# Assignment Model In Linear Programming

**Introduction:**

In linear programming model, which looks alike with transportation model with an objective function of minimizing the time or cost of manufacturing the products by allocating one job to one machine or one machine to one job or one destination to one origin or one origin to one destination only. This type of problem is given the name **ASSIGNMENT MODEL**.

**Approach to solution:**

**1. ****Solution by visual method**

a) In this method, first allocation is made to the cell having lowest element. (In case of maximization method, first allocation is made to the cell having highest element).

b) If there is more than one cell having smallest element, tie exists and allocation may be made to any one of them first and then second one is selected.

c) In such cases, there is a possibility of getting alternate solution to the problem. This method is suitable for a matrix of size 3 × 4 or 4 × 4. More than that, we may face difficulty in allocating.

**Example:**

**There are 3 jobs A, B, and C and three machines X, Y, and Z. All the jobs can be processed on all machines. The time required for processing job on a machine is given below in the form of matrix. Make allocation to minimize the total processing time.**

**Allocation: A to X, B to Y and C to Z and the total time = 11 13 12 = 36 hours. (Since 11 is least, Allocate A to X, 12 is the next least, Allocate C to Z)**

**2****. Solving the assignment problem by enumeration**

Let us take the same problem and workout the solution.

- Like this write all allocations and calculate the cost and select the lowest one.
- If more than one assignment has same lowest cost then the problem has alternate solutions.