Goal Programming

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Goal Programming
Mathematical Programming ModelingENGC 6362
Lecture 6
Dr. Rifat Rustom
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GP is a variation of Linear Programming that can
be used for problems that involve multiple
objectives.- Multiple objectives are referred
to as Goals.- In GP models, goals are expressed
as constraints.- In GP, the objective is to
satisfy the nongoal constraints (if any) and
achieve reasonably acceptable levels for the goal
constraints
Goal Programming
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A manager has formulated a goal for labor hours
Model Formulation
Minimize P1u1 P2v1 P3u2 Subject
to 2) 4x1 2x2 lt 40 (Nongoal
constraint) 3) 2x1 6x2 lt
60 4) 3x1 3x2 u1 - v1 75 (Goal
constraint) 5) x1 2x2 u2 - v2 50
(Goal constraint) x1, x2, u1, u2, v1, v2 gt
0 where x1, x2 Decision
variables u1, u2, v1, v2 Deviation variables
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Deviation Variables
To account for possible deviation from a goal,
deviation variables are incorporated into each
goal to represent the difference between actual
performance and target performance. vi amount
by which we numerically exceed the ith goal
being over the targeted amount (overachievement) u
i amount by which we numerically under the ith
goal being under the targeted amount
(underachievement) Some texts use S, S-
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Preemptive Goal Programming (Priority Models)
In priority models, the objective indicates which
deviation variables will be minimized and their
order of importance. The objective is to
minimize specified deviations (u or v) from
certain goals according to priority. Minimizing
the deviation with the highest priority is
treated as infinitely more desirable than
minimizing the deviation with the next highest
priority, and minimizing the next highest
priority is treated as infinitely more important
than the next highest priority, and so on. P1 gtgtgt
P2 gtgtgt P3 gtgtgt Pn
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Mathematical Representation of the Goal
Constraint
Nongoal constraint 4x1 2x2
lt 40
Goal constraint 4x1 2x2 u1 - v1 100
hours
Overtime hours (surplus)
Target (goal) amount
Decision variables
Underutilization of labor
Case 1 4x1 2x2 100 u1 0, v1
0 Case 2 4x1 2x2 80 u1 20, v1
0 Case 3 4x1 2x2 110 u1 0, v1 10
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Process to Formulate a Goal Model
Identify the Decision Variables
Identify the constraints (nongoals goals)
Formulate the goal constraints
Formulate the nongoal constraints
Formulate the Objective
Add the non-negativity requirement
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Example 1

• A company manufactures three products x1, x2,
  x3. Material and labor requirements per unit are
• Product x1 x2 x3 Availability
• Material 2 4 3 600 kg
• Assembly (min./unit) 9 8 7 900 min.
• Packing (min./unit) 1 2 3 300 min.
• The manger has listed the following objectives in
  order of priority
• 1. Minimize overtime in the assembly department.
• 2. Minimize undertime in the assembly department.
• 3. Minimize both undertime and overtime in the
  packing department.

• Minimize P1v1 P2u1 P3(u2 v2)
• Subject to
• 2) 2x1 4x2 3x3 lt 600
  (material constraint)
• 3) 9x1 8x2 7x3 u1 - v1 900
  (assembly constraint)
• 4) 1x1 2x2 3x3 u2 - v2 300
  (packing constraint)
• x1, x2, u1, u2, v1, v2 gt 0

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Graphical Representation of GP
X2
Minimize P1u1 Subject to x1 x2 u1 -
v1 40
60
50
Target (goal)
Goals are plotted one at a time, according to the
priorities given by the objective
40
over
P1
30
x1 x2 40
20
under
X1
20
40
60
80
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Graphical Solution of GP

• Example 2
• Minimize P1u1 P2u2 P3u3
• Subject to
• 2) 5x1 3x2 lt 150
• 3) 2x1 5x2 u1 - v1 100
• 4) 3x1 3x2 u2 - v2 180
• 5) x1 u3 - v3 40
• All variables gt 0

Optimal Solution x1 0 x2 50, u1 0 to
find u2 3x1 3x2 u2 - v2 180 3(0) 3(50)
u2 0 180 gt u2 30 u3 40
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Computer Solution to GP
Computer Solution
Solve the model that contains any nongoal
constraints and one goal constraint
Start with the goal constraint which contains
the deviation variable that has the highest
priority
Fix the value of the deviation variable for the
remainder of the analysis
Delete the variable from the model
Identify the next deviation variable that has
the highest priority
Is there any other deviation variables in the
objective
Yes
No
END
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Computer Solution Example 2

• Step 1
• Minimize 0x1 0x2 u1 0v1
• Subject to
• 2) 5x1 3x2 lt 150
• 3) 2x1 5x2 u1 - v1 100
• Solution 1
• s1 90 u1 0
• x2 20 v1 0
• Optimal z 0

• Minimize P1u1 P2u2 P3u3
• Subject to
• 2) 5x1 3x2 lt 150
• 3) 2x1 5x2 u1 - v1 100
• 4) 3x1 3x2 u2 - v2 180
• 5) x1 u3 - v3 40
• All variables gt 0

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Computer Solution Example 2 - Step 2
Step 2 - Omit u1 (substitute its value) Minimize
0x1 0x2 0v1 u2 v2 Subject to 2) 5x1
3x2 lt 150 3) 2x1 5x2 0 -
v1 100 4) 3x1 3x2 u2 - v2 180
Solution 2 x1 0 u2
30 x2 50 v1
150 Optimal z 30 v2 0

• Minimize P1u1 P2u2 P3u3
• Subject to
• 2) 5x1 3x2 lt 150
• 3) 2x1 5x2 u1 - v1 100
• 4) 3x1 3x2 u2 - v2 180
• 5) x1 u3 - v3 40
• All variables gt 0

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Computer Solution Example 2 - Step 3
Step 3 - Omit u2 (substitute its value) Min
0x1 0x2 0v1 0v2 u3 0v3 Subject to
2) 5x1 3x2 lt 150 3) 2x1 5x2
0 - v1 100 4) 3x1 3x2 30 - v2
180 5) x1 u3 - v3 40
Solution 3 x1 0 u3
40 x2 50 v1
150 v2 0 v3
0 Optimal z 40

• Minimize P1u1 P2u2 P3u3
• Subject to
• 2) 5x1 3x2 lt 150
• 3) 2x1 5x2 u1 - v1 100
• 4) 3x1 3x2 u2 - v2 180
• 5) x1 u3 - v3 40
• All variables gt 0

Final Solution x1 0
u1 0 x2 50 u2 30
u3
40 Optimal z 40
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Homework

• 1. Read pp. 282 and Sec. 6.6 pp 286
• 2. Project 1
• Group 1
• Optimize the use of three materials in road
  pavements (asphalt, base course, and kurkar).
  (criteria either min. cost or max stability)
• Group 2
• Optimize the use of three diameters of steel
  reinforcement in concrete footings of different
  loading (moment).
• Group 3
• Optimize the traffic signal timing in Al Jala
  Crossing.
• You should collect any necessary material and
  data as well as consult designers if necessary.
• Due date April 15, 2001