# Transportation Model

MAXIMIZATION IN A TRANSPORTATION PROBLEM

There are certain types of transportation problems where the objective function is to be maximized instead of being minimized. These problems can be solved by converting the maximization problem into a minimization problem.

Example : A manufacturing company has four plants situated at different locations, all producing the same product. The manufacturing cost varies at each plant due to internal and external factors. The size of each plant varies, and hence the production capacities also vary. The cost and capacities at different locations are given in the following table:

Cost and Capacity of Different Plants

The company has five warehouses. The demands at these warehouses and the transportation costs per unit are given in the Table below. The selling price per unit is Rs. 30/-

Transportation Problem

1. Formulate the problem to maximize profits.
2. Determine the solution using TORA.
3. Find the total profit

Solution:
The objective is to maximize the profits. Formulation of transportation problem as profit matrix table is shown in Table. The profit values are arrived as follows.

Profit = Selling Price – Production cost –Transportation cost

Profit Matrix

Converting the profit matrix to an equivalent loss matrix by subtracting all the profit values from the highest value 13. Subtracting all the values from 13, the loss matrix obtained is shown in the Table

Loss Matrix

(ii) To determine the initial solution using TORA

Input Screen:

TORA, Input Screen for TP Max Problem

Output Screen

TORA Output Screen (Vogel’s Method)

The first iteration itself is optimal, hence optimality is reached.

(iii) To find the total cost:

The total maximization profit associated with the solution is
Total Profit

= (6 × 10) + (4 × 20) + (10 × 120) + (3 × 180) + (9 × 70) + (10 × 20)+ (13 × 80) +  (15 × 70)
= 60 + 80 + 1200 + 540 + 630 + 200 + 1040 + 1050
= Rs 4800.00

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