Optimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs

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1 Applie Mathematics E-Notes, 8(2008), c ISSN Available free at mirror sites of amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items An Dynamic Costs Yacine Benhai, Lotfi Taj an Messaou Bounkhel Receive 11 March 2007 Abstract The objective of the present research is to make use of optimal control theory to solve a prouction inventory problem with eteriorating items an ynamic costs. 1 Introuction A great eal of effort has been focuse on the moelling of the prouction planning problem in a eterministic environment. Early static moels, such as the classical economic orer quantity (EOQ) moel, assume a constant eman rate an are solve using classical optimization tools. The ynamic counterparts, also known as generalize economic orer quantity (GEOQ) moels, assume a time varying eman rate. They are tackle using either classical optimization or optimal control theory. Which of the two theories is more appropriate to solve a particular problem an what are the possibilities an limitations of each of them are questions that are outsie the scope of this paper. For the intereste reaer see, for example, Axsäter [3] who provies such a critical evaluation. In this paper we are using optimal control theory, which is a branch of mathematics particularly well suite to fin optimal ways to control a ynamic system. It prove its efficiency in many areas of operations research such as finance [7, 9, 23], economics [2, 10, 16], marketing [11, 22], maintenance [18, 19], environment an transportation [4, 17], an the consumption of natural resources [1, 8]. We are especially intereste in the application of optimal control theory to the prouction planning problem. During the last two ecaes, various authors attacke this research irection. We state here some of them: Bounkhel an Taj [5], Hejar et al. [14], Khemlnitsky an Gerchak [17], Rials an Bennett [20], Salama [21], an Zhang et al. [25]. The problem of interest to us is relate to one of the assumptions of the classical EOQ moel an has receive very little attention in the literature. It concerns the various unit costs involve, which are assume to be known an constant. In a GEOQ moel, the eman rate, instea of being uniform, is assume to be ynamic, which is Mathematics Subject Classifications: 49J15, 90B30. College of Science, King Sau University, P.O.Box 2455, Riyah 11451, Saui Arabia. 194

2 Benhai et al. 195 a more realistic assumption. It seems natural to further generalize the GEOQ moel by relaxing the assumption of constant unit costs. Inee, except for the orering cost which is pai only once per cycle, costs can be affecte by time in various ways. For example, the unit cost of an item may ecrease with time as new proucts are prouce. Also, the holing cost may change as time goes by. Only very few papers that aress this issue of ynamic costs came to our attention. The first paper seems to be that of Giri et al. [12], who compute the optimal policy of an EOQ moel with ynamic costs. The moel they propose is very basic though, since they consier the very special case where the holing an orering costs are linear functions of time. The other shortcomings of that paper is that the items eterioration rate is also a linear function of time, an the algorithm they propose in orer to solve the problem is only vali as long as the eman rate is a linear function of time. Teng et al. [24], in a more recent paper, assume that only one of the unit costs, namely the purchasing or prouction cost, is time-varying. The goal of this paper is to generalize the moels of these papers in various ways. First, the costs in our moel are general functions of time instea of being linear functions. Secon, the on-han inventory eteriorates an the eterioration rate is a general function of time instea of being a linear function. Importance of items eteriorating in inventory moelling in now wiely acknowlege, as shown by the recent survey of Goyal an Giri [13]. Thir, the eman rate is a general function of time. Finally, an optimal control approach is use instea of an optimization approach. We aim primarily at eriving the necessary an sufficient conitions for the optimal solution of the moel uner consieration an from there introuce illustrative examples with their numerical verifications which explain the applications of the theoretical results to some real life problems. To construct the objective function of the problem, we assume that the firm has set an inventory goal level, a prouction goal rate, an a eterioration goal rate an penalties are incurre when the inventory level, the prouction rate, an the eterioration rate eviate from their respective goals. The paper eals with both continuous an perioic-review policies. We solve the first moel by applying Pontriyagin maximum principle. In the secon moel, the perioic-review moel, we use Lagrange multiplier metho to minimize an objective function subject to some ifference equation. In both moels we erive explicit optimal policies that can be use by managers to augment their capabilities in the ecisionmaking process. The rest of the paper is organize as follows. Following this introuction, all the notation neee in the sequel is state in Section 2. In Section 3, we buil an solve the continuous-review moel. Our main result in this section is state in Theorem 3.1. An illustrative example of this result is given. We o similarly for the perioic-review moel in Section 4 an the main result in this section is state in Theorem 4.1. The last section conclues the paper. 2 Notation Let us consier a manufacturing firm proucing a single prouct. We first introuce the notation that is inepenent of the time: T : length of the planning horizon, ρ :

3 196 Optimal Control of Prouction Inventory Systems constant nonnegative iscount rate, I 0 : initial inventory level. Next, the notation that epens on time, an that we ivie into two categories. The monetary parameters which are, as we mentione in the Introuction, ynamic: h(t) : holing penalty cost rate at time t, K(t) : prouction penalty cost rate at time t. The nonmonetary parameters that epen on time may also be ivie into two groups. The first group comprises the state variable, the control variable, an the exogenous functions: I(t) : inventory level at time t, P (t) : prouction rate at time t, D(t) : eman rate at time t, θ(t) : eterioration rate at time t. The secon group comprises the goals: Î(t) : inventory goal level at time t, ˆP(t) : prouction goal rate at time t. The interpretation of the goal rates are as follows: The inventory goal level Î is a safety stock that the company wants to keep on han. For example, Î coul be 200 units of the finishe prouct uring [0,t 1 ] an 180 units uring [t 1,T]. The prouction goal rate ˆP is the most efficient rate esire by the firm. All functions are assume to be non-negative, continuous an ifferentiable functions. As we mentione in the Introuction, we will be consiering both cases where the firms aopts a continuous-review an a perioic-review policy. The notation that oes not epen on time will be the same for both cases. Concerning the notation that epens on time, the variable t will be use in the continuous time case an the variable k will be use in the iscrete time case. 3 Moel Solution 3.1 Continuous-Review Policy We first assume that the firm aopts a continuous-review policy. The ynamics of the inventory level I(t) are governe by the following ifferential equation: I(t) =P (t) D(t) θ(t)i(t), (1) t with I(0) = I 0. The moel is presente as an optimal control problem with one state variable (inventory level) an one control variable (rate of manufacturing). The problem (P cr ) associate to this moel is to minimize the following objective function T min J(P, I) = F (t, I(t),P(t))t, P (t) 0 0 subject to the state equation (1) where an F (t, I(t),P(t)) = 1 2 e ρt { h(t) 2 I(t)+K(t) 2 P (t) }, (2) I(t) =I(t) Î(t), P (t) =P (t) ˆP(t). We aopt the wiely use quaratic objective function of Holt, Moigliani, Muth, an Simon (HMMS) [15]. The interpretation of this objective function is that penalties are incurre when the inventory level an prouction rate eviate from their respective

4 Benhai et al. 197 goals. Note that we are etermining the present value of future costs by iscounting them using the appropriate cost of capital. This is necessary because cash flows in ifferent time perios cannot be irectly compare since most people prefer money sooner rather than later. This is a simple interpretation of the fact that a ollar in your han toay is worth more than a ollar you may receive at some point in the future. The following is the main result of this section. THEOREM 1. Necessary conitions for the pair (P, I) to be an optimal solution of problem (P cr ) are [ 0 = 2 t 2 I(t)+ t K(t) ] ρ K(t) t I(t) + [ θ(t) θ(t) t [( t K(t) ) ] ρ θ(t) K(t) ] h(t) I(t) (3) K(t) an I(0) = I 0, P(T )= ˆP(T ), P(t) 0, t [0,T]. PROOF. We will use Pontryagin s maximum principle to solve the above problem (P cr ). The Hamiltonian function is H(t, I(t),P(t),λ(t)) = F (t, I(t),P(t)) + λ(t)f(t, I(t),P(t)), (4) where f(t, I(t), P(t)) is the right-han sie of the state equation (1) an λ is the ajoint function associate with this constraint. Assume (P, I) is an optimal solution to problem (P cr ), then H(t, I(t),P(t),λ(t)) H(t, I(t), P(t),λ(t)), for all P(t) 0, (5) t λ(t) = H(t, I(t),P(t),λ(t)), (6) I I(0) = I 0, λ(t )=0. (7) Equation (5) is equivalent to H(t, I(t),P(t),λ(t)) = 0, P which is equivalent to λ(t) =K(t)e ρt P (t). (8) Equation (6) is equivalent to t λ(t) =h(t)e ρt I(t)+λ(t)θ(t). (9) Now, combining Equation (8) an Equation (9) yiels P (t)+ t [ t K(t) ] ρ θ(t) K(t) P (t)= h(t) I(t). (10) K(t)

5 198 Optimal Control of Prouction Inventory Systems Inventory Level Inventory Level Goal Inventory Level Prouction rate Prouction Rate Goal Prouction Rate Time t Time t Figure 1: Optimal solution for Example 3.1. Using the fact that all the goal rates must satisfy the state equation we get tî(t) = ˆP (t) D(t) θ(t)î(t), (11) which yiels with the state equation (1) P (t) = I(t)+θ(t) I(t). (12) t Combining this equation with (10) yiels the following secon orer ifferential equation (3). The fact that P (T )= ˆP (T ) follows irectly from the equations (8) an (7). Thus the proof is complete. REMARK 1. The problem stuie in the present paper is convex an its convexity is ensure by the convexity of the function I F (t, I(t),P(t)) (for other convex an nonconvex problems see for instance [5]). EXAMPLE 1. We present an illustrative example of the obtaine results. Take T =5,ρ =0.001, I 0 =2,h(t) =K(t) =1+t, D(t) = 2 sin(t)+3, θ(t) = t, Î(t) =1+t + sin(t), an ˆP (t) is compute from the state equation. Figure 1 shows the variations of the optimal inventory level (left) an the optimal prouction rate (right). 3.2 Perioic-Review Policy We now assume that the firm aopts a perioic-review policy. The ynamics of the inventory level I(k) are governe by the following ifference equation: I(k +1) I(k) =P (k) D(k) θ(k)i(k), 1 k N, (1)

6 Benhai et al Prouction Rate Goal Prouction Rate Inventory rate Inventory Rate Goal Inventory Rate Prouction rate Time t Time t Figure 2: Optimal solution for Example 4.1. where N is the length of the planning horizon. Introuce the shifte variables I(k) an P (k) as I(k) =I(k) Î(k) an P (k) =P (k) ˆP (k). Note that the goal rates Î(k) an ˆP (k) must satisfy the equation Î(k +1) Î(k) = ˆP(k) D(k) θ(k)î(k), 1 k N, (2) an therefore equations (1) an (2) lea to I(k +1)= P (k)+(1 θ(k)) I(k). (3) The problem (P pr ) associate to this moel is to minimize the following objective function min J(P, I) = N F (k, I(k),P(k)), P 0 subject to the state equation (3) where k=1 F (k, I(k),P(k)) = 1 1 { h(k) 2 2 (1 + ρ) k 1 I(k)+K(k) 2 P (k) }. (4) The following is the main result of this section. THEOREM 2. A necessary conition for the pair (P, I) =(P (k),i(k)) 1 k N to be an optimal solution of problem (P pr )is I(k +1) = Î(k + 1) + [1 θ(k)] [1 α(k + 1)], (5) P (k) = α(k + 1) [1 θ(k)] I(k),

7 200 Optimal Control of Prouction Inventory Systems (1+ρ) where α(k +1)= k 1 s(k+1) K(k)+(1+ρ) k 1 s(k+1), an s(k) is given in the proof. PROOF. In orer to solve our perioic-review problem, we introuce the Lagrangian function N L(I,P,λ)= [F (k, I(k),P(k)) + λ(k +1)f(k, I(k),P(k))], (6) k=1 where f(k, I(k),P(k)) = I(k +1)+ P (k)+(1 θ(k)) I(k) an λ(k + 1) is the Lagrange multiplier associate with the ifference equation constraint (3). Thus, the necessary optimality conitions for (P, I) to be an optimal solution for (P pr ) are L P (k) = 0 an L I(k) =0 (1 k N). After some computations, these equations yiel respectively P (k) = (1+ρ) k 1 K(k) 1 h(k) λ(k+1) an λ(k) = I(k)+[1 θ(k)] λ(k+1) (1 + ρ) k 1 (7) The sweep metho (by Bryson an Ho [6]) assumes that λ(k) = s(k) I(k), with s(k) > 0, for k =0,...,N. Subtituting the last equation an (3) into (7) yiels P (k)= (1 + ρ)k 1 s(k +1)[1 θ(k)] K(k)+(1+ρ) k 1 I(k) an (8) s(k +1) s(k) =s(k +1)[1 θ(k)] [1 2 (1 + ρ) k 1 ] s(k +1) h(k) +. (9) K(k)+(1+ρ) k 1 s(k +1) (1 + ρ) k 1 By the fact that P (N) = 0 an by using the equations (8) an (9) we get s(n) = h(n) (1+ρ). The recursive equation (9) for s(k) is solve backwars starting from the N 1 en point N. Therefore, the esire optimal solutions are given by (5). Thus the proof is complete. REMARK 2. Note that it is not har to show that the necessary optimality conition (5) is also sufficient by the convexity of the function F as a function of I (see for instance [5]). EXAMPLE 2. Taking the same ata as in Example 1., Figure 2 shows the variations of the optimal inventory level (left) an the optimal prouction rate (right). References [1] R. Amit, Petroleum reservoir exploitation: switching from primary to seconary recovery, Operations Research, 34(4)(1986), [2] K. J. Arrow an M. Kurz, Public Investment, The rate of return, an Optimal Fiscal Policy, The John Hopkins Press, Baltimore, 1970.

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