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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
Introduction
Outline
1 Introduction
2 Oakwood Furniture Company Solving linear optimization problems Fractional variables? Sensitivity analysis General 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 some constraints, in order to minimize or maximize an objective function.
Main emphasis of course is on linear optimization: the constraints and 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 n Parameters: 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 Company Solving linear optimization problems Fractional variables? Sensitivity analysis General 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 a distributor, who pays $100 per table and $75 per chair. The distributor wants no more than 8 chairs. The distributor also wants at least twice as many chairs as tables.
How many tables and chairs should Oakwood produce to maximize its revenue?
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 produced xc : number of chairs produced
The total revenue is then
100xt + 75xc .
We want to maximize this linear function of xt and xc . This is our objective function.
Mitchell Math Models of OR: Introduction 6 / 20
Oakwood Furniture Company
Constraints We have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Since each chair requires 1 unit and each table requires 2 units, we have a constraint:
2xt + xc 12.5 upper bound on number of chairs: Distributor requires
xc 8. ratio of tables and chairs: The number of chairs has to be at least twice as large as the number of tables, so we require xc � 2xt . We can write this equivalently as:
2xt � xc 0. nonnegativity: We can’t produce negative numbers of tables or chairs. So we also have the constraints:
xt � 0, xc � 0. Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
Constraints We have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Since each chair requires 1 unit and each table requires 2 units, we have a constraint:
2xt + xc 12.5 upper bound on number of chairs: Distributor requires
xc 8. ratio of tables and chairs: The number of chairs has to be at least twice as large as the number of tables, so we require xc � 2xt . We can write this equivalently as:
2xt � xc 0. nonnegativity: We can’t produce negative numbers of tables or chairs. So we also have the constraints:
xt � 0, xc � 0. Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
Constraints We have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Since each chair requires 1 unit and each table requires 2 units, we have a constraint:
2xt + xc 12.5 upper bound on number of chairs: Distributor requires
xc 8. ratio of tables and chairs: The number of chairs has to be at least twice as large as the number of tables, so we require xc � 2xt . We can write this equivalently as:
2xt � xc 0. nonnegativity: We can’t produce negative numbers of tables or chairs. So we also have the constraints:
xt � 0, xc � 0. Mitchell Math Models of OR: Introduction 7 / 20

Oakwood Furniture Company
Constraints We have constraints on the variables:
raw material availablity: only 12.5 units of wood is available. Since each chair requires 1 unit and each table requires 2 units, we have a constraint:
2xt + xc 12.5 upper bound on number of chairs: Distributor requires
xc 8. ratio of tables and chairs: The number of chairs has to be at least twice as large as the number of tables, so we require xc � 2xt . We can write this equivalently as:
2xt � xc 0. nonnegativity: We can’t produce negative numbers of tables or chairs. So we also have the constraints:
xt � 0, xc � 0. Mitchell Math Models of OR: Introduction 7 / 20
Oakwood Furniture Company
Feasible solutions All the constraints are expressed as linear functions of the variables xt and xc . A production plan that simultaneously satisfies all the constraints is a feasible solution. For example, xt = 2, xc = 7 is a feasible 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 + 75xc subject to 2xt + xc 12.5
xc 8 2xt � xc 0 xt , xc � 0
Mitchell Math Models of OR: Introduction 9 / 20
Oakwood Furniture Company
Optimizing The problem is to find the feasible solution with the largest revenue. To find the optimal solution, we can look at contours of the objective function.
xt
xc
0
5
10
2 4 6
(2, 7)
100xt + 75xc = 725
Mitchell Math Models of OR: Introduction 10 / 20
Oakwood Furniture Company
Optimizing The problem is to find the feasible solution with the largest revenue. To find the optimal solution, we can look at contours of the objective function.
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
Optimizing The problem is to find the feasible solution with the largest revenue. To find the optimal solution, we can look at contours of the objective function.
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 solution is 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 Company Solving linear optimization problems Fractional variables? Sensitivity analysis General 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 the feasible region.
We will investigate the simplex algorithm, which is a method for solving linear optimization problems that moves systematically from one extreme point to a better neighboring extreme point 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 Company Solving linear optimization problems Fractional variables? Sensitivity analysis General form of a linear optimization problem
Mitchell Math Models of OR: Introduction 13 / 20
Oakwood Furniture Company Fractional variables?
Assumption of continuity The optimal solution is to build 8 chairs and 214 tables. May not want 14 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 which case the solution xc = 8, xt = 2.25 corresponds to building 8000 chairs and 2250 tables.
When we formulate a problem as a linear optimization problem, we are making an assumption of continuity in the variables.
Linear optimization problems are far easier to solve than integer optimization problems.
Mitchell Math Models of OR: Introduction 14 / 20
Oakwood Furniture Company Sensitivity analysis
Outline
1 Introduction
2 Oakwood Furniture Company Solving linear optimization problems Fractional