PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Goal ProgrammingHow do you find optimal solutions to the following?• Multiple criterion for measuring performance (car with low cost, good gas mileage, stylish, etc.. / attend school with good reputation, low tuition, close to home, right program…)• Multiple objectives / goals (e.g. Minimize service cost, maximize customer satisfaction)Answer: Use Goal ProgrammingGoal Programming – Example ProblemYou work for an Advertising agency. A customer has identifiedthree primary target audiences they are trying to reach, and has anAdvertising budget of $600,000. They have expressed their targets in the form of three goals:Goal 1 – Ads should be seen by at least 40 million high-income men (HIM)Goal 2 – Ads should be seen by at least 60 million low-income people (LIP)Goal 3 – Ads should be seen by at least 35 million high-income women (HIW)You recognize that advertising during football games and soap operas willcover the target audience. The table below indicates the number of viewers from the different categories that will be viewing these types of programming. HIM LIP HIW CostFootball ad (per min.) 7 million 10 million 5 million $100,000Soap Opera ad (/min) 3 million 5 million 4 million $60,000Goal Programming – Example ProblemExpressing the goals as an equation.Let: x1 – minutes of football ad x2 – minutes of soap opera adGoal 1 - HIM) 7 x1 + 3 x2 > 40 Goal 2 - LIP) 10 x1 + 5 x2 > 60 Goal 3 - HIW) 5 x1 + 4 x2 > 35Ad Budget) 100 x1 + 60 x2 < 600Goal Programming – Example ProblemFormulating the problem as an LP: Graphing the feasible regionMin (or Max) Z = somethings.t. HIM) 7 x1 + 3 x2 > 40 LIP) 10 x1 + 5 x2 > 60 HIW) 5 x1 + 4 x2 > 35Ad Bud.) 100 x1 + 60 x2 < 600 x1 , x2 > 0 Which constraints are real constraints versus “desired” constraints?Which constraints are “hard” constraints versus “soft” constraints?Goal Programming – Example ProblemSince the first three constraints are really goals, and not “hard”constraints, express these constraints in terms of deviati onalvariables.HIM) 7 x1 + 3 x2 + d1- - d1+ = 40 LIP) 10 x1 + 5 x2 + d2- - d2+ = 60 HIW) 5 x1 + 4 x2 + d3- - d3+ = 35 d1- , d1+ , d2- , d2+ , d3- , d3+ > 0Suppose each shortfall 0f 1,000,000 viewers from the goal translates to a cost of $200,000 for HIM, $100,000 for LIP and $50,000 for HIW. Then the objective function would be:Min Z = 200 d1- + 100 d2- + 50 d3-Goal Programming – Example ProblemThen in order to minimize the penalty for not reaching theviewing audience goal can be expressed as the following LP:Min Z = 200 d1- + 100 d2- + 50 d3- s.t.HIM) 7 x1 + 3 x2 + d1- - d1+ = 40 LIP) 10 x1 + 5 x2 + d2- - d2+ = 60 HIW) 5 x1 + 4 x2 + d3- - d3+ = 35Ad Bud.) 100 x1 + 60 x2 < 600 x1, x2, d1- , d1+ , d2- , d2+ , d3- , d3+ > 0The optimal solution to the above LP is:Z = 250, x1 = 6, x2 = 0, d1+ = 0 , d1- = 0 , d2+ = 0, d2- = 0 , d3+ = 0 , d3- = 5.Goal Programming: Weighted -vs-Preemptive GoalsIn the advertising example, the goals could readily be weighted by relative importance using the cost penalties ($200,000 for HIM, $100,000 for LIP and $50,000 for HIW).In many cases, the relative “weighting” of a goal is not easily determined, however the goals can be ranked from most important to least important. In this case, the most importantgoal pre-empts all the other goals. Once the most importantgoal is met, the second goal is addressed, and so on.Goal Programming: Preemptive GoalsSuppose the HIM constraint must be met first, followed by LIP and then HIW.First rewrite the LP as the following:Min Z = d1- s.t.HIM) 7 x1 + 3 x2 + d1- - d1+ = 40 LIP) 10 x1 + 5 x2 + d2- - d2+ = 60 HIW) 5 x1 + 4 x2 + d3- - d3+ = 35Ad Bud.) 100 x1 + 60 x2 < 600 x1, x2, d1- , d1+ , d2- , d2+ , d3- , d3+ > 0This LP solves to Z = 0, d1- = 0. So goal HIM is met.Goal Programming: Preemptive GoalsSince goal HIM is met, now make goal HIM a fixed constraint while trying to satisfy goal LIP.Min Z = d2- s.t.HIM) 7 x1 + 3 x2 + d1- - d1+ = 40 LIP) 10 x1 + 5 x2 + d2- - d2+ = 60 HIW) 5 x1 + 4 x2 + d3- - d3+ = 35Ad Bud.) 100 x1 + 60 x2 < 600 d1- = 0 x1, x2, d1- , d1+ , d2- , d2+ , d3- , d3+ > 0This LP solves to Z = 0, d2- = 0. So goal LIP is met.Goal Programming: Preemptive GoalsSince both goal HIM and LIP are met, make goal HIM and LIP fixed constraints while trying to satisfy goal HIW.Min Z = d3- s.t.HIM) 7 x1 + 3 x2 + d1- - d1+ = 40 LIP) 10 x1 + 5 x2 + d2- - d2+ = 60 HIW) 5 x1 + 4 x2 + d3- - d3+ = 35Ad Bud.) 100 x1 + 60 x2 < 600 d1- = 0 d2- = 0 x1, x2, d1- , d1+ , d2- , d2+ , d3- , d3+ > 0Goal Programming: Additional ExampleA company has two machines for manufacturing a product.Machine 1 make two units per hour, while machine 2 makesthree units per hour. The company has an order of 80 units.Energy restrictions dictate that only one machine can operateat one time. The company has 40 hours of regular machiningtime, but overtime is available. It costs $4.00 to run machine1 for one hour, while machine 2 costs $5.00 per hour. Thecompany has the following goals:1) Meet the demand of 80 units exactly.2) Limit machine