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### Transcript of Goal Programming - University of ... Goal Programming - Definition Goal programming (GP) is a branch

• Goal Programming

• Goal Programming - Definition

Goal programming (GP) is a branch of Multi Objective Optimization, which in turn is a branch of Multi-Criteria Decision Analysis (MCDA). This is an optimization program. It can be thought of as an extension or generalization of linear programming to handle multiple, normally conflicting objective measures. Each of these measures is given a goal or target value to be achieved. Unwanted deviations from this set of target values are then minimized in an achievement function. This can be a vector or a weighted sum dependent on the goal programming variant used. As satisfaction of the target is deemed to satisfy the decision maker(s), an underlying satisficing philosophy is assumed.

Goal programming is used to perform three types of analysis:

• Determine the required resources to achieve a desired set of objectives. • Determine the degree of attainment of the goals with the available resources. • Providing the best satisfying solution under a varying amount of resources and

priorities of the goals.

• Basic Definitions Decision Maker

: The decision maker(s) refer to the person(s), organization(s), or stakeholder(s) to whom the decision problem under consideration belongs.

Criterion : A criterion is a single measure by which the goodness of any solution to a decision problem can be measured. There are many possible criteria arising from different fields of application but some of the most commonly arising relate at the highest level to: • Cost • Profit • Time • Distance • Performance of a system • Company or organizational strategy • Personal preferences of the decision maker(s) • Safety considerations

A decision problem which has more than one criterion is therefore referred to as a multi-criteria decision making (MCDM) or multi-criteria decision aid (MCDA) problem. The space formed by the set of criteria is known as criteria space.

• Decision Variable

: A decision variable is defined as a factor over which the decision maker has control. An example is a manufacturing company which has to decide how many of a certain product to make in the next month. The set of decision variables fully describe the problem and form the decision to be made. The purpose of the goal programming model can be viewed as a search of all the possible combinations of decision variable values (known as decision space) in order to determine the point which best satisfies the decision maker’s goals and constraints.

Objective : An objective is be referred to as a criterion with the additional information of the direction (maximize or minimize) in which the decision maker(s) prefer on the criterion scale, for example minimize cost or maximize the performance of a system. A decision problem with a set of objectives to be maximized or minimized is referred to as a multi-objective optimization problem. In practice, these objectives will be conflicting, that is they cannot reach their optimal values simultaneously. If they could, then the model can be solved as a single-objective problem for any of the objectives. The space formed by the values of the set of objectives is known as objective space.

• Aspiration Level

: The numerical value specified by the decision maker that reflects his/her desire or satisfactory level with regard to the objective function under consideration

Goal : An objective function along with its aspiration level

Goal Deviation : The difference between what we actually achieve and what we desire to achieve. If the Goal Deviation is positive, it reflects over-achievement of a goal as the actual achieved value is more than the aspiration level. On the other hand, a negative value of Goal Deviation denotes under - achievement of the goal as the aspired level is more than what we could actually manage to achieve.

• ➢ Desirable vs. Undesirable Deviations: (depend on the objectives) • Max goals (≥) - the more the better - 𝑑𝑖

+ or 𝑝𝑖 desirable. • Min goals (≤) - the less the better - 𝑑𝑖

− or 𝑛𝑖 desirable. • Exact goals (=) - exactly equal - both 𝑑𝑖

+ (or 𝑝𝑖) and 𝑑𝑖 − (or 𝑛𝑖) undesirable

➢ In all the situations, we first identify the undesirable deviation of the expression in the goal and then attempt to minimize the same.

➢ In GP, the objective is to minimize the (weighted) sum of undesirable deviations (all undesirable 𝑑𝑖

+ (or 𝑝𝑖) and 𝑑𝑖 − (or 𝑛𝑖) →→ 0 ).

➢ For each goal, at least, one of 𝑑𝑖 + (or 𝑝𝑖) and 𝑑𝑖

− (or 𝑛𝑖) must be equal to "0“

➢ The actual constraints of the problem are called RIGID goals or HARD goals whereas, the goals representing the given objective functions are called SOFT goals. When a set of criteria has been determined it is necessary to distinguish between goals and hard constraints. Hard constraints are in variable space and are conditions that must be satisfied in order for the solution to be implementable. Any condition that does not fulfil this requirement should be included as a goal and not a hard constraint.

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:

Formulation of GP Problems

Non- preemptive goal programming

: No goal is pre-determined to dominate any other goal

Pre-emptive goal programming

: Goals are assigned different priority levels. Level 1 goal dominates level 2 goal, and so on.

• Examples

1. Beaver Creek Pottery Company Example

• Beaver Creek Pottery Company Example

• Beaver Creek Pottery Company is a small crafts operation run by a Native American tribal council. The company employs skilled artisans to produce clay bowls and mugs with authentic Native American designs and colors. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit.

The Problem (1 of 2)

• The two products have the following resource requirements for production and profit per item produced (i.e., the model parameters):

Resource Requirements

Product Labour

( Hr./Unit ) Clay

( Lb./Unit) Profit

( \$/Unit )

Bowl 1 4 40

Mug 2 3 50

There are 40 hours of labor and 120 pounds of clay available each day for production.

Now, the company (in order of importance, i.e. priorities):

Does not want to use fewer than 40 hours of labor per day.

Would like to achieve a satisfactory profit level of \$1,600 per day.

Prefers not to keep more than 120 pounds of clay on hand each day.

Would like to minimize the amount of overtime.

The Problem (2 of 2)

• ➢ Labor goals constraint (1, less than 40 hours labor; 4, minimum overtime):

• Minimize {P1d1 -, P4d1

+}

➢ Add profit goal constraint (2, achieve profit of \$1,600):

• Minimize {P1d1 -, P2d2

-, P4d1 +}

➢ Add material goal constraint (3, avoid keeping more than 120 pounds of clay on hand):

• Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

Goal Constraints and Objective Function (1 of 2)

• Complete Goal Programming Model:

Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

subject to: x1 + 2x2 + d1

- - d1 + = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+  0

Goal Constraints and Objective Function (2 of 2)

• Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

subject to: x1 + 2x2 + d1

- - d1 + = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+  0

Goal Constraints

Graphical Interpretation (1 of 6)

• The First-Priority Goal: Minimize

Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

subject to: x1 + 2x2 + d1

- - d1 + = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+  0

Graphical Interpretation (2 of 6)

• The Second-Priority Goal: Minimize

Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

subject to: x1 + 2x2 + d1

- - d1 + = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+  0

Graphical Interpretation (3 of 6)

• The Third-Priority Goal: Minimize

Minimize {P1d1 -, P2d2

-, P3d3 +, P4d1

+}

subject to: x1 + 2x2 + d1

- - d1 + = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+  0

Graphical Interpretation (4 of