Lecture 5 – Transportation Model and its variant

Introduction:

The transportation model is a special class of linear programs that deal with shipping a commodity from sources (e.g. factories) to destinations (e.g. warehouses). The objective is to determine the shipping schedule that minimizes the total shipping cost while satisfying supply and demand limits. The application of the transportation model can be extended to other areas of operation, including inventory control, employment scheduling and personal assignment.

Example of a transportation problem:

A cold drink manufacturing concern has m plants located in m different cities of England. The total supply potential of the manufactured product is absorbed by n retail shops in n different cities of the country. Determine the transportation schedule that minimizes the total cost of transporting cold drinks from various plant locations to various retail shops.

Mathematical Formulation of the problem:

Let us identify the m plant locations (origins) as O1, O2, …, Om and the n retail shops (destinations) as D1, D2, …, Dn respectively. Let ai >0, i = 1, 2, …, m be the amount of cold drinks available at the ith plant Oi and let the amount of cold drinks required at the jth retail shop Dj be bj > 0, j = 1, 2, …, n.

Let the cost of transporting one unit of cold drinks from plant O i to retail shop Dj be Cij (i = 1, 2, …, m ; j = 1, 2, …, n). if nij > 0 be the amount of cold drinks to be transported from origin Oi to destination Dj, then the problem is to determine xij so as to

Minimise

Z =

Subject to the constraints:

and xij ≥ 0 for all i and j.

1

The above formulation looks like a LP problem. This special LPP will be called a transportation problem (TP)

The Transportation Table (TP)

Since the TP is just a special case of a general LPP, the application of simplex method would no doubt, give an optimum solution to the problem. However, fortunately, a simplex like method for solving such problems have been developed. Whenever, it is possible to place the given LP in the transportation framework, it is far simple to solve it by the Transportation Model than by simplex method. Instead of using simplex table, the major computational vehicle in this method will be a different sort of table called the Transportation Table. A specimen of the transportation table for an m-origin, n-destination TP is given below:

D1 D2 D3 D4

C11 C12 … C1n

C21 C22 ...

C2n

. .

. .

Cm1 Cm2 …

Cmn

Example of formulating a transportation problem:

A dairy firm has two plants located in a metropolitan city. Daily cold drinks production at each plant is as follows:

Plant 1 - 8 million litresPlant 2 - 15 million litres

Each day the firm must fulfill the needs of its three distribution centers. Minimum requirement at each centre is as follows:

Distribution center 1 - 9 million litresDistribution center 2 - 11 million litresDistribution center 3 - 3 million litres.

Cost of shipping one million litres of cold drinks from each plant to each distribution centre is given in the following table in hundred of rupees:

2

Shipping CostPlant Distribution Centre

_______________1 2 3

_________________________________

1 2 3 3 2 2 4 2_________________________________

The dairy firm wishes to decide as to how much should be the shipment from which plant to which distribution centre so that the cost of shipment may be minimum.

You are required to formulate this problem as a linear programming problem.

Formulation as an LP problem:

Step1: The key decision to be made is to determine how much quantity of cold drinks from which plant to which distribution centre be shipped so as to satisfy the restrictions and minimize cost.

Step 2: Let xij be the decision variables (quantities to be found) when the cold drink is shipped from plant i (i = 1, 2) to distribution centre (j = 1, 2, 3).

Step 3: Feasible alternatives are the sets of values of x11 , x12, x13 , x21, x22, x23 where x11 , x12, x13 , x21, x22, x23 ≥ 0.

Step 4: The objective is to minimize the cost of transportation, that is, to minimize

Z = (2x11 + 3x12 + 3x13) + (2x21 + 4x22 + 2x23)

Step 5: Constraints are on the availability of cold drinks at the two plants and its requirements at the 3 distribution centres. These are:

1. Availability or supply : x11 + x12 + x13 = 8 x21 + x22 + x23 = 15

2. Requirements or demand: x11 + x21 = 9 x12 + x22 = 11 x13 + x23 = 3

3

Hence the transportation problem is:

4

MinimizeZ = 2x11 + 3x12 + 3x13 + 2x21 + 4x22 + 2x23

Subject to x11 + x12 + x13 = 8 x21 + x22 + x23 = 15 x11 + x21 = 9

x12 + x22 = 11 x13 + x23 = 3

x11 , x12, x13 , x21, x22, x23 ≥ 0.