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Chapter 3

Chapter 3. Introduction to optimization models. Linear Programming. The PCTech company makes and sells two models for computers, Basic and XP. Profits for Basic is $80/unit and for XP is $129/unit. Sales estimate is 600 Basics and 1200 XPs Making the computers involves two operations:

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Chapter 3

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  1. Chapter 3 Introduction to optimization models

  2. Linear Programming • The PCTech company makes and sells two models for computers, Basic and XP. • Profits for Basic is $80/unit and for XP is $129/unit. • Sales estimate is 600 Basics and 1200 XPs • Making the computers involves two operations: Assembly: Basic requires 5 hours and XP requires 6 hours Testing: Basic requires 1 hour and XP requires 2 hours • Available labor hours: Assembly: 10000 hours Testing: 3000 hours

  3. Linear Programming • PC Tech wants to know how many of each model it should produce (assemble and test) to maximize its net profit, but it cannot use more labor hours than are available, and it does not want to produce more than it can sell. • The problem objective: • Use LP to find the best mix of computer models that maximizes profit • Stay within the company’s labor availability • Don’t produce more than what can be sold

  4. Graphical Method x2 x1 = Number of basic computer model x2 = Number of XP computer model Net profit = 80x1 + 129x2 x1

  5. Graphical Method x2 x1 = Number of basic computer model x2 = Number of XP computer model 1800 1600 Net profit = 80x1 + 129x2 1400 If x1 = 1290, x2 = 0, Net profit = 103,200 1200 If x1 = 0, x2 = 800, Net profit = 103,200 1000 800 600 400 200 Net profit = $103,200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  6. Graphical Method x2 1800 Net profit = 80x1 + 129x2 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 If x1 = 1290, x2 = 0, Net profit = 103,200 1200 If x1 = 0, x2 = 800, Net profit = 103,200 1000 Iso-profit line 800 600 Net profit = $140,00 400 200 Net profit = $103,200 Net profit = $130,00 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  7. Constraints Labor hours constraints Sales constraints

  8. Assembly Hours Constraints x2 1800 X1 = 0, x2 = 1666.67 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 Assembly hours constraint: 5x1 + 6x2 <= 10,000 1200 If we make no XP model at all 5(2000) + 6(0) = 10,000 1800 If we make no Basic model at all 5(0) + 6(1666.67) = 10,000 800 600 X1 = 2000, x2 = 0 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  9. Testing Hours Constraints x2 1800 X1 = 0, x2 = 1500 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 Testing hours constraint: x1 + 2x2 <= 3,000 1200 If we make no XP model at all (3000) + 2(0) = 3,000 1800 If we make no Basic model at all (0) + 2(1500) = 3,000 800 600 X1 = 3000, x2 = 0 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  10. Maximum sales Constraints x2 1800 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 Maximum sales for basic model: x1 <= 600 1200 1800 800 600 X1 = 600, x2 = 0 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  11. Maximum sales Constraints x2 1800 X1 = 0, x2 = 1200 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 Maximum sales for XP model: x2 <= 1200 1200 1800 800 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  12. Feasible region x2 1800 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 x2 = 1200 1200 1800 800 x1 + 2x2 <= 3000 Feasible region Redundant constraint 600 x1 = 600 400 200 5x1 + 6x2 =10000 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  13. Optimum solution x2 1800 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 1200 1800 Redundant constraint x1 + 2x2 <= 3000 800 Feasible region 600 Iso-profit line 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  14. Optimum solution x2 1800 Optimum solution 1600 1400 x1 = Number of basic computer model x2 = Number of XP computer model 1200 1800 800 Feasible region 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  15. Optimum solution x2 1800 Optimum solution x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 x2 = 1200 1200 Optimum Solution is the intersection between: x2 = 1200, and 5x1 + 6x2 = 10000 Solve and x1 = 560 and x2= 1200 Profit = 80(560) + 129(1200) = $199,600 1800 800 Feasible region 600 400 5x1 + 6x2 =10000 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  16. The algebraic model Maximize 80x1 + 129x2 subject to: 5x1 + 6x2< 10000 x1 + 2x2< 3000 x1< 600 x2< 1200 x1, x2> 0

  17. Elements of LP model • Decision variables • The variable whose values must be determined • Objective function • A linear function of decision variables • The value of this function is to be optimized – minimized or maximized • Constraints • Linear functions of the variables • Represents limited resources or minimum requirements

  18. LP requirements • Proportionality of variables • Additivity of resources • Divisibility of variables • Non-negativity • Linear objective function • Linear constraints

  19. Scaling in LP • Poorly scaled model • model contains some very large numbers (e.g. 100,000 or more) and some very small numbers (e.g. 0.001 or less) • Solver may erroneously give an error that the linearity conditions are not satisfied • Three remedies for poorly scaled model • Use Automatic Scaling option in Solver/Options • Redefine the units in the model • Change the Precision setting in Solver's Options dialog box to a larger number, such 0.00001 or 0.0001. (The default has five zeros.)

  20. Solutions to LP problem • Feasible solution • Feasible region • Optimal solution • Unique • Multiple • Unbounded

  21. Multiple Optimum solution x2 1800 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 1200 1800 800 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  22. Multiple Optimum solution x2 1800 x1 = Number of basic computer model x2 = Number of XP computer model 1600 1400 1200 1800 800 600 400 Iso-profit line 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  23. Unbounded Solution x2 x1 = Number of basic computer model x2 = Number of XP computer model 1800 Constraint 2 1600 1400 1200 1000 Constraint 1 800 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  24. Unbounded Solution x2 x1 = Number of basic computer model x2 = Number of XP computer model 1800 Constraint 2 1600 1400 1200 1000 Constraint 1 800 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  25. Infeasible Solution x2 x1 = Number of basic computer model x2 = Number of XP computer model 1800 Constraint 2 1600 1400 1200 1000 Constraint 1 800 600 400 200 200 400 600 2200 3000 800 2400 2800 1200 1800 2000 1000 1400 1600 2600 x1

  26. Summary • An LP model may result in • an unique optimum solution • multiple optimum solutions • unbounded feasible region • infeasible region

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