# A Profit-Maximizing Production Lot sizing Decision Model with Stochastic Demand

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1 A Profit-Maximizing Production Lot sizing Decision Model with Stochastic Demand Kizito Paul Mubiru Department of Mechanical and Production Engineering Kyambogo University, Uganda Abstract - Demand uncertainty affects a significant portion of profits in production-inventory management. In this paper, a new mathematical model is developed to optimize production lot sizing policies of a single-item, finite horizon, periodic review inventory problem with stochastic demand. A special case of the model is where sales price and inventory replenishment periods are uniformly fixed over the planning periods. Adopting a Markov decision process approach, the states of a Markov chain represent possible states of demand for the production-inventory item. The production cost, holding cost, shortage cost and sales price are combined with demand and inventory positions in order to formulate the profit matrix. The matrix represents the long run measure of performance for the decision problem. The objective is to determine in each period of the planning horizon an optimal production lot sizing policy so that the long run profits are maximized for a given state of demand. Using weekly equal intervals, the decisions of when to produce additional units are made using dynamic programming over a finite period planning horizon. A numerical example demonstrates the existence of an optimal state-dependent production lot sizing policy and corresponding profits of the item. Keywords: Production lot sizing, profit maximizing, stochastic demand I. INTRODUCTION In production-inventory systems, proper planning and management of make-to-stock inventory items is vital for the growth and survival of manufacturing industries. This is however challenging when the item produced for stock is characterized by demand uncertainties if the industry looks at maximizing profits in the long run. Even when price of the item in question is static, optimality of production lot sizing decisions that enhance realistic profit margins are difficult to make when demand fluctuates from time to time. Two major problems are usually encountered: (i) (ii) Determining the most desirable period during which to produce additional units of the item in question Determining the optimal profits associated with a given production-inventory system when demand is uncertain In this paper, a production-inventory system is considered whose goal is to optimize profits associated with producing and holding inventory of an item. At the beginning of each period, a major decision has to be made: namely whether to produce additional units of the item in inventory or cancel production and utilize the available units in inventory with a view of maximizing profits. According to Goya[1],the EPQ may be computed each time a product is scheduled for production when all products are produced on a single machine. The distribution of lot size can be computed when the demand for each product is a stochastic variable with a known distribution. The distribution is independent for nonoverlapping time periods and identical for equal time epochs. In related work by Khouja[2],the EPQ is shown to be determined under conditions of increasing demand, shortages and partial backlogging. In this case the unit production cost is a function of the production process and the quality of the production process deteriorates with increased production rate. Tabucanno and Mario [3] also presented a production order quantity model with stochastic demand for a chocolate milk manufacturer. In this paper, production planning is made complex by the stochastic nature of demand for the manufacturer s product. An analysis of production planning via dynamic programming approach shows that the company s production order variable is convex. In the article presented by Kampf and Kochel[4], the stochastic lot sizing problem is examined where the cost of waiting and lost Volume 4 Issue 3 October ISSN:

2 demand is taken into consideration. In this model, production planning is made possible by making a trade-off between the cost of waiting and the lost demand. The four model presented have some interesting insights regarding economic production quantity in terms of optimization and stochastic demand. However, profit maximization is not a salient factor in the optimization models cited. The paper is organized as follows: After describing the mathematical model in 2, consideration is given to the process of estimating model parameters. The model is solved in 3 and applied to a special case study in 4. Some final remarks lastly follow in 5. II. MODEL FORMULATION An item is considered with varying size in a given production-inventory system whose demand during a chosen period over a fixed planning horizon is classified as either Favorable (state F) or Unfavorable (state U) and the demand of any such period is assumed to depend on the demand of the preceding period. The transition probabilities over the planning horizon from one demand state to another may be described by means on a Markov chain. Suppose one is interested in determining the optimal course of action, namely to produce additional units of a specific size of item (a decision denoted by Z=1) or not to produce additional units (a decision denoted by Z=0) during each time period over the planning horizon where Z is a binary decision variable. Optimality is defined such that the maximum expected profits are accumulated at the end of N consecutive time periods spanning the planning horizon. In this paper, a two-period (N=2) planning horizon is considered. 2.1 Model variables and parameters Varying demand is modeled by means of a Markov chain with state transition matrix where the entry Q Z ij in row i and column j of the transition matrix denotes the probability of a transition in demand from state i {F,U}to state j {U,F} under a given production lot sizing policy Z {0,1}.The number of customers observed in the system and the number units demanded during such a transition are captured by the customer matrix and demand matrix Furthermore, denote the number of units in inventory and the total (production, holding and shortage) cost during such a transition by the inventory matrix and the profit matrix Volume 4 Issue 3 October ISSN:

3 respectively. Also denote the expected future profits, the already accumulated profits at the end of period n when demand is in state i {F,U}for a given production lot sizing policy Z {0,1}by respectively v Z i and a Z i and let v Z =[v Z F, v Z U] T and a Z =[a Z F,a Z U ] T where T denotes matrix transposition. 2.2 Finite dynamic programming formulation Recalling that the demand can be in state F or in state U, the problem of finding an optimal production lot sizing policy may be expressed as a finite period dynamic programming model. Let g n (i) denote the expected total profits accumulated during periods n,n+1,...,n given that the state of the system at the beginning of period n is i {F,U}.The recursive equation relation g n and g n+1 is i {F,U},n=1,2,...N together with the final conditions (1) This recursive relationship maybe justified by noting that the cumulative total profits P Z ij + g n+1 (j) resulting from n+1 n occurs with probability Q Z ij. Clearly, v Z = Q Z (P Z ) T (2) Where T denotes matrix transposition, and hence the dynamic programming recursive equations n=1,2,...n-1 Z (3) result where (4) represents the Markov chain stable state. (4) III. COMPUTING Q Z AND P Z F, U } given production lot sizing policy Z {0,1} may be taken as the number of customers observed when demand is initially in state i and later with demand changing to state j, divided by the number of customers over all states. That is (5) Z When demand outweighs on-hand inventory, the profit matrix P Z may be computed by the relation: where c p denotes the unit production cost, c h denotes the unit holding cost and c s denotes the unit shortage cost. Volume 4 Issue 3 October ISSN:

4 Therefore (6) for all Z A justification for expression (6) is that D Z ij - I Z ij units must be produced in order to meet the excess demand. Otherwise production is cancelled when demand is less than or equal to on-hand inventory. The following conditions must however hold: 1. Z=1 when c p > 0, and Z=0 when c p = 0 2. c s > 0 when shortages are allowed and c s = 0 when shortages are not allowed. IV. COMPUTING AN OPTIMAL PRODUCTION LOT SIZING DECISION The optimal production lot sizing policy is found in this section for each time period separately, 4.1 Optimization during period 1 When demand is Favourable (i.e. in state F), the optimal production lot sizing decision during period 1 is The associated total profits are then Similarly, when demand is Unfavorable (i.e. in state U), the optimal production lot sizing decision during period 1 is In this case, the associated total profits are 4.2 Optimization during period 2 Using dynamic programming recursive equation (1), and recalling that a Z i denotes the already accumulated total profits at the end of period 1 as a result of decisions made during that period,it follows that: Volume 4 Issue 3 October ISSN:

5 Therefore when demand is favorable (i.e. in state F), the optimal production lot sizing decision during period 2 is while the associated total profits are Similarly, when demand is Unfavourable (i.e. in state U), the optimal production lot sizing decision during period 2 is while the associated total profits are V. CASE STUDY In order to demonstrate the use of the model in 2-4, a real case application from Nile Plastics; a manufacturer of plastic utensils in Uganda is presented in this section. Plastic pails are among the products manufactured, and the demand for pails fluctuates from month to month. The company wants to avoid mover-producing when demand is low or under-producing when demand is high, and hence seeks decision support in terms of an optimal production lot sizing decision and the associated profits over a two-week period. 5.1 Data Collection A sample of 30 customers was used. Past data revealed the following demand pattern and inventory levels of pails during the first week of the month when demand was Favorable (F) or Unfavorable (U): If additional pails are produced during week 1, the customer matrix, demand matrix and inventory matrix are given by If additional pails are not produced during week 1, these matrices are In either case the unit sales price(p s ) is \$2.00,the unit production cost (c p ) is \$1.50,the unit holding cost per week(c h ) is \$0.50,and the unit shortage cost per week(c s ) is \$ Computation of model parameters Using (5) and (6), the state transition matrix and the profit matrix for week 1 are and Volume 4 Issue 3 October ISSN:

6 respectively, for the case where additional pails are produced during week 1, while these matrices are given by respectively, for the case when additional pails are not produced during week 1. When additional pails are produced (Z=1), the matrices Q 1 and P 1 yield the profits (in US dollars) and However, when additional pails are not produced (Z=0), the matrices Q 0 and P 0 yield the profits (in US dollars) 5.3 The optimal production lot sizing decision Since > 12.5, it follows that Z=1 is an optimal production lot sizing decision for week 1 with associated total profits of \$49.22 when demand is favorable. Since 6.5 > 0.95, it follows that Z = 0 is an optimal production lot sizing policy for week 1 with associated total profits of \$6.5 when demand is unfavorable. If demand is favorable, then the accumulated profits at the end of week 1 are Since > 40.36, it follows that Z = 1 is an optimal production lot sizing decision for week 2 with associated accumulated profits of \$ for the case when favorable demand. However, if demand is unfavorable, then the accumulated profits at the end of week 1 are Since > 14.71, it follows that Z=0 is an optimal production lot sizing decision for week 2 with associated accumulated profits of \$27.10 for the case of favorable demand. When shortages are not allowed, the values of Z and g n (i) may be computed for F, U}in a similar fashion after substituting c s = 0 in the matrix function P S = p s [D Z ] (c p + c h + c s )[D Z I Z ] VI. CONCLUSION A production lot sizing decision model with static price and stochastic demand was presented in this paper. The model determines an optimal production lot sizing policy and profits of a given item with demand uncertainty. The decision of whether or not to produce additional inventory units is modeled as a multi-period decision problem using dynamic programming over a finite period planning horizon. The working of the model was demonstrated by means of a real case study. It is equally important to examine the behavior of production lot sizing policies of pails under non stationary demand conditions. In the same spirit, the model raises a number of salient issues to consider: Production disruptions in a typical manufacturing set up and customer response to abrupt changes in price of pails. Finally, special interest is sought in extending the model to optimize production lot sizing decisions using Continuous Time Markov Chains (CTMC) and dynamic programming. VII. ACKNOWLEDGMENT Volume 4 Issue 3 October ISSN:

7 The author wishes to thank the staff and management of Nile Plastics for all the assistance rendered during the data collection exercise. REFERENCES [1] Goyal S, 1974, Lot size scheduling on a single machine for stochastic demand, Management science, 19(11), [2] Khouja M & Mehrez A, 1994, Economic Production lot size model with varying production rate and imperfect quality, Journal of Operations Research Society, 45(2), [3] Tabuccano & Mario T, 1997, A Production order quantity model with stochastic demand for a chocolate milk manufacturer, Journal of Production Economics,21(2), [4] Kampf M & Kochel P, 2004, Scheduling and lot size optimization of a production and inventory system with multiple items using simulation and parallel genetic Algorithm, Operations Research Proceedings, Springer-Heidelberg. Volume 4 Issue 3 October ISSN:

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