Operations Management final

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? Correct!

(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

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

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

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.

120 20 40 70 Correct! None of the above

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

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

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

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 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)?

35 47.5 100 45 Correct! None of the above

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.

40 52 63 84 Correct! None of the above

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

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

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

The graph below represents a LP problem in which the objective function is "minimization" and parallel to the constraint 1. Which one of the following answers would best indicate how the best Z can be obtained in the problem below?

All the points from the segment of line "AB"

________ 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.

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).

False

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

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

False

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 Please indicate which one of the following options IS correct.

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

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 be affected by 1.6.

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.

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 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;

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.

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

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 graph below represents a LP problem in which the objective function is "Maximization" and parallel to the constraint 1. Which one of the following answers would best indicate how the best Z can be obtained in the problem below?

Point A Point B All the points in the segment of line: AB All the points in the segment of line: BC Correct! None of the above

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

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

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. Correct! None of the above.

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

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

True

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

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

X = sqrt(6r)

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:

X32 ≤ 350 X11 + X21 + X31 ≥ 350 X12 + X22 + X32 ≥ 350 X31 + X32 + X33 ≤ 350 Correct! None of the above

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)

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?

Z = 150; x1 = 50; and x2 = 0; Z = 300; x1 = 0; and x2 = 30; Z = 15; x1 = 5; and x2 = 0.5; Z = 30; x1 = 100; and x2 = 0; Correct! None of the above

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.

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

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

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 sensitivity range for the right-hand side of the constraint is the range of values over which the current __________________ of its respective "resource" remains the same. Please choose the option that would best fit the empty space above:

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

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.

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

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. Correct! None of the above

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

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.

will not be selected will be selected will always be selected is more likely to be selected Correct! None of the above

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

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 Correct! All of the above have a feasible region.

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

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