Developing a New Reformulation of Single Level Capacitated Lot Sizing Problem ( SLCLSP ) with Set up , Shortage and Inventory Costs

Formulation of SLCLSP given by Pochet and Wolsey [1] had set up, variables, inventory and shortage cost. We give a new reformulation where SLCLSP is reduced to set up and inventory variables. We find that this reformulation has less number of real variables than the reformulation of Pochet and Wolsey [1]. It is argued that this leads to computations advantages, and this is supported by the empirical investigation that we carried out.


Introduction and Literature Review
Capacitated lot sizing problem (CLSP) is well studied in literature, see Verma [2], and Verma and Sharma [3] [4] [5] for a summary of recent works on CLSP.For literature on reformulation of CLSP, see Pochet and Wolsey [1] and Miller and Nemhauser et al. [6] for a detailed exposition on reformulations of CLSP.In this paper we give a new approach which leads to a better reformulation of CLSP.

Formulation by Pochet and Wolsey [1]
Indices Used t: Set of the Time period from 1, •••, n, for which we are taking decisions; Definition of Constant t f : Fixed setup cost in time period "t"; t p : Per unit production cost in time period "t"; t d : Demand in time period "t", here demand is independent; t h : Per unit inventory carrying cost in time period "t"; t sh : Per unit shortage cost in time period "t"; t c : Production capacity in the time period "t"; Definition of Variables t x : Number of product produced in time period "t"; t y : Binary variable takes value "1" if machine setup to produce in time period "t", "0", otherwise; t I : In stock inventory at the end of time period "t"; t s : Backlog in the end of period "t"; Mathematical Model Production balance constraints Capacity constraints , 1 Pochet and Wolsey [1] gave the following constraint that lead to reformulation: Non-negativity constraints , , 0 SLCLSP as given by Pochet and Wolsey [1] is Model A1: Min (1); s.t. ( 2), (3), ( 4) and (6).By using (5) in place of (3) lead to reformulation (called Model A2: min (1); s.t. ( 2), (4), ( 5) and (6).Model A1 has less number of variables as variable "x" is eliminated.
Min 2 It can be seen that model A3 has least number of variables; it is followed by A2 that has less number of variables compared to model A1 which is well known reformulation (Pochet and Wolsey [1].We solve model A1, model A2, and

Preparing Test Problems and Results
We created problem instances with set up, inventory carrying, shortage and production cost are normally distributed with mean and variance given below: problem instances each for periods 50, 60 and 100.Models A1, A2 and A3 were coded in GAMS and were solved in GAMS; and they were run in branch and bound mode.The GAMS solver returns a satisfactory solution obtainable in reasonable time.It is to be noted that these problems are NP-HARD and will take few billion centuries to come to optimal solution.Detailed data are given in appendix see Tables 1-6; and consolidated results of "t" test are given in Tables 7-9 below.Models A1, A2 and A3 are compared on the criteria of execution time, cost: mean1500 and variance 100 Demand and capacity were chosen from uniform distribution in the range of 10,000 -15,000.In the case of infeasible solution, the capacity values are increased or demand values are decreased keeping other costs same.We created 50

Table 2 .
Problem for 50 time period (Iteration and execution time in GAMS).

Table 3 .
Problem for 60 time period (Z value and No. of nodes).

Table 4 .
Problem for 60 time period (Iteration and execution time).

Table 5 .
Problem for 100 time period (Z value and No. of nodes).

Table 6 .
Problem for 100 time period (Iteration and execution time).
*Significant at 0.05 level; **significant at 0.01 level; ***significant at 0.001 level; In Table 7: 2.742**means that model A3 takes less time than model A1 and is significant at 0.01 level.