CHAPTER 6:LINEAR PROGRAMMING INTRODUCTION

INPUTS: Resource: Raw material, Labour…and how much of each resource FACTORY: Production process: how much of each of the inputs per unit output. OUTPUTS: What products to manufacture. THE BUSINESS OBJECTIVE: To maximise profit, to minimise costs.

Example What production policy should the company adopt to make the maximum profit? Product Wood Machine-Time Polishing-Time Unit Profit Table 4 Kilos 2 Hours 1 hour £4 Chair 1 Kilo lHour £3 Resource 90 50 40  

OUTPUTS: Tables and Chairs INPUTS: The following resources with the availability as given: WOOD: 90 Kilograms available per week. MACHINE-TIME: 50 hours available per week. POLISHING-TIME: 40 hours available per week. FACTORY: One Table requires 4 Kilograms of Wood, 2 hours of Machine-Time and 1 hour of Polishing-Time. One Chair requires 1 Kilograms of Wood, 1 hour of Machine-Time and 1 hour of Polishing-Time. BUSINESS OBJECTIVE: How much is Tables and Chairs to be manufactured that leads to maximum profits.

SOLVING THIS PROBLEM Graphical Solution an Intuitive Approach Stage 1: The set of all possible production plans that meet all the factory input constraints are evaluated. Stage 2: From the set of all possible production plans the particular production plan that meets the business objective is found.

Wood resource constraints If all Wood is used to make only Tables then a maximum of 22.5 tables per week can be made. If all Wood is used to make only Chairs then a maximum of 90 Chairs per week can be made.

RESULT 1 : Any point that lies on a constraint line uses the exact amount of the resource that is available. RESULT 2: Any point under the constraint line uses less of the resource that is available. RESULT 3 : Any point above the constraint line uses more of the resource than is available.

Machine-Time resource constraint Using all the Machine-Time to make only Tables would enable a maximum of 25 Tables per week to be made. Using all the Machine-Time to make only Chairs then a maximum of 50 Chairs per week could be made.

Polishing-Time resource constraints If all the Polishing-Time is used to make only Tables then 40 Tables per week could be made. If all the Polishing-Time is used to make only Chairs the 40 Chairs per week could be made. Feasible Region: OABCD

Stage 2 Move vertically upwards: (15,20) Starting point: (15,15) Profit = £(4*15 + 3*15) = £105 Move vertically upwards: (15,20) Profit = £(4*15+ 3*20) = £120 Move down the Machine-Time line: (16,18) Profit = £(4*16+ 3*18) = £118 Move up the Machine-Time line: (14,22) Profit = £(4*14+ 3*22) = £122

The profit increasing direction: moving up along the Machine-Time line Point B: (10,30) Profit = £(4*10+ 3*30) = £130 Profit maximising production plan can reach at a corner point. Profit maximising production plan is: Make 10 Tables and 30 Chairs, this will give the largest possible profit of £130

WHAT IS LINEAR PROGRAMMING? To find the particular production plan that maximises a linear profit function, when production is constrained by a set of linear constraints.