Math Models of OR:...
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Math Models of OR: Introduction John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA August 2018 Mitchell Math Models of OR: Introduction 1 / 20
Transcript of Math Models of OR:...
Math Models of OR: Introduction
John E. Mitchell
Department of Mathematical SciencesRPI, Troy, NY 12180 USA
August 2018
Mitchell Math Models of OR: Introduction 1 / 20
Introduction
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 2 / 20
Introduction
What is optimization?
In optimization, we have to make a decision, subject to someconstraints, in order to minimize or maximize an objective function.
Main emphasis of course is on linear optimization: the constraintsand objective function are linear functions of the decision variables.
Mitchell Math Models of OR: Introduction 3 / 20
5.x , - I x ,
IzX , t ¥ X z
Decision variables: × , , X z , . . . , X nParameters: a , , a - , . . . , a n
Linear function: a , x ,t a r k e t . . - t a x ,-i§aixi
Oakwood Furniture Company
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 4 / 20
Oakwood Furniture Company
Oakwood Furniture
The Oakwood Furniture Company manufactures tables and chairs.Each table requires 2 units of wood, and each chair requires 1 unit.Oakwood has 12.5 units on hand. Oakwood sells its furniture to adistributor, who pays $100 per table and $75 per chair. The distributorwants no more than 8 chairs. The distributor also wants at least twiceas many chairs as tables.
How many tables and chairs should Oakwood produce to maximize itsrevenue?
Mitchell Math Models of OR: Introduction 5 / 20
u c = l × # B A #2 x e t X , E /Z 5
X c = 3 , x E - I
←x a
E 8 x . c a # -X c 3 2 x e
Oakwood Furniture Company
Decision variables
First: determine our decision variables:
xt : number of tables producedxc : number of chairs produced
The total revenue is then
100xt + 75xc .
We want to maximize this linear function of xt and xc . This is ourobjective function.
Mitchell Math Models of OR: Introduction 6 / 20
Oakwood Furniture Company
ConstraintsWe have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Sinceeach chair requires 1 unit and each table requires 2 units, we havea constraint:
2xt + xc 12.5upper bound on number of chairs: Distributor requires
xc 8.
ratio of tables and chairs: The number of chairs has to be at leasttwice as large as the number of tables, so we require xc � 2xt . Wecan write this equivalently as:
2xt � xc 0.
nonnegativity: We can’t produce negative numbers of tables orchairs. So we also have the constraints:
xt � 0, xc � 0.Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
ConstraintsWe have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Sinceeach chair requires 1 unit and each table requires 2 units, we havea constraint:
2xt + xc 12.5upper bound on number of chairs: Distributor requires
xc 8.
ratio of tables and chairs: The number of chairs has to be at leasttwice as large as the number of tables, so we require xc � 2xt . Wecan write this equivalently as:
2xt � xc 0.
nonnegativity: We can’t produce negative numbers of tables orchairs. So we also have the constraints:
xt � 0, xc � 0.Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
ConstraintsWe have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Sinceeach chair requires 1 unit and each table requires 2 units, we havea constraint:
2xt + xc 12.5upper bound on number of chairs: Distributor requires
xc 8.
ratio of tables and chairs: The number of chairs has to be at leasttwice as large as the number of tables, so we require xc � 2xt . Wecan write this equivalently as:
2xt � xc 0.
nonnegativity: We can’t produce negative numbers of tables orchairs. So we also have the constraints:
xt � 0, xc � 0.Mitchell Math Models of OR: Introduction 7 / 20
-
Oakwood Furniture Company
ConstraintsWe have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Sinceeach chair requires 1 unit and each table requires 2 units, we havea constraint:
2xt + xc 12.5upper bound on number of chairs: Distributor requires
xc 8.
ratio of tables and chairs: The number of chairs has to be at leasttwice as large as the number of tables, so we require xc � 2xt . Wecan write this equivalently as:
2xt � xc 0.
nonnegativity: We can’t produce negative numbers of tables orchairs. So we also have the constraints:
xt � 0, xc � 0.Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
Feasible solutionsAll the constraints are expressed as linear functions of the variablesxt and xc . A production plan that simultaneously satisfies all theconstraints is a feasible solution. For example, xt = 2, xc = 7 is afeasible solution, with value 2 ⇤ 100 + 7 ⇤ 75 = $725.
xt
xc
0
5
10
2 4 6
xc = 8
2xt + xc = 12.5
2xt � xc = 0
feasible region
(2, 7)
Mitchell Math Models of OR: Introduction 8 / 20
Oakwood Furniture Company
The linear optimization formulation
We can write the complete problem as a linear optimization problem:
maxx2IR2 100xt + 75xcsubject to 2xt + xc 12.5
xc 82xt � xc 0xt , xc � 0
Mitchell Math Models of OR: Introduction 9 / 20
Oakwood Furniture Company
OptimizingThe problem is to find the feasible solution with the largest revenue. Tofind the optimal solution, we can look at contours of the objectivefunction.
xt
xc
0
5
10
2 4 6
(2, 7)
100xt + 75xc = 725
Mitchell Math Models of OR: Introduction 10 / 20
Oakwood Furniture Company
OptimizingThe problem is to find the feasible solution with the largest revenue. Tofind the optimal solution, we can look at contours of the objectivefunction.
xt
xc
0
5
10
2 4 6
(2, 7)
100xt + 75xc = 725
100xt + 75xc = 1100
Mitchell Math Models of OR: Introduction 10 / 20
Oakwood Furniture Company
OptimizingThe problem is to find the feasible solution with the largest revenue. Tofind the optimal solution, we can look at contours of the objectivefunction.
xt
xc
0
5
10
2 4 6
(2, 7)
100xt + 75xc = 725
100xt + 75xc = 1100
100xt + 75xc = 825
(2.25, 8)
The unique optimal solutionis xc = 8, xt = 2.25.
The optimal value is $825.
Mitchell Math Models of OR: Introduction 10 / 20
Oakwood Furniture Company Solving linear optimization problems
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 11 / 20
Oakwood Furniture Company Solving linear optimization problems
Solving linear optimization problems
The optimal solution is a corner point or an extreme point of thefeasible region.
We will investigate the simplex algorithm,which is a method for solving linear optimization problems that movessystematically from one extreme point to a better neighboring extremepoint until it finds the optimal solution.
Mitchell Math Models of OR: Introduction 12 / 20
Oakwood Furniture Company Fractional variables?
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 13 / 20
Oakwood Furniture Company Fractional variables?
Assumption of continuityThe optimal solution is to build 8 chairs and 21
4 tables.May not want 1
4 of a table.So could use integer optimization to find the best integer solution,which is xt = 2, xc = 8, with revenue $800.
It may be that fractional values are meaningful.For example, perhaps Oakwood is measuring in units of 1000, in whichcase the solution xc = 8, xt = 2.25 corresponds to building 8000 chairsand 2250 tables.
When we formulate a problem as a linear optimization problem, we aremaking an assumption of continuity in the variables.
Linear optimization problems are far easier to solve than integeroptimization problems.
Mitchell Math Models of OR: Introduction 14 / 20
Oakwood Furniture Company Sensitivity analysis
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 15 / 20
Oakwood Furniture Company Sensitivity analysis
Sensitivity analysis
If the objective function or constrains are changed slightly, whathappens to the optimal solution?
Don’t necessarily have to solve the modified problem from scratch: canoften use information from the optimal solution of the original problem.
Mitchell Math Models of OR: Introduction 16 / 20
t
Oakwood Furniture Company Sensitivity analysis
A small decrease in revenue per chairFor example, if the revenue per chair changes slightly to 60, thecontour changes slightly, but the optimal solution remains the same,xc = 8, xt = 2.25, now with revenue of $705:
xt
xc
0
5
10
2 4 6
100xt + 60xc = 705
100xt + 75xc = 825
(2.25, 8)
Mitchell Math Models of OR: Introduction 17 / 20
n e w c o n t o u r
o l d contour
Oakwood Furniture Company Sensitivity analysis
A larger decrease in revenue per chairIf the revenue per chair drops further to below 50, then the optimalsolution changes. For example, if the revenue drops to $40 per chair,we get a new optimal solution:
xt
xc
0
5
10
2 4 6
100xt + 40xc = 562.5
100xt + 75xc = 825
(3.125, 6.25)
The optimal solution is the neighboring extreme point,xc = 6.25, xt = 3.125, now with revenue of $562.50.
Mitchell Math Models of OR: Introduction 18 / 20
(24,8) 1 0 0 × 1 50kA
= 6 2 T¥ multiple optimal2xetxc solutionse -12.5
Oakwood Furniture Company General form of a linear optimization problem
Outline
1 Introduction
2 Oakwood Furniture CompanySolving linear optimization problemsFractional variables?Sensitivity analysisGeneral form of a linear optimization problem
Mitchell Math Models of OR: Introduction 19 / 20
Oakwood Furniture Company General form of a linear optimization problem
General linear optimization problemsRecall we wrote the Oakwood Furniture problem as the linearoptimization problem
maxx2IR2 100xt + 75xcsubject to 2xt + xc 12.5
xc 82xt � xc 0xt , xc � 0
A general linear optimization problem has the form:
maximizeor a linear function of several variables in IRn
minimize
subject to linear inequality constraintsand / or
linear equality constraints
Mitchell Math Models of OR: Introduction 20 / 20
