OPTIMAL CONTROL OF A PRODUCTION SYSTEM WITH INVENTORY-LEVEL-DEPENDENT DEMAND

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1 Applie Mathematics E-Notes, 5(005), c ISSN Available free at mirror sites of amen/ OPTIMAL CONTROL OF A PRODUCTION SYSTEM WITH INVENTORY-LEVEL-DEPENDENT DEMAND Messaou Bounkhel,Lotfi Taj, Yacine Benhai Receive 18 April 004 Abstract Two prouction systems with inventory-level-epenent eman are consiere an Pontryagin maximum principle is use to etermine the optimal control. 1 Introuction It has been observe that neglecting the effect of system parameters on inventory systems leas to a poor performance an unsatisfactory management. Thus, the class of systems where some epenence among the system parameters exists have receive the attraction of many researchers. The epenence of the consumption rate on the on-han inventory is, without oubt, the epenence that receive the most attention an the literature on the subject is abunant. Among the most recent ones are [1, 9]. It has also been observe that the eterioration of items plays an important role in the inventory management. Thus, another class of systems was evelope taking into account items eterioration. The literature on this subject is immense an an excellent survey, in which eteriorating inventory systems are thoroughly classifie, has recently been one in [3]. Concerning cost parameters, the traitional approach in most moels is to keep them constant. This assumption is somewhat unrealistic. Nonlinear holing costs have been introuce in [5] an were then consiere in a few works such as []. All the moels cite above are EOQ-type or extene EOQ-type moels. EOQ-type moels assume that inventory items are unaffecte by time an replenishment is one instantaneously. Since this ieal situation is generally not applicable, extene (or generalize)-type moels have been introuce to stuy ynamic inventory systems. These moels assume that such system parameters as the eman rate, the prouction rate, or the eterioration rate vary with time; see the survey [3] for references on the subject. To cater for the ynamic behavior of prouction inventory systems, control theory has been successfully applie by some researchers; see for instance [4, 7, 8]. Mathematics Subject Classifications: 49J15, 90B30. College of Science, Department of Mathematics, P.O. Box 455, Riyah 11451, Saui Arabia College of Science, Department of Statistics an O.R., P.O. Box 455, Riyah 11451, Saui Arabia College of Science, Department of Mathematics, P.O. Box 455, Riyah 11451, Saui Arabia 36

2 Bounkhel et at. 37 In the present paper, we evelop a first moel in which the ynamic eman is a general functional of time an of the amount of on-han stock. We have focuse on the analysis of a prouction inventory system in which the nonlinear holing an prouction costs are treate as general functionals of the inventory level an prouction rate, respectively. We then exten this first moel to an even more general moel in which items eterioration is taken into account. The eterioration rate is also a general functional of time an of the amount of on-han stock. For both moels, we utilize optimal control theory to obtain an optimal control policy. The rest of the paper is organize as follows. Section escribes the first inventory moel an evelops the optimal control problem an its solution. A similar evelopment is conucte in Section 3 for the secon moel. Section 4 conclues the paper. Moel Without Item Deterioration Let us consier a manufacturing firm proucing a single prouct. We assume that the ecision horizon that the manager faces is finite, of length T.A finite planning horizon is interesting an appropriate because many firms are concerne with short an/or intermeiate term market activities. For t 0, let I(t) be the inventory level at, anletd(t, I(t)) an h(i(t)) be the corresponing eman rate an holing cost rate, respectively. Let K(P (t)) enote the cost rate corresponing to a prouction rate P (t) attimet. Let ρ 0 be the iscount rate. All functions are assume to be non-negative an continuous ifferentiable. Given T>0, the optimal control problem we are consiering is: (P) T min J(P, I) = P (t) D(t,I(t)) 0 e ρt h(i(t)) + K(P (t)) t t I(t) =P (t) D(t, I(t)), I(0) = I 0, I(T )=I T. The moel is represente as an optimal control problem with one state variable (inventory status) an one control variable (rate of manufacturing). Since eman occurs at rate D an prouction occurs at the controllable rate P, it follows that I(t) evolves accoring to the above ynamics (or state equation). The constraint P (t) D(t, I(t)) with the state equation ensure I(t) I 0 an I is nonecreasing. Therefore, shortages are not allowe in our stuy. Using the Pontryagin maximum principle (see [6]), the necessary conitions for (P,I ) to be an optimal solution of problem (P) are that there shoul exist a constant β, a continuous an piecewise continuously ifferentiable function λ an a piecewise continuous function µ, calle the ajoint an Lagrange multiplier functions, respectively, such that H(t, I,P, λ) H(t, I,P,λ), for all P (t) D(t, I (t)), (1) t λ(t) = L(t, I, P, λ,µ), () I

3 38 Optimal Control of a Prouction System I(0) = I 0, λ(t )=β, (3) L(t, I, P, λ,µ)=0, (4) P P (t) D(t, I(t)) 0, µ(t) 0, µ(t) P (t) D(t, I(t)) =0, (5) where H(t, I, P, λ) = e ρt h(i(t)) + K(P (t)) + λ(t) P (t) D(t, I(t)), (6) is the Hamiltonian function an ¾ L(t, I, P, λ,µ)= e ½h(I(t)) ρt + K(P (t)) ½ ¾ + λ(t)+µ(t) P (t) D(t, I(t)), (7) is the Lagrangian function. Equation () is equivalent to t λ(t) =e ρt I h(i(t)) + [λ(t)+µ(t)] D(t, I(t)). (8) I Equation (4) is equivalent to λ(t)+µ(t) =e ρt K(P (t)). (9) P Now, consier Equation (5). Then for any t, wehaveeitherp (t) D(t, I(t)) = 0 or P (t) D(t, I(t)) > 0. Case 1: P (t) D(t, I(t)) = 0 on some subset S of [0,T]. Then I(t) =0onS. In t this case I is obviously constant on S an Substituting Equation (9) into (8) yiels λ(t) =e ρt t P (t) =D(t, I (t)), for all t S. (10) I h(i (t)) + P K(P (t)) I D(t, I (t)) Integrating this equation we get an explicit form of the ajoint function λ an of the constant β. Then an explicit form of Lagrange multiplier function µ can be obtaine from Equation (9). Note that if the obtaine function µ is not nonnegative, then the solutions given in Equation (10) are not acceptable. Case : P (t) D(t, I(t)) > 0fort [0,T]\S. Thenµ(t) =0on[0,T]\S. Inthiscase the necessary conitions (3), (8), an (9) become. an t λ(t) =e ρt I h(i(t)) + λ(t) I D(t, I(t)), I(0) = I 0, λ(t )=β, λ(t) =e ρt K(P (t)). P

4 Bounkhel et at. 39 Combining these equations with the state equation yiels the following secon orer ifferential equation: P (t) t P K(P ) ρ + I) I D(t, P K(P )= I h(i),i(0) = I 0, P K(P (T )) = βeρt (11) For illustration purposes, let us assume K(P )= KP hi,h(i) =, an D(t, I) = 1 (t)+ I, where K, h, an are positive constants. For these functions the necessary conitions for (P,I ) to be an optimal solution of problem (P) become t I(t) ρ h t I(t) K + (ρ + ) I(t) =(ρ + ) 1 (t) t 1(t), I(0) = I 0, I(T )=I T. (1) This two-point bounary value problem (P TPBV ) is solve in the next proposition. PROPOSITION 1. The solution I of (P TPBV )isgivenby an its corresponing P is given by I (t) =a 1 e m1t + a e mt + Q(t), (13) P (t) =a 1 (m 1 + )e m1t + a (m + )e mt + t Q(t)+ Q(t)+ 1 (t) (14) where the constants a 1, a, m 1,anm are given in the proof below, an Q(t) isa particular solution of Equation (1). PROOF. We solve Equation (1) by the stanar metho. Its characteristic equation m ρm h K +(ρ + ) = 0, has two real roots of opposite signs, given by à s m 1 = 1! à s h ρ ρ +4 K +(ρ + ) < 0anm = 1! h ρ + ρ +4 K +(ρ + ) > 0, an therefore I (t) is given by (13), where Q(t) is a particular solution of (1). The initial an terminal conitions are use to etermine the constants a 1 an a as follows. From the initial an terminal conitions we have a 1 + a + Q(0) = I 0 an a 1 e m 1T + a e m T + Q(T )=I T. By putting b 1 = I 0 Q(0) an b = I T Q(T ), we obtain the following system of two linear equations in two unknowns which has the following unique solution a 1 + a = b 1 a 1 e m1t + a e mt = b, a 1 = b e mt b 1 e m1t e mt an a = b 1e m1t b e m1t e mt. The expression of P is euce using the expression of I an the state equation. From the above analysis we have the following theorem characterizing the optimal solution of (P).

5 40 Optimal Control of a Prouction System THEOREM 1. The optimal solution (P,I )of(p) has the form given in Equation (10) on S, an the form in Equations (13) (14) on [0,T]\S. EXAMPLE 1. Consier a prouction system with the following characteristics: initial an terminal inventory levels I(0) = 0,I(T ) = 10; unit costs an iscount factor h = 0.1,K = 5, anρ = 0, respectively. The planning horizon is T = 5, anthe stock-epenent eman is such that =0.1, 1 (t) =cos(t) + 1. Variations of the optimal prouction rate an optimal stock level are isplaye in Figure 1. The optimal cost was foun to be J = Changing the shape of the eman function by taking 1 (t) =e t an keeping all other parameters unchange yiele the graphs represente in Figure. The objective function value change to J = Moel With Item Deterioration In this section, we assume that the prouct eteriorates while in stock. For t 0, let θ(t, I(t)) be the eterioration rate at the inventory level I(t) at. Keeping the same notation as in the previous section, the optimal control problem becomes: (P θ ) 8 >< >: Z T min J(P, I) = e ρt {h(i(t)) + c [P (t) D(t, I(t))] + K(P (t))} t P (t) D(t,I(t))+θ(t,I(t)) 0 t I(t) =P (t) D(t, I(t)) θ(t, I(t)), I(0) = I 0, I(T )=I T, where c>0 is the unit cost. The necessary conitions (1)-(4) remain the same with H(t, I, P, λ) = e ρt h(i)+c [P (t) D(t, I)] + K(P ) + λ(t) P (t) D(t, I) θ(t, I), (15) L(t, I, P, λ,µ)=h(t, I, P, λ(t)) + µ(t) P (t) D(t, I) θ(t, I), (16) while Equation (5) becomes P (t) D(t, I) θ(t, I) 0, µ(t) 0, µ(t) P (t) D(t, I) θ(t, I) =0, (17) Equations (), (4), an (16) yiel λ(t) =e ρt t I h(i) c D(t, I) I +[λ(t)+µ(t)] I D(t, I)+ θ(t, I). (18) I λ(t)+µ(t) =e ρt K(P )+c. (19) P Now, consier Equation (17). Then, on some subset S of [0,T], we have P (t) D(t, I(t)) θ(t, I(t)) = 0 an the optimal control in this case is given by P (t) =D(t, I (t)) + θ(t, I (t)), for all t S. (0) On the set [0,T]\S, wehavep (t) D(t, I(t)) θ(t, I(t)) > 0. Using the same argument as in the previous section, we obtain the following secon orer ifferential equation:

6 Bounkhel et at. 41 P (t) t P K(P ) ρ + I D(t, I)+ I) I θ(t, K(P )+c = P I h(i) c D(t, I), (1) I an I(0) = I 0, P K(P (T )) = βeρt. Let us assume now K(P )= KP, h(i) = hi, D(t, I) = 1(t) + I, an θ(t, I) =θ 1 (t) +θ I where K, h,,anθ are positive constants. Then the previous ifferential equation in P becomes the following secon orer ifferential equation in I t I(t) ρ t I(t) h K +( + θ )(ρ + + θ ) I(t) =α(t), () with α(t) =(ρ+ +θ )( 1 (t)+θ 1 (t)) t 1(t) t θ 1(t) c,ani(0) = I 0,I(T)=I T. The solution of this two-point bounary value problem is given by Equation (13) with m 1 = 1 h ρ ρ +4 K +(ρ + + θ )( + θ ), m = 1 h ρ + ρ +4 K +(ρ + + θ )( + θ ), a 1 = I T Q(T ) (I 0 Q(0))e m T e m1t e mt, a = (I 0 Q(0))e m1t I T + Q(T ) e m 1T e m, T where Q(t) is a particular solution of (). The expression of P is euce using I along with the state equation. Finally, as in Theorem.1, the optimal solution (P,I )of(p θ ) is given on [0,T]\S by the solution of the ifferential equation () an its corresponing optimal prouction while on S, it has the form given in (0). EXAMPLE. Consier the prouction system of Example.1 an let I(T )= 0 an the unit cost c =0.1. The eterioration rate is such that θ 1 (t) =sin(t)+1, θ =0.1. The optimal control an state are isplaye in Figure 3. The optimal objective function value is J = To assess the effect of the eterioration rate on the value of the optimal objective function, we set θ 1 = 0 an varie the value of θ from to As shown by the table below, the resulting optimal cost increases as θ increases. θ J Conclusion Explicit optimal controls are obtaine for two general inventory-level-epenent eman prouction moels. These moels can be extene in various ways. For example,

7 4 Optimal Control of a Prouction System instea of minimizing the total cost, one may want to maximize the total profit where the unit revenue rate is both function of time an of the inventory level. 10 Inventory I(t) Prouction P(t) I(t) P(t) Figure 1. Figure 1. Variations of (P,I )asfunctionoftimet for 1 (t) =cos(t) Inventory I(t) 3.5 Prouction P(t) I(t) 6 P(t) Figure. Variations of (P,I )for 1(t) =e t. 0 Inventory I(t) Prouction P(t) I(t) 10 P(t) Figure 3. Variations of (P,I ) as function of.

8 Bounkhel et at. 43 References [1] K. J. Chung, An algorithm for an inventory moel with inventory-level-epenent eman rate, Computers an Operations Research, 30(003), [] B. C. Giri an K. S. Chauhuri, Deterministic moels of perishable inventory with stock-epenent eman rate an nonlinear holing cost, European Journal of Operational Research, 105(3)(1998), [3] S. K. Goyal an B. C. Giri, Recent trens in moeling of eteriorating inventory, European Journal of Operational Research, 134(001), [4] E. Khemlnitsky an Y. Gerchak, Optimal Control Approach to Prouction Systems with Inventory-Level-Depenent Deman, IIE Transactions on Automatic Control, (47)(3)(003), [5] E. Naor, Inventory Systems, John Wiley an Sons, New York [6] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize an E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley an Sons, New York 196. [7] Y. Salama, Optimal control of a simple manufacturing system with restarting costs, Operations Research Letters, 6(000), [8] H. A. Simon, On the application of servomechanism theory in the stuy of prouction control, Econometrica, 0(195), [9] J. T. Teng an C. T. Chang, Economic prouction quantity moels for eteriorating items with price- an stock-epenent eman, Computers an Operations Research, 3()(005),