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Introduction to Management Science, 13e (Taylor) Chapter 5 Integer Programming 1) The three types of integer programming models are total, 0-1, and mixed. Answer: TRUE Diff: 1 Page Ref: 188 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge 2) In a total integer model, all decision variables have integer solution values. Answer: TRUE Diff: 1 Page Ref: 188 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge 3) In a 0-1 integer model, the solution values of the decision variables are 0 or 1. Answer: TRUE Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge 4) In a mixed integer model, some solution values for decision variables are integer and others can be non-integer. Answer: TRUE Diff: 1 Page Ref: 190 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge 5) The college dean is deciding among three equally qualified (in their eyes, at least) candidates for his associate dean position. If this situation could be modeled as an integer program, the decision variables would be cast as 0-1 integer variables. Answer: TRUE Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: 0-1 variables, conditional constraints AACSB: Analytical thinking

6) The management scientist's fiancé informed him that if they were to be married, he would also have to welcome her mother into their home. The management scientist should model this decision as a contingency constraint. Answer: FALSE Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: corequisite constraints, 0-1 variables AACSB: Application of knowledge

7) In the classic game show Password, the suave, silver-haired host informed the contestants, "you can choose to pass or to play." This expression suggests a mixed integer model is most appropriate. Answer: FALSE Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, 0-1 variables AACSB: Analytical thinking 8) The production planner for Airbus showed his boss the latest product mix suggestion from their slick new linear programming model: 12.5 model 320s and 17.4 model 340s. The boss looked over his glasses at the production planner and reminded him that they had several unsold half airplanes from last year's production rusting in the parking lot. No one, it seems, is interested in half of an airplane. The production planner whipped out his red pen and crossed out the .5 and .4, turning the new plan into 12 model 320s and 17 model 340s. This production plan is definitely feasible. Answer: TRUE Diff: 1 Page Ref: 190 Section Heading: Integer Programming Graphical Solution Keywords: integer programming graphical solution AACSB: Analytical thinking 9) In a mixed integer model, all decision variables have integer solution values. Answer: FALSE Diff: 2 Page Ref: 190 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge 10) In a mixed integer model, the solution values of the decision variables are 0 or 1. Answer: FALSE Diff: 2 Page Ref: 190 Section Heading: Integer Programming Models Keywords: integer programming models AACSB: Application of knowledge

11) The branch and bound solution method cannot be applied to 0-1 integer programming problems. Answer: FALSE Diff: 2 Page Ref: 193 Section Heading: Integer Programming Models Keywords: integer programming models, branch and bound method AACSB: Application of knowledge 12) In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects. Answer: TRUE Diff: 2 Page Ref: 203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: capital budgeting, 0-1 variables AACSB: Analytical thinking

13) In a 0-1 integer programming problem involving a capital budgeting application (where xj = 1, if project j is selected, xj = 0, otherwise) the constraint x1 - x2 ≤ 0 implies that if project 2 is selected, project 1 cannot be selected. Answer: FALSE Diff: 2 Page Ref: 203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: capital budgeting, 0-1 variables AACSB: Analytical thinking 14) The divisibility assumption is violated by integer programming. Answer: TRUE Diff: 1 Page Ref: 188 Section Heading: Integer Programming Models Keywords: integer linear programming models, multiple choice constraint AACSB: Analytical thinking 15) One type of constraint in an integer program is a multiple-choice constraint. Answer: TRUE Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, multiple choice constraint AACSB: Application of knowledge 16) In Excel, a binary constraint in cell A1 is created using the =BIN($A$1) formula. Answer: FALSE Diff: 1 Page Ref: 195 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer linear programming models, constraint AACSB: Application of knowledge

17) Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution to an integer programming problem. Answer: TRUE Diff: 2 Page Ref: 192 Section Heading: Integer Programming Graphical Solution Keywords: integer programming models, graphical solution AACSB: Analytical thinking 18) A feasible solution to an integer programming problem is ensured by rounding down non- integer solution values. Answer: TRUE Diff: 2 Page Ref: 192 Section Heading: Integer Programming Graphical Solution Keywords: integer programming models, graphical solution AACSB: Application of knowledge 19) The feasible region in an integer programming graph is composed of a lattice of points. Answer: TRUE Diff: 2 Page Ref: 193 Section Heading: Integer Programming Graphical Solution Keywords: integer programming models, graphical solution, feasible region AACSB: Analytical thinking

20) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint. Answer: TRUE Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, 0-1 variables AACSB: Analytical thinking 21) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a mutually exclusive constraint. Answer: FALSE Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: 0-1 variables, multiple choice constraint AACSB: Analytical thinking 22) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a mutually exclusive constraint. Answer: FALSE Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: 0-1 variables, conditional constraints AACSB: Analytical thinking

23) If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a conditional constraint. Answer: FALSE Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: 0-1 variables, corequisite constraints AACSB: Analytical thinking 24) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint. Answer: TRUE Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: 0-1 variables, multiple choice constraint AACSB: Analytical thinking 25) Integer constraints are entered in the inequality dialog box within Excel's Solver routine. Answer: TRUE Diff: 1 Page Ref: 195 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer linear programming models, constraint AACSB: Application of knowledge

26) A mixed integer program has only integers as a solution; they are simply mixed, as opposed to an integer program where they are specific to the decision variables. Answer: FALSE Diff: 1 Page Ref: 190 Section Heading: Integer Programming Models Keywords: mixed integer AACSB: Analytical thinking 27) Rounding a noninteger solution ________ to the nearest integer guarantees a feasible, but perhaps suboptimal solution to an integer programming situation. Answer: down Diff: 2 Page Ref: 192 Section Heading: Integer Programming Models Keywords: integer programming models, mixed integer programming models AACSB: Application of knowledge 28) In a(n) ________ linear programming model, the solution values of the decision variables are zero or one. Answer: 0-1 integer Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming models, 0-1 integer programming models AACSB: Application of knowledge

29) The ________ method is based on the principle that the total set of feasible solutions can be partitioned into smaller subsets of solutions. Answer: branch and bound Diff: 3 Page Ref: 193 Section Heading: Integer Programming Graphical Solution Keywords: branch and bound method AACSB: Application of knowledge 30) "It's me or the cat!" the exasperated husband bellowed to his well-educated wife. "Hmmmm," she thought, "I could model this decision with a(n) ________ constraint." Answer: contingency or mutually exclusive Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: mutually exclusive constraints, 0-1 variables AACSB: Application of knowledge 31) A(n) ________ integer model allows for the possibility that some decision variables are not integers. Answer: mixed Diff: 1 Page Ref: 190 Section Heading: Integer Programming Models Keywords: mixed integer programming problem AACSB: Application of knowledge

32) In choosing four electives from the dazzling array offered by the Decision Sciences Department next semester, the students that had already taken the management science class were able to craft a model using a(n) ________ constraint. Answer: multiple-choice Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: multiple choice constraints, 0-1 variables AACSB: Application of knowledge 33) ________ variables are best suited to be the decision variables when dealing with yes-or-no decisions. Answer: 0-1 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, 0-1 variables AACSB: Application of knowledge

34) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a(n) ________ constraint. Answer: mutually exclusive Diff: 2 Page Ref: 189 Section Heading: Integer Programming Graphical Solution Keywords: integer linear programming models, 0-1 variables AACSB: Application of knowledge 35) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a(n) ________ constraint. Answer: multiple-choice Diff: 2 Page Ref: 189 Section Heading: Integer Programming Graphical Solution Keywords: 0-1 variables, multiple choice constraint AACSB: Application of knowledge 36) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a(n) ________ constraint. Answer: conditional Diff: 2 Page Ref: 189 Section Heading: Integer Programming Graphical Solution Keywords: 0-1 variables, conditional constraints AACSB: Application of knowledge 37) Rounding a noninteger solution ________ to the nearest integer value will likely result in an infeasible solution. Answer: up Diff: 2 Page Ref: 192 Section Heading: Integer Programming Models Keywords: integer linear programming models AACSB: Analytical thinking

38) In an integer program, if we were choosing between two locations to build a facility, this would be written as: ________. Answer: x1 + x2 = 1 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, constraint AACSB: Analytical thinking 39) In an integer program, if building one facility required the construction of another type of facility, this would be written as: ________. Answer: x1 = x2 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer linear programming models, constraint AACSB: Analytical thinking

The Deadbeats After months of broken promises, partial payments, and general stupidity, the landlord had no choice but to evict the long-term tenants that had become little more than squatters in his first rental property. As he surveyed the damage and pondered a mix of repairs and upgrades, he scoured the latest statistics on what different upgrades might be worth in terms of increased rent. Beautifully refinished wood floors could increase the monthly rent about $100 and an upgrade to the kitchen would fetch $80 per month. The garage door needed replacement, but even though it would receive daily use, it was almost an order qualifier, and wouldn't net more than $20 per month. The house had always suffered from lack of a back door – you had to access the backyard through the garage, so taking out a window and replacing it with a safety door would cost $250 and add only $15 to the monthly rent. The garage door would cost $350, the kitchen update would cost $1000 if he went with granite, and the floor refinish job would cost $400 to rent the buffer and buy the chemicals. It wouldn't be easy doing these upgrades; the garage door would take a half week, the back door one week, the floors two weeks and the tile three weeks. There was another way around these jobs though; instead of doing them himself, the landlord could always hire a professional in each field that could finish the job in half the time but would charge a pretty penny for that speed. Refinishing floors would cost $2700, upgrading the kitchen would cost $2500, replacing the back window with a door would cost $600, and installing a garage door opener would cost $350. 40) Obviously if the model wants to upgrade the kitchen, it should be done by either the landlord or a subcontractor. As he creates the IP model, the landlord wants to leave the choice of whether to actually upgrade the kitchen up to the optimization algorithm. How should this constraint be written if he uses the following scheme for decision variables? x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door A) x3 + x4 ≤ 1 B) x3 + x4 = 1 C) x3 - x4 ≤ 1 D) x3 - x4 = 1 Answer: A Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

41) Suppose the landlord really wants the back door to be installed. For too long he has had to cut through the garage and he figures when he retires, this house will be a perfect downsize home for him to move into. How should the constraint for the back door be written if he uses the following scheme for decision variables? x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door A) x5 + x6 ≤ 1 B) x5 + x6 = 1 C) x5 - x6 ≤ 1 D) x5 - x6 = 1 Answer: B Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking 42) The landlord uses the following scheme for decision variables: x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door What should the objective function be? A) Min Z = 2700x1 + 400x2 + 2500x3 + 1000x4 + 600x5 + 250x6 + 350x7 + 400x8 B) Min Z = 2700x1 - 400x2 + 2500x3 - 1000x4 + 600x5 - 250x6 + 350x7 - 400x8 C) Max Z = 100x1 + 100x2 + 80x3 + 80x4 + 15x5 + 15x6 + 20x7 + 20x8 D) Max Z = 100x1 - 100x2 + 80x3 - 80x4 + 15x5 - 15x6 + 20x7 - 20x8 Answer: C Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

43) The landlord uses the following scheme for decision variables: x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door Which of these constraints would not be appropriate for this scenario? A) 2700x1 + 400x2 + 2500x3 + 1000x4 + 600x5 + 250x6 + 350x7 + 400x8 ≤ 3000 B) 1x1 + 2x2 + 1.5x3 + 3x4 + 0.5x5 + 1x6 + 0.25x7 + 0.5x8 ≤ 4 C) x3 + x4 = 1 D) x1, x2, x3, x4, x5, x6, x7, x8 ≥ 0 and integer Answer: D Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking 44) The landlord ran the model in Excel and received the answer report contained in the table. Which of the following statements is correct? Variable Cells

Cell Name Original Value Final Value $G$4 wood floors contractor 1 0 $H$4 wood floors self 1 1 $G$5 kitchen tile contractor 1 1 $H$5 kitchen tile self 1 0 $G$6 back door contractor 1 0 $H$6 back door self 1 0 $G$7 garage door opener contractor 1 0 $H$7 garage door opener self 1 0

A) The rent will be $180 higher and the project will take 3.5 weeks to finish at a cost of $2900. B) The rent will be $195 higher and the project will take 2.5 weeks to finish at a cost of $2900. C) The rent will be $180 higher and the project will take 2.5 weeks to finish at a cost of $3700. D) The rent will be $195 higher and the project will take 3.5 weeks to finish at a cost of $3700. Answer: A Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

45) The landlord ran the model in Excel and received the sensitivity report for the constraints as contained in the table. Which of the following statements is correct?

Cell Name

Final Value

Shadow Price

Constraint R.H. Side

Allowable Increase

Allowable Decrease

$B$1 2 Total Time Weeks 4.00 13.64 4 1.1 3.1E-16 $E$10 Total Cost 3000.00 0.01 3000 550 3.1E-13 $I$4 wood floors Included? 1.00 67.27 1 1.27E-16 4.6E-01 $I$5 kitchen tile Included? 1.00 25.45 1 7.63E-17 2.8E-01 $I$6 back door Included? 0.00 0.00 1 1.00E+30 1.0E+00

$I$7 garage door opener Included? 1.00 11.82 1 5.09E-16 1.0E+00

A) If there are 2 more weeks available for work, it would be possible to increase rent by $27.28 per week B) Given the scenario, it is evident that a constraint is missing. C) If the landlord installed two garage doors, rent could be increased by $23.64. D) Given the scenario, it is evident that the objective function is misstated. Answer: B Diff: 3 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

The Cruise Their cruise would port out of New Orleans and promised seven days with a panoply of excursions in Jamaica, Cozumel, and Grand Cayman. A list of excursions at each site and key features of each appear in the table. The excursions were all day affairs, so it was possible to engage in only one per port. The cruise ship sailed at night and docked at each of these three ports at the crack of dawn. By dinner time, the ship was on its way to the next port and next set of excursions. The couple was energetic and active for a pair of 52-year-olds, and while enjoying an upper middle class lifestyle, they didn't want to spend money on excursions that might be better spent on tacky souvenirs. The couple therefore budgeted $250 for the excursions – the prices shown are per couple, so for example, the $60 will pay for both of them to fill up on jerk chicken and mannish water. For each of the duplicate excursions (e.g., snorkeling is offered in all three ports), the couple researched the quality of the activity and ranked the excursion among the available alternatives, with higher numbers indicating better quality. Thus, snorkeling in Jamaica is better than in Cozumel, and snorkeling in Cozumel is better than in Grand Cayman. For the unique experiences, i.e., the turtle farm, the default rating was the a 3. Site Rating Activity Cost Jamaica 3 snorkeling $100 Jamaica 1 party island $95 Jamaica 2 horseback ride $120 Jamaica 3 local cuisine $60 Cozumel 2 snorkeling $110 Cozumel 3 party island $55 Cozumel 1 horseback ride $70 Cozumel 2 local cuisine $90 Cozumel 3 tequila tasting $130 Grand Cayman 1 snorkeling $90 Grand Cayman 2 party island $60 Grand Cayman 3 horseback ride $110 Grand Cayman 1 local cuisine $130 Grand Cayman 3 turtle farm $95 (Note - data used in this test question should not be construed as vacation advice.)

46) What is an appropriate objective function for this vacation? A) Max Z = JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu B) Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu C) Max Z = 3JS + 1JP + 2JH + 3JL + 2CS + 3CP + 1CH + 2CL + 3CTe + 1GS + 2GP + 3GH + 1GL + 3GTu D) Min Z = 3JS + 1JP + 2JH + 3JL + 2CS + 3CP + 1CH + 2CL + 3CTe + 1GS + 2GP + 3GH + 1GL + 3GTu Answer: C Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking 47) Use the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu) to represent the decision variables. What of these sets of constraints appropriately limits the number of excursions based on the scenario? A) JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu ≤ 3 B) JS + JP + JH + JL = 1 CS + CP + CH + CL + CTe = 1 GS + GP + GH + GL + GTu = 1 C) JS + CS + GS ≤ 1 JP + CP + GP ≤ 1 JH + CH + GH ≤ 1 JL + CL + GL ≤ 1 D) JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu = 3 Answer: B Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

48) Use the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu) to represent the decision variables. What of these alternative sets of constraints ensures that no activities are duplicated at the ports of call? A) JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu ≤ 3 B) JS + JP + JH + JL = 1 CS + CP + CH + CL + CTe = 1 GS + GP + GH + GL + GTu = 1 C) JS + CS + GS ≤ 1 JP + CP + GP ≤ 1 JH + CH + GH ≤ 1 JL + CL + GL ≤ 1 D) JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu = 3 Answer: C Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

49) In all the excitement of waving to the longshoremen as the ship leaves the Port of New Orleans, the management scientist drops his wallet in the Mississippi River. Rather than maximize enjoyment for the three excursions, he must now adjust his model to select three inexpensive options. Which combinations of objective function and constraints are best if the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu) is used to represent the decision variables? A) Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu subject to: JS + JP + JH + JL = 1 CS + CP + CH + CL + CTe = 1 GS + GP + GH + GL + GTu = 1 B) Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu subject to: JS + JP + JH + JL ≤ 1 CS + CP + CH + CL + CTe ≤ 1 GS + GP + GH + GL + GTu ≤ 1 C) Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu JS + CS + GS ≤ 1 JP + CP + GP ≤ 1 JH + CH + GH ≤ 1 JL + CL + GL ≤ 1 D) Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu = 3 Answer: A Diff: 2 Page Ref: 202-203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

50) Types of integer programming models are: A) total. B) 0-1. C) mixed. D) total, 0-1, and mixed Answer: D Diff: 1 Page Ref: 188-190 Section Heading: Integer Programming Problems Keywords: integer programming models AACSB: Analytical thinking

51) In a ________ integer model, some solution values for decision variables are integers and others can be non-integer. A) total B) 0-1 C) mixed D) total, 0-1, and mixed Answer: C Diff: 2 Page Ref: 188 Section Heading: Integer Programming Problems Keywords: integer programming models, mixed integer models AACSB: Application of knowledge 52) In a ________ integer model, all decision variables have integer solution values. A) total B) 0-1 C) mixed D) total, 0-1, and mixed Answer: A Diff: 2 Page Ref: 188 Section Heading: Integer Programming Problems Keywords: integer programming models AACSB: Application of knowledge 53) In a ________ integer model, the solution values of the decision variables are 0 or 1. A) total B) 0-1 C) mixed D) total, 0-1, and mixed Answer: B Diff: 2 Page Ref: 184 Section Heading: Integer Programming Problems Keywords: integer programming models AACSB: Application of knowledge

54) Binary variables are: A) 0 or 1 only. B) any integer value. C) any continuous value. D) any negative integer value. Answer: A Diff: 1 Page Ref: 189 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming formulation AACSB: Application of knowledge

55) Which of the following is not an integer linear programming problem? A) pure integer B) mixed integer C) 0-1 integer D) continuous Answer: D Diff: 2 Page Ref: 188-190 Section Heading: Integer Programming Problems Keywords: integer programming models AACSB: Application of knowledge 56) If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is: A) always optimal and feasible. B) sometimes optimal and feasible. C) always feasible. D) never optimal and feasible. Answer: B Diff: 3 Page Ref: 189 Section Heading: Integer Programming Graphical Solution Keywords: integer programming models AACSB: Analytical thinking 57) The branch and bound method of solving linear integer programming problems is: A) an integer method. B) a relaxation method. C) a graphical solution. D) an enumeration method. Answer: D Diff: 2 Page Ref: 193 Section Heading: Integer Programming Problems Keywords: integer programming solution methods, branch and bound method AACSB: Application of knowledge

58) If a maximization linear programming problem consists of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will ________ result in a(n) ________ solution to the integer linear programming problem. A) always, optimal B) always, non-optimal C) never, non-optimal D) sometimes, optimal Answer: D Diff: 2 Page Ref: 192 Section Heading: Integer Programming Problems Keywords: integer programming solution, rounding off AACSB: Analytical thinking 59) If a maximization linear programming problem consists of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will ________ result in a feasible solution to the integer linear programming problem. A) always B) sometimes C) optimally D) never Answer: A Diff: 2 Page Ref: 192 Section Heading: Integer Programming Problems Keywords: integer programming solution, rounding off AACSB: Application of knowledge 60) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite Answer: B Diff: 2 Page Ref: 189 Section Heading: Integer Programming Problems Keywords: mutually exclusive constraints, 0-1 variables AACSB: Analytical thinking

The Exorbitant Course Fees The $75 per credit hour course fee tacked on to all the MBA classes has generated a windfall of $56,250 in its first semester. "Now we just need to make sure we spend it all," the Assistant Dean cackled. She charged the Graduate Curriculum Committee with generating a shopping list before their next meeting. Four months later, the chairman of the committee distributed the following. As the professor for the quantitative modeling course, he tended to think in terms of decision variables, so he added the left-most column for ease of use. Decision Variable

Item

Cost

Note

A iPads for everybody $750/unit Must get a cover if these are purchased B iPad covers with MBA logo $25/unit Not needed unless we buy iPads C Speaker series $15,000 Can't afford both this and the iPads

D Subscriptions to the Wall Street Journal $10/unit

Don't need if we have the electronic version

E Subscriptions to the electronic version of the Wall Street Journal $5/unit Worthless without the iPads

61) Which constraint best describes the situation with decision variables A and B? A) B - A ≤ 0 B) B - A = 0 C) B + A = 1 D) B + A ≤ 1 Answer: B Diff: 2 Page Ref: 188-190 Section Heading: Integer Programming Models Keywords: variables, multiple choice constraint AACSB: Analytical thinking 62) Which of the constraints best describes the relationship between the iPads for everyone and the speaker series? A) A - C ≤ 1 B) A + C = 1 C) A - C = 0 D) A + C = 2 Answer: B Diff: 2 Page Ref: 188-190 Section Heading: Integer Programming Models Keywords: integer programming AACSB: Analytical thinking

63) Which of the following constraints best describes the relationship between the electronic Wall Street Journal subscription and the iPads? A) E - A = 0 B) E - A = 1 C) E - A ≤ 0 D) E - A ≤ 1 Answer: C Diff: 2 Page Ref: 188-190 Section Heading: Integer Programming Models Keywords: variables, conditional constraints AACSB: Analytical thinking

64) Which of these formulations of the budget constraint is correct? Assume that there are 20 students in this semesters MBA class. A) $750A + $25B + $15,000C + $10D + $5E ≤ $56,250 B) 20A + 20B + C + 20D + 20E ≤ $56,250 C) A + B + C + D + E ≤ 20 D) $15,000A + $500B + $15,000C + $200D + $100E ≤= $56,250 Answer: D Diff: 2 Page Ref: 188-190 Section Heading: Integer Programming Models Keywords: integer linear programming formulation, capital budgeting AACSB: Analytical thinking

67) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite Answer: C Diff: 2 Page Ref: 189 Section Heading: Integer Programming Problems Keywords: conditional constraints, 0-1 variables AACSB: Application of knowledge 68) If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite Answer: D Diff: 2 Page Ref: 189 Section Heading: Integer Programming Problems Keywords: corequisite constraints, 0-1 variables AACSB: Application of knowledge 69) For a maximization integer linear programming problem, a feasible solution is ensured by rounding ________ non-integer solution values if all of the constraints are the less-than-or-equal- to type. A) up and down B) up C) down D) up or down Answer: C Diff: 2 Page Ref: 192 Section Heading: Integer Programming Graphical Solution Keywords: integer linear programming solution AACSB: Analytical thinking

70) In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation? A) x1 + x2 + x5 ≤ 1 B) x1 + x2 + x5 ≥ 1 C) x1 + x5 ≤ 1, x2 + x5 ≤ 1 D) x1 - x5 ≤ 1, x2 - x5 ≤ 1 Answer: C Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking

71) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, the constraint for the first restriction is: A) S1 + S3 + S7 ≥ 1. B) S1 + S3 + S7 ≤ 1. C) S1 + S3 + S7 = 2. D) S1 + S3 + S7 ≤ 2. Answer: D Diff: 3 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: integer programming formulation, constraint AACSB: Analytical thinking

Due to increased sales, a company is considering building three new distribution centers (DCs) to serve four regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (in thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region

DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3

The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. Assume that Xij = quantity shipped from distribution i to region j. i = 1,2,3 and j = 1, 2, 3, 4. Assume that Yi = 0 or 1 where i = distribution center 1, 2 or 3. 72) The constraint for distribution center 1 is: A) X11 + X12 + X13 + X14 - 500y1 ≤ 0. B) X11 + X12 + X13 + X14D + 500y1 ≤ 0. C) X11 + X12 + X13 + X14 ≤ 500. D) X11 + X12 + X13 + X14 ≥ 500. Answer: A Diff: 2 Page Ref: 202 Section Heading: Integer Programming Graphical Solution Keywords: facility location example, mixed integer programming AACSB: Analytical thinking

73) The objective function is

A) Min Z = +

B) Min Z = +

C) Min Z = +

D) Min Z = +

Answer: A Diff: 3 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: facility location example, mixed integer programming AACSB: Analytical thinking 74) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint(s) for the second restriction. A) S2 + S5 ≤ 1 B) S4 + S5 ≤ 1 C) S2 + S5 + S4 + S5 ≤ 2 D) S2 + S5 ≤ 1, S4 + S5 ≤ 1 Answer: D Diff: 3 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking

The Potluck The study tour group had four weeks to recover from their digestive issues and put together their videos for the class assignment, which were to be shared at a final meeting over a potluck supper. There were ten students in the class and each was tasked with preparing two dishes to share with the rest of the group — either an entrée, a side, or a dessert. In order to make this meal optimal, their tour guide, who doubled as their management science professor, decided to model it as an integer program. He asked the students to rate themselves on their ability to prepare sweet or savory dishes and proteins or vegetables. Their ratings appear in the table. A reasonably balanced meal with plenty of choices calls for at least five entrées, at least six desserts, and at least six sides. The students agreed to bring two dishes apiece and breathlessly awaited the results of the optimization model. Entrée Side Dessert Aaron 0.70 0.41 0.68 Alex 0.56 0.52 0.03 Josh 0.40 0.73 0.39 Erin 0.70 0.62 0.66 Julie 0.18 0.16 0.32 Kassie 0.88 0.37 0.11 Lindy 0.90 0.06 0.44 MacGregor 0.43 0.20 0.99 Marlene 0.86 0.81 0.02 Tracy 0.83 0.01 0.96 75) The constraint for the entrées is: A) AaE + AlE + JOE + EE + JuE + KE + LE + McE + MrE + TE ≥ 5. B) AaE + AlE + JOE + EE + JuE + KE + LE + McE + MrE + TE ≤ 5. C) AaE + AlE + JOE + EE + JuE + KE + LE + McE + MrE + TE > 5. D) AaE + AlE + JOE + EE + JuE + KE + LE + McE + MrE + TE < 5. Answer: A Diff: 2 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking 76) Aaron's constraint is: A) AaE + AaS + AaD ≤ 2. B) AaE + AaS + AaD = 2. C) AaE + AaS + AaD ≥ 2 D) AaE + AaS + AaD < 3 Answer: B Diff: 2 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking

77) What is the maximum value for the potluck? A) 13.28 B) 12.44 C) 15.02 D) 10.86 Answer: D Diff: 2 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking 78) What difference would it make to the solution if this scenario was modeled as a 0-1 problem instead of an integer model? A) There would not be any difference. B) The 0-1 model would result in a higher value for the objective function. C) The 0-1 model would result in a lower value for the objective function. D) The 0-1 model would be infeasible given the constraints that exist in the scenario. Answer: C Diff: 3 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking 79) Which student experiences the greatest loss of contribution to the objective function value if this scenario was modeled as a 0-1 problem instead of an integer model? A) Lindy B) Marlene C) MacGregor D) There is no difference in any student's contribution. Answer: A Diff: 3 Page Ref: 203 Section Heading: Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking

80) Max Z = 5x1 + 6x2 Subject to: 17x1 + 8x2 ≤ 136 3x1 + 4x2 ≤ 36 x1, x2 ≥ 0 and integer What is the optimal solution? A) x1 = 6, x2 = 4, Z = 54 B) x1 = 3, x2 = 6, Z = 51 C) x1 = 2, x2 = 6, Z = 46 D) x1 = 4, x2 = 6, Z = 56 Answer: D Diff: 2 Page Ref: 191-199 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming solution AACSB: Analytical thinking

81) Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise. The constraint (x1 + x2 + x3 + x4 = 2) means that ________ out of the ________ projects must be selected. A) exactly 1, 2 B) exactly 2, 4 C) at least 2, 4 D) at most 1, 2 Answer: B Diff: 2 Page Ref: 201 Section Heading: Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking 82) In a 0-1 integer programming model, if the constraint x1 - x2 = 0, it means when project 1 is selected, project 2 ________ be selected. A) can also B) can sometimes C) can never D) must also Answer: D Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation AACSB: Analytical thinking

83) In a 0-1 integer programming model, if the constraint x1 - x2 ≤ 0, it means when project 2 is selected, project 1 ________ be selected. A) must always B) can sometimes C) can never D) is already Answer: B Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation AACSB: Analytical thinking 84) In formulating a mixed integer programming problem, the constraint x1 + x2 ≤ 500y1 where y1 is a 0-1 variable and x1 and x2 are continuous variables, then x1 + x2 = 500 if y1 is: A) 0. B) 1. C) 0 or 1. D) none of the above Answer: B Diff: 3 Page Ref: 189 Section Heading: Integer Programming Models Keywords: mixed integer programming problem AACSB: Analytical thinking

85) Max Z = 13x1 + 8x2 Subject to: 15x1 + 12x2 ≤ 144 7x1 + 9x2 ≤ 64 x1, x2 ≥ 0 and integer What is the optimal solution? A) x1 = 5, x2 = 6, Z = 113 B) x1 = 7, x2 = 7, Z = 147 C) x1 = 9, x2 = 0, Z = 117 D) x1 = 0, x2 = 15, Z = 120 Answer: C Diff: 2 Page Ref: 191-199 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming solution AACSB: Analytical thinking

86) Consider the following integer linear programming problem: Max Z = 3x1 + 2x2 Subject to: 3x1 + 5x2 ≤ 30 4x1 + 2x2 ≤ 28 x1 ≤ 8 x1, x2 ≥ 0 and integer The solution to the linear programming formulation is: x1 = 5.714, x2 = 2.571. What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function. Answer: x1 = 6, x2 = 2, Z = 22 Diff: 2 Page Ref: 192 Section Heading: Integer Programming Graphical Solution Keywords: integer linear programming solution AACSB: Analytical thinking 87) Consider the following integer linear programming problem: Max Z = 3x1 + 2x2 Subject to: 3x1 + 5x2 ≤ 30 5x1 + 2x2 ≤ 28 x1 ≤ 8 x1, x2 ≥ 0 and integer The solution to the linear programming formulation is: x1 = 5.714, x2 = 2.571. What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function. Answer: x1 = 4, x2 = 3, Z = 18 Diff: 2 Page Ref: 192 Section Heading: Integer Programming Graphical Solution Keywords: integer linear programming solution AACSB: Analytical thinking

88) Consider a capital budgeting example with five projects from which to select. Let x1 = 1 if project a is selected, 0 if not, for a = 1, 2, 3, 4, 5. Projects cost $100, $200, $150, $75, and $300, respectively. The budget is $450. Write the appropriate constraint for the following condition: Choose no fewer than 3 projects. Answer: x1 + x2 + x3 + x4 + x5 ≥ 3 Diff: 1 Page Ref: 201 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: formulation of 0-1 constraints AACSB: Analytical thinking

89) Write the appropriate constraint for the following condition: If project 3 is chosen, project 4 must be chosen. Answer: x3 - x4 ≤ 0 Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: formulation of 0-1 constraints AACSB: Analytical thinking 90) Write the appropriate constraint for the following condition: If project 1 is chosen, project 5 must not be chosen. Answer: x1 + x5 ≤ 1 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: formulation of 0-1 constraints AACSB: Analytical thinking

The Deadbeats After months of broken promises, partial payments, and general stupidity, the landlord had no choice but to evict the long-term tenants that had become little more than squatters in his first rental property. As he surveyed the damage and pondered a mix of repairs and upgrades, he scoured the latest statistics on what different upgrades might be worth in terms of increased rent. Beautifully refinished wood floors could increase the monthly rent about $100 and an upgrade to the kitchen would fetch $80 per month. The garage door needed replacement, but even though it would receive daily use, it was almost an order qualifier, and wouldn't net more than $20 per month. The house had always suffered from lack of a back door – you had to access the backyard through the garage, so taking out a window and replacing it with a safety door would cost $250 and add only $15 to the monthly rent. The garage door would cost $350, the kitchen update would cost $1000 if he went with granite, and the floor refinish job would cost $400 to rent the buffer and buy the chemicals. It wouldn't be easy doing these upgrades; the garage door would take a half week, the back door one week, the floors two weeks and the tile three weeks. There was another way around these jobs though; instead of doing them himself, the landlord could always hire a professional in each field that could finish the job in half the time but would charge a pretty penny for that speed. Refinishing floors would cost $2700, upgrading the kitchen would cost $2500, replacing the back window with a door would cost $600, and installing a garage door opener would cost $350. 91) Formulate an appropriate objective function for this scenario. Answer: If the variables are assigned as in the table: x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door then an appropriate objective function will be: Max Z = 100x1 + 100x2 + 80x3 + 80x4 + 15x5 + 15x6 + 20x7 + 20x8 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

92) Formulate an appropriate model for this scenario. Answer: If the variables are assigned as in the table: x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door then an appropriate objective function will be: Max Z = 100x1 + 100x2 + 80x3 + 80x4 + 15x5 + 15x6 + 20x7 + 20x8 Subject to: x1 + x2 ≤ 1 x3 + x4 ≤ 1 x5 + x6 ≤ 1 x7 + x8 ≤ 1 2700x1 + 400x2 + 2500x3 + 1000x4 + 600x5 + 250x6 + 350x7 + 400x8 ≤ 3000 1x1 + 2x2 + 1.5x3 + 3x4 + 0.5x5 + 1x6 + 0.25x7 + 0.5x8 ≤ 4 x1, x2, x3, x4, x5, x6, x7, x8 are 0,1 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking 93) Formulate an appropriate model for this scenario if the variables are assigned as in the table: x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door Answer: Max Z = 100x1 + 100x2 + 80x3 + 80x4 + 15x5 + 15x6 + 20x7 + 20x8 Subject to: x1 + x2 ≤ 1 x3 + x4 ≤ 1 x5 + x6 ≤ 1 x7 + x8 ≤ 1 2700x1 + 400x2 + 2500x3 + 1000x4 + 600x5 + 250x6 + 350x7 + 400x8 ≤ 3000 1x1 + 2x2 + 1.5x3 + 3x4 + 0.5x5 + 1x6 + 0.25x7 + 0.5x8 ≤ 4 x1, x2, x3, x4, x5, x6, x7, x8 are 0,1 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

94) I know it's hard to believe that his job as a management scientist doesn't comfortably keep him in Ferraris and caviar, but the insurance premiums on his ride are outrageous, and frankly it doesn't get good gas mileage in the city. Hence, the management scientist/landlord runs his model and receives this output. Provide an interpretation of exactly what will happen in the coming weeks. Cell Name Original Value Final Value Integer $G$4 wood floors contractor 1 0 Binary $H$4 wood floors landlord 1 1 Binary $G$5 kitchen tile contractor 1 1 Binary $H$5 kitchen tile landlord 1 0 Binary $G$6 back door contractor 1 0 Binary $H$6 back door landlord 1 0 Binary $G$7 garage door opener contractor 1 0 Binary $H$7 garage door opener landlord 1 0 Binary Answer: It appears that the landlord/management scientist will tackle the wood floors himself and have a contractor install kitchen tile. This will take 3.5 weeks, leaving one half week free, and will cost $2900 out of the maximum $3,000 budget, leaving $100 – enough for beer and pizza, or at least pizza. It is somewhat ambiguous whether the landlord has four weeks to work with and the contractors have four weeks or whether the entire model's time constraint should have only four weeks - if we knew whether there were wood floors in the kitchen and two work groups would get in each other's way, then we could set up constraints to capture this situation. Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

95) The management scientist was planning to leave his actions up to the wisdom of his model, but before he could press "Solve" on his Excel spreadsheet, his wife weighed in on the project. This was only reasonable, as it was her house before they tied the knot. It has been her dream to move back into this cottage once the management scientist goes to his great reward. She insists that the back door and garage door components of this project happen. If the variables have been chosen as in the table, construct an objective function and constraints that meet the management scientist's bride's requests. x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = contractor works on kitchen tile x5 = contractor works on back door x6 = contractor works on back door x7 = contractor works on garage door x8 = contractor works on garage door Answer: Max Z = 100x1 + 100x2 + 80x3 + 80x4 + 15x5 + 15x6 + 20x7 + 20x8 Subject to: x1 + x2 ≤ 1 x3 + x4 ≤ 1 x5 + x6 = 1 x7 + x8 = 1 2700x1 + 400x2 + 2500x3 + 1000x4 + 600x5 + 250x6 + 350x7 + 400x8 ≤ 3000 1x1 + 2x2 + 1.5x3 + 3x4 + 0.5x5 + 1x6 + 0.25x7 + 0.5x8 ≤ 4 x1, x2, x3, x4, x5, x6, x7, x8 are 0,1 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

96) The management scientist was planning to leave his actions up to the wisdom of his model, but before he could press "Solve" on his Excel spreadsheet, his wife weighed in on the project. This was only reasonable, as it was her house before they tied the knot. It has been her dream to move back into this cottage once the management scientist goes to his great reward. She insists that the back door and garage door components of this project happen. After making these changes, the management scientist ran the model and obtained the following answer report. Provide a full interpretation. Variable Cells

Cell Name

Original Value

Final Value

Integer

$G$4 wood floors Contractor? 0 0 Binary $H$4 wood floors Landlord? 1 1 Binary $G$5 kitchen tile Contractor? 1 0 Binary $H$5 kitchen tile Landlord? 0 0 Binary $G$6 back door Contractor? 0 1 Binary $H$6 back door Landlord? 0 0 Binary $G$7 garage door opener Contractor? 0 1 Binary $H$7 garage door opener Landlord? 0 0 Binary Constraints Cell Name Cell Value Formula Status Slack

$B$12 Total Time Weeks 2.75 $B$12<=$B$13 Not Binding 1.25

$E$10 Total Cost $1,350 $E$10<=$E$11 Not Binding 1650

$I$4 wood floors Included 1 $I$4<=$J$4 Binding 0

$I$5 kitchen tile Included 0 $I$5<=$J$5 Not Binding 1

$I$6 back door Included 1 $I$6=$J$6 Binding 0

$I$7 gar door open Included 1 $I$7=$J$7 Binding 0

$G$4:$H$7= Binary Answer: The landlord will be refinishing the wood floors and the contractor(s) will be tackling the garage door opener and back door installations. those projects will cost a total of $1350, leaving $1650 left in the budget. These projects will take 2.75 weeks out of the four weeks available. The landlord should be able to increase the monthly rent by $135 once these upgrades have been completed. The only upgrade not completed will be the kitchen tile. Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

97) The management scientist entered a trance-like state as he formulated his model. This state was interrupted by an integer-hating colleague, who insisted that the model should be run as a linear program rather than integer program. The management scientist sighed, relaxed the integer constraint, and ran the model, obtaining the following answer report. Provide a full interpretation and plan for action. Variable Cells Cell Name Original Value Final Value Integer $G$4 wood floors Contractor? 0 0 Contin $H$4 wood floors Landlord? 0 1 Contin $G$5 kitchen tile Contractor? 0 0.833 Contin $H$5 kitchen tile Landlord? 0 0.167 Contin $G$6 back door Contractor? 0 0.000 Contin $H$6 back door Landlord? 0 0 Contin $G$7 garage door opener Contractor? 0 1 Contin $H$7 garage door opener Landlord? 0 0 Contin Constraints Cell Name Cell Value Formula Status Slack $B$12 Total Time Weeks Contractor 4 $B$12<=$B$13 Binding 0 $E$10 Total Cost Contractor Price $3,000 $E$10<=$E$11 Binding 0 $I$4 wood floors Included? 1 $I$4<=$J$4 Binding 0 $I$5 kitchen tile Included? 1 $I$5<=$J$5 Binding 0

$I$6 back door Included? 2.78E-16 $I$6<=$J$6 Not Binding 1

$I$7 garage door opener Included? 1 $I$7<=$J$7 Binding 0 Answer: The landlord uses the entire budget, takes the entire four weeks and installs (or has the contractor install) the floors, garage door, and the kitchen tile, leaving only the back door off the improvement list. The kitchen tile project is split between the landlord, who does one-sixth of the work, and the contractor, who completes the remaining five-sixths. Although the model meets the constraints, this is not a practical solution in the real world. The management scientist should realize that moving to an integer solution within the LP feasible region will result in a feasible solution, albeit one with no guarantee of optimality. If the integer solution of wood floors and the garage door is chosen, there would be money and time both left over in the project budget, but the management scientist should reintroduce the integer constraint to confirm the best choice of action. Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming formulation, 0-1 integer programming AACSB: Analytical thinking

99) What is an appropriate objective function and constraints for this vacation? Answer: Using the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu), the model should be expressed as: Max Z = 3JS + 1JP + 2JH + 3JL + 2CS + 3CP + 1CH + 2CL + 3CTe + 1GS + 2GP + 3GH + 1GL + 3GTu subject to: 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu <= 250 JS + JP + JH + JL ≤ 1 CS + CP + CH + CL + CTe ≤ 1 GS + GP + GH + GL + GTu ≤ 1 JS + CS + GS ≤ 1 JP + CP + GP <= 1 JH + CH + GH <= 1 JL + CL + GL <= 1 JS, JP, JH, JL, CS, CP, CH, CL, CTe, GS, GP, GH, GL, GTu are 0,1 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking 100) The vacationing management scientist sat down at his computer keyboard, cracked his knuckles and decided on the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu), for his decision variables. He wrote constraints that read: JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu ≤ 3 JS + JP + JH + JL ≤ 1 CS + CP + CH + CL + CTe ≤ 1 GS + GP + GH + GL + GTu ≤ 1 What do these constraints accomplish in the model and which ones are necessary? Answer: The first constraint limits the total number of excursions to 3. JS + JP + JH + JL + CS + CP + CH + CL + CTe + GS + GP + GH + GL + GTu ≤ 3 The second through the fourth constraints limit the excursions at each one of the ports to 1. JS + JP + JH + JL ≤ 1 CS + CP + CH + CL + CTe ≤ 1 GS + GP + GH + GL + GTu ≤ 1 The first constraint doesn't help the model and can be eliminated. If it were used in place of constraints 2 through 4, the model might suggest taking all three excursions at one port of call, which would be impossible. Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

101) The first day on the cruise was a "day at sea" meaning no port of call, and only the amenities onboard for amusement. Restless and uncomfortably full after seven trips through the buffet, the management scientist gambled away most of his vacation money at the onboard casino. The excursions would be a necessity, but now it became less important to maximize the joy of the excursions and more vital to get off the boat as cheaply as possible while still staying busy at the three ports of call. What is an appropriate objective function for this modified cruise vacation? Answer: Using the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu), the objective function should be expressed as: Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu Diff: 1 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking 102) The first day on the cruise was a "day at sea" meaning no port of call, and only the amenities onboard for amusement. Restless and uncomfortably full after seven trips through the buffet, the management scientist gambled away most of his vacation money at the onboard casino. The excursions would be a necessity, but now it became less important to maximize the joy of the excursions and more vital to get off the boat as cheaply as possible while still staying busy at the three ports of call. What is an appropriate model for this modified cruise vacation? Answer: Using the scheme of location (J, C, or G) and excursion (S, P, H, L, Te or Tu), the objective function should be expressed as: Min Z = 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu subject to: 100JS + 95JP + 120JH + 60JL + 110CS + 55CP + 70CH + 90CL + 130CTe + 90GS + 60GP + 110GH + 130GL + 95GTu ≤ 250 JS + JP + JH + JL = 1 CS + CP + CH + CL + CTe = 1 GS + GP + GH + GL + GTu = 1 JS + CS + GS ≤ 1 JP + CP + GP ≤1 JH + CH + GH ≤1 JL + CL + GL ≤1 JS, JP, JH, JL, CS, CP, CH, CL, CTe, GS, GP, GH, GL, GTu are 0,1 Diff: 1 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming models, 0-1 integer programming models AACSB: Analytical thinking

Saba conducts regular tours of his favorite city in the world, Paris. Each semester he selects among the finest students in the university and escorts them to the City of Lights. In addition to a world-class education on conducting business in Europe, he arranges a number of cultural outings for them to help them immerse themselves in all that France has to offer. He collects an extra $100 from each student for this purpose and limits his tour group to ten lucky individuals. Some of the events (and their prices) he proposes to the students include: Eiffel Tower visit, $40 per student, E Paris Sewer spelunking, $20 per student, S Half day passes to the Louvre, $60 per student, L Bon Beret tour, $50 per student, B So much to do and so little time! 103) What would the constraints be if the Eiffel Tower visit needed to take place at the same time as the half day at the Louvre and if students taking the Paris Sewer tour had to wear the special sanitary beret available only from the Bon Beret tour? Answer: E + L ≤ 1 S - B ≤ 0 Diff: 2 Page Ref: 200-209 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: mutually exclusive constraints, 0-1 variables AACSB: Analytical thinking 104) The tour group has three days remaining in Paris and the opportunity to do three cultural events. It is important to soak up as much culture as possible, so Saba decides to model this as a 0-1 integer program mandating that the group does three events. A couple of students object, not to the integer program, but to the set of cultural events that they have to choose from. They would rather have the option to do up to three events but perhaps only one or two and spend the rest of their time doing some "retail benchmarking." What was Saba's original constraint and how does that constraint change to cater to the whims of the students? Answer: Saba's original constraint was E + L + S + B = 3, which directed the group to three tours. The new constraint is E + L + S + B ≤ 3, which allows for the possibility of no tours up to a maximum of three tours. Diff: 1 Page Ref: 200-209 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: 0-1 variables, multiple choice constraint AACSB: Analytical thinking

105) What is the full set of constraints if the following situations occur? The Eiffel Tower visit needed to take place at the same time has the half day at the Louvre. Students taking the Paris Sewer tour had to wear the special sanitary beret available only from the Bon Beret tour. Saba applies for university travel funds and supplements the students' accounts with an extra $30 each. Answer: E + L ≤ 1 S - B ≤ 0 40E + 20S + 60L + 50B ≤ 130 Diff: 2 Page Ref: 200-209 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: mixed integer programming problem, constraint formulation AACSB: Analytical thinking

Item

Cost

Note

D Subscriptions to the Wall Street Journal $10/unit

Don't need if we have the electronic version

E Subscriptions to the electronic version of the Wall Street Journal $5/unit Worthless without the iPads

106) What is a full set of constraints for this problem if there are 30 MBA students enrolled this semester? Answer: 22,500A + 750B + 15,000C + 300D + 150E ≤ 56,250 A + C ≤ 1 A - B = 0 E - A ≤ 0 D + E ≤ 1 Diff: 2 Page Ref: 202-203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer linear programming models, 0-1 variables AACSB: Analytical thinking

The Wiethoff Company has a contract to produce 10,000 garden hoses for a customer. Wiethoff has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.

107) This problem requires two different kinds of decision variables. Clearly define each kind. Answer: xa = the number of hoses produced on machine a; ya = 1 if machine a is used, 0 if not Diff: 2 Page Ref: 201 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer program prob formulation, decision variable definition AACSB: Analytical thinking

108) Write the objective function. Answer: MIN Z = 1.25X1 + 1.50X2 + 1.00X3 + 2.000X4 + 750Y1 + 500Y2 + 1000Y3 + 300Y4 Diff: 2 Page Ref: 201 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: integer programming problem formulation, model constraints AACSB: Analytical thinking 109) Write a constraint to ensure that if machine 4 is used, then machine 1 will not be used. Answer: y1 + y4 ≤ 1 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming problem formulation, model constraints AACSB: Analytical thinking 110) Write a constraint that will ensure that Weithoff purchases exactly two machines. Answer: Y1 + Y2 + Y3 + Y4 = 2 Diff: 2 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking

111) Max Z = x1 + 6x2 Subject to: 17x1 + 8x2 ≤ 136 3x1 + 4x2 ≤ 36 x1, x2 ≥ 0 and integer Find the optimal solution. Answer: x1 = 0, x2 = 9, Z = 54 Diff: 2 Page Ref: 191-199 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming problem computer solution AACSB: Analytical thinking 112) Max Z = 3x1 + 5x2 Subject to: 7x1 + 12x2 ≤ 136 3x1 + 5x2 ≤ 36 x1, x2 ≥ 0 and integer Find the optimal solution. Answer: x1 = 12, x2 = 0, Z = 36 Diff: 2 Page Ref: 191-199 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming problem computer solution AACSB: Analytical thinking

113) Solve the following integer linear program graphically. MAX Z = 5x1 + 8x2 s.t. x1 + x2 ≤ 6 5x1 + 9x2 ≤ 45 x1, x2 ≥ 0 and integer Answer: x1 = 0, x2 = 5, Z = 40 Note that feasible space lattice points have been shaded.

Diff: 3 Page Ref: 191-199 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: integer programming graphical solution AACSB: Analytical thinking 114) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint for the first restriction. Answer: S1 + S3 + S7 ≤ 2 Diff: 3 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking

115) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint(s) for the second restriction. Answer: S2 + S5 ≤ 1, S4 + S5 ≤ 1 Diff: 3 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking 116) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint for the third restriction. Answer: S1 + S2 + S3 +S4 + S5 + S6 + S7 ≥ 3 Diff: 1 Page Ref: 189 Section Heading: Integer Programming Models Keywords: integer programming formulation, constraint AACSB: Analytical thinking

DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3

The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. 117) Define the decision variables for this situation. Answer: y1 = 1 if DC1 is selected, 0 otherwise y2 = 1 if DC2 is selected, 0 otherwise y3 = 1 if DC3 is selected, 0 otherwise x1A = quantity shipped from DC 1 to Region A x1B = quantity shipped from DC 1 to Region B x1C = quantity shipped from DC 1 to Region C x1D = quantity shipped from DC 1 to Region D x2A = quantity shipped from DC 2 to Region A x2B = quantity shipped from DC 2 to Region B x2C = quantity shipped from DC 2 to Region C x2D = quantity shipped from DC 2 to Region D x3A = quantity shipped from DC 3 to Region A x3B = quantity shipped from DC 3 to Region B x3C = quantity shipped from DC 3 to Region C x3D = quantity shipped from DC 3 to Region D Diff: 2 Page Ref: 203-204 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: facility location example, mixed integer programming AACSB: Analytical thinking 118) Write the objective function for this problem. Answer: Min Z = 1x1A + 3x1B + 3x1C + 2x1D + 2x2A + 4x2B + 1x2C + 3x2D + 3x3A + 2x3B + 2x3C + 3x3D + 500y1 + 600y2 + 525y3 Diff: 2 Page Ref: 204 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: facility location example, mixed integer programming AACSB: Analytical thinking

119) Write the constraints for the three distribution centers. Answer: x1A + x1B +x1C - 500y1 ≤ 0 x2A + x2B +x2C - 500y2 ≤ 0 x3A + x3B +x3C - 500y3 ≤ 0 Diff: 2 Page Ref: 204 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: facility location example, mixed integer programming AACSB: Analytical thinking

120) Formulate this problem. Answer: Max Z = .7AaE + .56AlE + .4JoE + .7EE + .18JuE + .88KE + .90LE + .43McE + .86MrE + .83TE + .68AaD + .03AlD + .39JoD + .66ED + .32JuD + .11KD + .44LD + .99McD + .02MrD + .96 + .41AaS + .52AlS + .73JoS + .62ES + .16JuS + .37KS + .06LS + .20McS + .81MrS + .01TS subject to: AaE + AlE + JoE + EE + JuE + KE + LE + McE + MrE + TE ≥ 5 AaD + AlD + JoD + ED + JuD + KD + LD + McD + MrD + TD ≥ 6 AaS + AlS + JoS + ES + JuS + KS + LS + McS + MrS + TS ≥ 6 AaE + AaS + AaD = 2 AlE + AlS + AlD = 2 JoE + JoS + JoD = 2 EE + ES + ED = 2 JuE + JuS + JuD = 2 KE + KS + KD = 2 LE + LS + LD = 2 McE + McS + McD = 2 MrE + MrS + MrD = 2 TE + TS + TD = 2 all decision variables integer Diff: 3 Page Ref: 188 Section Heading: Integer Programming Models Keywords: mixed integer linear programming problems AACSB: Analytical thinking 121) What is the optimal solution to the potluck problem? Answer: The potluck reaches a value of 15.02 with the students bringing these items. Student Entrée Aaron 2 entrées Alex 2 sides Josh 2 sides Erin 2 entrées Julie 2 desserts Kassie 2 entrées Lindy 2 entrées MacGregor 2 desserts Marlene 2 sides Tracy 2 desserts Diff: 2 Page Ref: 195 Section Heading: Computer Solution of Integer Programming Problems with Excel and QM for Windows Keywords: mixed integer linear programming problems AACSB: Analytical thinking

122) What difference would it make to the solution if this scenario was modeled as a 0-1 problem instead of an integer model? Explain. Answer: Each student would be bringing two different categories of items and the optimal answer would drop dramatically. The student would bring one of their best items from the original integer formulation and would then bring their second-best category. The meal would have a value of 10.86 and the assignments would be: Entrée Side Dessert Aaron 1 1 0 Alex 1 0 1 Dan 1 1 0 Erin 0 1 1 Julie 1 0 1 Kassie 1 1 0 Lindy 1 1 0 MacGregor 1 0 1 Marlene 0 1 1 Tracy 1 0 1 Diff: 3 Page Ref: 203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking

123) Which student experiences the greatest loss of contribution to the objective function value if this scenario was modeled as a 0-1 problem instead of an integer model? Answer: Lindy drops from two entrées and a 1.8 contribution to one entrée and a side, which is a 0.96 contribution, a reduction of 0.84. Diff: 3 Page Ref: 203 Section Heading: 0-1 Integer Programming Modeling Examples Keywords: mixed integer linear programming problems AACSB: Analytical thinking