Goal Programming and Multiple Objective LP

1. Goal Programming and Multiple Objective LP • The real-life problems occasionally have multiple (and perhaps conflicting) objectives. • Often, there are goals or targets that the management would like to achieve. • These objectives often conflict with one another. • Absolute requirements are implemented as “hard constraints” whereas targets or goals can be implemented as “soft constraints.“ • Both goal programming (GP) and multiple objective linear programming (MOLP) problems can be solved using: • weighted goals or objectives • or • absolute priorities

2. GP Example - Myrtle Beach Hotel Expansion • For any GP problem: • Define the decision variables • Determine the soft (goal) constraints • Determine the hard constraints (if any) • Develop the objective function (many ways to do this) • Setup with Excel & solve using the weighted goals approach (more . • Davis McKeown wants to expand the convention center at his hotel in Myrtle Beach, South Carolina. He is considering three types of conference rooms: Small, Medium, and Large.

3. Size (sq ft) Unit Cost Small 400 $18,000 Medium 750 $33,000 Large 1,050 $45,150 GP example cont’d • Davis would like to add 5 small, 10 medium and 15 large conference rooms. • He would also like the total expansion to be 25,000 square feet and to limit the cost to $1,000,000. • The room specs are: • Therefore, the decision variables: • Xi = number of i type rooms to add (i=small, medium, large)

4. GP example – summary • Goals: • G1 = expansion…approx. 5 small conf. rooms • G2 = …approx. 10 medium … • G3 = …approx. 15 large… • G4 = …approx. 25,000 sq. ft. • G5 = …cost approx. $1,000,000 • Define the goal (soft constraints): • X1 = 5 simply won’t work (very hard constraint) • Similarly X1 >= 5 or X1 <= 5 won’t work also, so, • X1 = 5 + Variableformoresmallrooms – Variableforlesssmallrooms • Usually the following notation is used: • X1 = 5 + d1+- d1- where subscript denotes the goal no. + and – superscripts denote positive and negative deviations from the goal. • We repeat the same process for all goal constraints by adding and subtracting pos/neg deviational variables.

5. Defining the Goal ConstraintsAlgebraicallySimplified • For each room size, space, and budget constraints:

6. GP Objective Function: A few alternatives • Assume • It is undesirable to underachieve any of the first three room goals • It is undesirable to overachieve or underachieve the 25,000 sq. ft. expansion goal • It is undesirable to overachieve the $1,000,000 total cost goal • GP objective function options: • Min deviations vs Min % deviations vs. Min weighted sum of deviations, etc. • Generally min (or max) weighted sum of deviations is a better choice for most GP formulations. • MiniMax objective is also (occasionally) a good choice to minimize maximum deviation from any goal.

7. More on GP Objective Functions • Minimize the sum of the deviations: • Problem: The deviations measure different things, so what does this objective represent? • Solution: Apply weights (not easy but can be done) • Minimize the sum of percentage deviations where ti represents the target value of goal i • Problem: Suppose the first goal is underachieved by 1 small room and the fifth goal is overachieved by $20,000. • Solution: Minimize the weighted sum of % deviations

8. Defining the Objective • Assume: • It is undesirable to underachieve any of the first three room goals • It is undesirable to overachieveor underachieve the 25,000 sq ft expansion goal • It is undesirable to overachieve the $1,000,000 total cost goal

9. MOLP example • Blackstone Mining operates two coal mines in Southwest Virginia. Monthly production by a shift of workers at each mine is summarized as follows: Type of Coal Wythe Mine Giles Mine High-grade 12 tons 4 tons Medium-grade 4 tons 4 tons Low-grade 10 tons 20 tons Cost per month $40,000 $32,000 Gallons of toxic water produced 800 1,250 Life-threatening accidents 0.20 0.45 Blackstone needs to produce 48 more tons of high-grade, 28 more tons of medium-grade, and 100 more tons of low-grade coal.

10. MOLP example continued • Assume the following goals: • 1: Minimize total production costs • 2: Minimize toxic water production • 3: Minimize life threatening accidents • Setup as an MOLP and solve using: • Absolute priorities approach --------------> • Weighted priorities approach • Define the decision variables: • Xi= No. of months to schedule an extra shift at mine i • Setup the objective functions and the constraints

11. Mining example formulation • There are three objectives: Min: $40 X1 + $32 X2 } Production costs Min: 800 X1 + 1250 X2 } Toxic water Min: 0.20 X1 + 0.45 X2 } Accidents • Three constraints: • 12 X1 + 4 X2 >= 48 • 4 X1 + 4 X2 >= 28 • 10 X1 + 20 X2 >= 100 • And, Xi >= 0 i=1,2 • First, find the target (best) values for each objective (one at a time) • Second, define the “MINIMAX” objective

12. MINIMAX Objective • The concept is to MINImize the MAXimum percent deviation from any goal • The fist goal is cost minimization for which we found the best possible value of $244. Therefore, we can write the percent deviation from this goal as follows: • We would like to have the above % deviation less than or equal to some unknown value, which we call “Q”, then we can write the first goal as a constraint as follows: • Repeat the same concept for all goals and then the objective becomes: Min Q • Let’s implement and experiment with the model…

13. Comments About GP • GP involves making trade-offs among the goals until the most satisfying solution is found • GP objective function values should not be compared because the weights are changed in each iteration. Compare the solutions! • An arbitrarily large weight will effectively change a soft constraint to a hard constraint. • Hard constraints can be placed on deviational variables. • Suggested practice problems: 11, 12, 14, 21, 29 • Group Exercise: • Mecklinburg County - GP version

14. #24