OPS Final

Parts of the Monte Carlo Simulation

-Analyzing results -Analyzing a real problem -Evaluating the Results NOT: -finding an optimal solution

1) Validation of a simulation model occurs when the true ___________ average results have been reached.

steady state

Which of the following combinations of constraints has no feasible region? x1 + x2>= 15 and x1 - x2 <=10 x1 + x2 >=5 and x1 >=10 x1 + x2 >=100 and x1 + x2 >=50 x1 + x2 >=10 and x1 >=5

ALL have a feasible region (No feasible region is when solutions go in opposite direction AND do not meet)

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?

x1 + x5 ≤ 1, x2 + x5 ≤ 1

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

False ( The three types are: 1)total integer 2)binary 3)mixed

You cannot vary both objective coefficients at once (c1 and c2) in a sensitivity analysis. TrueFalse

True

Consider the following linear programming problem: Max Z = 4xv1 + 2xv2 Subject to: x1 <=4 x2<=2 x1, x2 ≥ 0 The Z obtained from the best combination of x2 and x1 is:

20 (plug in max possible numbers for x1 and x2 which is 4 and 2)

For a "Dual maximization problem" with Z equal to 2,570, its respective "Primal minimization problem " would also have a Z of 2,570.

True (optimal solution of primal is also show price - both have the same set up) (a minimization problem can have a primal and max will be the dual and vice versa - when you transform this, you will see the relationship between p and q)

For a minimization integer linear programming problem, a feasible solution is ensured by rounding ________ non-integer solution values if all of the constraints are the greater-than-or-equal-to type.

up (for maximization problems round DOWN for feasible solution but this does not mean it will always be the optimal solution)

The Pinewood Furniture Company produces chairs and tables from two resources: labor and wood. The company has 120 hours of labor and 100 board-ft. of wood available each day. Demand for table is limited to 8 per day. Each chair requires 3 hours of labor and 4 board-ft. of wood, whereas a table requires 20 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 + 20x2 ≤ 120 4x1 + 9x2 ≤ 100 x2 ≤ 8 x1, x2 ≥ 0

unbounded

value of the obj. function increases indefinitely (it is unreal) ex. Max z=4x1+2x2 St. x1>=4 x2<=2 x1,x2>=0 **ONLY APPLIES TO MAX PROBLEMS** -USUALLY OCCURS BECAUSE OF ERROR SUCH AN UNLIMITED SUPPLY

1) If f(x) = x/8, what is the equation for generating x, given the random number r?

x=4root r

If f(x) = 2x, what is the equation for generating x, given the random number r?

x=root r

In formulating a mixed integer programming problem, the constraint 3x1 + 2x2 ≤ 100y1 where y1 is a 0-1 variable and x1 and x2 are continuous variables, then 3x1 + 2x2 = 0 if y1 is:

y1=0

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

All the points in the segment of line AB (these are the points parallel to OBJ function)

Simulation results will always equal analytical results if 1000 trials of the simulation have been conducted.

False (as the values in simulation results are only for certain parameter whereas analytical results gives a general system description for any number of parameters.)

The "certainty" linear programming (LP) hypothesis (LP are deterministic models) is not violated by integer programming.

True (Integer programming certainty hypothesis is just an extension of linear programming hypothesis regarding certainty as the parameter of objective function as well as their coefficients and coefficient of the constraints inequalities are basically known as certainty. Integer programming can be used for tracking on fractional values in which technique can be specifically used to make assumptions of divisibility that does not hold.)

If Xba = 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 250, we write:

X11 + X12 + X13 ≤ 250 (If this was Xab it would be x11 + x22 + x33 <=250)

In a 0-1 integer programming model, if the constraint -x1+x2<=0, it means when project x1 is selected, project x2________ be selected.

can sometimes

What does it mean if an LP problem is unbounded?

IF the feasible region of a system can be enclosed within the circle. An unbounded means that the feasible region does extend indefinitely in any direction

a scientific approach to solve management problems

Management Science ~ Operations Research ~ Quantitative Methods ~ Quantitative Analysis ~ Decision Sciences ~ Business Analytics

Random numbers are equally likely to occur.

True (Random numbers refers to numbers drawn from a pool of numbers with equal chance of selection. For example consider the series of natural numbers from 0 to 100. In this the probability of selection of one number will always remain equal to 1/100 irrespective of what one selects. Hence, one can say that random numbers are equally likely to occur because they have same probability of selection.

A table of random numbers must be:

efficiently generated

Conditional Constraint

x2<=x1 the construction of one facility is conditional upon the construction of another However, even if the pool is selected, there is no guarantee that the tennis center will also be selected

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:

1 (it can only be either 0 or 1 since its binary and when you plug in 0 it violates x1+x2=500)

Administrators at a university are planning to offer a summer seminar. It costs $4000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $45 per student for the administrators to provide the course materials. If we know that 100 people will attend, what price should be charged per person to break even?

85

The break-even point is the volume that the profit is positive (greater than zero).

False (BE point = volume where profit = 0 OR TOTAL REVENUE = TOTAL COST)

When building a mini-boat, a boat builder uses 2 pounds of wood and 3 ounces of glue. If he/she has 3 pounds of wood and 6 ounces of glue, how many mini-boat(s) will he/she be able to build?

None of the above (answer is 1 but it is not an option)

In what step do you identify a problem that exists in the system or organization?

Observation

Consider the following linear programming problem: Max Z = $200x1 + $100x2 Subject to: 8x1 + 5x2 ≤ 80 2x1 + x2 ≤ 100 x1, x2 ≥ 0 What is maximum Z and the value of x1 and x2 at the optimal solution?

Z=2000; x1=10 and x2=0

The constraint (x1 + x2 + x3 + x4 + x5 >= 3) means that ________ out of the ________ projects must be selected.

at least 3, 5

Corequisite constraint

if one facility is constructed, the other one will also be constructed and vice versa. x2=x1 Suppose the council worked out a political deal among themselves, wherein if the pool is accepted, the tennis center must also be selected and vice versa.

Assume that x1, x2 and x3 are the dollars invested in three different common stocks from New York Stock Exchange. The investing company requires that no more than 60% of the dollars invested should be in "stock 1". The constraint for this requirement can be written as:

0.4x1 - 0.6x2 - 0.6x3 ≤ 0

___________________ refers to the ability of an organization to sell products in a market. Please choose the option that best fits the empty space above.

Business Competitiveness

The constraint (x1 + x2 + x3 + x4 + x5 <= 4) means that ________ out of the ________ projects must be selected.

at most 4 out of the 5 projects can be selected

In a 0-1 integer programming model, if the constraint x1-x2<=0, it means when x1 is selected, x2 ________ be selected.

can sometimes (if it was =0 then it would be must)

Infeasible

every possible solution violates at least one constraint ex. Max Z = 5x1 + 3x2 st. 4x1 + 2x2 <=8 x1>=4 x2>=6 x1,x2 >=0

If by processing the same amount of inputs used in the past a company is now capable to produce a(n) ________ amount of outputs, it means that an improvement of productivity was achieved.

greater

multiple optimal solution

the obj. function is parallel to a constraint line ex. Max Z = 40x1 +30x2 s.t. 1x1 + 2x2 <=40 4x1 + 3x2 <=120 x1, x2 >=0 BC is also called "alternate optimal solutions"

A long period of real time cannot be represented by a short period of simulated time.

False (A longer period of time can be run as a simulation for the smallest time frame. This happens by executing the simulation tests within a shorter period of time quickly.)

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

None of the above (It is because the given graph is incorrect. The graph for a maximization problem is always towards origin. This is a minimization graph. Hence, none of the given option is correct. Had, the graph been inward faced, the correct answer would have been Point B.)

Australian road freight company Linfox uses aerodynamic trucks and trailers to reduce fuel consumption. This is a case of generating higher

productivity

Consider the following linear programming problem: Max Z = $15x + $20y Subject to: 8x + 5y ≤ 40 0.4x + y ≥ 4 x, y ≥ 0 Considering that the values for x and y that will maximize revenue are respectively x = 0 and y = 8, what is the amount of slack/surplus associated with the first constraint - in the optimal point?

0 (for this problem plug in x and y values and see what you get, in this case you get 0 surplus/slack bc the answer is <=40

Steps:

-Observation: identification of a problem that exists in the system or organization; -Definition of the Problem: problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organization; -Model Construction: development of the functional mathematical relationship that describes the decision variables, objective function and constraints of the problem; -Model Solution: Models solved using Management Sciences techniques; -Model Implementation: Actual use of the model or its solution;

The constraint x1 + x2 ≤ 1 is named as "conditional constraint" in 0-1 integer programming problems.

False (It's a mutually exclusive constraint case .Mathematically, this can be accomplished by only through restricting the sum of the 0-1 variables in such a way that the set to be also less than or equal to 1..) A conditional constraint would be: x1-x2 >=0 or x2-x1<=2 because x2 is conditional of x1.

Slack variables are only associated with maximization problems.

False (The statements 'slack variables are only associated with maximization problems' and 'surplus variables are only associated with minimization problems' are FALSE because: Slack variable is added to an inequality constraint in order to transform it to an equality. This variable replace an inequality constraint with non-negativity constraint. Surplus variable depicts that the solution value has exceeded the resource.They carry zero coefficient

The advertising budget is $3500, but there is no requirement that all the money be spent. The newspaper has only four issues before the end of the semester, but the radio is a 24/7 operation and has two dozen 30 second slots available. Facebook postings must be alternated with the rest of the mindless drivel posted on the college page; thus there is space for only three postings before the end of the semester. Twitter is complicated by the 140 character requirement. The communications director feels she needs five tweets to convey a single message about tours and semesters abroad, so for one message, the cost would be $25 for each of the five components of the single ad. Due to thumb fatigue, she feels that she has only 2800 characters left in her thumbs before the end of the semester. (A side note - During the intersession period, she plans to embark on a strict regimen of thumb yoga to prepare for the coming semester.) What is an appropriate objective function for this scenario?

Max Z = 5,000N + 3,000R + 700T + 200F (this problem focuses on maximizing exposure, not budget so focus on what the problem is focusing on maximizing or minimizing)

For a "Primal minimization problem" with Z equal to 2,150, its respective "Dual maximization problem " would also have a Z of 2,150.

True

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

True A true random numbers are generated by a physical process like a dice throw, counting the particles emitted by the decay of a radioactive element.Pseudo-random numbers are generated by software functions. They are referred to as "pseudo-random" because the sequence of numbers is deterministic. Given a particular function and a "seed" value, the same sequence of numbers will be generated by the function. If the pseudo-random number generation function is well designed, the sequence of numbers will appear to be statistically random. However, there is no question that these numbers were generated by a deterministic process (e.g., the pseudo-random number function).