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Linear Programming Assumptions

Linearity is a requirement of the model in both objective functionand constraints

Homogeneity of products produced (i.e., products must theidentical) and all hours of labor used are assumed equallyproductive

Divisibility assumes products and resources divisible (i.e., permitfractional values if need be)

While IntegerLinear Programming (ILP) removes this assumption,use integer variables can greatly increase the computing times

Every LP problem involves the following:

decisions that must be made

an objective

a set of restrictions (or constraints)

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Formulating LP Models, The Steps

X Given a problem, first, determine the objective or goal. Maximize(or minimize) what?

Y Identify & define the decision variables (unknowns).

What should they represent and how many do we need?

Z State the objective as a linearfunction of the decision variables.

[ Translate the requirements, restrictions, or wishes, that are innarrative form to linear functions.

\ Identify any lower or upper bounds on the decision variables(non-negativity constraints are very common)

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General form of an LP problem

MAX (or MIN): f0(X1, X2, , Xn)Subject to: f1(X1, X2, , Xn) = bk:

fm(X1, X2, , Xn) = bmXi >= 0

Note:

If all the functions in an optimization are linear, the problem is a LinearProgramming (LP) problem

If all the variables are integer, then the problem is an Integer LinearProgramming (ILP) problem

If some of the variables are integer, then the problem is an Mixed IntegerLinear Programming (MILP) problem

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An example

Blue Ridge Hot

Tubs, Inc. produces two types of hot tubs:

Aqua-Spas & Hydro-Luxes. The critical resources are available labor

hours, on hand tubing and pumps for the next production cycle.The following table outlines usage factors and unit profit:

Aqua-Spa Hydro-Lux

Pumps 1 1

Labor 9 hours 6 hours

Tubing 12 feet 16 feet

Unit Profit $350 $300

There are 200 pumps, 1566 hours of labor, and 2880 feet of tubingavailable.

Formulate an LP model, solve graphically, and solve via solver.

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Formulating LP Models

XGiven a problem, first, determine the objective orgoal. Maximize (or minimize) what?

Y Identify & define the decision variables (unknowns).

What should they represent and how many do we need?

Z State the objective as a linearfunction of the decisionvariables.

Maximize profits

Max 350 X1 + 300 X2

X1=number of Aqua-Spas to produceX2=number of Hydro-Luxes to produce

or abbreviate as follows

Xi = no. of product i to make, where i=1,2

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Formulating LP Models

[ Translate the requirements, restrictions, or wishes,that are in narrative form to linear functions.

\ Identify any lower or upper bounds on the decisionvariables (non-negativity constraints are v. common).

1X1 + 1X2 = 0 i=1,2

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The Complete LP Model

MAX: 350X1 + 300X2S.T.: 1X1 + 1X2

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Graphical solution approachX2

X1

250

200

150

100

50

0

0 50 100 150 200 250240

Feasible Region

boundary line of tubing constraint

12X1 + 16X2 = 2880

boundary line of pump constraintX1 + X2 = 200

boundary line of labor constraint

9X1 + 6X2 = 1566

261

174

180

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Enumerating the corner points

X2

X1

250

200

150

100

50

0

0 50 100 150 200 250

o.f.v. = $54,000

(0, 180)

o.f.v. = $64,000

(80, 120)

o.f.v. = $66,100

(122, 78)

o.f.v. = $60,900

(174, 0)o.f.v. = $0

(0, 0)

$15,000

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How solver views the model

Target cell - the cell in thespreadsheet that represents theobjective function

Changing cells - the cells in the

spreadsheet representing thedecision variables

Constraint cells - the cells in thespreadsheet representing the LHS

formulas on the constraints

Click on Options and checkAssume Linear Model andAssume Non-negative, then

Solve!

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Optimal Solution andSensitivity Analysis

In class problem: Solved problem 1 Formulate

Setup in Excel and solve

Interpret the solution and the sensitivity report

Formulating the business problem as an LP is the most critical and

value added part. There are many commercial solvers that can solve huge problems in

minutes or even seconds.

We can use solver to gain insights to scaled down versions of realproblems.

Group exercises:

Suggested problems: 3, 4, 5, 6, 9