linear programming models: graphical methods 5/4/1435 (1-3 pm)noha hussein elkhidir

Linear Programming Models:Graphical Methods

5/4/1435 (1-3 pm) noha hussein elkhidir

• Requirements of a Linear Programming Requirements of a Linear Programming ProblemProblem

• Formulating Linear Programming Formulating Linear Programming Problems.Problems.

• Graphical Solution to a Linear Graphical Solution to a Linear Programming ProblemProgramming Problem- Graphical Representation of ConstraintsGraphical Representation of Constraints

- Corner-Point Solution MethodCorner-Point Solution Method


Linear Programming Linear Programming ApplicationsApplications

Production.Production.

Diet Problem ExampleDiet Problem Example

Labor Scheduling Example.Labor Scheduling Example.


When you complete this module you should be able to:

- Formulate linear programming models, - Formulate linear programming models, including an objective function and including an objective function and constraintsconstraints

- Graphically solve an LP problem with the Graphically solve an LP problem with the corner-point method.corner-point method.

- Construct and solve a minimization and - Construct and solve a minimization and maximization problemmaximization problem


- Formulate production-mix, diet, and labor - Formulate production-mix, diet, and labor scheduling problemsscheduling problems


What about Linear What about Linear Programming?Programming?

- A mathematical technique to help plan and - A mathematical technique to help plan and make decisions relative to the trade-offs make decisions relative to the trade-offs necessary to allocate resourcesnecessary to allocate resources

- Will find the minimum or maximum value of - Will find the minimum or maximum value of the objectivethe objective

- Guarantees the optimal solution to the - Guarantees the optimal solution to the model formulatedmodel formulated


Requirements of an Requirements of an LP ProblemLP Problem

1.1. LP problems seek to maximize or LP problems seek to maximize or minimize some quantity (usually profit or minimize some quantity (usually profit or cost) expressed as an objective functioncost) expressed as an objective function

2.2. The presence of restrictions, or The presence of restrictions, or constraints, limits the degree to which we constraints, limits the degree to which we can pursue our objectivecan pursue our objective


3.3. There must be alternative courses of There must be alternative courses of action to choose fromaction to choose from

4.4. The objective and constraints in linear The objective and constraints in linear programming problems must be programming problems must be expressed in terms of linear equations or expressed in terms of linear equations or inequalitiesinequalities


Steps in Developing a Linear Programming Steps in Developing a Linear Programming (LP) Model(LP) Model

1)1) FormulationFormulation

2)2) SolutionSolution


Properties of LP Models

1)1) Seek to minimize or maximizeSeek to minimize or maximize

2)2) Include “constraints” or limitationsInclude “constraints” or limitations

3)3) There must be alternatives availableThere must be alternatives available

4)4) All equations are linearAll equations are linear


Example LP Model Formulation:Example LP Model Formulation:The Product Mix ProblemThe Product Mix Problem

1)1) Decision: How much to make of > 2 Decision: How much to make of > 2 products?products?

2)2) Objective: Maximize profitObjective: Maximize profit

3)3) Constraints: Limited resourcesConstraints: Limited resources


Example: Flair Furniture Co.

Two products: Chairs and TablesTwo products: Chairs and Tables

Decision:Decision: How many of each to make this How many of each to make this month? month?

Objective: Maximize profitObjective: Maximize profit


Flair Furniture Co. DataTables

(per table)

Chairs(per chair)

Hours Available

Profit Contribution

$7 $5

Carpentry 3 hrs 4 hrs 2400

Painting 2 hrs 1 hr 1000

Other Limitations:Other Limitations:

Make no more than 450 chairsMake no more than 450 chairs

Make at least 100 tablesMake at least 100 tables5/4/1435 (1-3 pm) noha hussein elkhidir

Decision Variables:Decision Variables:

T = Num. of tables to makeT = Num. of tables to make

C = Num. of chairs to makeC = Num. of chairs to make

Objective Function: Maximize ProfitObjective Function: Maximize Profit

Maximize $7 T + $5 CMaximize $7 T + $5 C


Constraints:

• Have 2400 hours of carpentry time available

3 T + 4 C < 2400 (hours)

• Have 1000 hours of painting time available

2 T + 1 C < 1000 (hours)


More Constraints:• Make no more than 450 chairs

C < 450 (num. chairs)

• Make at least 100 tables T > 100 (num. tables)

Nonnegativity:Cannot make a negative number of chairs or tables

T > 0C > 0


Model Summary

Max 7T + 5C (profit)

Subject to the constraints:

3T + 4C < 2400 (carpentry hrs)

2T + 1C < 1000 (painting hrs)

C < 450 (max # chairs)

T > 100 (min # tables)

T, C > 0 (nonnegativity)


Graphical Solution

• Graphing an LP model helps provide insight into LP models and their solutions.

• While this can only be done in two dimensions, the same properties apply to all LP models and solutions.


Carpentry

Constraint Line

3T + 4C = 2400

Intercepts

(T = 0, C = 600)

(T = 800, C = 0)

0 800 T

C

600

0

Feasible

< 2400 hrs

Infeasible

> 2400 hrs

3T + 4C = 2400


Painting

Constraint Line

2T + 1C = 1000

Intercepts

(T = 0, C = 1000)

(T = 500, C = 0)

0 500 800 T

C1000

600

0

2T + 1C = 1000


0 100 500 800 T

C1000

600

450

0

Max Chair Line

C = 450

Min Table Line

T = 100

Feasible

Region


0 100 200 300 400 500 T

C

500

400

300

200

100

0

Objective Function Line

7T + 5C = Profit

7T + 5C = $2,100

7T + 5C = $4,040

Optimal Point(T = 320, C = 360)7T + 5C

= $2,800


linear programming models: graphical methods 5/4/1435 (1-3 pm)noha hussein elkhidir

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