ADM 2302 Study Guide - Final Guide: List Of Gasoline Additives, Feasible Region, Profit Maximization
3 May 2018
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Business Analytics Notes
WEEK 2
Special Cases in Linear Programming
1. Redundant constraints:
A constraint that does not form a unique boundary of the feasible solution space; its
removal would not alter the feasible solution space.
Example: x <= 10
x <= 12
The second constraint is redundant
2. No feasible solution
Occurs in problems where to satisfy one of the constraints, another constraint must
be violated.
Maximize Z = 5x1 + 3x2
subject to: 4x1 + 2x2 8
x1 4
x2 6
x1, x2 0
3. Multiple Optimal Solutions
Problems in which different combinations of values of the decision variables yield
the same optimal value.
Max 2T + 2C
Subject to:
T + C < 10
T < 5
C < 6
T, C > 0
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The objective function can be
parallel to a constrain line.
Maximize Z=$40x1 + 30x2
subject to: 1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
Where:
x1 = number of bowls
x2 = number of mugs
4. Unbounded Solution
When nothing prevents the solution from becoming infinitely large
Max 2T + 2C
Subject to:
2T + 3C > 6
T, C > 0
Value of the objective function increases indefinitely.
Maximize Z = 4x1 + 2x2
subject to: x1 4
x2 2
x1, x2 0
Characteristics of Linear Programming
• Feasible Region: The set of points that satisfies all constraints
• Corner Point Property: An optimal solution must lie at one or more corner
points
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• Optimal Solution: The corner point with the best objective function value is
optimal
**Sample Midterm Question**
Consider the following linear program:
Max Z = X1 – 2X2
Subject to
– 4X1 + 3X2 <= 3
X1 – X2 <= 3
X1, X2 >= 0
a) Graph the feasible region for the problem.
b) Is the feasible region unbounded? Explain.
c) Find the optimal solution.
d) Does an unbounded feasible region imply that
the optimal solution to the linear program
will be unbounded?
WEEK 3 / WEEK 4
Example:
RMC, Inc. is a firm that produces chemical based products. In a particular process
three raw materials are used to produce two products. The Material requirements
per ton are:
Product Material 1 Material 2 Material 3
Fuel additive 2/5 0 3/5
Solvent base 1/2 1/5 3/10
For the current production period RMC has available the following quantities of
each raw material. Because of spoilage, any materials not used for current
production must be discarded.
Number of Tons
Material Available for Production
Material 1 20
Material 2 5
Material 3 21
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Document Summary
4x1 + 2x2 8 x1 4 x2 6 x1, x2 0. Special cases in linear programming: redundant constraints: A constraint that does not form a unique boundary of the feasible solution space; its removal would not alter the feasible solution space. The second constraint is redundant: no feasible solution. Occurs in problems where to satisfy one of the constraints, another constraint must be violated. Maximize z = 5x1 + 3x2 subject to: multiple optimal solutions. Problems in which different combinations of values of the decision variables yield the same optimal value. 4x1 + 3x2 120 x1, x2 0. The objective function can be parallel to a constrain line. Where: x1 = number of bowls x2 = number of mugs: unbounded solution. When nothing prevents the solution from becoming infinitely large. Maximize z = 4x1 + 2x2 subject to: x1 4 x2 2 x1, x2 0.
