© 2015 mcgraw-hill education. all rights reserved. chapter 3 introduction to linear programming

© 2015 McGraw-Hill Education. All rights reserved.


Frederick S. Hillier Gerald J. Lieberman

Chapter 3

Introduction to Linear Programming


Introduction

• Linear programming– Programming means planning

– Model contains linear mathematical functions

• An application of linear programming– Allocating limited resources among competing

activities in the best possible way

– Applies to wide variety of situations

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3.1 Prototype Example

• Wyndor Glass Co.– Produces windows and glass doors

– Plant 1 makes aluminum frames and hardware

– Plant 2 makes wood frames

– Plant 3 produces glass and assembles products

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Prototype Example

• Company introducing two new products– Product 1: 8 ft. glass door with aluminum

frame

– Product 2: 4 x 6 ft. double-hung, wood-framed window

• Problem: What mix of products would be most profitable?– Assuming company could sell as much of

either product as could be produced

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Prototype Example

• Products produced in batches of 20

• Data needed– Number of hours of production time available

per week in each plant for new products

– Production time used in each plant for each batch of each new product

– Profit per batch of each new product

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Prototype Example


Prototype Example

• Formulating the modelx1 = number of batches of product 1 produced per week

x2 = number of batches of product 2 produced per week

Z = total profit per week (thousands of dollars) from producing these two products

• From bottom row of Table 3.1

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Prototype Example

• Constraints (see Table 3.1)

• Classic example of resource-allocation problem

– Most common type of linear programming problem 8


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Prototype Example

• Problem can be solved graphically– Two dimensional graph with x1 and x2 as the

axes

– First step: identify values of x1 and x2 permitted by the restrictions

• See Figures 3.1 and Figure 3.2

– Next step: pick a point in the feasible region that maximizes value of Z

• See Figure 3.3


Prototype Example

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Prototype Example

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Prototype Example

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• General problem terminology and examples– Resources: money, particular types of

machines, vehicles, or personnel

– Activities: investing in particular projects, advertising in particular media, or shipping from a particular source

• Problem involves choosing levels of activities to maximize overall measure of performance

3.2 The Linear Programming Model


The Linear Programming Model

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• Standard form

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• Other legitimate forms– Minimizing (rather than maximizing) the

objective function

– Functional constraints with greater-than-or-equal-to inequality

– Some functional constraints in equation form

– Some decision variables may be negative

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• Feasible solution– Solution for which all constraints are satisfied

– Might not exist for a given problem

• Infeasible solution– Solution for which at least one constraint is

violated

• Optimal solution– Has most favorable value of objective function

– Might not exist for a given problem17



• Corner-point feasible (CPF) solution– Solution that lies at the corner of the feasible

region

• Linear programming problem with feasible solution and bounded feasible region– Must have CPF solutions and optimal

solution(s)

– Best CPF solution must be an optimal solution

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3.3 Assumptions of Linear Programming

• Proportionality assumption– The contribution of each activity to the value

of the objective function (or left-hand side of a functional constraint) is proportional to the level of the activity

– If assumption does not hold, one must use nonlinear programming (Chapter 13)

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Assumptions of Linear Programming

• Additivity– Every function in a linear programming model

is the sum of the individual contributions of the activities

• Divisibility– Decision variables in a linear programming

model may have any values• Including noninteger values

– Assumes activities can be run at fractional values

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Assumptions of Linear Programming

• Certainty– Value assigned to each parameter of a linear

programming model is assumed to be a known constant

– Seldom satisfied precisely in real applications• Sensitivity analysis used

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3.4 Additional Examples

• Example 1: Design of radiation therapy for Mary’s cancer treatment– Goal: select best combination of beams and

their intensities to generate best possible dose distribution

• Dose is measured in kilorads

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Example 1: Radiation Therapy Design

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• Linear programming model– Using data from Table 3.7

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• A type of cost-benefit tradeoff problem

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Example 2: Reclaiming Solid Wastes

• SAVE-IT company collects and treats four types of solid waste materials– Materials amalgamated into salable products

– Three different grades of product possible

– Fixed treatment cost covered by grants

– Objective: maximize the net weekly profit• Determine amount of each product grade

• Determine mix of materials to be used for each grade

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• Decision variables (for i = A, B, C; j = 1,2,3,4)

number of pounds of material j allocated to product grade i per week

• See Pages 56-57 in the text for solution

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3.5 Formulating and Solving Linear Programming Models on a Spreadsheet

• Excel and its Solver add-in– Popular tools for solving small linear

programming problems

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Formulating and Solving Linear Programming Models on a Spreadsheet

• The Wyndor example– Data entered into a spreadsheet

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• Changing cells– Cells containing the decisions to be made

– C12 and D12 in the Wyndor example below

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3.6 Formulating Very Large Linear Programming Models

• Actual linear programming models– Can have hundreds or thousands of functional

constraints

– Number of decision variables may also be very large

• Modeling language– Used to formulate very large models in

practice

– Expedites model management tasks

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Formulating Very Large Linear Programming Models

• Modeling language examples– AMPL, MPL, OPL, GAMS, and LINGO

• Example problem with a huge model– See Pages 73-78 in the text

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3.7 Conclusions

• Linear programming technique applications– Resource-allocation problems

– Cost-benefit tradeoffs

• Not all problems can be formulated to fit a linear programming model– Alternatives: integer programming or

nonlinear programming models

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© 2015 mcgraw-hill education. all rights reserved. chapter 3 introduction to linear programming

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