E:\UZIV\SARKA\Clanky\RM_25\Chaudha\final\RM_25_3_.dvi RATIO MATHEMATICA 25 (2013), 29–46 ISSN:1592-7415 Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market Kuldeep Chaudhary, Yogender Singh, P. C. Jha Department of Operational Research, University of Delhi, Delhi-110007, India chaudharyiitr33@gmail.com, aeiou.yogi@gmail.com, jhapc@yahoo.com Abstract In this paper, we use market segmentation approach in multi- product inventory - production system with deteriorating items. The objective is to make use of optimal control theory to solve the pro- duction inventory problem and develop an optimal production policy that maximize the total profit associated with inventory and produc- tion rate in segmented market. First, we consider a single production and inventory problem with multi-destination demand that vary from segment to segment. Further, we described a single source production multi destination inventory and demand problem under the assump- tion that firm may choose independently the inventory directed to each segment. This problem has been discussed using numerical ex- ample. Key words: Market Segmentation, Production-Inventory System, Optimal Control Problem MSC2010: 97U99. 1 Introduction Market segmentation is an essential element of marketing in industrialized countries. Goods can no longer be produced and sold without considering customer needs and recognizing the heterogeneity of these needs [1]. Earlier 29 K. Chaudhary, Y. Singh, P. C. Jha in this century, industrial development in various sectors of economy induced strategies of mass production and marketing. Those strategies were manu- facturing oriented, focusing on reduction of production costs rather than satisfaction of customers. But as production processes become more flexible, and customer’s affluence led to the diversification of demand, firms that iden- tified the specific needs of groups of customers were able to develop the right offer for one or more submarkets and thus obtained a competitive advantage. Segmentation has emerged as a key planning tool and the foundation for effective strategy formulation. Nevertheless, market segmentation is not well known in mathematical inventory-production models. Only a few papers on inventory-production models deal with market segmentation [2, 3]. Optimal control theory, a modern extension of the calculus of variations, is a math- ematical optimization tool for deriving control policies. It has been used in inventory-production [4, 6] to derive the theoretical structure of optimal policies. Apart from inventory-production, it has been successfully applied to many areas of operational research such as Finance [7, 8], Economics [9, 10, 11], Marketing [12, 13, 14, 15], Maintenance [16] and the Consump- tion of Natural Resources [17, 18, 19] etc. The application of optimal control theory in inventory-production control analysis is possible due to its dynamic behaviour. Continuous optimal control models provide a powerful tool for understanding the behaviour of production-inventory system where dynamic aspect plays an important role. Several papers have been written on the application of optimal control theory in Production-Inventory system with deteriorating items [20, 21, 22, 23]. In this paper, we assume that firm has defined its target market in a segmented consumer population and that it develop a production-inventory plan to attack each segment with the objective of maximizing profit. In ad- dition, we shed some light on the problem in the control of a single firm with a finite production capacity (producing a multi-product at a time) that serves as a supplier of a multi product to multiple market segments. Seg- mented customers place demand continuously over time with rates that vary from segment to segment. In response to segmented customer demand, the firm must decide on how much inventory to stock and when to replenish this stock by producing. We apply optimal control theory to solve the prob- lem and find the optimal production and inventory policies. The rest of the paper is organized as follows. Following this introduction, all the nota- tions and assumptions needed in the sequel is stated in Section 2. In section 3, we described the single source production-inventory problem with multi- destination demand that vary from segment to segment and developed the optimal control theory problem and its solution. In section 4 of this paper we introduce optimal control formulation of a single source production- multi 30 Optimal Control Policy of a Production and Inventory System for ... destination demand and inventory problem and discussion of solution. Nu- merical illustration is presented in the section 5 and finally conclusions are drawn in section 6 with some future research directions. 2 Notations and Assumptions Here we begin the analysis by stating the model with as few notations as possible. Let us consider a manufacturing firm producing m product in segmented market environment. We introduce the notation that is used in the development of the model: Notations: T : Length of planning period, Pj (t) : Production rate for j th product, Ij (t) : Inventory level for j th product, Iij (t) : Inventory level for j th product in ith segment, Dij (t) : Demand rate for j th product in ith segment, hj (Ij (t)) : Holding cost rate for j th product, (single source inventory) hij (Iij (t)) : Holding cost rate for j th product in ith segment, (multi destination) cj : The unit production cost rate for j th product, θj (t, Ij (t)) : Deterioration rate for j th product, (single source inventory) θij (t, Iij (t)) : The deterioration rate for j th product in ith segment, (multi destination) Kj (Pj(t)) : cost rate corresponding to the production rate for j th product, rij : The revenue rate per unit sale for j th product in ith segment, ρ : Constant non-negative discount rate. The model is based on the following assumptions: We assume that the time horizon is finite. The model is developed for multi-product in segmented market. The production, demand, and deterioration rates are function of time. The holding cost rate is function of inventory level & production cost rate depends on the production rate. The functions hij (Iij(t)) (in case of single source hj (Ij (t)) and θij (t, Iij(t)) (in case of single source θj (t, Ij(t))) are convex. All functions are assumed to be non negative, continuous and differentiable functions. This allows us to derive the most general and robust conclusions. Further, we will consider more specific cases for which we obtain 31 K. Chaudhary, Y. Singh, P. C. Jha some important results. 3 Single Source Production and Inventory- Multi-Destination Demand Problem Many manufacturing enterprises use a production-inventory system to manage fluctuations in consumers demand for the product. Such a system consists of a manufacturing plant and a finished goods warehouse to store those products which manufactured but not immediately sold. Here, we assume that once a product is made and put inventory into single warehouse, and demand for all products comes from each segment. Let there be m products and n segments. (i.e., j = 1, . . . , m and i = 1, . . . , n). Therefore, the inventory evolution in segmented market is described by the following differential equation: d dt Ij (t) = Pj(t) − n ∑ i=1 Dij(t) − θj (t, Ij (t)), ∀j = 1, . . . , m. (1) So far, firm want to maximize the total Profit during planning period in segmented market. Therefore, the objective functional for all segments is defined as max Pj (t)≥ P m i=1 Dij (t)+θj (t,Ij (t)) J = ∫ T 0 e−ρt m ∑ j=1 [ n ∑ i=1 rijDij (t) − Kj(Pj (t)) − hj (Ij (t)) ] dt + ∫ T 0 e−ρt m ∑ j=1 [ cj ( n ∑ i=1 Dij (t) − Pj(t) )] dt (2) Subject to the equation (1).This is the optimal control problem with m- control variable (rate of production) with m-state variable (inventory states). Since total demand occurs at rate ∑n i=1 Dij (t) and production occurs at controllable rate Pj(t) for j th, it follows that Ij(t) evolves according to the above state equation (1). The constraints Pj(t) ≥ ∑m i=1 Dij(t) − θj (t, Ij(t)) and Ij (0) = Ij0 ≥ 0 ensure that shortage are not allowed. Using the maximum principle [10], the necessary conditions for (P ∗j , I ∗ j ) to be an optimal solution of above problem are that there should exist a piecewise continuously differentiable function λ and piecewise continuous function µ , 32 Optimal Control Policy of a Production and Inventory System for ... called the adjoint and Lagrange multiplier function, respectively such that H(t, I∗, P ∗, λ) ≥ H(t, I∗, P, λ), for all Pj (t) ≥ n ∑ i=1 Dij (t) − θj (t, Ij(t) (3) d dt λj (t) = − ∂ ∂Ij L(t, Ij , Pj, λj, µj) (4) Ij (0) = Ij0, λj(T ) = 0 (5) ∂ ∂Pj L(t, Ij , Pj, λj, µj) = 0 (6) Pj (t) − n ∑ i=1 Dij(t) − θj (t, Ij (t)) ≥ 0, µj(t) ≥ 0, µj(t) [ Pj(t) − n ∑ i=1 Dij(t) − θj (t, Ij (t)) ] = 0 (7) Where, H(t, I, P, λ) and L(t, I, P, λ, µ) are Hamiltonian function and La- grangian function respectively. In the present problem Hamiltonian function and Lagrangian function are defined as H = m ∑ j=1 [ n ∑ i=1 rijDij (t) + cj ( n ∑ i=1 Dij (t) − Pj(t) ) − Kj (Pj(t)) − hj (Ij(t)) ] + m ∑ j=1 [ λj(t) ( Pj(t) − n ∑ i=1 Dij (t) − θj (t, Ij(t)) )] (8) L = m ∑ j=1 [ n ∑ i=1 rijDij (t) + cj ( n ∑ i=1 Dij (t) − Pj(t) ) − Kj (Pj(t)) − hj (Ij(t)) ] + m ∑ j=1 [ (λj(t) + µj (t)) ( Pj(t) − n ∑ i=1 Dij(t) − θj (t, Ij (t)) )] (9) A simple interpretation of the Hamiltonian is that it represents the overall profit of the various policy decisions with both the immediate and the future effects taken into account and the value of λj(t) at time t describes the future effect on profits upon making a small change in Ij(t) . Let the Hamiltonian H for all segments is strictly concave in Pj(t) and according to the Mangasarian sufficiency theorem [4, 10]; there exists a unique Production rate. 33 K. Chaudhary, Y. Singh, P. C. Jha From equation (4) and (6), we have following equations respectively d dt λj(t) = ρλj (t) − { − ∂hj (Ij(t)) ∂Ij − (λj(t) + µj (t)) ∂θj (t, Ij(t)) ∂Ij } , (10) for all j = 1, · · · , m λj (t) + µj(t) = cj + d dPj Kj (Pj(t)). (11) Now, consider equation (7). Then for any t, we have either Pj (t) − n ∑ i=1 Dij (t) − θj (t, Ij (t)) = 0 or Pj (t) − n ∑ i=1 Dij (t) − θj (t, Ij (t)) > 0 ∀ j = 1, · · · , m. 3.1 Case 1: Let S is a subset of planning period [0, T ], when Pj (t) − ∑n i=1 Dij (t) − θj (t, Ij(t)) = 0. Then d dt Ij (t) = 0 on S, in this case I ∗(t) is obviously constant on S and the optimal production rate is given by the following equation P ∗j (t) = n ∑ i=1 Dij (t) − θj (t, I ∗ j (t)), for all t ∈ S (12) By equation (10) and (11), we have d dt λj(t) = ρλj(t) − { − ∂hj (Ij(t)) ∂Ij − ( cj + d dPj Kj (Pj(t)) ) ∂θj (t, Ij (t)) ∂Ij } (13) After solving the above equation, we get a explicit from of the adjoint func- tion λj(t). From the equation (10)), we can obtain the value 0f Lagrange multiplier µj(t). 3.2 Case 2: Pj(t) − ∑n i=1 Dij(t) − θj (t, Ij (t)) > 0, for t ∈ [0, T ]\S. Then µj (t) = 0 on t ∈ [0, T ]\S. In this case the equation (10) and (11) becomes d dt λj(t) = ρλj (t) − { − ∂hj (Ij(t)) ∂Ij − λj(t) ∂θj (t, Ij (t)) ∂Ij } , ∀ j = 1, · · · , m (14) λj (t) = cj + d dPj Kj(Pj (t)) (15) 34 Optimal Control Policy of a Production and Inventory System for ... Cobining these equation with the state equation, we have the following second order differential equation: d dt Pj(t) d2 dP 2j Kj (Pj) − [ ρ + ∂θj (t, Ij (t)) ∂Ij ] d dPj Kj (Pj) =cj ( ρ + ∂θj (t, Ij(t)) ∂Ij ) + ∂hj (t, Ij (t)) ∂Ij (16) and Ij(0) = Ij0, cj + d dPj Kj (Pj(T )) = 0. For illustration purpose, let us assume the following forms the exogenous functions Kj(Pj ) = kj P 2 j /2, hj(t, Ij (t)) = hjIj (t) and θj (t, Ij (t)) = θj Ij(t), where kj hj θj are positive constants for all j = 1, · · · , m. For these functions the necessary conditions for (P ∗j , I ∗ j ) to be optimal solu- tion of problem (2) with equation (1) becomes d2 dt2 Ij (t) − ρ d dt Ij (t) − (ρ + θj )θ1j Ij (t) = ηj (t) (17) with Ij(0) = Ij0, cj + d dPj Kj(Pj (T )) = 0. Where, ηj(t) = − ∑n i=1 ( d dt Dij (t) ) + (ρ + θ1j ) ( ∑n i=1 Dij(t) ) + (cj (ρ+θ1j )+hj ) kj . This problem is a two point boundary value problem. Proposition 3.1. The optimal solution of (P ∗j , I ∗ j ) to the problem is given by I∗j (t) = a1j e m1j t + a2j e m2j t + Qj (t), (18) and the corresponding P ∗j is given by P ∗j (t) =a1j (m1j + θ1j )e m1j t + a2j (m2j + θj )e m2j t + d dt Qj (t) + θ1j Qj (t) + n ∑ i=1 Dij. (19) The values of the constant a1j , a2j , m1j , m2j are given in the proof, and Qj(t) is a particular solution of the equation (17). Proof. The solution of the two point boundary value problem (17) is given by standard method. Its characteristic equation m2j − ρmj − (ρ + θj )θ1j = 0, has two real roots of opposite sign, given by m1j = 1 2 ( ρ − √ ρ2 + 4(ρ + θ1j )θ1j ) < 0, m2j = 1 2 ( ρ + √ ρ2 + 4(ρ + θ1j )θ1j ) > 0, 35 K. Chaudhary, Y. Singh, P. C. Jha and therefore I∗j (t) is given by (18), where Qj (t) is the particular solution. Then initial and terminal condition used to determineed the values of con- stant a1j and a2j as follows a1j + a2j + Qj (0) = Ij0, a1j (m1j + θ1j )e m1j T + a2j (m1j + θ1j )e m2j T + ( cj kj + d dt Qj (T ) + θ1j Qj(T ) + n ∑ i=1 Dij(T ) ) = 0. By putting b1j = Ij0 − Qj(0) and b2j = −( cj kj + d dt Qj (T ) + θ1j Qj (T ) + ∑n i=1 Dij(T )), we obtain the following system of two linear equation with two unknowns a1j + a2j = b1j a1j (m1j + θ1j )e m1j T + a2j (m1j + θ1j )e m2j T = b2j (20) The value of P ∗j is deduced using the values of I ∗ j and the state equation. 4 Single Source Production- Multi Destina- tion Demand and Inventory Problem We assume the single source production and multi destination demand- inventory system. Hence, the inventory evolution in each segmented is de- scribed by the following differential equation: d dt Iij(t) = γijPj(t) − Dij (t) − θij (t, Iij(t)), ∀ j = 1, · · · , m; i = 1, · · · , n. (21) Here, γij > 0, ∑n i=1 γij = 1, ∀ j = 1, · · · m with the conditions Iij (0) = I 0 ij and γijPj (t) ≥ Dij (t) − θij (t, Iij(t)). We called γij > 0 the segment produc- tion spectrum and γijPj(t) define the relative segment production rate of j th product towards ith segment. We develop a marketing-production model in which firm seeks to maximize its all profit by properly choosing production and market segmentation. Therefore, we defined the profit maximization 36 Optimal Control Policy of a Production and Inventory System for ... objective function as follows: max γij Pj (t)≥Dij (t)−θij (t,Iij (t)) J = = ∫ T 0 e−ρt m ∑ j=1 [ n ∑ i=1 rij Dij(t) + cj ( n ∑ i=1 (Dij (t) − γijPj(t)) )] dt − ∫ T 0 e−ρt m ∑ j=1 [ n ∑ i=1 hij (Iij(t)) − Kj(Pj (t)) ] dt (22) subject to the equation (21). This is the optimal control problem (production rate) with m control variable with mn state variable (stock of inventory). To solve the optimal control problem expressed in equation (21) and (22), the following Hamiltonian and Lagrangian are defined as H = m ∑ j=1 [ n ∑ i=1 rijDij (t) + cj ( n ∑ i=1 (Dij(t) − γijPj (t)) )] − m ∑ j=1 [ n ∑ i=1 hij (Iij(t)) + Kj (Pj(t)) ] + m ∑ j=1 n ∑ i=1 λij (t)[γiPj(t) − Dij (t) − θij (t, Iij(t))] (23) L = m ∑ j=1 [ n ∑ i=1 rij Dij(t) + cj ( n ∑ i=1 (Dij (t) − γijPj(t)) )] − m ∑ j=1 [ n ∑ i=1 hij (Iij (t)) + Kj(Pj (t)) ] + m ∑ j=1 n ∑ i=1 (λij + µij(t))[γiPj(t) − Dij (t) − θij (t, Iij(t))] (24) Equation (4), (6) and (21) yield d dt λij (t) = ρλij (t) − { − ∂hij (Iij (t)) ∂Iij − λij(t) ∂θij (t, Iij (t)) ∂Iij } , (25) for all i = 1, · · · , n, j = 1, · · · , m n ∑ i=1 (λij (t) + µij(t))γi = cj + d dPj Kj (Pj(t)) (26) In the next section of the paper, we consider only case when γijPj(t) − Dij(t) − θij (t, Iij (t)) > 0, ∀ i, j. 37 K. Chaudhary, Y. Singh, P. C. Jha 4.1 Case 2: γijPj(t)−Dij (t)−θij (t, Iij(t)) > 0 ∀ i, j, for t ∈ [0, T ]\S. Then µij(t) = 0 on t ∈ [0, T ]\S. In this case, the equation (25) and (26) becomes d dt λij (t) = ρλij (t) − { − ∂hij (Iij (t)) ∂Iij − λij(t) ∂θij (t, Iij(t)) ∂Iij } (27) n ∑ i=1 γijλij (t) = cj + d dPj Kj (Pj(t)) (28) Cobining these equation with the state equation, we have the following second order differential equation: d dt Pj(t) d2 dP 2j Kj(Pj ) − 1 n n ∑ i=1 ( ρ + ∂θi(t, Iij (t)) ∂Iij ) d dPj Kj (Pj) = n ∑ i=1 cjγi ( ρ + ∂θij (t, Iij(t)) ∂Iij ) + n ∑ i=1 γi ∂hij (t, Iij(t)) ∂Iij (29) with Ij (0) = I 0 ij , ∑n i=1 γij λij(T ) = 0 → λij(T ) = 0 ∀ i and j, cj + d dPj Kj (Pj(T )) = 0. For illustration, let us assume the following forms the ex- ogenous functions Kj(Pj) = kj P 2 j /2, hij (t, Iij(t)) = hij Iij (t) and θij (t, Iij (t)) = θij Iij(t), where kj hij θij are positive constants. For these functions the necessary conditions for (P ∗j , I ∗ ij) to be optimal solu- tion of problem (19) with equation (18) becomes d2Iij (t) dt2 + (θij − A) dIij(t) dt − AθiIij (t) = ηij (t) (30) with Iij(0) = I 0 ij , λij (T ) = 0 ∀ i, cj + d dPj Kj(Pj (T )) = 0. Where, ηij (t) = −Dij (t)A + γj kj [ ∑n i=1 γi(hij + cj (ρ + θij )) ] + dDij (t) dt , A = ∑n i=1 (ρ+θij ) n . This problem is a two point boundary value problem. The above system of two point boundary value problem (29) is solved by same method that we used in to solve (17). 5 Numerical Illustration In order to demonstrate the numerical results of the above problem, the discounted continuous optimal problem (2) is transferred into equivalent dis- crete problem [24] that is solved to present numerical solution. The discrete 38 Optimal Control Policy of a Production and Inventory System for ... optimal control can be written as follows: J = T ∑ k=1 ( m ∑ j=1 [ n ∑ i=1 (rij(k − 1)Dij(k − 1)) ])( 1 (1 + ρ)k−2 ) + T ∑ k=1 ( m ∑ j=1 cj ( n ∑ i=1 Dij(k − 1) − Pj (k − 1) ))( 1 (1 + ρ)k−2 ) − T ∑ k=1 ( m ∑ j=1 [Kj (Pj(k − 1) + hj(Ij (k − 1)))] )( 1 (1 + ρ)k−2 ) such that Ij(k) = Ij(k − 1) + pj(k − 1) − n ∑ i=1 Dij (k − 1) − θj (k − 1, Ij(k − 1)) for all j = 1, · · · , m. Similar discrete optimal control problem can be written for single source production multi destination and inventory control problem. These discrete optimal control problems are solved by using Lingo11. We assume that the duration of all the time periods are equal and demand are equal from segment for each product. The number of market segmentsis 4 and the number of products is 3. The value of parameters are ri1 = 2.55, 2.53, 2.53, 2.54; Table 1: The Optimal production and inventory rate in segment market T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 P1 100 86 80 73 64 53 39 21 5 0 P2 110 81 76 70 62 52 38 21 5 0 P3 140 79 75 69 61 51 38 21 5 0 I1 20 98 154 199 232 254 262 255 231 193 I2 20 107 156 194 222 238 241 231 205 166 I3 20 137 179 211 233 244 244 231 203 161 ri2 = 2.52, 2.53, 2.54, 2.53; ri3 = 2.51, 2.54, 2.54, 2.52 for segments i = 1 to 4; cj = 1; kj = 2; θj = 0.10, 0.12, 0.13; hj = 1; for all the three products. The optimal production rate and inventory for every product for each segment is shown in Table 1 and their corresponding total profit is $177402.70. 39 K. Chaudhary, Y. Singh, P. C. Jha The optimal trajectories of production and inventory rate for every product for each segment are shown in Fig1, Fig2 and Fig3 respectively (Appendix). In case of single source production-multi destination demand and inventory, the number of market segments M is 4 and the number of products is 3. The values of additional parameters are each segment is shown in Table 2. Table 2: The values of parameter of deteriorating rate and holding cost rate constant Segment θi1 θi2 θi3 hi1 hi2 hi3 M1 0.10 0.11 0.11 1.0 1.1 1.0 M2 0.11 0.12 0.12 1.1 1.2 1.1 M3 0.13 0.11 0.11 1.2 1.1 1.2 M4 0.11 0.13 0.11 1.1 1.0 1.3 Table 3: Values of the parameter for single source production-multi destina- tion demand and inventory problem in each segment T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 P1 100 85 79 73 65 54 41 23 5 0 P2 110 82 77 70 62 52 38 21 5 0 P3 140 83 77 71 63 52 38 21 5 0 I11 20 98 153 197 231 254 263 258 236 197 I12 20 97 152 195 227 247 255 248 225 185 I13 20 97 150 192 223 242 247 239 214 173 I14 20 97 149 190 218 236 240 230 204 162 I21 20 108 158 198 227 245 250 242 217 178 I22 20 107 157 195 222 238 242 232 206 167 I23 20 108 158 198 227 245 250 242 217 178 I24 20 107 156 192 218 232 235 223 196 156 I31 20 138 186 223 250 265 268 258 231 191 I32 20 138 184 220 244 258 260 248 219 178 I33 20 138 185 223 250 265 268 258 231 191 I34 20 138 186 223 250 265 268 258 231 191 The optimal production rate and inventory for every product for each segment is shown in Table 3 with production spectrum γ11 = 0.10, γ12 = 0.10, γ13 = 0.77, γ14 = 0.03; γ21 = 0.12, γ22 = 0.12, γ23 = 0.75, γ24 = 0.01; γ31 = 0.14, γ32 = 0.14, γ33 = 0.72, γ34 = 0.04. The optimal value of 40 Optimal Control Policy of a Production and Inventory System for ... total profit for all products is $185876.90. In case of single source production- multi destination demand and inventory, The optimal trajectories of produc- tion and inventory rate for every product for each segment are shown in Fig4, Fig5, Fig6 and Fig7 respectively (Appendix). 6 Conclusion In this paper, we have introduced market segmentation concept in the production inventory system for multi product and its optimal control for- mulation. We have used maximum principle to determine the optimal pro- duction rate policy that maximizes the total profit associated with inventory and production rate. The resulting analytical solution yield good insight on how production planning task can be carried out in segmented market environment. 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Jha 0 50 100 150 200 250 300 1 2 3 4 5 6 7 8 9 10 I 1 I 2 I 3 Fig-2 � �� ��� ��� ��� ��� ��� � � � � � � � � �� P � P � P � � Fig-3 � �� � �� �� ��� ��� � � ��� � � � � � � � � �� � � � � � � Fig-4 44 Optimal Control Policy of a Production and Inventory System for ... 0 50 100 150 200 250 300 1 2 3 4 5 6 7 8 9 10 I 11 I 12 Fig-5 � �� ��� ��� ��� ��� ��� � � � � � � � � �� �� �� �� �� Fig-6 � �� �� �� ��� ��� ��� � � � � � � � � � � �� � �� � �� � �� Fig-7 45 46