CHAPTER 7: DEVELOPING THE LINEAR PROGRAMME AND SOLVING GRAPHICALLY

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CHAPTER 7 DEVELOPING THE LINEAR PROGRAMME
AND SOLVING GRAPHICALLY
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TOOLS REQUIRED TO SOLVE A LINEAR PROGRAMMING

• linear functions
• graphs and co-ordinate systems
• representing linear functions graphically
• solving simultaneous linear equations
• Software packageMS-Excel Solver

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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 4X 3Y

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• The set of constraints
• 4X 1Y ? 90 Constraint due to Wood
• 2X 1Y ? 50 Constraint due to Machine-Time
• 1X 1Y ? 40 Constraint due to Polishing-Time
  
• X?0, Y ? 0 non-negative constraint

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• SUMMARY
• Let X the number of Tables made per week,
• Let Y the number of Chairs made per week,
• Maximise Profit 4X 3Y Objective Function
• Subject to
• 4X1Y ? 90 Wood
• 2X1Y ? 50
  Machine-Time
• 1X 1Y ? 40
  Polishing-Time
• X, Y ? 0

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SOLVING A LINEAR PROGRAMME

• Stage 1
• 4X1Y ? 90 Wood constraints

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• 2X1Y ? 50 Machine-Time

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• 1X 1Y ? 40 Polishing-Time

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

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• 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.

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

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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.

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