v ; ~ ; ~v ; : : :) with the associated inventory process ~x() such that v M for all. Furthermore, the average cost of the alternate strategy ~ U does not exceed that of U. Proof. Let bvc denote the largest integer smaller than v. In view of Lemma., we can restrict our attention to strategies which do not order when the inventory level is nonzero. Suppose the strategy U orders v i v in its i-th order. First we observe that due to Lemma., the policy U does not order in the time interval ( i ; i + v). We shall introduce a strategy ~ U which splits the order in period i into bvc orders in periods i through i + bvc. More precisely, the policy ~ U is dened by ~ ; ~v v for < i; ~ i + (? i); ~v for i < i + bvc; ~ i + bvc; ~v v? bvc + ~?bvc ; ~v v?bvc for > i + bvc: We shall show that there is an M independent of i such that the average cost of ~ U does not exceed that of U if v M. 5

6 The trajectories of the inventory processes corresponding to the strategies U and ~ U are the same for t < i and t i+. For a given v, the total cost incurred by ~ U in the time interval [ i ; i+ ) is s(v) (bvc? ) c() + f(? )d + c(v? bvc + ) + It is easy to see that s(v) is of linear growth. On the other hand, let S(v) c(v) + v f(v? )d (v?bvc+) f(v? bvc +? )d: be the cost incurred by U in the same time interval. Since lim v! f(v), S(v) grows faster than any linear function. Therefore, there is an M < such that s(v) < S(v) for all v M: Consequently, for v M, we obtain J(x; U) J(x; ~ U). By the same toen, given a strategy U that orders more than M in some periods, we can construct a new policy ~ U that always orders less than M and incurs an average cost which does not exceed that of the original policy U. Denition. Let U ( ; v ; ; v ; : : :) be an admissible strategy and x() the inventory process associated with it. U is called bounded if x( ) and < v < M for ; ; ; : : :, where M is dened in Lemma.. It follows from Lemma. that we can restrict our search for optimal strategies to the class of bounded strategies. For such strategies it also holds that? M. Denition. An admissible policy U is called stable with respect to the function w(), if w(x( )) lim ;! where x() is the inventory process resulting from U. It follows immediately from Denition. that a bounded strategy is stable with respect to any function bounded on [; M]. 6

7 Theorem. Let (w; g) be a solution of (). Let U ( ; v ; ; v ; : : :) be an admissible policy stable with respect to w. Then J(x; U) g. If w is bounded on [; M] then J(x; V ) g for all V U. Proof. From () and the admissibility of U, we have for ; ; : : :, w(x( +)) R f(x( +)? )d + w(x( +)? (? ))? g(? ); w(x( )) c(v ) + w(x( ) + v ); which is equivalent to w(x( +)) R f(x( +)? )d + w(x( ))? g(? ); w(x( )) c(v ) + w(x( +)): Combining both inequalities we obtain g(? ) f(x( +)? )d + w(x( +))? w(x( +)) + c(v ): Summing over ; ; : : : ; n and dividing by n+ yields g n+ nx 6 f(x( +)? )d + c(v ) 5? c(v ) + w(x( n++)) > >; > >;? w(x(+)) : The last three terms tend to zero as!, due to the admissibility and the stability of U. Therefore, we obtain g lim sup! nx 6 (5) (6) f(x( +)? )d + c(v ) 5 J(x; U); () which proves the rst part of the theorem. The second part is obvious since it follows from Lemmas. and. that for any unstable policy there is a bounded (and therefore stable) policy with lower or equal cost. Theorem. Let (w; g) be a solution of () such that w(x) t f(x? )d + w(x? t)? gt; 8t x : (8)

8 Let v : R +! R + be a function that satises w(x) c(v(x)) + w(x + v(x)): () Let furthermore V be the strategy that orders v(x) if the inventory level is equal to x. Assume V to be admissible and stable with respect to w. Then g lim! nx 6 f(x( +)? )d + c(v ) 5 J(x; V ): Proof. Fix an initial inventory x() x. Since V is admissible, there are sequences ( ) and (v ) such that V ( ; v ; ; v ; : : :). Then it follows from (8) and () that w(x( +)) R f(x( +)? )d + w(x( +)? (? ))? g(? ); w(x( )) c(v ) + w(x( ) + v ): Proceeding as in the proof of Theorem. and using the stability and the admissibility of V yields g nx 6 lim! f(x( +)? )d + c(v ) 5? c(v ) nx 6 + w(x( n++))? w(x(+)) f(x( +)? )d + c(v ) 5 J(x; V ): Note that due to the above equality, the 'lim sup' in () can be replaced by 'lim'. Theorems. and. together prove that the policy V dened in Theorem., if it exists, is optimal. In the next theorem, we obtain a policy that does satisfy Theorems. and., and therefore is optimal. Theorem. Let g min v> v v C f(v? )d + c(v) A and 8

9 v argmin v> v v C f(v? )d + c(v) A : Then the policy V which orders v whenever the inventory level is zero is optimal, and J(x; V ) g for any x : Proof. We start by proving that (w ; g ) with satises (8). Let t x. Then w (x) x t t t w (x) x f(x? )d? g x f(x? )d? g t + f(x? )d? g t + f(x? )d? g x x t x?t f(x? )d? g t + w (x? t): It follows from the denition of g that for all v >, g v v f(x? )d? g ( x? t ) f(x? t? )d? g ( x? t ) C f(v? )d + c(v) A : Using this inequality and the denition of w, we obtain w (x + v)? w (x) (x+v) (x+v) x v f(x + v? )d? g x + v f(x + v? )d? g v f(v? )d? g v?c(v):? x f(x? )d + g x

10 Therefore, w satises (). Let v(x) 8 < : if x > v if x : Then it is easy to verify that () is satised by w. Also, it is obvious that V is stable w.r.t. to w. We can now apply Theorem. to complete the proof. Special Case: The EOQ Formula The model studied in Sections and reduces to the simple economic lotsize model or the EOQ model when f(x) hx and c(x) K(x) + cx; where (x) for x > and (). For this special case, v in Theorem. reduces to the classical EOQ formula: v argmin v> v! s hv K + K + cv h : The minimal average cost is g hv + c + K v : The corresponding potential function is w (x) hx? gx : () One does not see formula () for the potential function in the heuristic derivation of the EOQ formula in OR texts. However, the QVI approach requires the formulation of the potential function. Importantly, the potential function at the inventory level x represents the holding cost less the average cost for the remainder of the optimal ordering cycle. Since the potential function in our case is unique up to a constant, formula () assumes that the potential function is zero when x is equal to zero. 5 Conclusion In this paper, we have provided a rigorous derivation of the classical EOQ formula. We use the method of QVIs, which is a general methodology to treat dynamic problems

11 with xed costs. It is possible to derive other classical optimal lotsize formulas in cases when instantaneous production is not possible or when baclogging is allowed. More importantly, the methodology can be used to address continuous-time stochastic inventory models with xed ordering cost. References [] Bensoussan, A., Crouhy M. and Proth, J. (8): Mathematical Theory of Production Planning. North-Holland, Amsterdam, The Netherlands. [] Chand, S., Sethi, S. and Proth, J. (): Existence of Forecast Horizons in Undiscounted Discrete Time Lot Size Models, Oper. Res. 8, [] Erlenotter, D. (8): An Early Classic Misplaced: Ford W. Harris's Economic Order Quantity Model of 5. Mgmt. Sci. 5(8), No.. [] Harris, F.W. (5): Operations and Cost. Factory Management Series, Ch. IV, pp.8{5, A.-W. Shaw Company, Chicago, IL

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