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Modeling Summary
Types of Constraints
Resource Limitations
Minimum Requirements
Supply and Demand Balance
Ratio Control
Bounds
Accounting Relations
Deviation Constraints
Approximation or Convexity Constraints
Modeling Summary
Types of Variables
Production Variables
Sales Variables
Purchase Variables
Transformation Variables
Slack Variables
Surplus Variables
Artificial Variables
Step Variables
Deviation Variables
Accounting Variables
Modeling Summary
Violation of Assumptions
Proportionality
Max 10Y
s.t.
Y

 6X1
Z1
 8Z 2
 1.2X 2
3X1

3X 2
4X1

4X 2
 0

 0
Z1

Z2
 4
X1
 4
X2
Y,
X1 ,
 0
X2 ,
Z1 ,
Z2 ,
 0
Are the algorithmic assumptions violated? Yes and no. They are not
mathematically violated but they are conceptually violated
- Decreasing Returns to Scale
Modeling Summary
Violation of Assumptions
Non-Additive
Max 3Y
s.t.
Y
Y,
 2X1
 2X 2
 2X 3

0
4X1
 2X 2


r1
X1
 2X 2
 4X 3
 r2
X1 ,
X2 ,
X3
 0.
X3
We get more out of using the inputs together, f(X + Y), than we do separately, f(X) +
f(Y).
The additivity assumption always holds for the individual variables. It may not
hold for the model through the combination of variables.
Modeling Summary
Violation of Assumptions
Certainty
Max  3Y  .3(5X 1 )  .7(4X 2 )
s.t.
Y 
X1

0
X1

2
X2

0
X2

3
X2
 0.
Y
Y,

X1 ,
Are the returns to X certain?
Modeling Summary
Violation of Assumptions
Continuous
Max 3X1
 2X 2
s.t.
X1

X2
 10,000
2X1

X2
 4,000
Y2

X1 ,
0 and integer
The solution without the requirement that X1 and X2
be integer is X1 = 4,666 2/3 and X2 = 5,333 1/3.
Since X is so large can we round?