EOQ when holding costs grow with the stock level: well-posedness and solutions 1

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1 AMO - Advnced Modeling nd Optimiztion, Volume 1, Number 2, 28 EOQ when holding costs grow with the stock level: well-posedness nd solutions 1 Giovnni Mingri Scrpello nd Dniele Ritelli Diprtimento di Mtemtic per le scienze economiche e socili vile Filopnti, Bologn, Itly E-mil: giovnni.mingri@unibo.it E-mil: dniele.ritelli@unibo.it Abstrct. An existence-uniqueness theorem is proved bout minimum cost order for clss of inventory models whose holding costs grow, following power lw, with the stock level. The theorem requires to perform check of convergence of some improper integrls, nd constitutes the rticle s min theoreticl contribution to the subject. As ppliction, severl cses of demnd re considered s functions of the stock level. 1. Bckground nd motivtion Economic Order Quntity (EOQ) is set of models defining the optiml quntity of single item which shll be ordered for minimizing the totl cost: ordering nd inventory holding. These mthemticl models hve been in existence long before the computer, going their origin bck in time to [Hrris, 1913] even though [Wilson, 1934] is credited for his erly in-depth nlysis, on the subject. Bsic underlying ssumptions: 1. the monthly (nnul or, generlly: relevnt to unit time) demnd for the item is known, nd deterministic; 2. no led time (between order nd rrivls) is tken into ccount; 3. the receipt of the order occurs in single instnt nd immeditely fter ordering it; 4. quntity discounts re not clculted s prt of the model; 5. the ordering cost A is constnt. 1 AMO - Advnced Modeling nd Optimiztion. ISSN:

2 Severl extensions cn be mde to EOQ model: the items deterministic demnd cn chnge with the instntneous stock level or with time; the model cn include bckordering costs nd multiple items. Should they undergo deteriortion, the perishbility cn be modelled either constnt or vrible with the stock level. Finlly, the bove determinism could be relesed, leding to probbilistic view, which we will keep out of. The pproch followed hereinfter is of geometricl nture in the sense tht qudrture reltionships re obtined providing vi definite integrls: the reordering time, the globl cost function nd the minimum cost (optimlity) condition. In such wy no previous pproximtion is inserted, such tht of [Giri nd Chudhuri, 1998] where t the beginning lineriztion is done by truncted series development. In our tretment numericl pproximtions rise t the end, in order to evlute the economic order quntity Q. Finlly, it shll be highlighted we mrk out the foundtions to ll the subject obtining-in rther generl frme- some sufficient conditions ensuring the inventory cost function ttins minimum nd tht it is unique, nmely: the EOQ-problem well-posedness. The MAB models generliztion: vrible inventory costs Let it be q continuous function describing the instntneous stock level of our inventory t time t. Such q-level is ruled by blowdown dynmics: ( q(t) = f(q(t)) (1) q() = Q > where f : [, [ R is continuous nd positive function describing whtever cn get empty the store. We chose, for shortness of mthemtics lnguge, to describe the inventory blowdown through only one function of the instntneous stock level q. Such depletion (which cn depend upon multiple cuses) is here ssumed to occur prtly to meet the mrket demnd, nd prtly for items deteriortion ring the period of positive inventory. The solution to (1) meets < q(t) Q for ech t, nd, e to the utonomous (1) nture, we cn solve by qudrtures. If: F (q) := q 1 = t (2) f(u) then q(t) = F 1 (t) solves (1). We will tret models with the ssumptions of the previous section, plus the q-monotonic blowdown ( q < for ech t) nd then nmed MAB (Monotonic Autonomous Blowdown) for the bsence of ny imposed exogenous timedepending forcing. This pper comes out fter previous one (which is outstnding) nd releses the pst ssumption of fixed inventory costs, ssuming they cn chnge s growing function of q itself, s seen in the prctice. The MAB version which will be nlyzed here hs: invrint specific delivery costs (A > ), nd holding cost considered s power function of the on-hnd inventory: h(q) = ĥqα, α >, ĥ > following [Giri nd Chudhuri, 1998], subsection 3.2 (Model B) pge 471, nd will be referred s generlized MAB model. As usully, we men reordering time generted by Q, the rel vlue T (Q) > cpble of getting zero the solution of (1): T (Q) = F () = 1 f(u). 234

3 Minding A nd h menings, then the totl cost for (delivering + holding) whichever Q > mount of item will be: C(Q) = A T (Q) + ĥ T (Q) Z T (Q) [q(t)] α dt. (3) The erly Wilson model [Wilson, 1934] is found gin when the instntneous stock depletion rte q is ssumed to hve constnt mgnitude: f(q) = δ >, nd α = 1. The further ones e to [Goh, 1994] nd to [Giri nd Chudhuri, 1998], pge 471, will correspond (lwys for α = 1), to f(q) = δq β with < β < 1 nd to f(q) = θ q + δ q β with δ > nd < θ, β < The existence of minimum cost In our previous pper, [Mingri nd Ritelli, to pper], we obtined some conditions sufficient to ensure tht cost function like (3) ttins minimum. The improvement of this rticle to the previous one is to increse our nlysis ssuming h not constnt ny more, but growing, s e, with q itself. If in the integrl t right hnd side of (3) one mkes the chnge t = F (u), notice tht t = u = Q, t = T (Q) u =, nd dt = (1/f(u)), remembering q(t) = F 1 (t) one finds: C(Q) = A T (Q) + ĥ T (Q) A + ĥ = u α f(u). 1 f(u) ˆF 1 (F (u)) α f(u) = The (4) well-posedness requires, if f() = the integrbility t the origin of the functions: 1 f(u), u α f(u). Formul (4) will llow to infer tht C(Q) ttins n bsolute minimum for some Q > t one point only. In fct one cn see soon tht: lim C(Q) =. Q Furthermore, ssuming: f(u) = then, the cost function will diverge gin if Q s one cn immeditely check through De l Hospitl rule: ĥ Q α f(q) lim C(Q) = lim Q Q 1 f(q) =. The double divergence nd the continuity of C(Q) s well, imply tht C(Q) is bounded from below nd then it shll hve somewhere t lest criticl point. Then minimum does exist nd, in ddition, it hs to be unique. In fct, the first derivtive of C(Q) becomes zero if nd only if Q is root of the eqution: ĥ Q α f(u) j A + ĥ 235 u α f(u) (4) ff =. (5)

4 But the function N (Q) := ĥ Qα j Z Q ff f(u) A + ĥ u α f(u) is difference of two incresing functions, then the criticl point is unique. At the sme conclusion one rrives through nlogous rgument, whenever: nd: f(u) R, f(u) R, u f(u) = u R. f(u) Wht bove is not e to the h growth lw like q α : in fct, ssuming for it n rbitrry function k : [, [ R continuous, positive, nd so tht k() =, one rrives t: C(Q) = A T (Q) + ĥ T (Q) = A T (Q) + whose relevnt condition becomes: ĥ k(q) ĥ T (Q) f(u) k `F 1 (F (u)) k(u) f(u) 3. Some f(q) lws of interest f(u) (6) j ff A + ĥ k(u) f(u) =. (7) The bsence of direct Q formul took in the literture the effect tht more long pth shll be followed in performing n EOQ nlysis: f(q) is ssumed, the utonomous ODE to q(t) is solved with the initil condition q() = Q, the reordering time T so tht q(t ) = is computed. Successively the cost function C(Q) (3) is formed, nd, putting dc/dq to zero, the trnscendentl Q-eqution is finlly written, to be numericlly solved to Q. The existence- uniqueness for this kind of models defines which conditions re sufficient to ensure tht minimum cost exists nd is unique for (generlized) MAB model. Conversely, let us now present severl cses of f(q) > which cn be considered, writing down (5) immeditely. Notice tht f(q) could lso be known not nlyticlly. In fct experimentl dt set could be fitted in some relible nlyticl expression: this explins the theoreticl f(q) demnd lws we re going to study. We will show hereinfter some of them with the holding cost α-power lw, nd with ll the following (rther obvious) ssumptions: ĥ >, α >, Q >, A > ; < β < 1, < ε < 1, < δ < 1, >, b >, p <, r <. For ech of the f(q), t lest one of the integrl sufficient conditions is met, nd being this possible for more thn one, we decided of omitting t ll such elementry sequence of checks: nywy we re sure the Economic Order Quntity does exist nd is unique. In order to know it, we provide the trnscendentl equtions, more complicted thn in [Mingri nd Ritelli, to pper], nd then to be solved numericlly in ny cse. For shortness, only the trnscendentl optimlity condition will be displyed, omitting t ll the re-ordering time nd the globl cost function C(Q). 236

5 Wilson For the bsic model we hve f(q) = δ, so tht (7) gives: δa ĥ = 1 2 Q2 + Q 1+α, which, nothwithstnding the bsic f(q) expression, requires numericl tretment, unless α = 1. If α = 2 crdnic formule cn be n lterntive tool. Goh s model In [Goh, 1994] being f(q) = δq β we obtin: Giri-Chudhuri model δa ĥ = Q1+α β 1 β Q2 β 2 β. Setting f(q) = θq + δq β s in [Giri nd Chudhuri, 1998], where θ is the rte of deteriortion, the optimlity condition leds to n integrl not expressible through known functions. We provide numericl simultion, obtined vi Mthemtic R tking: A = 1, ĥ = 3, β = 1/3. The picture below shows the globl cost Q function, for which we get the minimum Q = Figure 1: The globl cost function of Giri-Chudhuri with A = 1, ĥ = 3, β = 1/3, θ =.1, δ = 1 Unnmed model Setting f(q) = δ + εq β, the slightly less complicted nture of this problem leds to trctble optimlity condition:! δa ĥ = 1 2 1, 2 Q2 β 2F ε 1 δ Qβ + Q 1+α 2F, 1! β ε δ Qβ β β which will provide Q through numericl pproch. Notice tht 2F 1 is the Guss hypergeometric x -power series, x < 1: «, b X 2F 1 c x () = n (b) n x n (c) n n!, n= 237

6 where () k is Pochhmmer symbol: () k = (+1) (+k 1). For 2F 1 powerful integrl representtion theorem is vilble: «Z, b 2F 1 c x Γ(c) 1 t 1 (1 t) c 1 = dt, Γ(c )Γ() (1 xt) b whose vlidity rnges re: Re > Re c >, x < 1. Affine If f(q) = δ + εq we obtin: ε A «δ ĥ = Q + ε + Qα ln 1 + Qε «, ε which shll be numericlly solved in spite of the f(q) esiness. Rtionl (first) When the inventory depletion is rtionl: f(q) = b + q, the optimum condition leds to the Q eqution: A b ĥ = 2 + Q «Q to be solved through numericl pproch. Rtionl (second) b + Q 2 «Q 1+α, If f(q) = b 2 + q, 2 then: ««A b ĥ = 2 + Q2 Q 2 b Q2 Q 1+α, 3 which cn be solved lgebriclly if α = 1, numericlly for α 1 Qudrtic Let the instntneous inventory stock level be ruled by (1) with < f(q) = (q p)(q r). In such wy the optimum condition (5) will specilize in: «A ĥ = Qα r(q p) p r ln 1 p(q r) p r p ln 1 Q ««r + r ln. p Q r Exponentil The inventory mnger is fced with n periodic demnd which either is lwys incresing, or decresing, nmely f(q) = e q, or f(q) = e q. Even if the integrls in (5) re ll elementry for the exponentil sitution, the relevnt Q-equtions: ` 1 + e Q f(q) = e q ĥq α ĥ `e Q (Q 1) + 1 = A = `1 e Q f(q) = e q ĥq α ĥ `1 e Q (Q + 1) = A = 238

7 re trnscendentl yet. In ddition the f(q) exponentil nture is not n nlyticl oddness, but hs deep mrket mening. In this lst sitution, we provide numericl simultion, obtined using Mthemtic R tking: A = 1, ĥ = 2, = 1, α = 1/3. The figure below gives the inventory globl cost s function of Q: we get Q = Figure 2: The globl cost function to exponentil incresing demnd with A = 9, ĥ = 5, = 1, α = 1/3 4. Conclusions An existence-uniqueness theorem is proved bout minimum cost btch for clss of inventory MAB models, leding to set of sufficient conditions. This rticle enlrges our previous nlysis ssuming h not constnt ny more, but growing, s e, with q itself. The nlyzed MAB version hs therefore invrint specific delivery costs (A > ) nd holding costs vrible s: h(q) = ĥqα, α >, ĥ >. The sufficient conditions require to check the convergence of some improper integrls, nd form the rticle s min theoreticl effort. As ppliction, severl cses hve been considered of demnd f(q) ssumed s continuous function of the stock level q. Being one of the sufficient conditions met in ny cse, the economic order quntity is unique, nd the relevnt computtions led to trnscendentl equtions. In some cses the plot of the globl costs is given, nd, even if the optimlity condition cn be written in closed (but trnscendentl) form, its solution shll in (lmost) ny cse be fced numericlly. References [1] Giri, B.C., nd Chudhuri, K.S., (1998) Deterministic models of perishble inventory with stock-dependent demnd rte nd nonliner holding cost. Europen Journl of Opertionl Reserch, vol. 15 pp [2] Goh, M., (1994) EOQ models with generl demnd nd holding cost function. Europen Journl of Opertionl Reserch, vol. 73 pp [3] Hrris, F.W., (1913) How mny prts to mke t once. Fctory The Mgzine of Mngement, vol. 1 pp

8 [4] Mingri Scrpello, G., nd Ritelli, D., (to pper) EOQ problem well-posedness: sufficient conditions nd pplictions. [5] Wilson, R.H:, (1934) A Scientific Routine for Stock Control. Hrvrd Business Review, vol. 13 pp