CHAPTER 7 DEVELOPING THE LINEAR PROGRAMME AND SOLVING

CHAPTER 7: DEVELOPING THE LINEAR PROGRAMME AND SOLVING GRAPHICALLY

TOOLS REQUIRED TO SOLVE A LINEAR PROGRAMMING ¨ linear functions ¨ graphs and co-ordinate systems ¨ representing linear functions graphically ¨ solving simultaneous linear equations ¨ Software package—MS-Excel Solver

FORMULATING THE LINEAR PROGRAMMING ¨ The definition of the decision variables – Let X = the number of Tables manufactured per week – Let Y = the number of Chairs manufactured per week ¨ The objective function – Maximise Profit = 4 X + 3 Y

¨ The set of constraints – 4 X + 1 Y 90 [Constraint due to Wood] – 2 X + 1 Y 50 [Constraint due to Machine-Time] – 1 X + 1 Y 40 [Constraint due to Polishing-Time] – X 0, Y 0 [non-negative constraint]

¨ SUMMARY – Let X = the number of Tables made per week, Let Y = the number of Chairs made per week, – Maximise Profit = 4 X + 3 Y Subject to 4 X+1 Y 90 2 X+1 Y 50 1 X +1 Y 40 X, Y 0 Objective Function Wood Machine-Time Polishing-Time

SOLVING A LINEAR PROGRAMME ¨ Stage 1 – 4 X+1 Y 90 Wood constraints

– 2 X+1 Y 50 Machine-Time

– 1 X +1 Y 40 Polishing-Time

– FEASIBLE REGION (O, A, B. C, D)--the set of all possible solutions to satisfy all the constraints.

¨ Stage 2 – The conclusion is that the optimal solution, the value of X and Y that maximises the profit function, must lie at one of the corner points. X Y Profit 0 0 0 Read off graph 0 40 120 Read off graph 10 30 130 Solving, see i) below. 20 10 110 Solving, see ii) below. 22. 5 0 90 Read off graph

– i) This corner point is the intersection of the Machine-Time constraint line and the Polishing -Time constraint line 2 X + Y = 50 X + Y = 40 (X, Y) = (10, 30) – ii) This corner point is the intersection of the Wood and the Machine-Time constraints 4 X + Y = 90 2 X + Y = 50 (X, Y) = (20, 10)

INTERPRETING THE SOLUTION ¨ The Solution to the linear programming can be stated is: The firm should manufacture 10 Tables and 30 Chairs per week to make the maximum possible weekly profit of £ 130.