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**MAT 540**

**Quantitative Methods**

** **

**WEEK 1**

Quiz 1 (2 Sets)

**WEEK 2**

Discussion 1 and 2

**WEEK 5**

Quiz 3 (2 Sets)

**WEEK 6**

Discussion 1 and 2

Quiz 4 (2 Sets)

Midterm Exam

**WEEK 7**

Assignment 3, Case Problem, Julia’s Food Booth

Discussion 1 and 2

**WEEK 8**

Discussion 1 and 2

**WEEK 9**

Discussion 1 and 2

Quiz 5 (2 Sets)

**Assignments:**

**Week 3 Assignment 1****:**** ****JET Copies Case Problem**

Read the JET Copies Case Problem on pages 678-679 of the text. Using simulation estimate the loss of revenue due to copier breakdown for one year, as follows:

- In Excel, use a suitable method for generating the number of days needed to repair the copier, when it is out of service, according to the discrete distribution shown.
- In Excel, use a suitable method for simulating the interval between successive breakdowns, according to the continuous distribution shown.
- In Excel, use a suitable method for simulating the lost revenue for each day the copier is out of service.
- Put all of this together to simulate the lost revenue due to copier breakdowns over 1 year to answer the question asked in the case study.
- In a word processing program, write a brief description/explanation of how you implemented each component of the model. Write 1-2 paragraphs for each component of the model (days-to-repair; interval between breakdowns; lost revenue; putting it together).
- Answer the question posed in the case study. How confident are you that this answer is a good one? What are the limits of the study? Write at least one paragraph.

**Week 4 Assignment 2: Internet Field Trip – Forecasting Methods**

1. Research: Research at least six (6) information sources on forecasting methods; take notes and record and interpret significant facts, meaningful graphics, accurate sounds and evaluated alternative points of view.

2. Preparation: Produce as storyboard with thumbnails of at least ten (10) slides. Include the following elements:

Title of slide, text, background color, placement & size of graphic, fonts – color, size, type for text and headings

Hyperlinks (list URLs of any site linked from the slide), narration text, and audio files (if any)

o Number on slides clear

o Logical sequence to the presentation

3. Content: Provide written content with the following elements:

o introduction that presents the overall topic (clear sense of the projects main idea) and draws the audience into the presentation with compelling questions or by relating to the audience’s interests or goals.

o accurate, current

o clear, concise, and shows logical progression of ideas and supporting information

o motivating questions and advanced organizers

o drawn mainly from primary sources

4. Text Elements: Slides should have the following characteristics:

o fonts are easy-to-read; point size that varies appropriately for headings and text

o italics, bold, and indentations enhance readability

o background and colors enhance the readability of text

o appropriate in length for the target audience; to the point

5. Layout: The layout should have the following characteristics:

o visually pleasing

o contributes to the overall message

o appropriate use of headings, subheadings and white space

6. Media: The graphics, sound, and/or animation should

o assist in presenting an overall theme and enhance understanding of concept, ideas and relationships

o have original images that are created using proper size and resolution; enhance the content

o have a consistent visual theme.

7. Citations: The sources of information should:

o properly cited so that the audience can determine the credibility and authority of the information presented

o be properly formatted according to APA style

**Week 7 Assignment 3 Case Problem – Julias Food Booth**

Julia is a senior at Tech, and shes investigating different ways to finance her final year at school. She is considering leasing a food booth outside the Tech stadium at home football games. Tech sells out every home game, and Julia knows, from attending the games herself, that everyone eats a lot of food. She has to pay $1,000 per game for a booth, and the booths are not very large. Vendors can sell either food or drinks on Tech property, but not both. Only the Tech athletic department concession stands can sell both inside the stadium. She thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items she would sell.

Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Julia to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For $600 she can lease a warming oven for the six-game home season. The oven has 16 shelves, and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.

Julia has negotiated with a local pizza delivery company to deliver 14-inch cheese pizzas twice each game 2 hours before the game and right after the opening kickoff. Each pizza will cost her $4.50 and will include 6 slices. She estimates it will cost her $0.50 for each hot dog and $1.00 for each barbecue sandwich if she makes the barbecue herself the night before. She measured a hot dog and found it takes up about 16 in2 of space, whereas a barbecue sandwich takes up about 25 in2. She plans to sell a piece of pizza for $1.50 and a hot dog for $1.60 each and a barbecue sandwich for $2.25. She has $1,500 in cash available to purchase and prepare the food items for the first home game; for the remaining five games she will purchase her ingredients with money she has made from the previous game.

Julia has talked to some students and vendors who have sold food at previous football games at Tech as well as at other universities. From this she has discovered that she can expect to sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined. She also anticipates that she will probably sell at least twice as many hot dogs as barbecue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand.

If Julia clears at least $1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth.

A. Formulate a linear programming model for Julia that will help you to advise her if she should lease the booth. Formulate the model for the first home game. Explain how you derived the profit function and constraints and show any calculations that allow you to arrive at those equations.

B. Solve the linear programming model using a computer for Julia that will help you advise her if she should lease the booth. In this solution, determine the number of pizza slices, hot dogs and barbecue sandwiches she should sell at each game. Also determine the revenues, cost and profit; and do an analysis of how much money she actually will make each game given the expenses of each game.

Do an analysis of the profit solution and what impact it has on Julias ability to have sufficient funds for the next home game to purchase and prepare the food. What would you recommend to Julia?

C. If Julia were to borrow some money from a friend before the first game to purchase more ingredients, she feels she can increase her profits. What amount, if any, would you recommend to Julia to borrow?

D. Food prices have been rising lately. Assume purchase costs for the food is now $6.00 for each pizza, $0.75 for each hot dog, and $1.25 for each barbecue sandwich. Repeat the analysis of Part B. What would you recommend to Julia to do at this point?

E. Julia seems to be basing her analysis on the assumptions that everything will go as she plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Julias analysis? Given these uncertainties and the results in (B), (C), and (D), what do you recommend that Julia do? Take into consideration her profit margin for each game.

**Week 10 Assignment 4 Case Problem – Stateline Shipping and Transport Company**

Read the Stateline Shipping and Transport Company Case Problem on pages 273-274 of the text. Analyze this case, as follows:

1. In Excel, or other suitable program, develop a model for shipping the waste directly from the 6 plants to the 3 waste disposal sites.

2. Solve the model you developed in #1 (above) and clearly describe the results.

3. In Excel, or other suitable program, Develop a transshipment model in which each of the plants and disposal sites can be used as intermediate points.

4. Solve the model you developed in #3 (above) and clearly describe the results.

5. Interpret the results and draw conclusions that address the question posed in the case problem. What are the limits of the study? Write at least one paragraph.

*There are two deliverables for this Case Problem, 1)the Excel spreadsheet, with the different solutions given in separate worksheets, and 2) an accompanying written description/explanation (submitted as a Word document). Please submit both of them electronically via the drop box.*

**Discussion Questions:**

**Week 2:**

- What are some benefits of using decision trees? In what ways can decision trees be used for business decisions? Name some real-world examples.

- How does the science of probability affect decisions? Why?

**Week 3:**

- Why do we use pseudorandom numbers in simulations? How do pseudorandom numbers affect the accuracy of a simulation?

- What is the role of statistical analysis in simulation?

**Week 4:**

- Choose one of the forecasting methods and explain the rationale behind using it in real-life. Describe how a domestic fast food chain with plans for expanding into China would be able to use a forecasting model.

- What is the difference between a causal model and a time- series model? Give an example of when each would be used. What are some of the problems and drawbacks of the moving average forecasting model? How do you determine how many observations to average in a moving average model? How do you determine the weightings to use in a weighted moving average model?

**Week 6:**

- What are some business uses of a linear programming model? Provide an example.

- In the graphical method, how do you know when a problem is infeasible, unbounded, or when it has multiple optimal solutions? What are the essential ingredients of an LP model? Why is it helpful to understand the characteristics of LP models?

- Not very many real world examples use only two variables, and those that do can usually be solved much more easily by guess and check methods rather than LP models. Why then do we study the graphical method? Be specific.

- Distinguish between a minimization and maximization LP model. How do you know which of these to use for any given problem?

- Give examples of both a minimization LP model and a maximization LP model. Every minimization model has a related maximization model. In what way do you think they are related?

**Week 7:**

- What does the shadow price reflect in a maximization problem? Please explain. How do the graphical and computer-based methods of solving LP problems differ? In what ways are they the same? Under what circumstances would you prefer to use the graphical approach?

- How does sensitivity analysis affect the decision making process? How could it be used by managers?

- In many ways, shadow prices are far more important results of an LP model then the optimal solution. Explain. Make sure that your answer provides context to the nature and utility of shadow prices.

**Week 8:**

- What is the relationship between decision variables and the objective function? What is the difference between an objective function and a constraint?

- Does the linear programming approach apply the same way in different applications? Explain why or why not using examples.

- Many LP models can be viewed through the lens of a Transpotation model. Choose one of the following types of problem and explain how to see it as a Transportation problem. That is, explain what the “goods” we are shipping are, what the “roads” we are shipping along are, what the costs of “shipping” are, what the supply” and “demand” are:

**Week 9:**

- Explain how the applications of Integer programming differ from those of linear programming. Why is rounding-down an LP solution a suboptimal way to solve Integer programming problems?

- Explain the characteristics of integer programming problems. Give specific instances in which you would use an integer programming model rather than an LP model. Provide real-world examples.

- Do you think Integer Programming or Linear Programming has more real world applications? Why? If Integer Programming is more prevalent, why do we focus so much on Linear Programming in this course? If Linear Programming is more prevalent, what do you think the challenges are facing Linear Programmers (since software like QM can handle all the computations)?

- Explain the characteristics of integer programming problems. Give specific instances in which you would use an integer programming model rather than an LP model. Provide real-world examples.

**Quizzes:**

**Week 1 Quiz 1:**

- Which of the following is incorrect with respect to the use of models in decision making?
- The probabilities of mutually exclusive events sum to zero.
- The term decision analysis refers to testing how a problem solution reacts to changes in one or more of the model parameters.
- Variable costs are independent of volume and remain constant.
- A frequency distribution is an organization of numerical data about the events in an experiment.
- Objective probabilities that can be stated prior to the occurrence of an event are
- A set of events is collectively exhaustive when it includes _______ the events that can occur in an experiment.
- Which of the following is not an alternative name for management science?
- Which of the following is an equation or an inequality that expresses a resource restriction in a mathematical model?
- ____________ techniques include uncertainty and assume that there can be more than one model solution.
- A ___________ probability is the probability that two or more events that are not mutually exclusive can occur simultaneously.
- Total cost equal the fixed cost plus the variable cost per unit divided by volume.
- Profit is the difference between total revenue and total cost.
- The events in an experiment are mutually exclusive if only one can occur at a time.
- In a given experiment, the probabilities of all mutually exclusive events sum to one.
- An experiment is an activity that results in one of several possible outcomes.
- Fixed costs depend on the number of items produced.
- The steps of the scientific method are:
- A model is a functional relationship and include:
- In a given experiment the probabilities of mutually exclusive events sum to

**Week 3 Quiz 2:**

- The coefficient of optimism is a measure of the decision maker’s optimism.
- A payoff table is a means of organizing a decision situation, including the payoffs from different decisions given the various states of nature.
- The maximin criterion results in the maximum of the minimum payoffs.
- A state of nature is an actual event that may occur in the future.
- The ______________ minimizes the maximum regret.
- The maximin criterion results in the
- A dominant decision is one that has better payoff than another decision under each state of nature.
- The maximin approach to decision making refers to
- The maximax criterion results in the maximum of the minimum payoffs.
- Determining the worst payoff for each alternative and choosing the alternative with the best worst is called
- The Hurwicz criterion is a compromise
- The maximax criterion results in the
- Regret is the difference between the payoff from the best decision and all other decision payoffs.
- Regret is the difference between the payoff from the
- The minimax regret criterion
- The Hurwicz criterion is a compromise between the maximax and maximin criteria.
- The term opportunity loss is most closely related to
- The equal likelihood criterion multiplies the decision payoff for each state of nature by an equal weight.
- The basic decision environment categories are
- Expected opportunity loss is the expected value of the regret for each decision.

**Week 5 Quiz 3:**

- The terms in the objective function or constraints are not additive.
- A feasible solution violates at least one of the constraints.
- Which of the following could not be a linear programming problem constraint?
- In a linear programming problem, a valid objective function can be represented as
- Non-negativity constraints restrict the decision variables to negative values.
- A graphical solution is limited to linear programming problems with
- A constraint is a linear relationship representing a restriction on decision making.
- Linear programming is a model consisting of linear relationships representing a firm’s decisions given an objective and resource constraints.
- Non-negativity constraints
- Decision models are mathematical symbols representing levels of activity.
- Which of the following could be a linear programming objective function?
- Decision variables
- A linear programming model consists of decision variables, constraints, but no objective function.
- The region which satisfies all of the constraints in a graphical linear programming problem is called the
- The _______________ property of linear programming models indicates that the decision variables cannot be restricted to integer values and can take on any fractional value.
- The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of the limited resources are production time (8 hours = 480 minutes per day) and syrup limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
- Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the objective function?
- In a linear programming model, the numberof constraints must be less than the number of decision variables.
- Proportionality means the slope of a constraint or objective function line is not constant.
- The objective function is a linear relationship reflecting the objective of an operation.

**Week 6 Quiz 4:**

- The standard form for the computer solution of a linear programming problem requires all variables to the right and all numerical values to the left of the inequality or equality sign
- _________ is maximized in the objective function by subtracting cost from revenue.
- A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit?
- In an unbalanced transportation model, supply does not equal demand and supply constraints have signs.
- The production manager for Liquor etc. produces 2 kinds of beer: light and dark. Two of his resources are constrained: malt, of which he can get at most 4800 oz per week; and wheat, of which he can get at most 3200 oz per week. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the objective function?
- In a media selection problem, the estimated number of customers reached by a given media would generally be specified in the _________________. Even if these media exposure estimates are correct, using media exposure as a surrogate does not lead to maximization of ______________.
- The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50. Which of the following is not a feasible production combination?
- When formulating a linear programming model on a spreadsheet, the measure of performance is located in the target cell.
- In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.
- In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure.
- ____________ solutions are ones that satisfy all the constraints simultaneously.
- The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8 hours = 480 minutes per day) and syrup (1 of her ingredient) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?
- Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem.
- When using linear programming model to solve the “diet” problem, the objective is generally to maximize profit.
- Profit is maximized in the objective function by
- Linear programming model of a media selection problem is used to determine the relative value of each advertising media.
- Media selection is an important decision that advertisers have to make. In most media selection decisions, the objective of the decision maker is to minimize cost.
- The dietician for the local hospital is trying to control the calorie intake of the heart surgery patients. Tonight’s dinner menu could consist of the following food items: chicken, lasagna, pudding, salad, mashed potatoes and jello. The calories per serving for each of these items are as follows: chicken (600), lasagna (700), pudding (300), salad (200), mashed potatoes with gravy (400) and jello (200). If the maximum calorie intake has to be limited to 1200 calories. What is the dinner menu that would result in the highest calorie in take without going over the total calorie limit of 1200.
- In a multi-period scheduling problem the production constraint usually takes the form of :
- A constraint for a linear programming problem can never have a zero as its right-hand-side value.

**Week 9 Quiz 5:**

- In a _______ integer model, some solution values for decision variables are integer and others can be non-integer.
- In a total integer model, some solution values for decision variables are integer and others can be non-integer.
- In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects.
- If a maximization linear programming problem consist 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.
- The branch and bound method of solving linear integer programming problems is an enumeration method.
- In a mixed integer model, all decision variables have integer solution values.
- For a maximization integer linear programming problem, feasible solution is ensured by rounding _______ non-integer solution values if all of the constraints are less-than -or equal- to type.
- In a total integer model, all decision variables have integer solution values.
- The 3 types of integer programming models are total, 0 – 1, and mixed.
- The branch and bound method of solving linear integer programming problems is ________________.
- The linear programming relaxation contains the _______ and the original constraints of the integer programming problem, but drops all integer restrictions.
- The branch and bound method can only be used for maximization integer programming problems.
- The solution value (Z) to the linear programming relaxation of a minimization problem will always be less than or equal to the optimal solution value (Z) of the integer programming minimization problem
- The implicit enumeration method
- Types of integer programming models are _____________.
- In a 0 – 1 integer model, the solution values of the decision variables are 0 or 1.
- Which of the following is not an integer linear programming problem?
- In a mixed integer model, some solution values for decision variables are integer and others can be non-integer.
- Rounding small values of decision variables to the nearest integer value causes ______________ problems than rounding large values.
- In using rounding of a linear programming model to obtain an integer solution, the solution is

**Week 6 Midterm Exam:**

- ___________ is a technique for selecting numbers randomly from a probability distribution.
- Monte Carlo is a technique for selecting numbers randomly from a probability distribution.
- Analogue simulation replaces a physical system with an analogous physical system that is _____________ to manipulate.
- Variable costs are independent of volume and remain constant.
- Regret is the difference between the payoff from the best decision and all other decision payoffs.
- The maximin criterion results in the
- A payoff table is a means of organizing a decision situation, including the payoffs from different decisions given the various states of nature.

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