Goal Weighted Goal and Preemptive Programming Ivan G

Goal, Weighted Goal, and Preemptive Programming Ivan G. Guardiola Ph. D

What is Goal Programming? l l Mathematical model similar to Linear Programming, however it allows for multiple goals to be satisfied at the same time. Allows for the multiple goals to be prioritized and weighted to account for the DM’s utility for meeting the various goals.

Assumptions l Similar to LP: l l l Non-negative variables Conditions of certainty Variables are independent Limited resources Deterministic

Components l Economic Constraints l l Physical Concerned with resources Cannot be violated Example: # of production hours each week

Components l Goal Constraints l l Variable Concerned with target values Can be changed/modified Example: Desire to achieve a certain level of profit

Introduction l l Most of the optimization problems considered to this point have had a single objective. Often, more than one objective can be identified for a given problem. l l Maximize Return or Minimize Risk Maximize Profit or Minimize Pollution These objectives often conflict with one another. This chapter describes how to deal with such problems.

Goal Programming (GP) l Most LP problems have hard constraints that cannot be violated. . . l l l In some cases, hard constraints are too restrictive. . . l l l There are 1, 566 labor hours available. There is $850, 00 available for projects. You have a maximum price in mind when buying a car (this is your “goal” or target price). If you can’t buy the car for this price you’ll likely find a way to spend more. We use soft constraints to represent such goals or targets we’d like to achieve.

Components l Objective Function l l Minimizes the sum of the weighted deviations from the target values – this is ALWAYS the objective for Goal Programming Not the same as LP (which was maximize revenue/minimize costs)

Goal Programming Steps l l l Define decision variables Define Deviational Variable for each goal Formulate Constraint Equations l l l Economic constraints Goal constraints Formulate Objective Function

Goal Programming Terms l l Decision Variables are the same as those in LP formulations (represent products, hours worked) Deviational Variables represent overachieving or underachieving the desired level of each goal l l d+ Represents overachieving level of the goal d- Represents underachieving level of the goal

Goal Programming Constraints l Economic Constraints l l Stated as <=, >=, or = Linear (stated in terms of decision variables) Example: 3 x + 2 y <= 50 hours Goal Constraints l General form of goal constraint: Decision Variables - d+ + d- = Desired Goal Level

Goal Programming Example l l l Microcom is a growth oriented firm which establishes monthly performance goals for its sales force Microcom determines that the sales force has a maximum available hours per month for visits of 640 hours Further, it is estimated that each visit to a potential new client requires 3 hours and each visit to a current client requires 2 hours

Goal Programming Example l Microcom establishes two goals for the coming month: l l l Contact at least 200 current clients Contact at least 120 new clients Overachieving either goal will not be penalized

Goal Programming Example l Steps Required: 1. Define the decision variables 2. Define the goals and deviational variables 3. Formulate the GP Model’s Parameters: § § § Economic Constraints Goal Constraints Objective Function 4. Solve the GP using the graphical approach

Goal Programming Example l Step 1: Define the decision variables: l l l X 1 = the number of current clients visited X 2 = the number of new clients visited Step 2: Define the goals: l l Goal 1 – Contact 200 current clients Goal 2 – Contact 120 new clients

Goal Programming Example l Step 3: Define the deviational variables l l d 1+ = the number of current clients visited in excess of the goal of 200 d 1 - = the number of current clients visited less than the goal of 200 d 2+ = the number of new clients visited in excess of the goal of 120 d 2 - = the number of new clients visited less than the goal of 120

Goal Programming Example l l Formulate the GP Model: Economic Constraints: l l 2 X 1 + 3 X 2 <= 640 (note: can be <, =, >) X 1, X 2 => 0 d 1+, d 1 -, d 2+, d 2 - => 0 Goal Constraints: l l Current Clients: X 1 + d 1 - - d 1+ = 200 New Clients: X 2 + d 2 - - d 2+ = 120 Must be =

Goal Programming Example l Web. Net establishes two goals for the coming month: l l l Contact at least 200 current clients Contact at least 120 new clients Overachieving either goal will not be penalized

Goal Programming Example l Objective Function: l l Minimize Weighted Deviations Minimize Z = d 1 - + d 2 -

Goal Programming Example l Complete formulation: l l Minimize Z = d 1 - + d 2 Subject to: l 2 X 1 + 3 X 2 <= 640 l X 1 + d 1 - - d 1+ = 200 l X 2 + d 2 - - d 2+ = 120 l X 1, X 2 => 0 l d 1+, d 1 -, d 2+, d 2 - => 0

Goal Programming Example l Graph constraint: l 2 X 1 + 3 X 2 = 640 l l l If X 1 = 0, X 2 = 213 If X 2 = 0, X 1 = 320 Plot points (0, 213) and (320, 0)

Graphical Solution X 2 200 (0, 213) 2 X 1 150 +3 X 2 100 =6 40 50 (320, 0) 0 50 100 150 200 250 300 350 X 1

Goal Programming Example l Graph deviation lines l l l X 1 + d 1 - - d 1+ = 200 (Goal 1) X 2 + d 2 - - d 2+ = 120 (Goal 2) Plot lines for X 1 = 200, X 2 = 120

Goal Programming Example X 2 (0, 213) 200 2 X 1 Goal 1 +3 X 2 150 <= d 164 0 d 2+ (140, 120) 100 d 1+ Goal 2 d 2 - (200, 80) 50 (320, 0) 0 50 100 150 X 1 200 250 300 350

Solving Graphical Goal Programming l l Want to Minimize d 1 - + d 2 So we evaluate each of the candidate solution points: For point (140, 120) d 1 - = 60 and d 2 - = 0 Z = 60 + 0 = 60 Optimal Point For point (200, 80) d 1 - = 0 and d 2 - = 40 Z = 0 + 40 = 40 Contact at least 200 current clients Contact at least 120 new clients

Goal Programming Solution l X 1 = 200 X 2 = 80 d 1+ = 0 d 1 - = 0 l Z = 40 l l l Goal 1 achieved Goal 2 not achieved d 2+ = 0 d 2 - = 40

Lets do this in Excel l Please watch this carefully

A Goal Programming Example: Myrtle Beach Hotel Expansion l l Davis Mc. Keown wants to expand the convention center at his hotel in Myrtle Beach, SC. The types of conference rooms being considered are: Size (sq ft) Unit Cost Small 400 $18, 000 Medium 750 $33, 000 1, 050 $45, 150 Large § Davis would like to add 5 small, 10 medium and 15 large conference rooms. § He also wants the total expansion to be 25, 000 square feet and to limit the cost to $1, 000.

Defining the Decision Variables X 1 = number of small rooms to add X 2 = number of medium rooms to add X 3 = number of large rooms to add

Defining the Goals l Goal 1: The expansion should include approximately 5 small conference rooms. l Goal 2: The expansion should include approximately 10 medium conference rooms. l Goal 3: The expansion should include approximately 15 large conference rooms. l Goal 4: The expansion should consist of approximately 25, 000 square feet. l Goal 5: The expansion should cost approximately $1, 000.

Defining the Goal Constraints-I l Small Rooms § Medium Rooms § Large Rooms where

Defining the Goal Constraints-II l Total Expansion § Total Cost (in $1, 000 s) where

GP Objective Functions l l There are numerous objective functions we could formulate for a GP problem. Minimize the sum of the deviations: § Problem: The deviations measure different things, so what does this objective represent?

GP Objective Functions (cont’d) l Minimize the sum of percentage deviations MIN where ti represents the target value of goal i § Problem: Suppose the first goal is true? underachieved Is this Only thegoal decision maker by 1 small room and the fifth is overachieved by $20, 000. can say for sure. – We underachieve goal 1 by 1/5=20% – We overachieve goal 5 by 20, 000/1, 000= 2% – This implies being $20, 000 over budget is just as undesirable as having one too few small rooms.

GP Objective Functions (cont’d) § Weights can be used in the previous objectives to allow the decision maker indicate – desirable vs. undesirable deviations – the relative importance of various goals l Minimize the weighted sum of deviations MIN § Minimize the weighted sum of % deviations MIN

Defining the Objective l Assume l It is undesirable to underachieve any of the first three room goals l It is undesirable to overachieve or underachieve the 25, 000 sq ft expansion goal l It is undesirable to overachieve the $1, 000 total cost goal Initially, we will assume all the above weights equal 1.

Implementing the Model See file Fig 7 -1. xlsm in the text student website We will walk through it now.

Comments About GP l GP involves making trade-offs among the goals until the most satisfying solution is found. l GP objective function values should not be compared because the weights are changed in each iteration. Compare the solutions! l An arbitrarily large weight will effectively change a soft constraint to a hard constraint. l Hard constraints can be place on deviational variables.

The Mini. Max Objective l Can be used to minimize the maximum deviation from any goal. MIN: Q etc. . .

Summary of Goal Programming 1. Identify the decision variables in the problem. 2. Identify any hard constraints in the problem and formulate them in the usual way. 3. State the goals of the problem along with their target values. 4. Create constraints using the decision variables that would achieve the goals exactly. 5. Transform the above constraints into goal constraints by including deviational variables. 6. Determine which deviational variables represent undesirable deviations from the goals. 7. Formulate an objective that penalizes the undesirable deviations. 8. Identify appropriate weights for the objective. 9. Solve the problem. 10. Inspect the solution to the problem. If the solution is unacceptable, return to step 8 and revise the weights as needed.

The Dewright Company l l l The Dewright Company is one of the largest producers of power tools in the United States. The company is preparing to replace its current product line with the next generation of products—three new power tools. Management needs to determine the mix of the company’s three new products to best meet the following three goals: 1. 2. 3. Achieve a total profit (net present value) of at least $125 million. Maintain the current employment level of 4, 000 employees. Hold the capital investment down to no more than $55 million.

Penalty Weights Goal Factor Penalty Weight for Missing Goal 1 Total profit 5 (per $1 million under the goal) 2 Employment level 4 (per 100 employees under the goal) 2 (per 100 employees over the goal) 3 Capital investment 3 (per $1 million over the goal) 17. 42 © The Mc. Graw-Hill Companies, Inc. , 2008

Data for Contribution to the Goals Unit Contribution of Product Factor 1 2 3 Goal Total profit (millions of dollars) 12 9 15 ≥ 125 Employment level (hundreds of employees) 5 3 4 = 40 Capital investment (millions of dollars) 5 7 8 ≤ 55 17. 43 © The Mc. Graw-Hill Companies, Inc. , 2008

Weighted Goal Programming l l A common characteristic of many management science models (linear programming, integer programming, nonlinear programming) is that they have a single objective function. It is not always possible to fit all managerial objectives into a single objective function. Managerial objectives might include: l l l l Maintain stable profits. Increase market share. Diversify the product line. Maintain stable prices. Improve worker morale. Maintain family control of the business. Increase company prestige. Weighted goal programming provides a way of striving toward several objectives simultaneously.

Weighted Goal Programming l With weighted goal programming, the objective is to l l Minimize W = weighted sum of deviations from the goals. The weights are the penalty weights for missing the goal. Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal. The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal

Weighted Goal Programming Formulation for the Dewright Co. Problem Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize W = 5(Under Goal 1) + 2 Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3) subject to Level Achieved Deviations Goal 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = 125 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 55 and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

Weighted Goal Programming Spreadsheet

Weighted vs. Preemptive Goal Programming l l Weighted goal programming is designed for problems where all the goals are quite important, with only modest differences in importance that can be measured by assigning weights to the goals. Preemptive goal programming is used when there are major differences in the importance of the goals. l l The goals are listed in the order of their importance. It begins by focusing solely on the most important goal. It next does the same for the second most important goal (as is possible without hurting the first goal). It continues the following goals (as is possible without hurting the previous more important goals).

Preemptive Goal Programming l Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal. l The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal l Start with the objective of achieving the first goal (or coming as close as possible): l l Continue with the next goal, but constrain the previous goals to not get any worse: l l Minimize (Amount Over/Under Goal 1) Minimize (Amount Over/Under Goal 2) subject to Amount Over/Under Goal 1 = (amount achieved in previous step) Repeat the previous step for all succeeding goals.

Preemptive Goal Programming for Dewright The goals in the order of importance are: 1. 2. 3. 4. l Start with the objective of achieving the first goal (or coming as close as possible): l l Achieve a total profit (net present value) of at least $125 million. Avoid decreasing the employment level below 4, 000 employees. Hold the capital investment down to no more than $55 million. Avoid increasing the employment level above 4, 000 employees. Minimize (Under Goal 1) Then, if for example goal 1 is achieved (i. e. , Under Goal 1 = 0), then l Minimize (Under Goal 2) subject to (Under Goal 1) = 0

Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 1) Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 1) subject to Level Achieved Goal 1: 12 P 1 + 9 P 2 + 15 P 3 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Goal 125 40 55

Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 2) Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 2) subject to Level Achieved Goal 1: 12 P 1 + 9 P 2 + 15 P 3 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Goal 125 40 55

Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 3) Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 3) subject to Level Achieved Goal 1: 12 P 1 + 9 P 2 + 15 P 3 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Goal 125 40 55

Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 4) Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 2) subject to Level Achieved Goal 1: 12 P 1 + 9 P 2 + 15 P 3 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) (Over Goal 3) = (Level Achieved in Step 3) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Goal 125 40 55

Preemptive Goal Programming Spreadsheet Step 1: Minimize (Under Goal 1)

Preemptive Goal Programming Spreadsheet Step 3: Minimize (Over Goal 3)

Preemptive Goal Programming Spreadsheet Step 4: Minimize (Over Goal 2)

Multi-Objective Decision Making l Many problems have multiple objectives: l Planning the national budget l l Admitting students to college l l raise money, minimize tax burden on low-income, minimize flight of business Planning an investment portfolio l l budget, reach, expenses, target groups Choosing taxation levels l l high SAT or GMAT, high GPA, diversity Planning an advertising campaign l l save social security, reduce debt, cut taxes, build national defense maximize expected earnings, minimize risk Techniques l l Preemptive goal programming Weighted goal programming

Trade-offs Between Objectives & Dominated Decision Alternatives Profit C B A Toxic Waste

Multiple Objective Linear Programming (MOLP) l An MOLP problem is an LP problem with more than one objective function. l MOLP problems can be viewed as special types of GP problems where we must also determine target values for each goal or objective. l Analyzing these problems effectively also requires that we use the Mini. Max objective described earlier.

An MOLP Example: The Blackstone Mining Company l l Blackstone Mining runs 2 coal mines in Southwest Virginia. Monthly production by a shift of workers at each mine is summarized as follows: Type of Coal High-grade Medium-grade Low-grade Cost per month Gallons of toxic water produced Life-threatening accidents Wythe Mine 12 tons 4 tons 10 tons $40, 000 800 0. 20 Giles Mine 4 tons 20 tons $32, 000 1, 250 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.

Defining the Decision Variables X 1 = number of months to schedule an extra shift at the Wythe county mine X 2 = number of months to schedule an extra shift at the Giles county mine

Defining the Objective l There are three objectives: Min: $40 X 1 + $32 X 2 } Production costs Min: 800 X 1 + 1250 X 2 } Toxic water Min: 0. 20 X 1 + 0. 45 X 2 } Accidents

Defining the Constraints l High-grade coal required 12 X 1 + 4 X 2 >= 48 l Medium-grade coal required 4 X 1 + 4 X 2 >= 28 l Low-grade coal required 10 X 1 + 20 X 2 >= 100 l Nonnegativity conditions X 1, X 2 >= 0

Handling Multiple Objectives l If the objectives had target values we could treat them like the following goals: Goal 1: The total cost of productions cost should be approximately t 1. Goal 2: The amount of toxic water produce should be approximately t 2. Goal 3: The number of life-threatening accidents should be approximately t 3. l We can solve 3 separate LP problems, independently optimizing each objective, to find values for t 1, t 2 and t 3.

Implementing the Model See file Fig 7 -8. xlsm

Summarizing the Solutions X 2 12 11 Feasible Region 10 9 8 7 Solution 1 (minimum production cost) Solution 2 (minimum toxic water) 6 5 4 3 2 Solution 3 (minimum accidents) 1 0 0 1 2 3 4 5 6 7 Solution X 1 X 2 Cost 1 2 3 2. 5 4. 0 10. 0 4. 5 3. 0 0. 0 $244 $256 $400 8 9 10 11 12 X 1 Toxic Water Accidents 7, 625 6, 950 8, 000 2. 53 2. 15 2. 00

Defining The Goals l l l Goal 1: The total cost of productions cost should be approximately $244. Goal 2: The gallons of toxic water produce should be approximately 6, 950. Goal 3: The number of life-threatening accidents should be approximately 2. 0.

Defining an Objective l We can minimize the sum of % deviations as follows: § It can be shown that this is just a linear combination of the decision variables. § As a result, this objective will only generate solutions at corner points of the feasible region (no matter what weights are used).

Defining a Better Objective MIN: Q Subject to the additional constraints: § This objective will allow the decision maker to explore non-corner point solutions of the feasible region.

Implementing the Model See file Fig 7 -14. xlsm

Possible Mini. Max Solutions X 2 12 11 Feasible Region 10 9 8 7 6 w 1=10, w 2=1, w 3=1, x 1=3. 08, x 2=3. 92 5 4 w 1=1, w 2=10, w 3=1, x 1=4. 23, x 2=2. 88 3 2 1 w 1=1, w 2=1, w 3=10, x 1=7. 14, x 2=1. 43 0 0 1 2 3 4 5 6 7 8 9 10 11 12 X 1

Comments About MOLP l Solutions obtained using the Mini. Max objective are Pareto Optimal. l Deviational variables and the Mini. Max objective are also useful in a variety of situations not involving MOLP or GP. l For minimization objectives the percentage deviation is: (actual - target)/target l For maximization objectives the percentage deviation is: (target - actual)/target l If a target value is zero, use the weighted deviations rather than weighted % deviations.

Summary of MOLP 1. Identify the decision variables in the problem. 2. Identify the objectives in the problem and formulate them as usual. 3. Identify the constraints in the problem and formulate them as usual. 4. Solve the problem once for each of the objectives identified in step 2 to determine the optimal value of each objective. 5. Restate the objectives as goals using the optimal objective values identified in step 4 as the target values. 6. For each goal, create a deviation function that measures the amount by which any given solution fails to meet the goal (either as an absolute or a percentage). 7. For each of the functions identified in step 6, assign a weight to the function and create a constraint that requires the value of the weighted deviation function to be less than the MINIMAX variable Q. 8. Solve the resulting problem with the objective of minimizing Q. 9. Inspect the solution to the problem. If the solution is unacceptable, adjust the weights in step 7 and return to step 8.