Deck 9: Multicriteria Decision Making

Question 1

The different objectives in a goal programming problem are referred to as goals.

Answer 1

True

Question 2

A negative deviational variable is the amount by which a goal level is exceeded.

Answer 2

False

Question 3

The objective function in a goal programming model seeks to minimize the deviation from goals in the order of the goal priorities.

Answer 3

True

Question 4

One goal is never achieved at the expense of another higher-priority goal.

Question 5

All goal constraints are inequalities that include deviational variables.

Question 6

Goal programming solutions do not always achieve all goals.

Question 7

A pairwise comparison matrix summarizes the pairwise comparisons for a criterion.

Question 8

Goal constraints can include all deviational variables.

Question 9

Objective function terms are not summed in goal programming because not all goals are achievable.

Question 10

Goal programming provides a method for simultaneously striving to achieve several objectives.

Question 11

A goal can be achieved at the expense of another lower-priority goal.

Question 12

In a pairwise comparison, two alternatives are compared according to a criterion and one is preferred.

Question 13

In goal programming, terms are summed in the objective function in order to make consistent decisions.

Question 14

In a goal programming model, the terms in the objective function are summed to determine the maximum profit or minimum cost.

Question 15

A preference scale assigns numerical values to different levels of preference.

Question 16

At least one or both deviational variables in a goal constraint must equal 0.

Question 17

A requirement for the solution procedure for the goal programming problem is that all goals must be achieved.

Question 18

Goal programming violates the divisibility property of linear programming.

Question 19

In goal programming, problems cannot have two or more goals at the same priority level.

Question 20

On a preference scale for pairwise comparisons, the number "1" indicates that two objects are equally preferred.

Question 21

A ________ vector is the average of the values in each row of the normalized matrix in the AHP process.

Question 22

The analytic hierarchy process, scoring models, and goal programming are all considered to be ________ decision making techniques because they incorporate decisions based on more than one objective.

Question 23

A(n) ________ model is a similar to AHP, but simpler and easier to understand.

Question 24

A consistency index measures the degree of inconsistency in pairwise comparisons.

Question 25

In synthesization, decision alternatives are prioritized within each criterion.

Question 26

A preference scale assigns numerical values to different levels of preferences.

Question 27

At least one or both deviational variables in a goal constraint must equal ________.

Question 28

A(n) ________ measures the degree of inconsistency in pairwise comparisons.

Question 29

A(n) ________ variable is the amount by which a goal level is exceeded.

Question 30

If the CI/RI ratio is more than ________, then there are probably serious inconsistencies in the AHP results.

Question 31

A(n) ________ variable is the amount by which a goal level is underachieved.

Question 32

Consider the following constraint: 2x1 + 3x2 =60. Assume that we convert this constraint into a goal constraint with the value of x1 = 15 and the value of x2 = 15. Then the values of d1+ and d1- are ________ and ________, respectively.

Question 33

In a facility location problem, if location A receives a score of 45 for the criteria "traffic congestion", which has a weight of .30; a score of 50 for the criteria "labor force", which has a weight of .45; and a score of 60 for "utilities", which has a weight of .25, the overall score for location A is ________.

Question 34

A(n) ________ assigns numerical values to different levels of preferences.

Question 35

In a given AHP problem, the consistency index is .15, and the random index is .90. In this instance, there are probably serious inconsistencies and, therefore, the AHP result may not be meaningful.

Question 36

A scoring model is more subjective than the analytical hierarchy process.

Question 37

Scoring models use consistency indexes to measure the degree of consistency between choices.

Question 38

Deviational variables that occur in the objective function and the constraints of a goal programming model indicate the difference between all actual and ________ values.

Question 39

In scoring models, the two alternatives with the highest scores are compared to make the final decision.

Question 40

The objective function formulation for a goal programming model is as follows:
Min P₁ d₁ -, P₂ d₂ -, P₃ d₁ +, P₄ d₃ -
Which one of the variables would the program designers like to have as the second goal such that its deviation from zero is minimized?

Question 41

Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period:
$$\begin{array} { l c c c } & \text { Quilts } \left( x _ { \mathbf { 1 } } \right) & \text { Blinds } \left( \boldsymbol { x } _ { \mathbf { 2 } } \right) & \text { Pillows } \left( x _ { 3 } \right) \\\text { Fabric required (yards) } \left( \boldsymbol { d } _ { \mathbf { 1 } } \right) & 7 & 4 & 9 \\\text { Time required (hours) } \left( \boldsymbol { d } _ { \mathbf { 2 } } \right) & 1.5 & 2 & 0.5 \\\text { Packaging material (ounces) } \left( \boldsymbol { d } _ { \mathbf { 3 } } \right) & 3 & 2 & 1 \\\text { Profit } \left( \boldsymbol { d } _ { \mathbf { 4 } } \right) & 12 & 10 & 8\end{array}$$ Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley's highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours, and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item are given in the table.

-What is the profit constraint?

Question 42

An investor has $80,000 to invest in three stocks, stock A costs $100, stock B costs $120 and stock C costs $80. Each stock A has a risk factor of 8, each stock B has a risk factor of 10 and each stock C has a risk factor of 7. The investor believes that the sum of the risk factors for all stocks purchase should not exceed 6000. The projected annual growth rate for the three stocks are 9%, 13% and 8% respectively. The projected annual dividend income from these stocks are as follows: Stock A: $14/stock, Stock B: $15/stock, and Stock C: $20/stock. The investor desires an annual dividend income of $10,000. The investor has established the following goals in order of their importance:

(1) The investor believes that the budget cannot be exceeded. (d₁)
(2) The risk factor should not exceed the target amount of 6000. (d₂)
(3) The average annual growth rate in stock prices must be at least 10%. (d₃)
(4) The investor desires a dividend income of at least $10,000. (d₄)


-Write the risk factor constraint

Question 43

Using the normalized matrix given below, calculate the row averages.

Question 44

Assume that a decision maker using analytical hierarchy process generated the following pairwise comparison matrix.
Calculate the normalized matrix.

Question 45

Assume that a plant manager has to decide where to locate its warehouse. The decision has been narrowed down to choices among the following three cities: Detroit, Cleveland, and St. Louis. The following pairwise comparison matrix summarizes the preferences of the plant manager.

Determine the normalized matrix.

Question 46

A production of 300 units of Twiddle Bugs, an educational toy for children, must be completed within 1 week by BugU Manufacturing. Two production lines are available, each for 30 hours during the week. Production line 1 can produce five units per hour and production line 2 can produce four units an hour. Line 1 costs $50 per hour to operate and line 2 costs $55 per hour. Overtime is available for line 1 at $15 per hour and for line 2 at $12 per hour. Management goals in decreasing order are:
P1: Produce 300 units.
P2: Maximum allowable overtime of 6 hours for line 1.
P3: Cost of overtime must not exceed $750.
P4: Avoid the underutilization of either production line. Assign weights that are proportional to their production capability.
P5: Producing more than 300 units is 1-1/2 times as undesirable as producing under 300 units.

-What are the constraints for this problem?

Question 47

Assume that a plant manager has to decide where to locate its warehouse The decision has been narrowed down to choices among the following three cities: Detroit,, Cleveland, and St. Louis. The company has weighted each of these criteria in terms of its relative importance in the decision making process, and it has analyzed each potential warehouse location and graded them according to each criteria as shown in the following table.

Calculate the scores for each location and state where the warehouse should be located.

Question 48

Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period:
$$\begin{array} { l c c c } & \text { Quilts } \left( x _ { \mathbf { 1 } } \right) & \text { Blinds } \left( \boldsymbol { x } _ { \mathbf { 2 } } \right) & \text { Pillows } \left( x _ { 3 } \right) \\\text { Fabric required (yards) } \left( \boldsymbol { d } _ { \mathbf { 1 } } \right) & 7 & 4 & 9 \\\text { Time required (hours) } \left( \boldsymbol { d } _ { \mathbf { 2 } } \right) & 1.5 & 2 & 0.5 \\\text { Packaging material (ounces) } \left( \boldsymbol { d } _ { \mathbf { 3 } } \right) & 3 & 2 & 1 \\\text { Profit } \left( \boldsymbol { d } _ { \mathbf { 4 } } \right) & 12 & 10 & 8\end{array}$$ Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley's highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours, and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item are given in the table.

-What is the packaging material constraint?

Question 49

An investor has $80,000 to invest in three stocks, stock A costs $100, stock B costs $120 and stock C costs $80. Each stock A has a risk factor of 8, each stock B has a risk factor of 10 and each stock C has a risk factor of 7. The investor believes that the sum of the risk factors for all stocks purchase should not exceed 6000. The projected annual growth rate for the three stocks are 9%, 13% and 8% respectively. The projected annual dividend income from these stocks are as follows: Stock A: $14/stock, Stock B: $15/stock, and Stock C: $20/stock. The investor desires an annual dividend income of $10,000. The investor has established the following goals in order of their importance:

(1) The investor believes that the budget cannot be exceeded. (d₁)
(2) The risk factor should not exceed the target amount of 6000. (d₂)
(3) The average annual growth rate in stock prices must be at least 10%. (d₃)
(4) The investor desires a dividend income of at least $10,000. (d₄)


-State the goal programming objective function.

Question 50

The analytical hierarchy process was used to determine which car to purchase. Four criteria were used: price, miles per gallon, comfort, and durability. Consider the following normalized matrix and the preference vector for durability. $$~~~~~~~~~~~~~~~~~~~~~~~~~~{ \text { Durability Matrix } } ~~~~~~~~~ \text { Preference } \\\begin{array} { l c c c c } & \text { Car A } & \text { Car B } & \text { Car C } & \text { Vector } \\\hline \text { Car A } & 1 & 0.25 & 4 & 0.36 \\\text { Car B } & 4 & 1 & 0.5 & 0.59 \\\text { Car C } & 0.25 & 2.00 & 1.00 & 0.41 \\\hline\end{array}$$
Multiplying the normalized matrix by the preference vector gives the following results:
$$\begin{array} { | l | } \hline 2.17 \\\hline 2.26 \\\hline 1.69 \\\hline\end{array}$$
Compute the consistency index.

Question 51

Assume that a decision maker has to make a choice between three types of cars. Chevy, Honda, and Ford. Based on three criteria: comfort, MPG, and style.
The row averages for criteria is summarized in the following column vector:
The preference matrix for the three types of cars is given as follows:
Compute the overall score for each decision alternative (car).

Question 52

The objective function formulation for a goal programming model is as follows:
Min P₁ d₁ -, P₂ d₂ -, (4 P₄ d₃ - + 6P₄ d₂ +)
At the priority level 4, which one of the deviational variables is most important?

Question 53

A production of 300 units of Twiddle Bugs, an educational toy for children, must be completed within 1 week by BugU Manufacturing. Two production lines are available, each for 30 hours during the week. Production line 1 can produce five units per hour and production line 2 can produce four units an hour. Line 1 costs $50 per hour to operate and line 2 costs $55 per hour. Overtime is available for line 1 at $15 per hour and for line 2 at $12 per hour. Management goals in decreasing order are:
P1: Produce 300 units.
P2: Maximum allowable overtime of 6 hours for line 1.
P3: Cost of overtime must not exceed $750.
P4: Avoid the underutilization of either production line. Assign weights that are proportional to their production capability.
P5: Producing more than 300 units is 1-1/2 times as undesirable as producing under 300 units.

-What is the objective function?

Question 54

An investor has $80,000 to invest in three stocks, stock A costs $100, stock B costs $120 and stock C costs $80. Each stock A has a risk factor of 8, each stock B has a risk factor of 10 and each stock C has a risk factor of 7. The investor believes that the sum of the risk factors for all stocks purchase should not exceed 6000. The projected annual growth rate for the three stocks are 9%, 13% and 8% respectively. The projected annual dividend income from these stocks are as follows: Stock A: $14/stock, Stock B: $15/stock, and Stock C: $20/stock. The investor desires an annual dividend income of $10,000. The investor has established the following goals in order of their importance:

(1) The investor believes that the budget cannot be exceeded. (d₁)
(2) The risk factor should not exceed the target amount of 6000. (d₂)
(3) The average annual growth rate in stock prices must be at least 10%. (d₃)
(4) The investor desires a dividend income of at least $10,000. (d₄)


-Write the budget constraint

Question 55

Centerville city council is in the process of developing city tax rates. The annual tax base for real estate property is $750 million and for general sales, $80 million. Annual local gas consumption is estimated at 12 million gallons. They have three goals, listed in order of priority:
1. Tax revenues must be at least $25 million to meet the city's financial commitments.
2. Sales tax cannot exceed 25% of all taxes collected.
3. Gasoline tax cannot exceed 8 cents per gallon.
Formulate as a goal programming problem.

Question 56

Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period:
$$\begin{array} { l c c c } & \text { Quilts } \left( x _ { \mathbf { 1 } } \right) & \text { Blinds } \left( \boldsymbol { x } _ { \mathbf { 2 } } \right) & \text { Pillows } \left( x _ { 3 } \right) \\\text { Fabric required (yards) } \left( \boldsymbol { d } _ { \mathbf { 1 } } \right) & 7 & 4 & 9 \\\text { Time required (hours) } \left( \boldsymbol { d } _ { \mathbf { 2 } } \right) & 1.5 & 2 & 0.5 \\\text { Packaging material (ounces) } \left( \boldsymbol { d } _ { \mathbf { 3 } } \right) & 3 & 2 & 1 \\\text { Profit } \left( \boldsymbol { d } _ { \mathbf { 4 } } \right) & 12 & 10 & 8\end{array}$$ Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley's highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours, and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item are given in the table.

-What is the objective function?

Question 57

A business is trying to decide which restaurant to hold its annual awards banquet. Use the information below to determine the best choice.
Restaurant

Question 58

Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period:
$$\begin{array} { l c c c } & \text { Quilts } \left( x _ { \mathbf { 1 } } \right) & \text { Blinds } \left( \boldsymbol { x } _ { \mathbf { 2 } } \right) & \text { Pillows } \left( x _ { 3 } \right) \\\text { Fabric required (yards) } \left( \boldsymbol { d } _ { \mathbf { 1 } } \right) & 7 & 4 & 9 \\\text { Time required (hours) } \left( \boldsymbol { d } _ { \mathbf { 2 } } \right) & 1.5 & 2 & 0.5 \\\text { Packaging material (ounces) } \left( \boldsymbol { d } _ { \mathbf { 3 } } \right) & 3 & 2 & 1 \\\text { Profit } \left( \boldsymbol { d } _ { \mathbf { 4 } } \right) & 12 & 10 & 8\end{array}$$ Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley's highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours, and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item are given in the table.

-What is the production time constraint?

Question 59

Consider the following normalized matrix for car comfort that was computed using the analytical hierarchy process:

Which car is the most preferred based on comfort?

Question 60

Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period:
$$\begin{array} { l c c c } & \text { Quilts } \left( x _ { \mathbf { 1 } } \right) & \text { Blinds } \left( \boldsymbol { x } _ { \mathbf { 2 } } \right) & \text { Pillows } \left( x _ { 3 } \right) \\\text { Fabric required (yards) } \left( \boldsymbol { d } _ { \mathbf { 1 } } \right) & 7 & 4 & 9 \\\text { Time required (hours) } \left( \boldsymbol { d } _ { \mathbf { 2 } } \right) & 1.5 & 2 & 0.5 \\\text { Packaging material (ounces) } \left( \boldsymbol { d } _ { \mathbf { 3 } } \right) & 3 & 2 & 1 \\\text { Profit } \left( \boldsymbol { d } _ { \mathbf { 4 } } \right) & 12 & 10 & 8\end{array}$$ Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley's highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours, and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item are given in the table.

-What is the fabric constraint?

Question 61

Deviational variables in a goal programming model constraint represent the

A) underachievement or overachievement of a goal level.
B) resource constraint as defined by the linear model.
C) probability of an optimal solution.
D) probabilistic variables in the objective constraints.

Question 62

The objective function in a goal programming model seeks to ________ the deviation from goals in the order of the goal priorities.

A) free
B) even out
C) maximize
D) minimize

Question 63

Consider the following constraint: 2x1 + 3x2 =60. Assume that we convert this constraint into a goal constraint with the value of x1 = 15 and the value of x2 = 15. Then the values of d1+ and d1- are ________ and ________, respectively.

A) 0, 15
B) 15, 0
C) 20, 0
D) 0, 20

Question 64

Goal constraints can include ________ deviational variables.

A) no
B) some
C) all
D) multiple

Question 65

A ________ deviational variable is the amount by which a goal level is underachieved.

A) negative
B) positive
C) positive or negative
D) positive and negative

Question 66

Deviation variables

A) represent the actual value of the function part of a goal constraint.
B) represent the difference between the target and actual values.
C) must sum to 1.
D) equal the difference between actual and predicted values of the decision variables.

Question 67

Two or more goals at the same priority level can be assigned weights to indicate their relative

A) meaning.
B) slack.
C) difference.
D) importance.

Question 68

A decision with more than one objective

A) should be decomposed into a separate model for each objective .
B) depends on the probability of satisfying each objective.
C) requires the decision maker to put the objectives in some order of importance.
D) cannot have an optimal solution.

Question 69

The objective function min P₁d₁^- , P₂d₂⁺

A) attempts to avoid being below target for the priority 1 goal.
B) attempts to avoid being below target for the priority 2 goal.
C) will not have any feasible solutions.
D) will have a solution only if P₁ > P₂.

Question 70

Goal programming solutions ________ achieve all goals.

A) always
B) don't always
C) sometimes
D) never

Question 71

If the constraint 3_x1 + 4_x2 + (d_1-) - (d₁₊) = 250 measures hours, then

A) 3_x1 + 4_x2 is equal to the actual time spent.
B) if d₁- is equal to 25, then 3_x1 + 4_x2 must equal 25.
C) overtime could be represented by d_1-.
D) hours can take only an integer value.

Question 72

A company has three goals, listed in order of importance.
1) Achieve a total profit of at least $240 million.
2) Maintain the current employment level of 3,000 employees
3) Invest no more than $70 million in capital.
What is the objective function for a goal program for this situation?

A) Min P₁d₁^- , P₂d₂^- , P₃d₃⁺
B) Min P₁d₁^- , P₂d₂⁺ , P₂d₂^- , P₃d₃^-
C) Min P₁d₁^- , P₂d₂^- , P₂d₂^- , P₃d₃⁺
D) Min P₁d₁⁺ , P₂d₂^- , P₂d₂^- , P₃d₃⁺

Question 73

At least one or both deviational variables in a goal constraint must equal:

A) 1
B) -1
C) 0
D) 2

Question 74

The objective function in all goal programming models is to ________ from the goal constraint levels.

A) maximize alternatives
B) minimize alternatives
C) minimize deviation
D) maximize deviation

Question 75

A company has a goal of calling on at least 400 customers a month (goal 1), but no fewer than 260 customers a month (goal 2). If X represents the number of customers contacted, which pair of constraints listed below is the appropriate representation in a goal programming problem?

A) X + d₁^-- d₁⁺ = 400, X + d2^- - d2⁺ = 260
B) X - d₁^-+ d₁⁺ = 400, X - d2^- - d2⁺ = 260
C) X - d₁^- + d₁⁺ = 400, X + d2^- - d2⁺ = 260
D) X + d₁⁺ - d₁⁺ = 400, X - d2^- + d2⁺ = 260

Question 76

A ________ deviational variable is the amount by which a goal level is exceeded.

A) negative
B) positive
C) positive or negative
D) positive and negative

Question 77

A positive deviational variable is the amount by which a goal level is

A) underachieved.
B) exceeded.
C) minimized.
D) maximized.

Question 78

The majority of the goal constraints are equalities that include ________ variables.

A) only deviational
B) deviational and decision
C) only decision
D) neither deviational nor decision variables

Question 79

A company has a goal to maintain the current employment level, but places more importance on laying employees off than on hiring new employees. In other words, the company would rather hire than lay off employees. If the company assigns a "penalty weight" of 4 for the amount under the employment goal (goal 2) and a weight of 2 for the amount over the employment goal, how would this be expressed in the objective function?

A) Min 4P₂d₂^- + 2P₂d₂⁺
B) Min 2P₂d₂^-+ 4P₂d₂⁺
C) Max 4P₂d₂^- + 2P₂d₂⁺
D) Min 4P₂d₂+ ⁺2P₂d₂^-

Question 80

One goal ________ achieved at the expense of another higher-priority goal.

A) is never
B) can sometimes be
C) is always
D) is under certain circumstances