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Goal Programming. Multiple-Objectives Problem: In most practical cases, decision makers are faced a situation where they must achieve more than two objectives (those may even be in conflict) at same time. Or more than two criteria must be used to evaluate a decision. Examples:
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Goal Programming • Multiple-Objectives Problem: In most practical cases, decision makers are faced a situation where they must achieve more than two objectives (those may even be in conflict) at same time. Or more than two criteria must be used to evaluate a decision. • Examples: Production Planning - Maximize Profit/Maximize Market Share Location Selection - Maximize Sales/Minimize Delivery Cost Personal Schedule - Maximize GPA/Maximize Income
Formulation of GP Problems • Deviations: the amount away from the desired standards or objectives: • Overachievement (d+i ≥ 0) vs. Underachievement (d-i ≥ 0) • Desirable vs. Undesirable Deviations: (depend on the objectives) • Max goals (≥) - the more the better - d+i desirable. • Min goals (≤) - the less the better - d-i desirable. • Exact goals (=) - exactly equal - both d+i and d-i undesirable • In GP, the objective is to minimize the (weighted) sum of undesirable deviations (all undesirable d+i and d-i →→ 0 ). • For each goal, at least, one of d+i and d-i must be equal to "0"
Formulation of GP Problems • Goals are prioritized in some sense, and their level of aspiration is stated. • An optimal solution is attained when all the goals are reached as close as possible to their aspiration level, while satisfying a set of constraints. • There are two types of goal programming models: • Nonpreemptive goal programming - no goal is pre-determined to dominate any other goal. • Preemptive goal programming - goals are assigned different priority levels. Level 1 goal dominates level 2 goal, and so on.
NONPREEMPTIVE GOAL PROGRAMMINGAn Advertisement Example • A company is considering three forms of advertising. • Goals • Goal 1: Spend no more $25,000 on advertising. • Goal 2: Reach at least 30,000 new potential customers. • Goal 3: Run at least 10 television spots.
An Advertisement Example LP Model: 3000X1 + 800X2 + 250X3£25,0001000X1 + 500X2 + 200X3³ 30,000 X1³ 10
An Advertisement Example • Detrimental variables Ui = the amount by which the left hand side falls short of (under) its right hand side value. Ei = the amount by which the left hand side exceeds its right hand side value. • The goal equations3000X1 + 800X2 + 250X3 + U1 – E1 = 25,0001000X1 + 500X2 + 200X3 + U2 – E2 = 30,000 X1 + U3 – E3 = 10
An Advertisement Example • The objective is to minimize the penalty of not meeting the goals, represented by the detrimental variables E1 U2 U3 £25,000 ³ 30,000 ³ 10
An Advertisement Example • The penalties are estimated to be as follows: • Each extra dollar spent on advertisement above $25,000 cost the company $1. • There is a loss of $5 to the company for each customer not being reached, below the goal of 30,000. • Each television spot below 10 is worth 100 times each dollar over budget.
An Advertisement Example – The goal programming model • It is assumed that no advantage is gained by overachieving a goal.Minimize 1E1 + 5U2 + 100U3s.t.3000X1 + 800X2 + 250X3 + U1 – E1 = 25,0001000X1 + 500X2 + 200X3 + U2 – E2 = 30,000 X1 + U3 – E3 = 10 All variables are non-negative.
NONPREEMPTIVE GOAL PROGRAMMING Conceptual Products Conceptual Products is a computer company that produces the CP400 and the CP500 computers. The computers use different mother boards produced in abundant supply by the company, but use the same cases and disk drives. The CP400 models use two floppy disk drives and no zip disk drives whereas the CP500 models use one floppy disk drive and one zip disk drive. The disk drives and cases are bought from vendors. There are 1000 floppy disk drives, 500 zip disk drives, and 600 cases available to Conceptual Products on a weekly basis. It takes one hour to manufacture a CP400 and its profit is $200 and it takes one and one-half hours to manufacture a CP500 and its profit is $500.
Goals: • Goal 1: Produce at least 200 CP400 computers each week. • Goal 2: Produce at least 500 total computers each week. • Goal 3: Reach at least $250 (in thousands) on profit. • Goal 4: Consume no more than 400 total man-hours each week.
