Question

The demand for an item over the next four quarters is 300,400, 450, and 250 units, respectively. The price per unit starts at Rs. 20 in the first quarter and increased by Rs. 3 each quarter thereafter. Although we can take advantage of lower prices in early quarters, a storage cost of Rs. 3.80 is incurred per unit per quarter. In addition, the number of units that can be held over from one quarter to the next cannot exceed 100. List all the decision variables to develop a linear program formulation and hence write down the formulation to determine the optimum schedule for purchasing the item to meet the demand. Label each constraint appropriately

Answer 1

To determine the optimum schedule for purchasing an item to meet demand over four quarters with changing prices and storage costs, a linear programming model with decision variables for purchase quantities and storage capacities should be formulated. The goal is to minimize total costs while adhering to demand fulfillment and storage constraints.

Step-by-step explanation:

Linear Programming Formulation

To solve this linear programming problem for finding the optimum schedule for purchasing the item to meet demand, we need to consider several decision variables and constraints. The decision variables will represent the number of units to purchase in each quarter, while taking into account the price increases, storage costs, and maximum storage capacity of 100 units that can be carried over to the next quarter.

Let X1, X2, X3, and X4 be the number of units purchased in quarters 1, 2, 3, and 4 respectively. Let S1, S2, and S3 be the number of units stored from quarter 1 to 2, 2 to 3, and 3 to 4, respectively.

The objective function is to minimize the total cost, which includes the purchase cost at an increasing price per quarter and the storage cost per unit per quarter.

The storage limitation constraint ensures that at the end of each quarter, the maximum number of units stored does not exceed 100.

The demand constraints ensure that the demand is met for each quarter.

The linear program can be written as follows (ignoring any carrying costs or other complexities; actual formulations would be more detailed and specific to context):

Minimize Z = 20*X1 + 23*X2 + 26*X3 + 29*X4 + 3.80*(S1 + S2 + S3)

Subject to:

X1 - S1 = 300 (Demand in Q1)

X2 + S1 - S2 = 400 (Demand in Q2)

X3 + S2 - S3 = 450 (Demand in Q3)

X4 + S3 = 250 (Demand in Q4)

S1, S2, S3 <= 100 (Storage constraints)

X1, X2, X3, X4, S1, S2, S3 >= 0 (non-negativity constraints)

Each constraint should be labeled accordingly, and all decision variables should be clearly defined, considering the increments in unit price and the incurred storage cost.