Question

5. A manufacturing firm produces widgets and distributes them to five wholesalers at a fixed delivered price of $2.50 per unit. Sales forecasts indicate that monthly deliveries will be 2700, 2700, 9000, 4500 and 3600 widgets to wholesalers 1-5 respectively. The monthly production capacities are 4500, 9000 and 11,250 at plants 1, 2 and 3, respectively. The direct costs of producing each widget are $2 at plant 1, $1 at plant 2 and $1.80 at plant 3. The transport cost of shipping a widget from a plant to a wholesaler is given below.

Plant/wholesaler 1 2 3 4 5
Plant 1 0.05 0.07 0.11 0.15 0.16
Plant 2 0.08 0.06 0.10 0.12 0.15
Plant 3 0.10 0.09 0.09 0.09 0.16

Required:
A) Formulate an LP model for this production and distribution problem.
B) Find the Initial Basic Feasible Solution through VAM.
C) Test the optimality of VAM’s solution through MODI method.

Answer 1

A) Formulating the LP model for this problem would involve the following steps:

Identifying the decision variables: The decision variables in this problem would be the number of widgets to be produced at each plant and the number of widgets to be delivered from each plant to each wholesaler.

Defining the objective function: The objective of this problem is to minimize the total cost of production and distribution, which would be the sum of the direct costs of producing each widget and the transport costs of shipping the widgets from the plants to the wholesalers.

Defining the constraints: The constraints in this problem would include production capacity constraints, which specify the maximum number of widgets that can be produced at each plant, and demand constraints, which specify the number of widgets that must be delivered to each wholesaler.

The LP model can be written as:

Minimize Z = 2x1 + x2 + 1.8x3 + 0.05x11 + 0.07x12 + 0.11x13 + 0.15x14 + 0.16x15 + 0.08x21 + 0.06x22 + 0.1x23 + 0.12x24 + 0.15x25 + 0.1x31 + 0.09x32 + 0.09x33 + 0.09x34 + 0.16x35

Subject to:

x1 <= 4500 (Production constraint at plant 1)

x2 <= 9000 (Production constraint at plant 2)

x3 <= 11,250 (Production constraint at plant 3)

x11 + x12 + x13 + x14 + x15 = 2700 (Demand constraint for wholesaler 1)

x21 + x22 + x23 + x24 + x25 = 2700 (Demand constraint for wholesaler 2)

x31 + x32 + x33 + x34 + x35 = 9000 (Demand constraint for wholesaler 3)

x41 + x42 + x43 + x44 + x45 = 4500 (Demand constraint for wholesaler 4)

x51 + x52 + x53 + x54 + x55 = 3600 (Demand constraint for wholesaler 5)

x1, x2, x3, xij >= 0 (Non-negativity constraints)

B) Finding the Initial Basic Feasible Solution through VAM (Variable Addition Method) would involve the following steps:

Setting up the initial feasible solution by assigning 0 to all the decision variables.

Determining the entering variable by selecting the non-basic variable with the most negative reduced cost.

Determining the leaving variable by selecting the basic variable with the smallest positive ratio of the right-hand side constraint to the coefficient of the entering variable.

Updating the basic feasible solution by replacing the leaving variable with the entering variable.

Repeating steps 2 to 4 until all the reduced costs are non-negative.

C) Testing the optimality of VAM’s solution through MODI method (Modified Distribution Method) would involve the following steps:

Forming the inverse matrix by calculating the inverse of the matrix of coefficients of the basic variables.

Calculating the vector of reduced costs by multiplying the inverse matrix with the objective function coefficients of the non-basic variables.

Checking if all the reduced costs are non-negative, if so, the solution is optimal.

If not, select the entering variable with the most negative reduced cost and repeat steps 1 to 3 until all the reduced costs are non-negative