final exam operations

For the given objective function which of the following combinations of constraints would produce multiple optimal solutions? Max Z = 10x1 +6x2

5x1 + 3x2 <= 15 and x1 + x2 <= 10

For a "Dual minimization problem" with optimal point equal to (x1 = 5, x2 = 1) and best Z = 56, its respective "Primal maximization problem " would have __________. Please choose the option that would best fit the empty space above.

Its respective best Z also equal to 56.

For the the Clayton County Rescue Squad and Ambulance Service (Facility Location Nonlinear Model) example from Lecture 6, the number of estimated trips to Dunning has changed to 15. Which one of the options below, best represents the new optimal point (x,y) - best location to place the facility?

(19.24,16.30)

Consider the following distribution and random numbers: Demand Frequency 0 0.15 1 0.25 2 0.30 3 0.15 4 0.15 Random Numbers are represented by ri. Where 0 <= ri <= 1 i = 1, 2, 3, 4, ... The first 6 elements of a sequence of random numbers are: r1 = 0.82 ; r2 = 0.23; r3 = 0.95 ; r4 = 0.20 ; r5 = 0.14 ; r6 = 0.48 ; If a simulation begins with the fifth random number (r5), the first demand value would be ___________.

0

Administrators at a university are planning to offer a summer seminar. It costs $6000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $75 per student for the administrators to provide the course materials. If we know that 60 people will attend, what price should be charged per person to break even?

175

After reviewing the market prices, the investors Frijo-Lane Food Products realized that the prices of the six farms (that they were considering to purchase) have changed to: Farms Annual fixed costs ($1000) 1 100 2 450 3 150 4 250 5 350 6 300 Therefore, the model version was updated to: minimize Z = 18x1A +15x1B + 12x1C + 13x2A + 10x2B +17x2C + 16x3A + 14x3B + 18x3C + 19x4A + 15x4B + 16x4C + 17x5A + 19x5B + 12x5C + 14x6A + 16x6B + 12x6C + 100y1 + 450y2 + 150y3 + 250y4 + 350y5 + 300y6 subject to x1A + x1B + x1C - 11.2y1 <= 0 x2A + x2B + x2C - 10.5y2 <= 0 x3A + x3B + x3C - 12.8y3 <= 0 x4A + x4B + x4C - 9.3y4 <= 0 x5A + x5B +x5C - 10.8y5 <= 0 x6A + x6B + x6C - 9.6y6 <= 0 x1A + x2A + x3A + x4A + x5A + x6A = 12 x1B + x2B + x3B + x4B + x5B + x6B = 10 x1C + x2C + x3C + x4C + x5C + x6C = 14 xij >= 0 yi = 0 or 1 Please indicate which one of the following options IS correct. (hint: use the original excel file of this model and implement the changes, adjusting it in order to properly represent the current scenario)

Farm 2 (y2) should NOT be purchased, according to the model solution.

Choose the right format of definition for the decision variable of 'Transportation Problem' with 3 warehouses (whole sail stores or storages named as '1', '2', '3') and 4 stores (retail stores named as 'A', 'B', 'C', 'D').

Xij = the number that you transport from warehouse i to store j. (for i = 1,2,3 & j = A,B,C,D)

Answer the following statement with TRUE or FALSE: the correct way to implement a sensitivity analysis for the second coefficient of the objective function is to vary the second coefficient and the next one at the same time (eg.: coefficient #2 and #3).

shadow price

The proportionality is a property of linear programming problems that plays a very important role. This property ________ . Please choose the option that best fit the empty space above.

states that the rate of change along the line is constant.

When building a mini-boat, a boat builder uses 1 pound of wood and 2 ounces of glue. If he/she has 9 pounds of wood and 6 ounces of glue, how many mini-boat(s) will he/she be able to build?

three

The company produces 3 different products. If Xab = the production of product a in period b, then to indicate that the limit on production of the company's 3 products in period 1 is 350, we write:

none of the above- ANSWER X32 ≤ 350 X11 + X21 + X31 ≥ 350 X12 + X22 + X32 ≥ 350 X31 + X32 + X33 ≤ 350

For the following constraint: 10x1 + 8x2 =500y1, The decision variables x1 and x2 are continuous, while the decision variable y1 is a binary type (0-1). The scenario bellow is true only if the variable y1 _______ . 10x1 + 8x2 = 0 Please choose one of the following options that best fit the empty space above.

not selected

In a 0-1 integer programming model, if the constraint x2 + x3 <= 1. It means that if project 2 is not selected, project 3 ______________. Choose the option below that best represents the empty space above.

none of the above- ANSWER will not be selected will be selected will always be selected is more likely to be selected

Consider the following linear programming problem: Max Z = $12x1 + $5x2 Subject to: 8x1 + 5x2 ≤ 30 4x1 + 3x2 ≤ 12 x1, x2 ≥ 0 Considering that the values for x1 and x2 that will maximize revenue are respectively x1 = 3 and x2 = 0, what is the amount of slack/surplus associated with the first constraint ("8x1 + 5x2 ≤ 30") in the optimal point?

6

Consider the following linear programming problem: Min Z = 7x1 + 5x2 Subject to: 4x1 + 3x2 ≥ 200 (constraint 1) 2x1 + x2 ≥ 70 (constraint 2) x1, x2 ≥ 0 What is the lower limit of the right-hand side of the second constraint (q2)?

none of the above- ANSWER 35 47.5 100 45

Compared to blending and product mix problems, transportation problems are unique because ________ . Please choose the option that would best fit the empty space above.

None of the above- ANSWER they minimize the transportation cost. the constraints are all equality constraints with no "≤" or "≥" constraints. they contain many different variables. the solution values are always fractional values.

As a project manager, Mary is managing 12 different projects. She realized something very interesting related to them. Sometimes they present some unique and particular rules, like the ones for projects #: 3, 6, and 8. She concluded that if either project 3 or project 6 is selected, then project 8 must be selected. For integer programming problems, which one of the alternatives listed below correctly models this situation?

x6 ≤ x8 , x3 ≤ x8

The drying rate in an industrial process is dependent on many factors and varies according to the following distribution. Minutes Relative Frequency 13 0.18 14 0.36 15 0.28 16 0.10 17 0.08 If this following sequence represents a simulation of 5 random numbers trials, r (for 0 <= r <= 1), what is the average drying time: r1 = 0.17 ; r2 = 0.22 ; r3 = 0.29, r4 = 0.31 , and r5 = 0.42.

13.8

Consider the following linear programming problem: Min Z = 21x1 + 15x2 Subject to: 4x1 + 3x2 ≥ 200 (constraint 1) 2x1 + x2 ≥ 70 (constraint 2) x1, x2 ≥ 0 What is the lower limit of the right hand side of the first constraint (q1)?

140

Consider the following linear programming problem: Min Z = 14x1 + 10x2 Subject to: 4x1 + 3x2 ≥ 200 (constraint 1) 2x1 + x2 ≥ 70 (constraint 2) x1, x2 ≥ 0 What is the upper limit of the first coefficient of the objective function (c1)?

20

Consider the following linear programming problem: Max Z = 4x1 + 2x2 s.t. x1 ≤ 4 x2 ≤ 8 x1, x2 ≥ 0 The Z obtained from the best combination of x1 and x2 is:

32

Consider the following linear program, which maximizes profit for two products: regular (R) and super (S): MAX 5R + 75S s.t. 1.2 R + 1.6 S ≤ 600 assembly (hours) 0.8 R + 0.5 S ≤ 300 paint (hours) 0.16 R + 0.4 S ≤ 100 inspection (hours) R, S >= 0 See the sensitivity report provided below: Regarding the resource "assembling (hours)" _________ . Please choose the option that best fit the empty space above.

If the company has only 150 hours of this process they will no longer be willing to pay nothing (zero) for this resource.

Consider the following linear programming problem: Max Z = $30x1 + $10x2 s.t. 30x1 + 70x2 ≤ 210 20x1 + 10x2 ≤ 100 x1, x2 ≥ 0 What is maximum Z and the value of x1 and x2 at the optimal solution?

None of the above- ANSWER

If a firm's profit is modeled by the following function: Z = - 3x2 +12x + 25, Then the maximum profit is ________ . Please choose the option that properly fit the empty space above.

none of the above- ANSWER 40 52 63 84

In a 0-1 integer programming model, which of the alternatives listed below correctly models the following statement? "There is no chance for x3 to be constructed unless you construct x5."

x3 ≤ x5

Assume that x1, x2 and x3 are the dollars invested by investor A in three different common stocks from Chicago Stock Exchange. The investor A requires that at least 50% of the dollars invested should be in "stock 1". The constraint for this requirement can be written as:

0.5x1 - 0.5x2 - 0.5x3 ≥ 0

Answer the following statement with TRUE or FALSE:Sensitivity ranges cannot be computed for the coefficients of the objective functions.

False

Random numbers generated by a physical process instead of a mathematical process are pseudorandom numbers.

False

For 1000 trials of simulation, the simulation result will not always be equal to the analytical results.

True

If f(x) = x/3, what is the equation for generating x, given the random number r?

X =square root 6r

Which of the following combinations of constraints has no feasible region?

x1 + x2 >= 50 and x1 - x2 <= 100 x1 + x2 >= 10 and x1 >= 15 x1 >= 20 and x2 <= 10 x1 - x2 >= 5 and x1 - x2 <= 35

The Western Out fitters Store specializes in denim jeans. The variable cost of the jeans varies according to several factors, including the cost of the jeans from the distributor, labor costs, handling, packaging, and so on. Price also is a random variable that varies according to competitors' prices. Sales volume also varies each month. The probability distributions for volume, price, and variable costs each month are as follows: The fixed cost of the store is $11,000 per month. Simulate 20 months of store sales and compute the probability that the store will at least break even and the average profit (or loss). IMPORTANT - read the instructions below. For the random numbers, please use the excel file named as "seq100trials33" - it is available on Canvas. Use the first 20 random numbers of the: "sequence 14" for the "RN1", "sequence 15" for the "RN2", and "sequence 16" for the "RN3". Hint: use the Excel file of the Problem 3 from Lecture 7. What is the average profit (+) or loss (-) computed, regarding the 20 months of store sales simulation?

-3,370

If a firm's profit is modeled by the following function: Z = 16x - 4x2 + 24, Then the level of x that maximizes profit is __________ . Please choose the option that best fit the empty space above.

2

The break-even point is the volume of production that allows the company to have a greater revenue comparing to its total cost.

False

Regarding the "Investment Portfolio Selection problem" from lecture 6, in which Jessica Todd has identified four stocks that she wanted to include in her investment portfolio. Initially she was seeking to obtain a total annual return (total portfolio return) of at least 0.11, but she changed her mind and now she is looking for a total annual return of at least 0.05. Please identify the correct option below, considering this new scenario.

If she marginally increase her desired total portfolio return, the Z will not be affected.

The Pinewood Furniture Company produces chairs and tables from two resources: labor and wood. The company has 145 hours of labor and 100 board-ft. of wood available each day. Demand for chair is limited to 8 per day. Each chair requires 3 hours of labor and 4 board-ft. of wood, whereas a table requires 22 hours of labor and 9 board-ft. of wood. The profit derived from each chair is $100 and from each table is $500. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. The correct linear programming model formulation of this problem is:

Max Z = 100x1 + 500x2 Subject to: 3x1 + 22x2 ≤ 145 4x1 + 9x2 ≤ 100 x1 ≤ 8 x1, x2 ≥ 0

Data Envelope Analysis (DEA) compares a number of service units of the same type based on their inputs (resources) and outputs. The result indicates if a particular unit is less productive (efficient) than other units. Consider a town with four elementary schools—Alton (x1), Beeks (x2), Carey (x3), and Delancey (x4). The state has implemented a series of standards of learning (SOL) tests in reading, math, and history that all schools are required to administer to all students in the fifth grade. The average test scores are a measurable output of the school's performance. The school board has identified three key resources, or inputs, that affect a school's SOL scores—the teacher-to-student ratio, supplementary funds per student (i.e., funding generated by the PTA and other private sources over and above the normal budget), and the average educational level of the parents (where 12 = high school graduate, 16 = college graduate, etc.). Therefore, the elementary school comparison is based on the following type of variables: Input 1 = teacher to student ratio; Input 2 = supplementary funds/student; Input 3 = average educational level of parents; Output 1 = average reading SOL score; Output 2 = average math SOL score; and Output 3 = average history SOL score. The school board wants to identify the school or schools that are less efficient in converting their inputs to outputs relative to the other elementary schools in the town. The DEA LP model will compare one particular school with all the others. Which one of the following model formulations best describe the Beeks efficiency as compared with the other schools? Note that the efficiency of the school must be less than or equal to 1, value of the school's output / value of the school's input where xi = 1, 2, 3 while yi = a price per unit of each input where i = 1, 2, 3.

Maximize Z = 82x1 + 72x2 + 67x3 subject to: 0.05 y1 + 320y2 + 10.5y3 = 1 86x1 + 75x2 + 71x3 ≤ 0.06y1 + 260y2 + 11.3y3 82x1 + 72x2 + 67x3 ≤ 0.05y1 + 320y2 + 10.5y3 81x1 + 79x2 + 80x3 ≤ 0.08y1 + 340y2 + 12.0y3 81x1 + 73x2 + 69x3 ≤ 0.06y1 + 460y2 + 13.1y3 xi, yi ≥ 0

The sensitivity range of c1 (profit per x1, bowls) in the Beaver Creek Pottery example was 25 ≤ c1 ≤ 66.67. For whatever reason, it became less profitable to produce bowls, so c1 decreased from US$40 to US$32. Considering that c1 is now US$32, what is the correct statement regarding the current sensitivity range of c1?

The lower limit didn't change

Considering the following LP problem: Max Z = 5x1 + 12x2 s.t. x1 + x2 >= 25 3x1 + x2 >= 45 5x1 + 7x2 >= 20 7x1 + 13x2 <= 83 x1, x2 >= 0 Which one of the following options is correct?

The problem is infeasible

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint.

True

The Monte Carlo simulation is a technique that can be used to represent a long period of real-time by a short period of simulated time.

True

The three types of integer programming models are total, 0-1, and mixed.

True

A company X has 34 pounds of the resource Y available today. The shadow price of this resource Y is US$14 per pound. The Z obtained if the company implement the optimal solution is US$ 74. The upper limit of this resource Y is 58 pounds. How much would be the Z if the company gains access to 3 more pounds of the resource Y (totalizing 37 pounds of this resource Y available)?

US$116

If by processing a greater amount of input used in the past a company is now producing a(n) ________ amount of output, it means that an increase of productivity was not achieved. Please choose the option that best fits the empty space above.

exactly the same

In a 0-1 integer programming model, if the constraint x1 - x2 ≤ 0, it means if project 1 is selected, project 2 ________________. Choose the option below that best represents the empty space above.

must be selected

In a nonlinear programming model, the Lagrange multiplier reflects the appropriate change in the _________________ due to a marginal change in the right-hand side of a constraint. Please choose the option that best fit the empty space above.

the "Z" (from the objective function)

If one of the coefficients of the objective function is changed to a value inside of its respective sensitivity range (lower than the upper limit or greater than the lower limit) ___________ . Please choose the option that best fit the empty space above.

the original optimal solution will remain the same.

The Rutgers Small Bakery store has the resources to produce 5 different kinds of muffins. In order to satisfy their customers, they have decided to sell at least 3 different kinds of these muffins. Which one of the inequalities listed below best represents the above restriction/constraint?

x1 + x2 + x3 + x4 + x5 >= 3

________ is not part of a particular pattern of Monte Carlo method. Please choose the option that would better fit the empty space above.

Calculate optimal solutions for the given deterministic domain.

Consider the following linear program, which maximizes profit for two products: regular (R) and super (S): MAX 5R + 75S s.t. 1.2 R + 1.6 S ≤ 600 assembly (hours) 0.8 R + 0.5 S ≤ 300 paint (hours) 0.16 R + 0.4 S ≤ 100 inspection (hours) R, S >= 0 See the sensitivity report provided below: The "assembly (hours)" process is _________ . Please choose the option that best fit the empty space above.

as "valuable" as "Paint (hours)" process for this company.

Consider the following linear programming problem: Min Z = 7x1 + 5x2 Subject to: 4x1 + 3x2 ≥ 200 2x1 + x2 ≥ 70 x1, x2 ≥ 0 What is the upper limit of the second coefficient of the objective function (c2)?

5.25

Consider the following linear program, which maximizes profit for two products: regular (R) and super (S): MAX 5R + 75S s.t. 1.2 R + 1.6 S ≤ 600 assembly (hours) 0.8 R + 0.5 S ≤ 300 paint (hours) 0.16 R + 0.4 S ≤ 100 inspection (hours) R, S >= 0 See the sensitivity report provided below: The lower limit of the super product (S) objective function coefficient (profit of S) is _________ . Please choose the option that best fit the empty space above.

12.5

The time between arrivals of oil tankers at a loading clock at Prudhoe Bay is given by the following probability distribution: (hint: use the Excel file of the Problem 1 from Lecture 7) Time Between Ship Arrivals (days) Probability 1 0.05 2 0.1 3 0.2 4 0.3 5 0.2 6 0.1 7 0.05 The original time required to fill a tanker (service time) with oil and prepare it for sea is given by the following probability distribution: Service time (day) Probability 3 0.1 4 0.2 5 0.4 6 0.3 After implementing a training program aiming to improve the productivity of the employees, the probability distribution to fill a tanker changed to the following one: P(y) Cumulative Service Time, y 0.03 0.00 3 0.08 0.03 4 0.16 0.11 5 0.73 0.27 6 IMPORTANT - read the instructions below. For the random numbers, please use the excel file named as "seq100trials33" - it is available on Canvas. Please consider the same "starting condition" that was applied in the original problem resolved in class. Use the first 32 random numbers of the "sequence 16" for the "time between ship arrivals", and the whole "sequence 17" for the "time to fill and prepare". Which one of the following options below is NOT correct, regarding the scenario created after the implementation of the training program?

Average number of days spent in the "system" 15.20

The time between arrivals of oil tankers at a loading clock at Prudhoe Bay is given by the following probability distribution: (hint: use the Excel file of the Problem 1 from Lecture 7) Time Between Ship Arrivals (days) Probability 1 0.05 2 0.1 3 0.2 4 0.3 5 0.2 6 0.1 7 0.05 The original time required to fill a tanker (service time) with oil and prepare it for sea is given by the following probability distribution: Service time (day) Probability 3 0.1 4 0.2 5 0.4 6 0.3 After implementing a training program aiming to improve the productivity of the employees, the probability distribution to fill a tanker changed to the following one: P(y) Cumulative Service Time, y 0.03 0.00 3 0.08 0.03 4 0.16 0.11 5 0.73 0.27 6 IMPORTANT - read the instructions below. For the random numbers, please use the excel file named as "seq100trials33" - it is available on Canvas. Please consider the same "starting condition" that was applied in the original problem resolved in class. Use the first 32 random numbers of the "sequence 16" for the "time between ship arrivals", and the whole "sequence 17" for the "time to fill and prepare". Which one of the following options below is correct?

The program was not successful, since the number of tankers waiting to be loaded would diverge.

Regarding the "Investment Portfolio Selection problem" from lecture 6, in which Jessica Todd has identified four stocks that she wanted to include in her investment portfolio. Initially she was seeking to obtain a total annual return (total portfolio return) of at least 0.11, but she changed her mind and now she is looking for a total annual return of at least 0.05. Considering this new scenario, her total portfolio return will be _______ Please choose one of the options below that best fit the empty space above.

none of the above- ANSWER 0.6 0.65 0.7 0.75

Consider the following linear programming problem, which maximizes profit for two products: regular (R) and super (S): MAX 5R + 75S s.t. 1.2 R + 1.6 S ≤ 600 assembly (hours) 0.8 R + 0.5 S ≤ 300 paint (hours) 0.16 R + 0.4 S ≤ 100 inspection (hours) R, S >= 0 See the sensitivity report provided below: The upper limit of the regular product (R) objective function coefficient (profit of R) is _________ . Please choose the option that best fit the empty space above.

none of the above- ANSWER 120 20 40 70

Please choose the correct option below regarding the set covering example described in lecture 5.

none of the above- ANSWER The optimal strategy is to build a hub in each one of the 12 cities. Boston will always be selected as a city to construct the hub. Nashville will always be selected as a city to construct the hub. The city with the "highest number of cities nearby (within the 300 miles restriction)" will certainly be chosen.

By changing the layout of its manufacturing plant, a company was able to produce more products per day, by using the same amount of resources. This is a case of generating higher __________. Please choose the option that best fits the empty space above.

productivity

The Western Out fitters Store specializes in denim jeans. The variable cost of the jeans varies according to several factors, including the cost of the jeans from the distributor, labor costs, handling, packaging, and so on. Price also is a random variable that varies according to competitors' prices. Sales volume also varies each month. The probability distributions for volume, price, and variable costs each month are as follows: The fixed cost of the store is $11,000 per month. Simulate 20 months of store sales and compute the probability that the store will at least break even and the average profit (or loss). IMPORTANT - read the instructions below. For the random numbers, please use the excel file named as "seq100trials33" - it is available on Canvas. Use the first 20 random numbers of the: "sequence 14" for the "RN1", "sequence 15" for the "RN2", and "sequence 16" for the "RN3". Hint: use the Excel file of the Problem 3 from Lecture 7. What is the probability that the store will at least break even, regarding the 20 months of store sales simulation? Please write it in %. If it is "56%", you write: 56

10

Rutgers Mills produces working cloth that it sells to jeans manufacturers. It is negotiating a contract with RK company to provide working cloth on a weekly basis. Rutgers Mills has established its monthly available production for this contract to be between 0 and 500 yards, according to the following probability distribution f(x) = x / 125000, 0 <= x <= 500 yd RK Company's weekly demand for working cloth varies according to the following probability distribution. Probability of Demand: P(x) Cumulative Demand (yds) 0.05 0 0 0.10 0.05 100 0.20 0.15 200 0.30 0.35 250 0.20 0.65 350 0.15 0.85 450 1.00 For the random numbers, please use the excel file named as "seq100trials33" - it is available on Canvas. Please consider the same "starting condition" that was applied in the original problem resolved in class. Use the first 20 random numbers of the: "sequence 2" for the "Capacity" and "sequence 3" for the "Demand". Simulate RK company's orders for 20 weeks and determine the average weekly capacity and demand. What is the probability that Rutgers Mills have sufficient capacity to meet demand?

70%

The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount: Exposure (people/ad or commercial) Cost (US$) TV commercial 40,000 16,000 Radio commercial 28,000 14,000 Newspaper ad 29,000 8,000 The company must consider the following resource constraints: The budget limit for advertising is $200,000. The television station has time available for 6 commercials. The radio station has time available for 12 commercials. The newspaper has space available for 5 ads. The advertising agency has time and staff available for producing no more than a total of 20 commercials and/or ads. If this problem is formulated as an integer and also a linear programming problem (separately), the maximization ("Z", the optimal solution) of the exposure would be, respectively:

Integer Programming: 497,000 ; Linear Programming: 513,000 ;

The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount: Exposure (people/ad or commercial) Cost (US$) TV commercial 40,000 16,000 Radio commercial 28,000 14,000 Newspaper ad 29,000 8,000 The company must consider the following resource constraints: The budget limit for advertising is $200,000. The television station has time available for 6 commercials. The radio station has time available for 12 commercials. The newspaper has space available for 5 ads. The advertising agency has time and staff available for producing no more than a total of 20 commercials and/or ads. If this problem is formulated as an integer and also a linear programming problem (separately), the combination of commercial and advertisements that would maximize the exposure would be, respectively:

Integer Programming: x1 = 6; x2 = 4; x3 = 5; Linear Programming: x1 = 6; x2 = 4.57 ; x3 = 5;