Topic – 8 Goal Programming (GP)

1. Topic – 8Goal Programming (GP)

2. Goal Programming (GP) 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 Key Point: In reality, it may be impossible to satisfy all objectives at their highest degree at the same time. Instead, tradeoff and compromise must be made. That is, there is only "satisfied" solutions, no "optimal" solutions. GP: is an extension of LP to handle multi-objective problems to: • Achieve a set of goals under given resources. • Determine the degree of goal attainments (acceptable ranges) to offer flexibility to Management for their priority preference. • Find a satisfying solution from Management specified priorities.

3. Formulation of GP Problems 1. 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". • Positive desirable deviations are in fact the indication of "slack" for Min Goals (≤), or the "surplus" for Max Goals (≥). 2. Ranking Objectives: (Ordinal/Cardinal/Mix of Two) • Ordinal: only indicate the importance of goals by ranking order (Pi) - ("Absolute Priorities"). • Cardinal: express importance of goals by assigning scaled weights (Wi) - ("Relative Preference"). Different objectives may be measured in different scales [($)(%)]. Equal weights can be used if all goals are viewed equal important.

4. Differences Between GP and LP

5. Elements of GP Problems Decision Variables: Xi (as in LP) and d+i/d-i (Deviation variables). • Xi only in constraint set, while • d+i/d-i in both the objective function and constraint set. 2. Constraints: (System Constraints vs. Goal Constraints) • System Constraints: no deviations allowed (as in LP, original resource or demand type constraints). • Goal Constraints: acceptable ranges (an one-sided value or a two-side interval range) for each goal with allowable deviations. 3. Objective Function: Minimize the Sum of Undesirable Deviations. • Single Variable vs. Multiple Variables • Ordinal Ranked (Given Pi) vs. Cardinal Ranked (Given Wi) Formulation Procedure of GP Problem: • 1. Determine decision variables (Xi) and deviation variables. • 2. Identify system (hard) constraints set (no deviation variables). • 3. Identify goal (soft) constraints set (with deviation variables). • 4. Ranking (ordinal or cardinal) goal set with given priorities. • 5. Formulate the objective function (with deviation variables only). • 6. Solve GP problem with a modified LP Simplex method. • 7. Evaluate the solution with what-if questions.

6. Example: Construct Goal Constraints Given a production planning problem: Three products: A, B, C -- three decision variables: X1, X2, X3. Knowing: net unit profit: (5, 10, 15) One Objective: Total Profit = 5X1 + 10X2 + 15X3 ≥ 300,000 • Considering resource limitation and other objectives, managers plan for a possible short on this $300,000 profit objective. Let: d+1 - the amount of profit exceeds 300,000 ("overachievement"), d-1 - the amount of profit short of 300,000 ("underachievement"). Writing this objective as a goal constraint: 5X1 + 10X2 + 15X3 + d-1 - d+1 = 300,000 • In the final solution, there are three possibilities: a. (X1, X2, X3 = 10,000, 12,000, 10,000) Equation (1): 320,000 + (d-1 - d+1) = 300,000 So, d+1 = 20,000, d-1 = 0 b. (X1, X2, X3 = 10,000, 8,000, 10,000) Equation (1): 280,000 + (d-1 - d+1) = 300,000 So, d-1 = 20,000, d+1 = 0 c. (X1, X2, X3 = 10,000, 10,000, 10,000) Equation (1): 300,000 + (d-1 - d+1) = 300,000 So, d-1 = 0, d+1 = 0 To ensure the nonnegativity of d+i/d-i, when writing on the left-side of goal constraint equation, always: (d-i - d+i)

7. GP: Objective Function and Solution Methods 1. Cardinal Ranked: (with given weights Wi) Min: Z = 9d-1 + 5d+2 + 0.2d-1 [2 (≥) & 1 (≤) goal constraints] • Use standard LP Simplex method to solve the problem. • Useful when different goals have different scales to measure and known relative allowable undesirable deviations. • The weights must be determined carefully. 2. Ordinal Ranked: (with given priorities Pi) Min: Z = P1Pd-1 + P2Pd+2 + .... + PnPd-n [n goal constraints] • Based on a modified Simplex method, sequentially search solution from the highest priority goal (P1) to the lowest priority goal (Pn). • At each step, only the system constraints and the limits on the solution value of Xi from the previous steps (higher priority levels) are considered in minimizing the current deviation variables. • For an exact goal (=), since both d-i and d+i are undesirable, both are needed in the objective function and to be considered in the same search step. A cardinal rank among the two may be needed. (e.g., job due-date, both "early" and "late" are undesirable.) 3. Mixed with both Ordinal and Cardinal Ranking: (given Pi and Wj) Min: Z = P1Pd+1 + P2P(3d+2+2d-2) + P3P(5d-3+4d+4+3d-5) • Goal 2 is an exact type (=). d+2 and d-2 have weights of 3 to 2. • Goals 3, 4, and 5 are in the same priority level but have weights of 5/4/3. • The solution process is the same as in (2).

8. Multicriteria Decision Problems Goal Programming Problem Description RMC, Inc. is a firm that produces chemical-based products. In a particular process three raw materials are used to produce two products. The material requirements per ton are shown below.

9. For the current production period RMC has available the following quanities of each raw material. Because of spoilage, any material not used for current production must be discarded. Management would like to achieve the following P1 priority level goals: Goal 1: Produce at least 30 tons of fuel additive Goal 2: produce at least 15 tons of solvent base

14. Sensitivity Analysis of GP Problem • For a given GP problem, assigning different priority ranks or priority weights among different goals will result in different solutions. So, GP is viewed as a heuristic approach to multicriteria Linear problems. • There is no sensitivity analysis information provided along computer solution printouts. To answer managerial what-if questions: • Trail-and-error (interactive resolving) methods are usually used to check whether there is an impact on the solution values when: • -Changing in priority ranking of goal settings, • -Changing in the target value of goals, and • -Changing in the weights assigned to each goal deviation. • To resolve the problem when there are significant undesirable deviations in solution values, common methods are: • Relaxing system resources constraints (add new capacity), • Making tradeoffs among different goals (goal seeking), or • Setting up a minimum level to guarantee acceptable goal values.