Vietnam Jounal of Mathematics 33:4 005) 437 44 9LHWQDP -RXUQDO RI 0$7+0$7,&6 9$67 A Stochastic EOQ Policy of Cold-Dink-Fo a Retaile Shib Sanka Sana 1 Kipasindhu Chaudhui 1 Depatment of Math., Bhanga Mahavidyalaya Univesity of Calcutta Viil. +P.O.+ P. S.-Bhanga, Dist.-4PGSSouth) West Bengal, India Depatment of Mathematics Jadavpu Univesity, Calcutta-70003 West Bengal, India Received June, 005 Abstact. This pape extends a stochastic EOQ economic ode quantity) model both fo discete continuous distibution of dems of cold-dink. A geneal chaacteization of the optimal inventoy policy is developed analytically. An optimal solution is obtained with pope numeical illustation. 1. Intoduction A well-known stochastic extension of the classical EOQ economic ode quantity) model bases the e-ode decision o the stock level see Hadley Whitin [4], Wagne [13]). Models of stoage systems with stochastic supply dem have been widely analysed in the models of Faddy [3], Haison Resnick [5], Mille [8], Moan [9], Pliska [10], Puteman [11], Meye, Rothkopf Smith [7], Teisbeg [1], Chao Manne [1], Hogan [6] Devaangan Weine []. In this pape, an optimal inventoy policy is chaacteised by conditions: a) dem ate is stochastic that depends upon tempeatue as om vaiable; b) supply ate is instanteneously infinite ode is placed in the begining of the cycle; c) inventoy cost is a linea function of tempeatue.. Fundamental Assumptions Notations 1. Model is developed on single-item poducts.
438 Shib Sanka Sana Kipasindhu Chaudhui. Lead time is negligible. 3. Dem is unifom ove the peiod a function of tempeatue that follows a pobability distibutions. 4. Poduction ate is instanteneously infinite. 5. Reode-time is fixed known. Thus the set-up cost is not included in the total cost. Let the holding cost pe item pe unit time be C h, the shotage cost pe item pe unit time be C s, the inventoy level be Q of item, is the dem ove the peiod, T is the cycle length. 3. The Model In this model, we conside dem ate of the poduct ) inventoy holding cost pe item pe unit time C h )ae: = a C h = C 1 + C μ). whee, a = d 0) = maginal esponse of cold-dink consumption to a change in d tempeatue) C 1 = oppotunity cost of money tied up in inventoy. C = ate of change of inventoy cost with espect to tempeatue. μ = optimum tempeatue fo a buye, accoding to thei dem. Geneally μ is 5 C. Now, the govening equations ae as follows: Case 1. When Shotage does not occu dq dt = T, 0 t T 1) with Q0) = Q 0. Fom Eq. 1), we have Qt) =Q 0 t, 0 t T. T Hee QT ) 0 Q 0 T T 0 Q 0. Theefoe, the inventoy is T 0 Q 0 T t)dt =Q 0 )T, fo Q 0. Case. When Shotage occus: dq dt = T, 0 t t 1 ) with Q0) = Q 0,Qt 1 )=0,
A Stochastic EOQ Policy of Cold-Dink -Fo a Retaile 439 with QT ) < 0. Fom Eq. ), we have dq dt = T,t 1 t T 3) Qt) =Q 0 T t, 0 t t 1. Now Qt 1 )=0 t 1 = Q 0T. The Eq. 3) gives us Qt) = T t t 1), t 1 t T. So QT ) < 0 T T t 1) < 0 T>t 1 T> Q0T the inventoy duing 0,t 1 )is Q 0 <. Theefoe, t 1 The shotage duing t 1,T)is 0 Q 0 T t)dt = Q 0t 1 T t 1 = 1 T t 1 Qt)dt = T T t 1) Q 0 T. = 1 T 1 Q 0 ), > Q 0. Since, Q 0 Q 0 a 1 a Q 0 = say). i.e., Q 0 = a. Also, Q 0 < >. Case I. Unifom dem discete units. is om vaiable with pobability p) such that p) 0. Theefoe the expected aveage cost is Eac )= 1 T C h + 1 C s Q 0 ) Tp)+ 1 = +1 T 1 Q ) 0 p) = +1 = C 1 C μ + C )a ) p) + 1 p) ac 1 C μ + C ) + 1 C s = +1 = +1 a 1 ) p) p) =1 C h Q 0 p)t
440 Shib Sanka Sana Kipasindhu Chaudhui Now, Eac +1)=Eac )+ )a p)+ + 1 ) +1 C s + ac p)+ + 1 ) ) p). = +1 p) ) In ode to find the optimum value of Q 0 i.e., so as to minimize Eac ), the following conditions must hold: Eac +1) >Eac )Eac 1) > Eac ) i.e., Eac +1) Eac ) > 0Eac 1) Eac ) > 0. Now, Eac +1) Eac ) > 0 implies I )+ C ) > C s, whee I )= )= p)+ + 1 ) p)+ + 1 ) +1 p) = +1 p). Similaly Eac 1) Eac ) > 0 implies I 1) + C 1) < C s. Theefoe fo minimum value of Eac ), the following condition must be satisfied I C )+ ) C s > > I C 1)+ 1) 4) Case II. Unifom dem continuous units. When uncetain dem is estimated as a continuous om vaiable, the cost equation of the inventoy involves integals instead of summation signs. The discete point pobabilities p) ae eplaced by the pobability diffeential f) fo small inteval. In this case 0 f) d =1f) 0. Poceeding exactly in the same manne as in Case I, The total expected aveage cost duing peiod 0, T)is
A Stochastic EOQ Policy of Cold-Dink -Fo a Retaile 441 Eac )=a a C 1 C μ + C )f)d Now, d Eac ) d + a deac ) d C 1 C μ + C ) { = a ) 0 f) d + a C s 0 + C f)d + 0 { = a ) f) C 1 C μ + C )f)d f)d + ) f) d 5) ) } f)d C s d + C f) ) d 6) } f)d > 0. 7) Fo minimum value of Eac ), deac ) d =0 d Eac ) d > 0mustbe satisfied. The equation, deac ) d = 0 being nonlinea can only be solved by any numeical method Bisection Method) fo given paamete values. 4. Numeical Examples Example 1. Fo discete case: In this case, we conside C 1 =0.135, C =0.001, C s =5.0, μ=5.0, a=0.8 in appopiate units also conside the pobability of tempeatue in a week such that in C: 35 36 37 38 39 40 41 p) : 0.05 0.15 0.14 0.10 0.5 0.10 0.1 Then the optimal solution is =38 o C i.e., Q = a =30.4 units. Example. Fo continuous case: We take the values of the paametes in appopiate units as follows: f) =0.04 0.0008, 0 50 =0, elsewhee
44 Shib Sanka Sana Kipasindhu Chaudhui C 1 =0.135, C =0.001, C s =5.0,μ=5.0, a=0.8. Then the optimal solution is: =9.30 C i.e., Q = a =3.44 units. 5. Conclusion Fom physical phenomenon, it is common belief that the consumption of cold dinks depend upon tempeatue. Tempeatue is also om in chaacte. Geneally the pocuement cost of cold dinks is smalle than thei selling pice. Consequently, supply of cold dinks to the etiale is sufficiently lage. Inventoy holding cost is boken down into two components: i) the fist is the oppotunity cost of money tied up in inventoy that is consideed hee as C 1 ii)thend is C μ), whee μ is optimum tempeatue fo a buye accoding to thei dem. Geneally, μ is 5 C. So the cost of declining tempeatue μ) has a emakable effect on the inventoy cost. In eality, the discete case is moe ealistic than the continuous one. But we discuss both the cases. As fa as the authos ae infomed, no stochastic EOQ model of this type has yet been discussed in the inventoy liteatue. Refeences 1. H. Chao A. S. Manne, It oil stock-piles impot eductions: A dynamic pogamming appoach, Opns. Res. 31 1983) 63 651.. S. Devaangan R. Weine, Stockpile Behavio as an Intenational Game, Havad Univesity, 1983. 3. M. J. Faddy, Optimal contol of finite dams, Adv. Appl. Pob. 6 1974) 689 710. 4. G. Hadly, T. Whitin, Analysis of Inventoy System. Pentice-Hall, Englewood Cliffs, NJ, 1963. 5. J. M. Haison S. I. Resnick, The stationay distibution fist exit pobabilities of a stoage pocess with geneal elease ules, Math. Opns. Res. 1 1976) 347 358. 6. W. W. Hogan, Oil stockpiling: help thy neighbo, Enegy. J. 4 1983) 49 71. 7. R. R. Meye, M. H. Rothkopf, S. A. Smith, Reliability inventoy in a poduction-stoage system, Mgmt. Sci. 5 1979) 799 807. 8. R. G. Mille, J., Continuous time stochastic stoage pocesses with om linea inputs outputs, J. Math. Mech. 1 1963) 75 91. 9. P. A. Moan, The Theoy of Stoage, Metuen, London, 1959. 10. S. R. Pliska, A diffusion pocess model fo the optimal opeations of a esevoi system, J. Appl. Pob. 1 1975) 859 863. 11. M. L. Puteman, A diffusion pocess model fo the optimal opeations of a esevoi system, Noth-Holl/Times studies in the management sevice 1 1975) 143 159. 1. T. J. Teisbeg, A dynamic pogamming model of the U. S. stategic petoleum eseve, Bell. J. Econ. 1 1981) 56 546. 13. H. M. Wagne, Statistical Management of Inventoy Systems, John Wiley & Sons, 196.