Ch. 9 1) The different objectives in a goal programming problem are referred to as goals. Answer: TRUE 2) All goal constraints are inequalities that include deviational variables. Answer: FALSE 3) A negative deviational variable is the amount by which a goal level is exceeded. Answer: FALSE 4) At least one or both deviational variables in a goal constraint must equal 0. Answer: TRUE 5) The objective function in a goal programming model seeks to minimize the deviation from goals in the order of the goal priorities. Answer: TRUE 6) In goal programming, terms are summed in the objective function in order to make consistent decisions. Answer: FALSE 7) Goal constraints can include all deviational variables. Answer: TRUE 8) In goal programming, problems cannot have two or more goals at the same priority level. Answer: FALSE 9) Goal programming provides a method for simultaneously striving to achieve several objectives. Answer: TRUE 10) A requirement for the solution procedure for the goal programming problem is that all goals must be achieved. Answer: FALSE 11) One goal is never achieved at the expense of another higher-priority goal. Answer: TRUE 12) A goal can be achieved at the expense of another lower-priority goal. Answer: TRUE 13) Goal programming solutions do not always achieve all goals. Answer: TRUE 14) In a goal programming model, the terms in the objective function are summed to determine the maximum profit or minimum cost. Answer: FALSE 15) Objective function terms are not summed in goal programming because not all goals are achievable. Answer: FALSE
16) Goal programming violates the divisibility property of linear programming. Answer: FALSE 17) In a pairwise comparison, two alternatives are compared according to a criterion and one is preferred. Answer: TRUE 18) A preference scale assigns numerical values to different levels of preference. Answer: TRUE 19) On a preference scale for pairwise comparisons, the number "1" indicates that two objects are equally preferred. Answer: TRUE 20) A pairwise comparison matrix summarizes the pairwise comparisons for a criterion. Answer: TRUE 21) In synthesization, decision alternatives are prioritized within each criterion. Answer: TRUE 22) A consistency index measures the degree of inconsistency in pairwise comparisons. Answer: TRUE 23) A preference scale assigns numerical values to different levels of preferences. Answer: TRUE 24) 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. Answer: TRUE 25) A scoring model is more subjective than the analytical hierarchy process. Answer: TRUE 26) Scoring models use consistency indexes to measure the degree of consistency between choices. Answer: FALSE 27) In scoring models, the two alternatives with the highest scores are compared to make the final decision. Answer: FALSE Ch 10 1) The slope of a curve at any point is equal to the derivative of the curves function. Answer: TRUE 2) The slope of a curve at its highest point equals 1. Answer: FALSE
3) A profit function of Z = 3x2 - 12x + 5 reaches maximum profit at x = 2 units of output. Answer: TRUE 4) Classical optimization uses calculus to determine the optimal values of a variable. Answer: TRUE 5) A firm has a cost function of 3x2 - 25x + 374. Without having two examples of their output volume and total cost, we cannot determine their fixed cost. Answer: FALSE 6) Maximum profit is achieved everywhere the first derivative of the profit function equals zero. Answer: FALSE 7) The first derivative of a cost function equals zero at the point V = 100. This is definitely the worst output volume for the producer to choose. Answer: FALSE 8) Decision variables cannot be multiplied by each other in the objective function of a nonlinear program. Answer: FALSE 9) Both linear and nonlinear programming models are examples of constrained optimization models. Answer: TRUE 10) An optimal solution to a nonlinear programming problem will always occur at the boundary of the feasible solution space formed by the constraint. Answer: FALSE 11) The Lagrange multiplier is analogous to the dual variables in a linear programming problem. Answer: TRUE 12) The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value. Answer: TRUE 13) Both linear and nonlinear programming models have the general form of an objective function subject to more than 1 constraint. Answer: FALSE 14) Classical optimization is the use of calculus to determine the optimal value of a variable. Answer: TRUE 15) If a nonlinear program has been correctly formulated, procedures guarantee a solution. Answer: FALSE 16) In an unconstrained nonlinear programming problem, we have a single nonlinear objective function and no constraints.. Answer: TRUE
17) Constraints for nonlinear programs are usually nonlinear. Answer: FALSE 18) In portfolio selection problems, risk is measured by the variance of the return on the portfolio. Answer: TRUE 19) In solving the facility location problem, the objective is to locate a centralized facility that serves customers or other facilities such that the distance traveled between the facility and customers or other facilities is minimized. Answer: TRUE FILL IN THE BLANK CH 9
28) At least one or both deviational variables in a goal constraint must equal ________. Answer: zero 29) 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. Answer: 15; 0 30) A(n) ________ variable is the amount by which a goal level is underachieved. Answer: negative deviational 31) A(n) ________ variable is the amount by which a goal level is exceeded. Answer: positive deviational 32) A(n) ________ assigns numerical values to different levels of preferences. Answer: preference scale 33) Deviational variables that occur in the objective function and the constraints of a goal programming model indicate the difference between all actual and ________ values. Answer: target 34) A(n) ________ measures the degree of inconsistency in pairwise comparisons. Answer: consistency index 35) A ________ vector is the average of the values in each row of the normalized matrix in the AHP process. Answer: preference 36) If the CI/RI ratio is more than ________, then there are probably serious inconsistencies in the AHP results. Answer: .10 37) 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. Answer: multiple criteria 38) A(n) ________ model is a similar to AHP, but simpler and easier to understand. Answer: scoring 39) 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 ________. Answer: 51 40) The objective function formulation for a goal programming model is as follows: Min P1d1-, P2d2-, P3d1+, P4d3Which one of the variables would the program designers like to have as the second goal such that its deviation from zero is minimized? Answer: d241) The objective function formulation for a goal programming model is as follows: Min P1d1-, P2d2-, (4P4d3- + 6P4d2+) At the priority level 4, which one of the deviational variables is most important? Answer: d2+
Ashley's manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period: Quilts (x1) Fabric required (yards) (d1) 7 Time required (hours) (d2) 1.5 Packaging material (ounces) (d3) 3 Profit (d4) 12 Blinds (x2) 4 2 2 10 Pillows (x3) 9 0.5 1 8 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. 42) What is the objective function? Answer: Min P1d4-, P2d3+, P3d2-, P4d1+ 43) What is the fabric constraint? Answer: 7x1+4x2+9x3+d1--d1+=3000 44) What is the profit constraint? Answer: 12x1+10x2+8x3+d4--d4+=3200 45) What is the production time constraint? Answer: 1.5x1 + 2x2 + 0.5x3 + d2- - d2+ = 500 46) What is the packaging material constraint? Answer: 3x1+2x2+1x3+d3--d3+=400 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) 47) State the goal programming objective function. Answer: Min P1d1+, P2d2+, P3d3-, P4d448) Write the budget constraint Answer: 100A + 120B + 80C + d1- - d1+ = 80,000 49) Write the risk factor constraint Answer: 8A + 10B + 7C + d2- - d2+ = 6000 50) 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. Answer: Let xp = tax rate for property; xs = sales tax rate, and xg = gas tax in cents per gallon. Min P1d1-, P2d2-, P3d3+ Subject to: 750xp + 80xs + .12xg ≥ 25 80xs ≤ .25(750xp + 80xs + .12xg) xg ≤ 8 Simplify the second constraint: 187.5xp - 60xs +.03xg ≥ 0. If this constraint is written as -187.5xp + 60xs -.03xg = 0, then the objective function should be: Min P1d1-, P2d2+, P3d3+. 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