Question (3): Solve the following Problem: (10 marks)

Consider the following linear program:

MAX.Z = 5X + 6Y

Subject to:

.2X + Y<12(1)

2X + 4Y< 24(2)

a-Solve this linear program graphically. (2 marks)

b-Determine the optimal quantities of X, Y, and the value of Z.

c-What is the slack for constraint (1)?

d-Find the change in the objective function coefficient when profit is raised in (y) from 6 to 12.

Question (4): Based on what you have studied about the Assignment model, solve the following problem:

The given table contains the cost matrix of workers and jobs. Workers: Hussein, Ahmed, Karim are assigned to Jobs: Project1, Project2, Project3. Given that each worker needs to perform only one project, each project must be assigned to only one worker.

project

Worker

Project 1

Project 2

Project 3

Hussein

$11

$14

$6

Ahmed

$8

$10

$11

Karim

$9

$12

$7

Required:

a-Formulate as a linear program with clarification of:

-Decision variables.

-Objective function.

-Constraints. (2 marks)

b-Use the Hungarian method to find the optimal assignment of workers to projects. (4 Marks)

c-Calculate the total cost. (1 Mark)