Goal Programming In many linear programming problems, the Strategic Resource Allocation and Planning MGMT E-5050 In many linear programming problems, the objective function or goal extends beyond just maximizing total profit or minimizing total cost. Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD

Goal Programming Maximizing market share Maintaining full employment POSSIBLE ECONOMIC GOALS Maximizing market share Maintaining full employment Minimizing production idle time Restricting overtime labor Adhering to limited storage space Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions

Goal Programming Maximizing land usage POSSIBLE NON-ECONOMIC GOALS Maximizing land usage Minimizing federal grant overspending Maximizing the number of people reached by adverti-sing Minimizing noise levels in the neighborhood Minimizing staff shortages

Goal Programming Regardless of their nature and number, these multiple and diverse goals share several common characteristics: THEY CANNOT BE MEASURED ON THE SAME SCALE SOME INVOLVE SQUARE FOOTAGE, MONEY, TIME, COMPLIANCE, AND QUALITY OF LIFE ISSUES THEY ARE USUALLY IN CONFLICT THE ACHIEVEMENT OF ONE OBJECTIVE THREATENS DIMINISHMENT OR ABANDONMENT OF ANOTHER

Multiple Objectives 1st Approach TWO APPROACHES DEVELOP A SINGLE OBJECTIVE FUNCTION THAT ADDRESSES ALL GOALS SIMULTANEOUSLY THIS IS ACCOMPLISHED BY CONVERTING THE VALUES OF ALL THE GOALS TO A COMMON MEASURE OF VALUE OR UTILITY

Multiple Objectives 2nd Approach TWO APPROACHES ACKNOWLEDGE THE DIFFERENCES IN THE VARIOUS GOALS AND USE A PROCEDURE, EITHER GRAPHICAL, COMPUTER-BASED, OR SIMPLEX, TO ADDRESS THEM INDIVIDUALLY THIS IS CALLED GOAL PROGRAMMING

Goal Programming Not to develop an optimal solution as such, THE PRIMARY PURPOSE Not to develop an optimal solution as such, but to ‘satisfice’, that is, meet the desired level of achievement set for each goal, and if that is not entirely possible, to minimize the actual deviations from those desired levels.

Goal vs. Linear Programming THE DIFFERENCES Goal programming attempts to minimize the deviations between the various established goals and what can be actually achieved, given the available resources. Variables called deviational variables are typically the ONLY variables in the objective function. The objective function is formulated to MINIMIZE the sum of the differences between the deviational variables.

Goal Programming THERE MAY BE GOALS THAT MUST BE MET , AS OPPOSED TO GOALS THAT THE DECISION MAKER ATTEMPTS TO MEET THE FORMER WILL ALWAYS BE REPRESENTED BY REGULAR LINEAR PROGRAMMING CONSTRAINTS !

Goal Programming THE OBJECTIVE FUNCTION A function of the deviational variables, it represents non-achievement of the various established goals, and must always be minimized - d deviational variable that measures under achievement + d deviational variable that measures over achievement

Goal Programming Words such as : INDICATORS should must ought avoid exactly ….or others that imply that there is a level of performance , below which, or above which, one does not want to be.

Goal Programming * * If overachievement is undesired, the deviational OBJECTIVE FUNCTION FORMULATION If overachievement is undesired, the deviational variable d+ will be placed in the objective function to be minimized. If underachievement is undesired, the deviational variable d- will be placed in the objective function If both overachievement and underachievement are undesired, the deviational variables d+ and d- will be placed in the objective function to be minimized. * * THAT IS, THE GOAL IS TO BE MET EXACTLY

Goal Programming Example HARRISON ELECTRIC COMPANY The Harrison Electric Company produces two products popular with home renovators: old-fashioned chandeliers and ceiling fans. Both products require a two-step production process involving wiring and assembly. It takes about 2 hours to wire each chandelier and 3 hours to wire a ceiling fan. Final assembly of the chandeliers and fans requires 6 and 5 hours, respectively. The production capability is such that only 12 hours of wiring time and 30 hours of assem- bly time are available. Each chandelier nets the firm $7.00 and each fan $6.00 . REQUIREMENT: Formulate the objective function and resource constraints. PROBLEM STATEMENT

Goal Programming Example HARRISON ELECTRIC COMPANY OBJECTIVE FUNCTION Maximize Z = $7.00 X1 + $6.00 X2 subject to: 2X1 + 3X2 =< 12 wiring hours 6X1 + 5X2 =< 30 assembly hours MODEL FORMULATION RESOURCE CONSTRAINTS Where: X1 = CHANDELIERS PRODUCED X2 = CEILING FANS PRODUCED

Goal Programming Example HARRISON ELECTRIC COMPANY OBJECTION FUNCTION MODIFICATION Management now wants to achieve a profit goal of exactly $30.00. The new objective function is: - + Minimize deviations = d1 + d1 WHERE d1- REPRESENTS UNDERACHIEVEMENT OF THE PROFIT GOAL WHERE d1+ REPRESENTS OVERACHIEVEMENT OF THE PROFIT GOAL

Goal Programming Example HARRISON ELECTRIC COMPANY - + Minimize total deviation = d1 + d1 subject to: SINGLE GOAL MODEL FORMULATION - + 7X1 + 6X2 + d1 - d1 = $30.00 profit goal 2X1 + 3X2 =< 12 wiring hours 6X1 + 5X2 =< 30 assembly hours X1, X2, d1- , d1+ => 0 NON-NEGATIVITY CONSTRAINT

Goal Programming Example HARRISON ELECTRIC COMPANY SINGLE GOAL MODEL SOLUTION X1 ( chandeliers ) = 4.2857 units X2 ( ceiling fans ) = 0 units d1 = 0 ( no profit overachievement ) d1 = 0 ( no profit underachievement ) d2 = 3.4286 ( under-used wiring hours ) d3 = 4.2857 ( under-used assembly hrs ) Z = 0 ( objective function ) DETERMINED VIA COMPUTER + - - -

Goal Programming Example HARRISON ELECTRIC COMPANY X1 ( chandeliers ) = 4.2857 units X2 ( ceiling fans ) = 0 units d1 = 0 ( no profit overachievement ) d1 = 0 ( no profit underachievement ) d2 = 3.4286 ( under-used wiring hours ) d3 = 4.2857 ( under-used assembly hrs ) Z = 0 ( objective function ) IF THE TARGET GOAL OF $30.00 IS ACHIEVED EXACTLY, BOTH d1+ AND d1- WILL BE EQUAL TO ZERO SINGLE GOAL MODEL SOLUTION + - THE OBJECTIVE FUNCTION WILL ALSO BE MINIMIZED AT ZERO - -

Goal Programming Example HARRISON ELECTRIC COMPANY SOLUTION POSTSCRIPT If overachievement were acceptable, the “d+” deviational variable can be eliminated from the objective function. If underachievement were acceptable, the “d-”

Goal Programming Example HARRISON ELECTRIC COMPANY GOAL RANKING SCENARIO Goal #1 Produce as much profit above $30.00 as possible during the production period. Fully employ available wiring hours. Avoid overtime in assembly hours. Produce at least seven (7) ceiling fans. Goal #2 Goal #3 Goal #4

Goal Programming Example HARRISON ELECTRIC COMPANY GOAL RANKING THE VARIABLES P1d1+/- P2d2+/- P3d3+/- P4d4+/- profit target wiring hours used assembly hours used ceiling fans produced THE RANKINGS

Goal Programming Example HARRISON ELECTRIC COMPANY THE GOAL RANKING MODEL - - + - Minimize Σ deviations = P1d1 + P2d2 + P3d3 + P4d4 subject to: + - 7X1 + 6X2 + d1 – d1 = 30 ( profit ) 2X1 + 3X2 + d2 – d2 = 12 ( wiring hours ) 6X1 + 5X2 + d3 – d3 = 30 ( assembly hours ) 1X2 + d4 – d4 = 7 ( ceiling fans ) - + - + - + All Xi , di variables => 0

Goal Programming Example HARRISON ELECTRIC COMPANY WEIGHTED GOALS Sometimes, one goal is more important than another goal, but their priority levels are the very same. In these cases, the coefficients in the objective function for the deviational variables are the weights assigned to the goals.

Goal Programming Example HARRISON ELECTRIC COMPANY WEIGHTED GOALS If goal # 4 is the least important goal……………… its weight is “1” If goal # 3 is twice as important as goal # 4…….. ..its weight is “2” If goal # 2 is four times as important as goal # 4... its weight is “4” If goal # 1 is six times as important as goal # 4… ..its weight is “6” THE LEAST IMPORTANT GOAL IS ALWAYS WEIGHTED “1”

Goal Programming Example HARRISON ELECTRIC COMPANY WEIGHTED GOALS THE OBJECTIVE FUNCTION BECOMES…. - - + - Minimize Σ deviations = 6d1 + 4d2 + 2d3 + 1d4 THE RELATIVE WEIGHTS

Goal Programming Example HARRISON ELECTRIC COMPANY GOAL RANKING WITH WEIGHTS THE OBJECTIVE FUNCTION BECOMES…. THE RANKINGS - - + - Minimize Σ deviations = P1( 6d1 ) + P2( 4d2 ) + P3( 2d3 ) + P4( 1d4 ) THE WEIGHTS

Goal Programming with QM for WINDOWS Harrison Electric Company

Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions

Goal 1 : to produce profit of $30 Goal 1 : to produce profit of $30.00 if possible during the production period. Goal 2 : to fully utilize the available wiring department hours. Goal 3 : to avoid overtime in the assembly department. Goal 4 : to meet a contract requirement to produce at least seven ceiling fans.

Goal Programming Example Attaché Training Program Major Bill Bligh, Director of the Army War College’s new six-month attaché training program, is concerned about how the 20 officers taking the course spend their time while in his charge. Major Bligh recognizes that there are 168 hours per week and thinks that his students have been using them rather inefficiently. Bligh lets: X1 = number of hours of sleep needed per week X2 = number of personal hours ( eating, personal hygiene, handling laundry, and so on ). X3 = number of hours of class and studying. X4 = number of hours of social time off base ( dating, sports, family visits, and so on )

Goal Programming Example Attaché Training Program He thinks that students should study 30 hours a week to have time to absorb the material. This is his most important goal. Bligh feels that students need at most 7 hours sleep per night on average and that this goal is number 2. He believes that goal number 3 is to provide at least 20 hours per week of social time. Requirement: Formulate this as a goal programming problem. Solve the problem using computer software.

Goal Programming Example Attaché Training Program MODEL FORMULATION Let: d1- = underachievement of class and study goal d1+ = overachievement of class and study goal d2+ = overachievement of sleeping goal d3- = underachievement of social time goal

Goal Programming Example Attaché Training Program MODEL FORMULATION The objective function becomes: Minimize = d1- + d1+ + d2+ + d3- subject to constraints ( per week ) 1X3 + d1- - d1+ = 30 hours class / study 1X1 + d2- - d2+ = 49 hours sleep 1X4 + d3- - d3+ = 20 hours social time all variables => 0 1X1 + 1X2 + 1X3 + 1X4 =< 168 hours personal time is whatever is left ! ( “d4” can be omitted )

Goal Programming Example Attaché Training Program MODEL FORMULATION Since the goals have priority, they can be rewritten in this order, yielding the absolute completion of each goal before attempting to achieve the next goal. The objective function would become: Minimize = P1d1- + P1d1+ + P2d2+ + P3d3- where: P1 = meet class and study goal P2 = meet sleeping goal P3 = meet socializing goal

Goal Programming Example Attaché Training Program MODEL SOLUTION X1 = 49 hours, sleep X2 = 69 hours, personal X3 = 30 hours, class and studying X4 = 20 hours, social time All goals are fully met !

Goal Programming with QM for WINDOWS Attache Training Program

Goal 1 , 1st Priority : Class and Study time should be 30 hours Goal 2 , 2nd Priority : Sleep time must be, at most, 49 hours Goal 3 , 3rd Priority : Social time must be at least 20 hours

Goal Programming