Unit 3:

Transportation and Assignment Models:

Transportation Model:

Introduction: The transportation problem is one of the subclasses of LPPs in which the objective is to transport various quantities of a single homogeneous commodity that are initially stored at various orgins to different destinations in such a way that the transportation cost is minimum. To achieve this objective we must know the amount and location of available supplies and the quantities demanded. In addition we must know the costs that result from transporting one unit of commodity from various Orgins to various destinations. Initial Basic Feasible Solution:

3 methods:a. North West Corner Rule. (NWCR)b. Least Cost Method. (LCM) or Matrix minima Method.c. Vogel’s Approximation Method.

2. To find out Optimal Solution – MODI Method or Modified Distribution Method.3. Unbalanced Transportation Problem: If ∑ai ≠ ∑ bj, the problem is said to be Unbalanced Transportation Problem. Convert this into a balanced one by introducing Dummy source or with zero cost vectors and solve them. Maximisation in Transportation Problem.

Problems:

1. Solve the following transportation problem by NWCR.

Orgin / Destination

D1 D2 D3 D4 Supply

O1 12 25 15 17 100O2 20 6 12 10 125O3 14 18 13 15 175O4 6 9 12 18 50Demand 75 200 125 50 450

2. Obtain the Initial Basic Feasible Solution of a transportation problem whose cost and rim requirement table is given below:

Orgin / Destination

D1 D2 D3 Supply

O1 2 7 4 5O2 3 3 1 8O3 5 4 7 7O4 1 6 2 14Demand 7 9 18 34

Least Cost Method:

1. Solve the following Transportation Problem using L.C.M

Orgin / Destination

D1 D2 D3 D4 Supply

O1 12 25 15 17 100O2 20 6 12 10 125O3 14 18 13 15 175O3 6 9 12 18 50Demand 75 200 125 50 450

2. Obtain an IBFS to the following T.P Using LCM.

Orgin / Destination

D1 D2 D3 D4 Supply

O1 1 2 3 4 6O2 4 3 2 0 8O3 0 2 2 1 10Demand 4 6 8 6 24Vogel’s Approximation Method:

1. Find the IBFS using VAM:

Orgin / Destination

D1 D2 D3 D4 Supply

O1 3 3 4 1 100O2 4 2 4 2 125O3 1 5 3 2 75Demand 120 80 75 25 300

Home work:

1. Obtain the initial Solution for the following Transportation Problem using (1) NWCR (2) LCM (3) VAM

Orgin / Destination

D1 D2 D3 Supply

O1 2 7 4 5O2 3 3 1 8O3 5 4 7 7 O4 1 6 2 14Demand 7 9 18 34

Unbalanced Transportation Problem:

If ∑ai ≠ ∑ bj, the problem is said to be Unbalanced Transportation Problem. Convert this into a balanced one by introducing Dummy source or with zero cost vectors and solve them.1. Solve the following LPP by Using NCW, LCM, and VAM.

Orgin / Destination

D1 D2 D3 D4 Supply

O1 11 20 7 8 50O2 21 16 20 12 40O3 8 12 18 9 70Demand 30 25 35 40 Maximization in TP:

1. Solve the following TP to maximize the profit:

Orgin / Destination

D1 D2 D3 D4 Supply

O1 40 25 22 33 100O2 44 35 30 30 30O3 38 38 28 30 70Demand 40 20 60 30

Conversion of maximization to minimization:

The given problem is of maximization type. Let us convert this into minimization type by subtracting all the cost elements from the highest cost element. Then it become minimization one.