Decisions – that can be represented by decision variables (DVs). Typically represented as (X 1 , X 2 , X 3 , …. X n ); a set of n decision variables
Constraints – one or more functions of the DVs that must be satisfied. Typically should be thought of as restrictions (constraints) on the resources available to achieve the objective.
Objective - some function of the DVs the decision maker wants to MAXimize of MINimize.
As in, MAX or MIN some objective subject to a specific set of constraints based on the decision problem.
The 3 general forms of constraint relationships are represented here.
Note: If all the functions in an optimization problem are linear (the objective functions and the constraints) , then the problem is a Linear Programming (LP) problem
1. Plot the boundary line of each constraint in the model.
2. Identify the feasible region; i.e. the set of all points on the graph that simultaneously satisfy all the constraints.
3. Locate the optimal solution by either:
a. Plotting level curves for the objective function to identify the value of the coordinates for the DVs that satisfy the objective function (will occur at extreme point in the feasible region).
b. Enumerate all the extreme points and calculate the value of the objective function for each of these points.
Alternate Optimal Solutions ; instances where more than one point in the feasible region MAX (or MIN) the objective function
Redundant Constraints ; can occur if a constraint plays no role in determining the feasible region.
Unbounded Solutions ; the objective function can be made infinitely large (for MAX …small for MIN) and still satisfy the constraints. problem
Infeasibility ; instances where there is no way to satisfy all the constraints simultaneously.
9.
An Example LP Problem Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. Aqua-Spa Hydro-Lux Pumps 1 1 Labor 9 hours 6 hours Tubing 12 feet 16 feet Unit Profit $350 $300
10.
5 Steps In Formulating LP Models (Blue Ridge example):
1. Understand the problem.
2. Identify the decision variables.
X 1 =number of Aqua-Spas to produce
X 2 =number of Hydro-Luxes to produce
3. State the objective function as a linear combination of the decision variables.
MAX: 350X 1 + 300X 2
11.
5 Steps In Formulating LP Models ((Blue Ridge example …. continued)
4. State the constraints as linear combinations of the decision variables.
1X 1 + 1X 2 <= 200 } pumps
9X 1 + 6X 2 <= 1566 } labor
12X 1 + 16X 2 <= 2880 } tubing
5. Identify any upper or lower bounds on the decision variables.
X 1 >= 0
X 2 >= 0
12.
So, the formulation of the LP Model for Blue Ridge Hot Tubs is; MAX: 350X 1 + 300X 2 S.T.: 1X 1 + 1X 2 <= 200 9X 1 + 6X 2 <= 1566 12X 1 + 16X 2 <= 2880 X 1 >= 0 X 2 >= 0
The optimal solution occurs where the “pumps” and “labor” constraints intersect.
This occurs where:
X 1 + X 2 = 200 (1)
and 9X 1 + 6X 2 = 1566 (2)
From (1) we have, X 2 = 200 -X 1 (3)
Substituting (3) for X 2 in (2) we have,
9X 1 + 6 (200 -X 1 ) = 1566
which reduces to X 1 = 122
So the optimal solution is,
X 1 =122, X 2 =200-X 1 =78
Total Profit = $350*122 + $300*78 = $66,100
22.
Enumerating The Corner Points Note: This technique will not work if the solution is unbounded. X 2 X 1 250 200 150 100 50 0 0 50 100 150 200 250 (0, 180) (174, 0) (122, 78) (80, 120) (0, 0) obj. value = $54,000 obj. value = $64,000 obj. value = $66,100 obj. value = $60,900 obj. value = $0
23.
Example of Alternate Optimal Solutions X 2 X 1 250 200 150 100 50 0 0 50 100 150 200 250 450X 1 + 300X 2 = 78300 objective function level curve alternate optimal solutions
24.
Example of a Redundant Constraint X 2 X 1 250 200 150 100 50 0 0 50 100 150 200 250 boundary line of tubing constraint Feasible Region boundary line of pump constraint boundary line of labor constraint
25.
Example of an Unbounded Solution X 2 X 1 1000 800 600 400 200 0 0 200 400 600 800 1000 X 1 + X 2 = 400 X 1 + X 2 = 600 objective function X 1 + X 2 = 800 objective function -X 1 + 2X 2 = 400
26.
Example of Infeasibility X 2 X 1 250 200 150 100 50 0 0 50 100 150 200 250 X 1 + X 2 = 200 X 1 + X 2 = 150 feasible region for second constraint feasible region for first constraint
Be the first to comment