UNIT-4

TRANSPORTATION PROBLEM

Definition:Transportation problem:

The transportation or shipping problem involves determining the quantity of goods to be shipped down from a number of origins to a number of destinations.

The constraints pertain to the capacities at each origin and requirements at each destination. The transportation problem is a very specific case of a linear programming.

Objectives:

To minimize total shipping distances or costs.

Characteristics and Assumptions:

1. Supply : A limited quantity of items such as steel, cement, oil, is available at certain sources or origins such as factories, refineries, etc.

2. Demand : There is demand for items at several destinations such as warehouses, distribution centers or stores.

3. Quantities : The quantities of supply at each source and the demand or requirements at each destination are known.

4. Shipping Cost: The per unit cost of transporting the commodity from each source to each destination are assumed to be known.

5. It is assumed that no shipments are given in whole numbers. 6. All supply and demand quantities are given in whole numbers.7. The problem is to determine how many units should be shipped from each source to

each destination, so that all demands are satisfied if possible at the minimum total shipping cost.

Procedure for solving a Transportation Problem:

Step:1 : Arrange the data in a tabular form. Step:2: Balance the Matrix : Transportation technique requires the problem to be

balanced.i.e; the total supply must be equal to the total demand. If the matrix is not balanced it has to be balanced first.

E.g. : If the supply exceeds demand then such a matrix is balanced by adding an artificial destination column to absorb the excess supply, referred as dummy destination. The amount in dummy column equals the excess supply. When total of demand exceeds supply, a dummy source row is added to meet the extra demand.

Note: The cost of shipment from a dummy source or dummy destination is set to be zero. Step: 3: Initial feasible solution: There are 3 different methods to determine IBFS.

i) North – West Corner method (NWCM) ii) Least Cost Method (LCM) iii) Vogel Approximation method (VAM).

Step:4: Testing for optimality: The purpose of optimality is to check whether the proposed solution just generated can be improved or not. Two procedures for testing the optimality are i) Stepping stone method and ii) Modified distribution method (MODI)

In both the methods, the solution to be checked for optimality must be non-degenerate i.e; the number of occupied cells must be equal to (m+n-1). Where m= No. of. Sources and n= No.of.destinations.

Step:5 : Improving a non optimal solution:

Obtaining Initial Basic Feasible Solution (IBFS):

Method:1 : North-West Corner method:

Step:1: Starting with north west corner (left upper most cell in the matrix) allocate the smaller amount of either the row supply or the column demand.

Step: 2 : Subtract from the row supply and from the column demand the amount allocated. Step: 3: If the column demand is now zero, move to the next cell next to right. If the row

supply is zero, move down the cell in the next row. If both are zero, move first to the next cell on the right, then down one cell.

Step: 4: Once a cell is identified as per step3 it becomes north west cell. Allocate to it as per step 1.

Step: 5: Repeat the above steps 1 to Step 4 until all the remaining supply and demand is gone.

Total Cost =19*5+30*2+30*6+40*3+70*4+20*14 = Rs. 1015

Method:2: Least Cost Method:

Step:1: Select the lowest cost cell in the whole matrix(i.e.; out of all values of rows and columns of the given transportation matrix). In case of tie between any cells , select the cell to which maximum allocation can be made considering the supply and demand values.

Step: 2 : Allocate maximum possible units to the least cost cell depending upon the demand/ supply for that cell.

Step: 3: Eliminate the row or column satisfied fully by the above allocation, if feasible. Step: 4: Make allocation to the next lowest cost cell depending upon the remaining

demand/supply units for the row/ column containing that cell. Step: 5: Repeat step 1 to step 4 until all the remaining supply and demand is gone.

The total transportation cost obtained by this method= 8*8+19*5+20*10+10*2+40*7+60*2= Rs.779Here, we can see that Vogel’s Approximation Method involves the lowest cost than North-West Corner Method and Least Cost Method and hence is the most preferred method of finding initial basic feasible solution.

Assignment Problem:

It is a special type of linear programming problem where assignees are assigned to perform tasks.

It arises in a variety of decision making situations. In many management situations, management finds it necessary to assign personnel to jobs,

jobs to machines, machines to job locations within a plant, or sales persons to territories within the distribution area of business.

Distinguishing feature of assignment problem is that the one worker (or) job is assigned to one and only one task (or) machine.

Set of assignment s will optimize a stated objective such as maximize profit, minimize time or cost.

Assignment model is a square matrix as the objects and assignments are equal in number.

Characteristics of Assignment problem:

The objectives under consideration, such as service teams, jobs, employees or projects are finite in number.

The object s are assigned on a one-one basis to other objects. The result of each assignment is expressed in terms of pay-offs such as costs or profits. The aim is to assign all objects in such a way that the total cost is minimized or the total

profit is maximized.

Assumptions of a Assignment Problem:

The number of assignees and the number of jobs or tasks are the same. Each assignee is to be assigned to exactly one job or task. Each job or task is to be performed by exactly one assignee. There is a cost Cij associated with assignee “I” (i= 1, 2….n) performing task “j” (j=1, 2 …

n). The objective is to determine how all “n” assigned should be made to minimize the

total cost. When these assumptions are not satisfied it is possible to reformulate the problem to

make it fit.Eg: dummy assignees or dummy jobs or tasks can be used for this purpose.