Goal programming

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  • ADVANCED OPERATIONS

    RESEARCH

    By: - HakeemUrRehman

    IQTMPU 1

    R A O GOAL PROGRAMMING (GP)

  • GOAL PROGRAMMING: AN INTRODUCTION

    2

    Firms often have more than one goal They may want to achieve several, sometimes contradictory,

    goals In linear and integer programming methods the objective

    function is measured in one dimension only It is not possible for LP to have multiple goals unless they

    are all measured in the same units, and this is a highly unusual situation

    An important technique that has been developed to supplement LP is called goal programming

    Goal programming approach establishes a specific numeric goal for each of the objective and then attempts to achieve each goal sequentially up to a satisfactory level rather than an optimal level.

  • GOAL PROGRAMMING: AN INTRODUCTION (Cont)

    3

    GP; Channes and Cooper (1961); Suggested a method for solving an infeasible LP problem arising from various conflicting resource constraints (Goals). Examples of Multiple Conflicting Goals are:

    i. Maximize Profit and increase wages paid to employees ii. Upgrade product quality and reduce product cost

    Ijiri (1965) developed the concept of preemptive priority factors, assigning different

    priority levels to incommensurable goals and different weights to the goals at the same priority level.

    In GP, instead of trying to minimize or maximize the objective function directly, as in case of an LP, the deviations from established goals within given set of constraints are minimized.

    In GP, Slack and Surplus variables are known as Deviational Variables (di and di

    +) (means UNDERACHIEVEMENT & OVERACHIEVEMENT); These deviations from each goal or sub-goal.

    These deviational variables represent the extent to which target goals are not achieved. The objective function then becomes the minimization of a sum of these deviations, based on the relative importance within the preemptive structure assigned to each deviation.

  • GOAL PROGRAMMING Vs LINEAR PROGRAMMING

    4

    Multiple goals (instead of one goal)

    Deviational variables minimized (instead of maximizing profit or minimizing cost of LP)

    Satisficing (instead of optimizing)

    Deviational variables are real (& replace Slack & Surplus variables)

  • GOAL PROGRAMMING MODEL FORMULATION: LINEAR PROGRAMMING Vs GOAL PROGRAMMING

    (SINGLE GOAL)

    5

    The Company produces two products popular with home renovators, old-fashioned chandeliers and ceiling fans Both the chandeliers and fans 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 assembly time are available Each chandelier produced nets the firm $7 and each fan $6. Harrison is moving to a new location and feels that maximizing profit is not a realistic

    objective Management sets a profit level of $30 that would be satisfactory during this period The goal programming problem is to find the production mix that achieves this goal as closely

    as possible given the production time constraints

    We can now state the Harrison Electric problem as a single-goal programming model

    subject to $7X1 + $6X2 + d1 d1

    + = $30 (profit goal constraint)

    2X1 + 3X2 12 (wiring hours)

    6X1 + 5X2 30 (assembly hours)

    X1, X2, d1, d1

    + 0

    Minimize under or overachievement of profit target = d1 + d1

    +

  • GOAL PROGRAMMING MODEL FORMULATION: EQUALLY RANKED MULTIPLE GOALS

    6

    A manufacturing firm produces two types of products: A & B. The units profit from product A is Rs. 100 and that of product B is Rs. 50. The goal of the firm is to earn a total profit of exactly Rs. 700 in the next week. Also, wants to achieve a sales volume for product A and B close to 5 and 4, respectively. Formulate this problem as a Goal programming model.

    MODEL FORMULATION: Let X1 and X2 be the number of units of products A and B produced, respectively. The constraints of the problem can be stated as:

    100X1 + 50X2 = 700 (Profit target goal) X1 5 (Sales target Goal) X2 4 (Sales target Goal)

    GP MODEL FORMULATION: The problem can now be formulated as GP model as follows:

    Minimization Z = d1 + d1

    + + d2 + d3

    Subject to: 100X1 + 50X2 + d1

    d1+ = 700 (Profit target goal)

    X1 + d2 = 5 (Sales target Goal)

    X2 + d3 = 4 (Sales target Goal)

    X1, X2, d1+, d2

    , d3 0

  • GOAL PROGRAMMING MODEL FORMULATION: EQUALLY RANKED MULTIPLE GOALS

    7

    Now Harrisons management wants to achieve several goals of equal in priority Goal 1: to produce a profit of $30 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 The deviational variables are

    d1 = underachievement of the profit target

    d1+ = overachievement of the profit target

    d2 = idle time in the wiring department (underutilization)

    d2+ = overtime in the wiring department (overutilization)

    d3 = idle time in the assembly department (underutilization)

    d3+ = overtime in the assembly department (overutilization)

    d4 = underachievement of the ceiling fan goal

    d4+ = overachievement of the ceiling fan goal

    Because management is unconcerned about d1

    +, d2+, d3

    , and d4

    + these may be omitted from the objective function

    The new objective function and constraints are

    subject to 7X1 + 6X2 + d1 d1

    + = 30 (profit constraint)

    2X1 + 3X2 + d2 d2

    + = 12 (wiring hours)

    6X1 + 5X2 + d3 d3

    + = 30 (assembly hours)

    X2 + d4 d4

    + = 7 (ceiling fan constraint)

    All Xi, di variables 0

    Minimize total deviation = d1 + d2

    + d3+ + d4

  • GOAL PROGRAMMING MODEL FORMULATION: RANKING GOALS WITH PRIORITY LEVELS

    8

    A key idea in goal programming is that one goal is more important than another. Priorities are assigned to each deviational variable.

    Priority 1 is infinitely more important than Priority 2, which is infinitely more important than the next goal, and so on.

    Harrison Electric has set the following priorities for their four goals

    GOAL PRIORITY

    Reach a profit as much above $30 as possible P1

    Fully use wiring department hours available P2

    Avoid assembly department overtime P3

    Produce at least seven ceiling fans P4

    Minimize total deviation = P1d1 + P2d2

    + P3d3+ + P4d4

    The constraints remain identical to the previous ones

  • PROCEDURE TO FORMULATE GP MODEL

    9

    1. Identify the goals and constraints based on the availability of resources (or constraints) that may restrict achievement of the goals (targets).

    2. Determine the priority to be associated with each goal in such a way that goals with priority level p1 are most important, those with priority level P1 are next most important, and so on.

    3. Define the decision variables. 4. Formulate the constraints in the same manner as formulated

    in LP model. 5. For each constraint, develop a equation by adding

    deviational variables d-i and d+

    i . These variables indicate the possible deviations blow or above the target value (right-hand side of each constraint).

    6. Write the objective function in terms of minimizing a prioritized function of the deviational variables.

  • GENERAL GP MODEL

    10

    With m goals, the general goal linear programming model may be stated as:

    Where, Z is the sum of the deviations from all desired goals. The Wi are non-negative constants representing the relative weight to

    be assigned to the deviational variables d-i , d+

    i within a priority level. The Pi is the priority level assigned to each relevant goal in rank order

    (i.e. P1 > P2 ,,>Pn ). The aii are constants.

    Remark: two types of constraints may be formulated for a GP problem: a) System constraints that may influence but are not directly related to goals, and b) Goal constraints that are directly related to goals.

  • SOLVING GOAL PROGRAMMING PROBLEMS GRAPHICALLY

    11

    We can analyze goal programming problems graphically

    We must be aware of three characteristics of goal programming problems

    1. Goal programming models are all minimization problems

    2. There is no single objective, but multiple goals to be attained

    3. The deviation from the high-priority goal must be minimized to the greatest extent possible before the next-highest-priority goal is considered

  • SOLVING GOAL PROGRAMMING PROBLEMS GRAPHICALLY

    Recall the Harrison Electric goal programming model

    Minimize total deviation = P1d1 + P2d2

    + P3d3+ + P4d4

    subject to 7X1 + 6X2 + d1 d1

    + = 30 (profit )

    2X1 + 3X2 + d2 d2

    + = 12 (wiring )

    6X1 + 5X2 + d3 d3

    + = 30 (assembly )

    X2 + d4 d4

    + = 7 (ceiling fans)

    All Xi, di variables 0 (nonnegativity)

    where X1 = number of chandeliers produced X2 = number of ceiling fans produced

  • SOLVING GOAL PROGRAMMING PROBLEMS GRAPHICALLY

    To solve this we graph one constraint at a time starting with the constraint