PowerPoint Slides by Robert F. BrookerCopyright (c) 2001 by Harcourt, Inc. All rights reserved. Linear Programming Mathematical Technique for Solving Constrained.

PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.

Linear Programming

• Mathematical Technique for Solving Constrained Maximization and Minimization Problems

• Assumes that the Objective Function is Linear

• Assumes that All Constraints Are Linear


Applications of Linear Programming

• Optimal Process Selection

• Optimal Product Mix

• Satisfying Minimum Product Requirements

• Long-Run Capacity Planning

• Least Cost Shipping Route(Transportation Problems)


Applications of Linear Programming

• Airline Operations Planning

• Output Planning with Resource and Process Capacity Constraints

• Distribution of Advertising Budget

• Routing of Long-Distance Phone Calls

• Investment Portfolio Selection

• Allocation of Personnel Among Activities


Production Processes

Production processes are graphed as linear rays from the origin in input space.

Production isoquants are line segments that join points of equal output on the production process rays.



Processes Isoquants



Feasible Region Optimal Solution (S)


Formulating and Solving Linear Programming Problems

• Express Objective Function as an Equation and Constraints as Inequalities

• Graph the Inequality Constraints and Define the Feasible Region

• Graph the Objective Function as a Series of Isoprofit or Isocost Lines

• Identify the Optimal Solution


Profit MaximizationQuantities of Inputs

Available perTime Period

Input Product X Product Y TotalA 1 1 7B 0.5 1 5C 0 0.5 2

Quantities of InputsRequired perUnit of Output

Maximize

Subject to

(objective function)

(input A constraint)

(input B constraint)

(input C constraint)

(nonnegativity constraint)

= $30QX + $40QY

1QX + 1QY 7

0.5QX + 1QY 5

0.5QY 2

QX, QY 0



Profit MaximizationMultiple Optimal Solutions

New objective function has the same slopeas the feasible region at the optimum


Profit MaximizationAlgebraic Solution

Points of Intersection Between Constraintsare Calculated to Determine the Feasible Region


Profit MaximizationAlgebraic Solution

Profit at each point of intersection between constraintsis calculated to determine the optimal point (E)

Corner Point QX QY Profit

0 0 0 $0D 7 0 210*E 4 3 240F 2 4 220G 0 4 160


Cost Minimization

Minimize

Subject to

C = $2QX + $3QY

1QX + 2QY 14

1QX + 1QY 10

1QX + 0.5QY 6

QX, QY 0


(protein constraint)

(minerals constraint)

(vitamins constraint)


Meat (Food X) Fish (Food Y)Price per pound $2 $3

Minimum DailyRequirements

Nutrient Meat (Food X) Fish (Food Y) TotalProtein 1 1 7

Minerals 0.5 1 5Vitamins 0 0.5 2

Units of NutrientsPer Pound of


Cost Minimization

Feasible Region Optimal Solution (E)


Cost MinimizationAlgebraic Solution

Cost at each point of intersection between constraintsis calculated to determine the optimal point (E)

Corner Point QX QY Cost

D 14 0 $28*E 6 4 24F 2 8 28G 0 12 36


Dual of the ProfitMaximization Problem

Maximize

Subject to

= $30QX + $40QY

1QX + 1QY 7

0.5QX + 1QY 5

0.5QY 2

QX, QY 0


(input A constraint)

(input B constraint)

(input C constraint)


Minimize

Subject to

C = 7VA + 5VB + 2VC

1VA + 0.5VB $30

1VA + 1VB + 0.5VC $40

VA, VB, VC 0


Dual of the CostMinimization Problem

Maximize

Subject to

= 14VP + 10VM + 6VV

1VP + 1VM + 1VV $30

2VP + 1VM + 0.5VV $40

VP, VM, VV 0

Minimize

Subject to

C = $2QX + $3QY

1QX + 2QY 14

1QX + 1QY 10

1QX + 0.5QY 6

QX, QY 0


(protein constraint)

(minerals constraint)

(vitamins constraint)


PowerPoint Slides by Robert F. BrookerCopyright (c) 2001 by Harcourt, Inc. All rights reserved. Linear Programming Mathematical Technique for Solving Constrained.

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objective function input

profit maximization