Download - Chapter 2A Linear Prog HK
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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