CHAPTER 6: LINEAR PROGRAMMING NTRODUCTION INPUTS OUTPUTS FACTORY BUSNESS OBJECTTES Figure I
CHAPTER 6:LINEAR PROGRAMMING INTRODUCTION
5+ 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
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
W Example Product Wood Machine- Polishing Unit Time Time Pr Table 4 Kilos 2 Hours nour Chair I Kilo Resource 50 40 What production policy should the company adopt to make the maximum profit?
Example What production policy should the company adopt to make the maximum profit? Product Wood MachineTime PolishingTime Unit Profit Table 4 Kilos 2 Hours 1 hour £4 Chair 1 Kilo lHour 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 l FACTORY One Table requires 4 Kilograms of Wood, 2 hours of Machine-Time and I hour of Polishing-Time One Chair requires 1 Kilograms of Wood, I hour of Machine-Time and I hour of polishing-Time A BUSINESS OBJECTIVE: How much is Tables and Chairs to be manufactured that leads to maximum profits
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
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
ae 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 GRAPH I
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
– 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 WOCO& MACHINE TIME CONSTRAINTS Nu, Tables
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
l 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 ALL THREE CONSTRAINTS easible region ABCD 5
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 Starting point: (15, 15) Profit=f(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=f(4*16+3*18)=￡118 Move up the machine-Time line:(14, 22) ° Profit=f(4*14+3*22)=￡122
Stage 2 – 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