EOQ Problem Well-Posedness: an Alternative Approach Establishing Sufficient Conditions

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1 pplied Mathematical Sciences, Vol. 4, 2, no. 25, EOQ Problem Well-Posedness: an lternative pproach Establishing Sfficient Conditions Giovanni Mingari Scarpello via Negroli, 6, 226 Milano Italy Daniele Ritelli Dipartimento di Matematica per le scienze economiche e sociali viale Qirico Filopanti, Bologna, Italy daniele.ritelli@nibo.it bstract n existence-niqeness theorem is proved abot a minimm cost order for a class of inventory MB models, leading to a sfficient conditions. They consist of doing a check of some improper integrals, and form the article s main theoretical contribtion to the sbject. The reason of this article consists of the alternative approach to [5] of proving the existence/niqeness of the Economic Order Qantity for a generalized Wilson model by means of the integral of the inverse fnction. Mathematics Sbject Classification: 9B5 Keywords: Inventory, Cost Optimization, By Planning Backgrond and motivation When the first T Fords were rolling off the assembly line (93), manfactrers were already appraising the financial benefits of inventory management determining the minimm cost answers. s a matter of fact, long before JIT (Jst In Time), TQM (Total Qality Management), TOC (Theory Of Constraints), and MRP (Manfactring Resorces Planning), companies were sing these same (then nnamed) concepts in managing both their proction and inventory. Economic Order Qantity (EOQ) is a set of models defining the

2 24 G. M. Scarpello and D. Ritelli optimal qantity to order that minimizes the total variable costs reqired to both: order and hold inventory. Inventory models for calclating optimal order qantities and reorder points have been in existence long before the arrival of the compter: in fact the first of them was originally developed by Harris, [3] (93), thogh Wilson, [6] (934), is credited for his early in-depth analysis of the sbject. Basic nderlying assmptions:. the demand for the item is known, and deterministic; 2. no lead time (between order and arrivals) is taken into accont; 3. the receipt of the order occrs in a single instant and immediately after ordering it; 4. qantity disconts are not calclated as part of the model; 5. the ordering cost is a constant. Several extensions can be made to the EOQ model: the items deterministic demand can change with amont itself or with time; the model can inclde backordering costs and mltiple items. Shold they ndergo deterioration, perishability can be modelled either constant or variable with the items stock level. dditionally, the economic order qantity, and the order interval can be evalated from the EOQ so that the EPQ, namely the optimal proction qantity, can be determined in a similar fashion. The above determinism is not the only possible approach to the problem: there is the probabilistic one, bt we will keep ot of it. The idea exposed in this paper, even if born first, comes to the press, following an article already pblished, [5] (28), bt where the athors, throgh a different approach, generalized its analytic base. Nevertheless its original formlation keeps its interest as a readable contribtion as a primer in the athors research line on EOQ. 2 The MB models with fixed specific costs Throghot all this paper q(t) means the instantaneos stock level of or inventory at time t, which is rled by a blowdown dynamics: { q(t) = f(q(t)) (2.2.) q() = Q> where f :[, [ R is a continos and positive fnction, so that the soltion to (2.2.) meets q(t) Q for each t. De to the atonomos (2.2.) natre,

3 EOQ problem well-posedness 25 we solve by qadratres: if F (q) := = t (2.2.2) q f() then q(t) =F (t) solves (2.2.). We will treat models compliant with the assmptions of the previos section, with monotonic blowdown ( q < for each t) and then named MB (Monotonic tonomos Blowdown), whose version with delivery >and holding h>time-invariant specific costs will be analyzed here. We define reordering time generated by Q the real vale T (Q) > capable of getting zero the soltion of (2.2.): T (Q) =F () = f(). Minding and h meanings, then the total cost for (delivering + holding) a whichever Q> amont of goods will be: T (Q) T (Q) + h q(t)dt. (2.2.3) T (Q) The early Wilson model [6] is fond again when the instantaneos rate q of stock depletion is assmed to have a constant magnitde: f(q) = δ >. The frther ones e to Goh, [2] (994), and to Giri and Chadhri, [] (998), will correspond to f(q) =δq β with <β< and to f(q) =ϑq+ δq β with δ> and <ϑ,β<. 3 The existence of a minimm cost The well known MB models have not been stdied in their general featres so far. What we mean to do is what has not been done yet, namely to establish, on the above assmptions, some conditions sfficient to ensre that a cost fnction like (2.2.3) attains a minimm, and eventally its nicity. Sch a research arises adeqately motivated becase the minimm comptation reqires a certain transcendental eqation to be solved nmerically. Then it makes a little sense to go on withot proofs of existence and nicity of this aim. Nevertheless sch a research is neglected at all in literatre which faces single particlar problems withot making sre of their well-posedness. Let s start with the cost fnction (2.2.3). It will be recalled that for integrating an inverse fnction, say g, one can also operate on the direct one, see Key (994): g(a) g(b) g (y)dy = ag(a) bg(b)+ b a g(x)dx for g :[a, b] R strictly decreasing fnction. We will prove the following:

4 26 G. M. Scarpello and D. Ritelli Lemma 3.. ssme that both fnctions: f(), f(). are integrable at the origin. Then the cost fnction (2.2.3) relevant to a Q- amont of goods can be expressed by: + h f(). (3.3.) f() Proof. (3.3.) stems applying to (2.2.3) the aforementioned formla for inverse integration: T (Q) + h T (Q) F () F (Q) F (t)dt = T (Q) + h T (Q) F (q)dq. Then, minding (2.2.2), throgh the doble integrals rection, we get: T (Q) + and then (3.3.). h ( T (Q) q ) dq = f() Theorem 3.2. ssme that: f() or: or: f() R, f() R, T (Q) + h T (Q) ( ) dq f() =, (3.3.2) =, (3.3.3) f() R. (3.3.4) f() In any of sch three occrrences the cost fnction (3.3.) attains a minimm for exactly one Q > vale. Proof. We can immediately check that: lim. Q +

5 EOQ problem well-posedness 27 Sch a first divergence has its theoretical explanation: for a batch of zero consistency the reordering time will be zero, and then each of sb-costs being distribted on a zero time span will give specific infinite costs. Frthermore, if we assme (3.3.2), then the cost fnction will go to infinite if Q : lim lim Q Q hq f(q) f(q) = according to the De l Hospital rle. This second divergence complies with the fact that a cost fnction gets nlimited growing, if the batch consistency does so. The doble divergence and the continity of C(Q) as well, imply that C(Q) is bonded from below and then it shall have somewhere at least a critical point. Then a minimm does exist and, in addition, it has to be niqe. In fact, the first derivative of C(Q) becomes zero if and only if Q solves the eqation: hq { Q f() + h } f() =. (3.3.5) Notice that the fnction { Q } Q hq f() + h f() := N (Q) is a difference of two increasing fnctions, so that the critical point is niqe. The way to prove or statement is similar if we assme (3.3.3). Finally, if relation (3.3.4) holds, notice that N () = < and in or specific case we have: lim N (Q) = Q then only one critical point exists capable of minimizing C(Q). The existence and nicity of the minimm cost is then, in or assmptions, completely proved. The reader can check that for Wilson model, [6], corresponding to f() =δ, (3.3.) gives back the classic optimm condition hq 2 2δ =. For the Goh model, [2], the optimm condition gives hq 2 β (β 2 3β +2)δ =. Finally, in Giri-Chadhri model, [], the detection of the optimm batch means that: hq ( β)ϑ ln (+ ϑδ ) Q β = + hq ϑ [ 2 F (, /( β) (2 β)/( β) ϑ )] δ Q β

6 28 G. M. Scarpello and D. Ritelli to be solved to Q. nyway the above formla involving the Gass hypergeometric fnction 2 F is not present in the article []; it is fonded pon the integral identities: Q ϑ+ δ β = ϑ+ δ β = ϑ (+ ϑ( β) ln ϑβ ) Q β, [ (, /( β) Q Q 2 F (2 β)/( β) n= ϑ )] δ Q β. We limit here to recall that 2 F is the Gass hypergeometric x -power series, x < : ( ) a, b 2F c x (a) = n (b) n x n (c) n n!, where (a) k is a Pochhammer symbol: (a) k = a(a +) (a + k ). We sed the integral representation theorem for 2 F : ( ) a, b 2F c x Γ(c) t a ( t) c b = dt, Γ(c a)γ(a) ( xt) b Re c>re a>, x <. The IRT provides the way for extending the region where the (complex) hypergeometric fnction is defined, namely for its analytical contination to the (almost) whole complex plane exclding the half line ], [. 4 Conclsions n existence niqeness theorem 3.2 is proved abot a minimm cost batch for a class of inventory MB models, leading to a set of, completely new, sfficient conditions. They reqire to check some improper integrals, and form the article s main theoretical effort. In [5] an analogos existence/niqeness theorem has been established following a flly different approach and relevant applications have been provided. References [] B.C. Giri and K.S. Chadhri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, Eropean Jornal of Operational Research, 5 (998), [2] M. Goh, EOQ models with general demand and holding cost fnction, Eropean Jornal of Operational Research, 73 (994), 5-54.

7 EOQ problem well-posedness 29 [3] F.W. Harris, How many parts to make at once, Factory: The Magazine of Management, (93), [4] E. Key, Disks, Shells, and Integrals of Inverse Fnctions, College Mathematical Jornal, 25 (994), [5] G. Mingari Scarpello and D. Ritelli, EOQ when holding costs grow with the stock level: well-posedness and soltions, dvanced Modeling and Optimization, (28), [6] R.H. Wilson, Scientific Rotine for Stock Control, Harvard Bsiness Review, 3 (934), Received: September, 29