RATIO MATHEMATICA 25 (2013), 29 46 ISSN:1592-7415 Optimal Control Policy of a Prouction an Inventory System for multi-prouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational Research, University of Delhi, Delhi-110007, Inia chauharyiitr33@gmail.com, aeiou.yogi@gmail.com, jhapc@yahoo.com Abstract In this paper, we use market segmentation approach in multiprouct inventory - prouction system with eteriorating items. The objective is to make use of optimal control theory to solve the prouction inventory problem an evelop an optimal prouction policy that maximize the total profit associate with inventory an prouction rate in segmente market. First, we consier a single prouction an inventory problem with multi-estination eman that vary from segment to segment. Further, we escribe a single source prouction multi estination inventory an eman problem uner the assumption that firm may choose inepenently the inventory irecte to each segment. This problem has been iscusse using numerical example. Key wors: Market Segmentation, Prouction-Inventory System, Optimal Control Problem MSC2010: 97U99. 1 Introuction Market segmentation is an essential element of marketing in inustrialize countries. Goos can no longer be prouce an sol without consiering customer nees an recognizing the heterogeneity of these nees [1]. Earlier 29
K. Chauhary, Y. Singh, P. C. Jha in this century, inustrial evelopment in various sectors of economy inuce strategies of mass prouction an marketing. Those strategies were manufacturing oriente, focusing on reuction of prouction costs rather than satisfaction of customers. But as prouction processes become more flexible, an customer s affluence le to the iversification of eman, firms that ientifie the specific nees of groups of customers were able to evelop the right offer for one or more submarkets an thus obtaine a competitive avantage. Segmentation has emerge as a key planning tool an the founation for effective strategy formulation. Nevertheless, market segmentation is not well known in mathematical inventory-prouction moels. Only a few papers on inventory-prouction moels eal with market segmentation [2, 3]. Optimal control theory, a moern extension of the calculus of variations, is a mathematical optimization tool for eriving control policies. It has been use in inventory-prouction [4, 6] to erive the theoretical structure of optimal policies. Apart from inventory-prouction, it has been successfully applie to many areas of operational research such as Finance [7, 8], Economics [9, 10, 11], Marketing [12, 13, 14, 15], Maintenance [16] an the Consumption of Natural Resources [17, 18, 19] etc. The application of optimal control theory in inventory-prouction control analysis is possible ue to its ynamic behaviour. Continuous optimal control moels provie a powerful tool for unerstaning the behaviour of prouction-inventory system where ynamic aspect plays an important role. Several papers have been written on the application of optimal control theory in Prouction-Inventory system with eteriorating items [20, 21, 22, 23]. In this paper, we assume that firm has efine its target market in a segmente consumer population an that it evelop a prouction-inventory plan to attack each segment with the objective of maximizing profit. In aition, we she some light on the problem in the control of a single firm with a finite prouction capacity (proucing a multi-prouct at a time) that serves as a supplier of a multi prouct to multiple market segments. Segmente customers place eman continuously over time with rates that vary from segment to segment. In response to segmente customer eman, the firm must ecie on how much inventory to stock an when to replenish this stock by proucing. We apply optimal control theory to solve the problem an fin the optimal prouction an inventory policies. The rest of the paper is organize as follows. Following this introuction, all the notations an assumptions neee in the sequel is state in Section 2. In section 3, we escribe the single source prouction-inventory problem with multiestination eman that vary from segment to segment an evelope the optimal control theory problem an its solution. In section 4 of this paper we introuce optimal control formulation of a single source prouction- multi 30
Optimal Control Policy of a Prouction an Inventory System for... estination eman an inventory problem an iscussion of solution. Numerical illustration is presente in the section 5 an finally conclusions are rawn in section 6 with some future research irections. 2 Notations an Assumptions Here we begin the analysis by stating the moel with as few notations as possible. Let us consier a manufacturing firm proucing m prouct in segmente market environment. We introuce the notation that is use in the evelopment of the moel: Notations: T : Length of planning perio, P j (t) : Prouction rate for j th prouct, I j (t) : Inventory level for j th prouct, I ij (t) : Inventory level for j th prouct in i th segment, D ij (t) : Deman rate for j th prouct in i th segment, h j (I j (t)) : Holing cost rate for j th prouct, (single source inventory) h ij (I ij (t)) : Holing cost rate for j th prouct in i th segment, (multi estination) c j : The unit prouction cost rate for j th prouct, θ j (t, I j (t)) : Deterioration rate for j th prouct, (single source inventory) θ ij (t, I ij (t)) : The eterioration rate for j th prouct in i th segment, (multi estination) K j (P j (t)) : cost rate corresponing to the prouction rate for j th prouct, r ij : The revenue rate per unit sale for j th prouct in i th segment, ρ : Constant non-negative iscount rate. The moel is base on the following assumptions: We assume that the time horizon is finite. The moel is evelope for multi-prouct in segmente market. The prouction, eman, an eterioration rates are function of time. The holing cost rate is function of inventory level & prouction cost rate epens on the prouction rate. The functions h ij (I ij (t)) (in case of single source h j (I j (t)) an θ ij (t, I ij (t)) (in case of single source θ j (t, I j (t))) are convex. All functions are assume to be non negative, continuous an ifferentiable functions. This allows us to erive the most general an robust conclusions. Further, we will consier more specific cases for which we obtain 31
K. Chauhary, Y. Singh, P. C. Jha some important results. 3 Single Source Prouction an Inventory- Multi-Destination Deman Problem Many manufacturing enterprises use a prouction-inventory system to manage fluctuations in consumers eman for the prouct. Such a system consists of a manufacturing plant an a finishe goos warehouse to store those proucts which manufacture but not immeiately sol. Here, we assume that once a prouct is mae an put inventory into single warehouse, an eman for all proucts comes from each segment. Let there be m proucts an n segments. (i.e., j = 1,...,m an i = 1,..., n). Therefore, the inventory evolution in segmente market is escribe by the following ifferential equation: t I j(t) = P j (t) D ij (t) θ j (t, I j (t)), j = 1,...,m. (1) So far, firm want to maximize the total Profit uring planning perio in segmente market. Therefore, the objective functional for all segments is efine as max J = P j (t) P m D ij(t)+θ j (t,i j (t)) T 0 T + m [ e ρt 0 ] r ij D ij (t) K j (P j (t)) h j (I j (t)) t m [ ( )] e ρt c j D ij (t) P j (t) t (2) Subject to the equation (1).This is the optimal control problem with m- control variable (rate of prouction) with m-state variable (inventory states). Since total eman occurs at rate n D ij(t) an prouction occurs at controllable rate P j (t) for j th, it follows that I j (t) evolves accoring to the above state equation (1). The constraints P j (t) m D ij(t) θ j (t, I j (t)) an I j (0) = I j0 0 ensure that shortage are not allowe. Using the maximum principle [10], the necessary conitions for (Pj, I j ) to be an optimal solution of above problem are that there shoul exist a piecewise continuously ifferentiable function λ an piecewise continuous function µ, 32
Optimal Control Policy of a Prouction an Inventory System for... calle the ajoint an Lagrange multiplier function, respectively such that H(t, I, P, λ) H(t, I, P, λ), for all P j (t) D ij (t) θ j (t, I j (t) (3) t λ j(t) = L(t, I j, P j, λ j, µ j ) (4) I j I j (0) = I j0, λ j (T) = 0 (5) L(t, I j, P j, λ j, µ j ) = 0 P j (6) P j (t) D ij (t) θ j (t, I j (t)) 0, µ j (t) 0, [ µ j (t) P j (t) ] D ij (t) θ j (t, I j (t)) = 0 Where, H(t, I, P, λ) an L(t, I, P, λ, µ) are Hamiltonian function an Lagrangian function respectively. In the present problem Hamiltonian function an Lagrangian function are efine as H = [ + ( ) ] r ij D ij (t) + c j D ij (t) P j (t) K j (P j (t)) h j (I j (t)) [ ( λ j (t) P j (t) )] D ij (t) θ j (t, I j (t)) (7) (8) L = [ + ( ) ] r ij D ij (t) + c j D ij (t) P j (t) K j (P j (t)) h j (I j (t)) [ ( (λ j (t) + µ j (t)) P j (t) )] D ij (t) θ j (t, I j (t)) A simple interpretation of the Hamiltonian is that it represents the overall profit of the various policy ecisions with both the immeiate an the future effects taken into account an the value of λ j (t) at time t escribes the future effect on profits upon making a small change in I j (t). Let the Hamiltonian H for all segments is strictly concave in P j (t) an accoring to the Mangasarian sufficiency theorem [4, 10]; there exists a unique Prouction rate. 33 (9)
K. Chauhary, Y. Singh, P. C. Jha From equation (4) an (6), we have following equations respectively { t λ j(t) = ρλ j (t) h j(i j (t)) (λ j (t) + µ j (t)) θ } j(t, I j (t)), (10) I j I j for all j = 1,, m λ j (t) + µ j (t) = c j + K j (P j (t)). (11) P j Now, consier equation (7). Then for any t, we have either P j (t) D ij (t) θ j (t, I j (t)) = 0 or P j (t) 3.1 Case 1: D ij (t) θ j (t, I j (t)) > 0 j = 1,, m. Let S is a subset of planning perio [0, T], when P j (t) n D ij(t) θ j (t, I j (t)) = 0. Then t I j(t) = 0 on S, in this case I (t) is obviously constant on S an the optimal prouction rate is given by the following equation P j (t) = D ij (t) θ j (t, Ij (t)), for all t S (12) By equation (10) an (11), we have { t λ j(t) = ρλ j (t) h ( j(i j (t)) c j + ) } θj (t, I j (t)) K j (P j (t)) I j P j I j (13) After solving the above equation, we get a explicit from of the ajoint function λ j (t). From the equation (10)), we can obtain the value 0f Lagrange multiplier µ j (t). 3.2 Case 2: P j (t) n D ij(t) θ j (t, I j (t)) > 0, for t [0, T]\S. Then µ j (t) = 0 on t [0, T]\S. In this case the equation (10) an (11) becomes t λ j(t) = ρλ j (t) { h j(i j (t)) λ j (t) θ j(t, I j (t)) I j I j }, j = 1,, m (14) λ j (t) = c j + K j (P j (t)) (15) P j 34
Optimal Control Policy of a Prouction an Inventory System for... Cobining these equation with the state equation, we have the following secon orer ifferential equation: [ ρ + θ j(t, I j (t)) I j t P j(t) 2 K Pj 2 j (P j ) ] ( K j (P j ) =c j ρ + θ ) j(t, I j (t)) P j I j + h j(t, I j (t)) (16) I j an I j (0) = I j0, c j + P j K j (P j (T)) = 0. For illustration purpose, let us assume the following forms the exogenous functions K j (P j ) = k j P 2 j /2, h j (t, I j (t)) = h j I j (t) an θ j (t, I j (t)) = θ j I j (t), where k j h j θ j are positive constants for all j = 1,, m. For these functions the necessary conitions for (P j, I j ) to be optimal solution of problem (2) with equation (1) becomes 2 t 2I j(t) ρ t I j(t) (ρ + θ j )θ 1j I j (t) = η j (t) (17) with I j (0) = I j0, c j + P j K j (P j (T)) = 0. Where, η j (t) = ( ) n D t ij(t) + (ρ + θ 1j ) ( n This problem is a two point bounary value problem. ) D ij(t) + (c j(ρ+θ 1j )+h j ) k j. Proposition 3.1. The optimal solution of (P j, I j ) to the problem is given by an the corresponing P j is given by I j (t) = a 1je m 1jt + a 2j e m 2jt + Q j (t), (18) Pj (t) =a 1j(m 1j + θ 1j )e m1jt + a 2j (m 2j + θ j )e m2jt + t Q j(t) + θ 1j Q j (t) + D ij. (19) The values of the constant a 1j, a 2j, m 1j, m 2j are given in the proof, an Q j (t) is a particular solution of the equation (17). Proof. The solution of the two point bounary value problem (17) is given by stanar metho. Its characteristic equation m 2 j ρm j (ρ + θ j )θ 1j = 0, has two real roots of opposite sign, given by m 1j = 1 ( ) ρ ρ 2 2 + 4(ρ + θ 1j )θ 1j < 0, m 2j = 1 ( ) ρ + ρ 2 2 + 4(ρ + θ 1j )θ 1j > 0, 35
K. Chauhary, Y. Singh, P. C. Jha an therefore I j (t) is given by (18), where Q j (t) is the particular solution. Then initial an terminal conition use to eterminee the values of constant a 1j an a 2j as follows a 1j + a 2j + Q j (0) = I j0, a 1j (m 1j + θ 1j )e m1jt + a 2j (m 1j + θ 1j )e m 2jT ( cj + + ) k j t Q j(t) + θ 1j Q j (T) + D ij (T) = 0. By putting b 1j = I j0 Q j (0) an b 2j = ( c j k j + t Q j(t) + θ 1j Q j (T) + n D ij(t)), we obtain the following system of two linear equation with two unknowns a 1j + a 2j = b 1j a 1j (m 1j + θ 1j )e m 1jT + a 2j (m 1j + θ 1j )e m 2jT = b 2j (20) The value of P j is euce using the values of I j an the state equation. 4 Single Source Prouction- Multi Destination Deman an Inventory Problem We assume the single source prouction an multi estination emaninventory system. Hence, the inventory evolution in each segmente is escribe by the following ifferential equation: t I ij(t) = γ ij P j (t) D ij (t) θ ij (t, I ij (t)), j = 1,, m; i = 1,, n. (21) Here, γ ij > 0, n γ ij = 1, j = 1, m with the conitions I ij (0) = I 0 ij an γ ij P j (t) D ij (t) θ ij (t, I ij (t)). We calle γ ij > 0 the segment prouction spectrum an γ ij P j (t) efine the relative segment prouction rate of j th prouct towars i th segment. We evelop a marketing-prouction moel in which firm seeks to maximize its all profit by properly choosing prouction an market segmentation. Therefore, we efine the profit maximization 36
Optimal Control Policy of a Prouction an Inventory System for... objective function as follows: max γ ij P j (t) D ij (t) θ ij (t,i ij (t)) T [ = 0 T 0 e ρt m m [ e ρt J = ( )] r ij D ij (t) + c j ij (t) γ ij P j (t)) t (D ] h ij (I ij (t)) K j (P j (t)) t (22) subject to the equation (21). This is the optimal control problem (prouction rate) with m control variable with mn state variable (stock of inventory). To solve the optimal control problem expresse in equation (21) an (22), the following Hamiltonian an Lagrangian are efine as [ ( )] H = r ij D ij (t) + c j ij (t) γ ij P j (t)) (D L = + [ ] h ij (I ij (t)) + K j (P j (t)) λ ij (t)[γ i P j (t) D ij (t) θ ij (t, I ij (t))] (23) [ ( )] r ij D ij (t) + c j ij (t) γ ij P j (t)) (D + [ ] h ij (I ij (t)) + K j (P j (t)) (λ ij + µ ij (t))[γ i P j (t) D ij (t) θ ij (t, I ij (t))] (24) Equation (4), (6) an (21) yiel t λ ij(t) = ρλ ij (t) { h ij(i ij (t)) λ ij (t) θ ij(t, I ij (t)) I ij I ij }, (25) for all i = 1,, n, j = 1,, m (λ ij (t) + µ ij (t))γ i = c j + K j (P j (t)) (26) P j In the next section of the paper, we consier only case when γ ij P j (t) D ij (t) θ ij (t, I ij (t)) > 0, i, j. 37
K. Chauhary, Y. Singh, P. C. Jha 4.1 Case 2: γ ij P j (t) D ij (t) θ ij (t, I ij (t)) > 0 i, j, for t [0, T]\S. Then µ ij (t) = 0 on t [0, T]\S. In this case, the equation (25) an (26) becomes { t λ ij(t) = ρλ ij (t) h ij(i ij (t)) λ ij (t) θ } ij(t, I ij (t)) (27) I ij I ij γ ij λ ij (t) = c j + K j (P j (t)) (28) P j Cobining these equation with the state equation, we have the following secon orer ifferential equation: t P j(t) 2 K Pj 2 j (P j ) 1 n = ( ρ + θ ) i(t, I ij (t)) K j (P j ) P j c j γ i ( ρ + θ ij(t, I ij (t)) I ij I ij ) + γ i h ij (t, I ij (t)) I ij (29) with I j (0) = I 0 ij, n γ ijλ ij (T) = 0 λ ij (T) = 0 i an j, c j + P j K j (P j (T)) = 0. For illustration, let us assume the following forms the exogenous functions K j (P j ) = k j P 2 j /2, h ij (t, I ij (t)) = h ij I ij (t) an θ ij (t, I ij (t)) = θ ij I ij (t), where k j h ij θ ij are positive constants. For these functions the necessary conitions for (P j, I ij) to be optimal solution of problem (19) with equation (18) becomes 2 I ij (t) t 2 + (θ ij A) I ij(t) t Aθ i I ij (t) = η ij (t) (30) with I ij (0) = Iij, 0 λ ij (T) = 0 i, c j + P j K j (P j (T)) = 0. [ ] Where, η ij (t) = D ij (t)a + γ n j k j γ i(h ij + c j (ρ + θ ij )) n (ρ+θ ij ) n + D ij(t) t, A =. This problem is a two point bounary value problem. The above system of two point bounary value problem (29) is solve by same metho that we use in to solve (17). 5 Numerical Illustration In orer to emonstrate the numerical results of the above problem, the iscounte continuous optimal problem (2) is transferre into equivalent iscrete problem [24] that is solve to present numerical solution. The iscrete 38
Optimal Control Policy of a Prouction an Inventory System for... optimal control can be written as follows: J = T ( m [ ])( ) 1 (r ij (k 1)D ij (k 1)) (1 + ρ) k 2 k=1 + T ( m ( ))( ) 1 c j D ij (k 1) P j (k 1) (1 + ρ) k 2 T ( m )( ) 1 [K j (P j (k 1) + h j (I j (k 1)))] (1 + ρ) k 2 k=1 k=1 such that I j (k) = I j (k 1) + p j (k 1) for all j = 1,, m. D ij (k 1) θ j (k 1, I j (k 1)) Similar iscrete optimal control problem can be written for single source prouction multi estination an inventory control problem. These iscrete optimal control problems are solve by using Lingo11. We assume that the uration of all the time perios are equal an eman are equal from segment for each prouct. The number of market segmentsis 4 an the number of proucts is 3. The value of parameters are r i1 = 2.55, 2.53, 2.53, 2.54; Table 1: The Optimal prouction an inventory rate in segment market T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 P 1 100 86 80 73 64 53 39 21 5 0 P 2 110 81 76 70 62 52 38 21 5 0 P 3 140 79 75 69 61 51 38 21 5 0 I 1 20 98 154 199 232 254 262 255 231 193 I 2 20 107 156 194 222 238 241 231 205 166 I 3 20 137 179 211 233 244 244 231 203 161 r i2 = 2.52, 2.53, 2.54, 2.53; r i3 = 2.51, 2.54, 2.54, 2.52 for segments i = 1 to 4; c j = 1; k j = 2; θ j = 0.10, 0.12, 0.13; h j = 1; for all the three proucts. The optimal prouction rate an inventory for every prouct for each segment is shown in Table 1 an their corresponing total profit is $177402.70. 39
K. Chauhary, Y. Singh, P. C. Jha The optimal trajectories of prouction an inventory rate for every prouct for each segment are shown in Fig1, Fig2 an Fig3 respectively (Appenix). In case of single source prouction-multi estination eman an inventory, the number of market segments M is 4 an the number of proucts is 3. The values of aitional parameters are each segment is shown in Table 2. Table 2: The values of parameter of eteriorating rate an holing cost rate constant Segment θ i1 θ i2 θ i3 h i1 h i2 h i3 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 prouction-multi estination eman an inventory problem in each segment T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 P 1 100 85 79 73 65 54 41 23 5 0 P 2 110 82 77 70 62 52 38 21 5 0 P 3 140 83 77 71 63 52 38 21 5 0 I 11 20 98 153 197 231 254 263 258 236 197 I 12 20 97 152 195 227 247 255 248 225 185 I 13 20 97 150 192 223 242 247 239 214 173 I 14 20 97 149 190 218 236 240 230 204 162 I 21 20 108 158 198 227 245 250 242 217 178 I 22 20 107 157 195 222 238 242 232 206 167 I 23 20 108 158 198 227 245 250 242 217 178 I 24 20 107 156 192 218 232 235 223 196 156 I 31 20 138 186 223 250 265 268 258 231 191 I 32 20 138 184 220 244 258 260 248 219 178 I 33 20 138 185 223 250 265 268 258 231 191 I 34 20 138 186 223 250 265 268 258 231 191 The optimal prouction rate an inventory for every prouct for each segment is shown in Table 3 with prouction spectrum γ 11 = 0.10, γ 12 = 0.10, γ 13 = 0.77, γ 14 = 0.03; γ 21 = 0.12, γ 22 = 0.12, γ 23 = 0.75, γ 24 = 0.01; γ 31 = 0.14, γ 32 = 0.14, γ 33 = 0.72, γ 34 = 0.04. The optimal value of 40
Optimal Control Policy of a Prouction an Inventory System for... total profit for all proucts is $185876.90. In case of single source prouctionmulti estination eman an inventory, The optimal trajectories of prouction an inventory rate for every prouct for each segment are shown in Fig4, Fig5, Fig6 an Fig7 respectively (Appenix). 6 Conclusion In this paper, we have introuce market segmentation concept in the prouction inventory system for multi prouct an its optimal control formulation. We have use maximum principle to etermine the optimal prouction rate policy that maximizes the total profit associate with inventory an prouction rate. The resulting analytical solution yiel goo insight on how prouction planning task can be carrie out in segmente market environment. In orer to show the numerical results of the above problem, the iscounte continuous optimal problem is transferre into equivalent iscrete problem [24] that is solve using Lingo 11 to present numerical solution. In the present paper, we have assumption that the segmente eman for each prouct is a function of time only. A natural extension to the analysis evelope here is the consieration of segmente eman that is a general functional of time an amount of onhan stock (inventory). References [1] Kotler, P., Marketing Management, 11th Eition, Prentice Hall, Englewoo Cliffs, New Jersey, 2003. [2] Duran, S. T., Liu, T., Simcji-Levi, D., Swann, J. L., optimal prouction an inventory policies of priority an price ifferentiate customers, IIE Transactions,39(2007), 845 861. [3] Chen, Y., an Li, X., the effect of customer segmentation on an inventory system in the presence of supply isruptions, Proc. of the Winter Simulation Conference, December 4-7, Orlano, Floria, (2009), 2343 2352. [4] Sethi, S.P., an Thompson, G.L., Optimal Control Theory: Applications to Management Science an Economics, Kluwer Acaemic Publishers, Beston/Dorrecht/Lonon, 2000. 41
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K. Chauhary, Y. Singh, P. C. Jha 300 I 1 I 2 I 3 250 200 150 100 50 0 1 2 3 4 5 6 7 8 9 10 Fig-2 P P P Fig-3 Fig-4 44
Optimal Control Policy of a Prouction an Inventory System for... 300 I 11 I 12 250 200 150 100 50 0 1 2 3 4 5 6 7 8 9 10 Fig-5 Fig-6 Fig-7 45
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