Reference no: EM13943944

Case 1:

Parket Sisters - (Contributed by Jack Yurkiewicz, Lubin School of Business, Pace University, New York.)

Computers and word processors not withstanding, the art of writing by hand re- cently entered a boom era. People are buying fountain pens again, and mechanical pencils are becoming more popular than ever. Joe Script, the president and CEO of Parket Sisters, a small but growing pen and pencil manufacturer, wants to establish a better foothold in the market. The writing market is divided into two main sectors. One, dominated by Mont Blanc, Cross, Parker Brothers, Waterman, Schaffer, and a few others, caters to people who want writing instruments. The product lines from these companies consist of pens and pencils of elaborate design, lifetime warranty, and high price. At the other end of the market are manufacturers such as BIC, Pen- tel, and many companies from the Far East, offering good quality items, low price, few trims, and limited diversity. These pens and pencils are meant to be used for a limited time and disposed of when the ink in a ballpoint pen runs out or when the lead in a mechanical pencil won't retract or extend. In short, these items are not meant for repair.

Joe thinks that there must be a middle ground, and that is where he wants to position his company. Parket Sisters makes high-quality items, with limited trim and diversity, but also offers lifetime warranties. Furthermore, its pens and pencils are ergonomically efficient. Joe knows that some people want the status of the Mont Blanc Meisterstuck pen, for example, but he has never met a person who said that writing with such a pen is enjoyable. The pen is too large and clumsy for smooth writing. Parket Sisters' products, on the other hand, have a reputation for working well, are easy to hold and use, and cause limited "writer's fatigue."

Parket Sisters makes only three items-a ballpoint pen, a mechanical pencil, and a fountain pen. All are available in just one color, black, and are sold mostly in specialty stores and from better catalog companies. The per-unit profit of the items is $3.00 for the ballpoint pen, $3.00 for the mechanical pencil, and $5.00 for the fountain pen. These values take into account labor, the cost of materials, packing, quality control, and so on.

The company is trying to plan its production mix for each week. Joe believes that the company can sell any number of pens and pencils it produces, but production is currently limited by the available resources. Because of a recent strike and certain cash- flow problems, the suppliers of these resources are selling them to Parket Sisters in limited amounts. In particular, Joe can count on getting at most 1,000 ounces of plastic, 1,200 ounces of chrome, and 2,000 ounces of stainless steel each week from his suppli- ers, and these figures are not likely to change in the near future. Because of Joe's excel- lent reputation, the suppliers will sell Joe any amount (up to his limit) of the resources he needs when he requires them. That is, the suppliers do not require Joe to buy some fixed quantities of resources in advance of his production of pens and pencils; there- fore, these resources can be considered variable costs rather than fixed costs for the pens and pencils.

Each ballpoint pen requires 1.2 ounces of plastic, 0.8 ounces of chrome, and 2 ounces of stainless steel. Each mechanical pencil requires 1.7 ounces of plastic, no chrome, and 3 ounces of stainless steel. Each fountain pen requires 1.2 ounces of plastic, 2.3 ounces of chrome, and 4.5 ounces of stainless steel. Joe believes LP could help him decide what his weekly product mix should consist of Getting his notes and notebooks, Joe grapples with the LP formulation. In addition to the constraints of the available resources, he recognizes that the model should include many other constraints (such as labor time availability and materials for packing). How- ever, Joe wants to keep his model simple. He knows that eventually, he'll have to take other constraints into account, but as a first-pass model, he'll restrict the constraints to just the three resources: plastic, chrome, and stainless steel.

With only these three constraints, Joe can formulate the problem easily as:
MAX: 3.0X1 1 3.0X2 1 5.0X3
Subject to: 1.2X1 1 1.7X2 1 1.2X3 # 1000
0.8X1 1 0X2 1 2.3X3 # 1200
2.0X1 1 3.0X2 1 4.5X3 # 2000
X1, X2, X3 $ 0

where:

X1 5 the number of ballpoint pens
X2 5 the number of mechanical pencils X3 5 the number of fountain pens

Joe's knowledge of Excel and the Solver feature is limited, so he asks you to enter and solve the problem for him and then answer the following questions. (Assume each question is independent unless otherwise stated.)

1. What should the weekly product mix consist of, and what is the weekly net profit?

2. Is the optimal solution to question 1 degenerate? Explain your response.

3. Is the optimal solution from question 1 unique, or are there alternate answers to this question? Explain your response.

4. What is the marginal value of one more unit of chrome? Of plastic?

5. A local distributor has offered to sell Parket Sisters an additional 500 ounces of stain- less steel for $0.60 per ounce more than it ordinarily pays. Should the company buy the steel at this price? Explain your response.

6. If Parket Sisters buys the additional 500 ounces of stainless steel noted in question 5, what is the new optimal product mix, and what is the new optimal profit? Explain your response.

7. Suppose that the distributor offers to sell Parket Sisters some additional plastic at a price of only $1.00 over its usual cost of $5.00 per ounce. However, the distributor will sell the plastic only in lot sizes of 500 ounces. Should Parket Sisters buy one such lot? Explain your response.

8. The distributor is willing to sell the plastic in lots of just 100 ounces instead of the usual 500-ounce lots, still at $1.00 over Parket Sisters' cost of $5.00 per ounce. How many lots (if any) should Parket Sisters buy? What is the optimal product mix if the company buys these lots, and what is the optimal profit?

9. The R&D department at Parket Sisters has been redesigning the mechanical pencil to make it more profitable. The new design requires 1.1 ounces of plastic, 2.0 ounces of chrome, and 2.0 ounces of stainless steel. If the company can sell one of these pen- cils at a net profit of $3.00, should it approve the new design? Explain your response.

10. If the per-unit profit on ballpoint pens decreases to $2.50, what is the optimal prod- uct mix, and what is the company's total profit?

11. The marketing department suggested introducing a new felt-tip pen that requires

12 ounces of plastic, 0.5 ounces of chrome, and 1.3 ounces of stainless steel. What profit must this product generate in order to make it worthwhile to produce?

13. What must the minimum per-unit profit of mechanical pencils be in order to make them worthwhile to produce?

14. Management believes that the company should produce at least 20 mechanical pen- cils per week to round out its product line. What effect would this have on overall profit? Give a numerical answer.

15. If the profit on a fountain pen is $6.75 instead of $5.00, what is the optimal product mix and optimal profit?Hamilton & Jacobs

Hamilton & Jacobs (H&J) is a global investment company, providing start-up capital to promising new ventures around the world. Due to the nature of its business, H&J holds funds in a variety of countries and converts between currencies as needs arise in different parts of the world. Several months ago, the company moved $16 million into Japanese yen (JPY) when 1 U.S. dollar (USD) was worth 75 yen. Since that time, the value of the dollar has fallen sharply, where it now requires almost 110 yen to purchase 1 dollar.

Case 2:

Besides its holdings of yen, H&J also currently owns 6 million European euros and 30 million Swiss Francs (CHF). H&J's chief economic forecaster is predicting that all of the currencies it is presently holding will continue to gain strength against the dollar for the rest of the year. As a result, the company would like to convert all its surplus currency holdings back to U.S. dollars until the economic picture improves.
The bank that H&J uses for currency conversions charges different transaction fees for converting between various currencies. The following table summarizes the transaction fees (expressed as a percentage of the amount converted) for U.S. dollars (USD), Australian dollars (AUD), British pounds (GBP), European Euros (EURO), Indian Ru- pees (INR), Japanese yen (JPY), Singapore dollars (SGD), and Swiss Francs (CHF).

Transaction Fee Table

FROM\TO	USD	AUD	GBP	EUR	INR	JPY	SGD	CHF
USD	-	0.10%	0.50%	0.40%	0.40%	0.40%	0.25%	0.50%
AUD	0.10%	-	0.70%	0.50%	0.30%	0.30%	0.75%	0.75%
GBP	0.50%	0.70%	-	0.70%	0.70%	0.40%	0.45%	0.50%
EUR	0.40%	0.50%	0.70%	-	0.05%	0.10%	0.10%	0.10%
INR	0.40%	0.30%	0.70%	0.05%	-	0.20%	0.10%	0.10%
JPY	0.40%	0.30%	0.40%	0.10%	0.20%	-	0.05%	0.50%
SGD	0.25%	0.75%	0.45%	0.10%	0.10%	0.05%	-	0.50%
CHF	0.50%	0.75%	0.50%	0.10%	0.10%	0.50%	0.50%	-

Because it costs differing amounts to convert between various currencies, H&J deter- mined that converting existing holdings directly into U.S. dollars may not be the best strategy. Instead, it might be less expensive to convert existing holdings to an inter- mediate currency before converting the result back to U.S. dollars. The following table summarizes the current exchange rates for converting from one currency to another.

Exchange Rate Table

From\To	USD	AUD	GBP	EUR	INR	JPY	SGD	CHF
USD	1	1.29249	0.55337	0.80425	43.5000	109.920	1.64790	1.24870
AUD	0.77370	1	0.42815	0.62225	33.6560	85.0451	1.27498	0.96612
GBP	1.80710	2.33566	1	1.45335	78.6088	198.636	2.97792	2.25652
EUR	1.24340	1.60708	0.68806	1	54.0879	136.675	2.04900	1.55263
INR	0.02299	0.02971	0.01272	0.01849	1	2.5269	0.03788	0.02871
JPY	0.00910	0.01176	0.00503	0.00732	0.39574	1	0.01499	0.01136
SGD	0.60683	0.78433	0.33581	0.48804	26.3972	66.7031	1	0.75775
CHF	0.80083	1.03507	0.44316	0.64407	34.8362	88.0275	1.31969	1

The exchange rate table indicates, for instance, that 1 Japanese yen can be converted into 0.00910 U.S. dollars. So 100,000 yen would produce $910 U.S. dollars. However, the bank's 0.40% fee for this transaction would reduce the net amount received to $910 3 (1 2 0.004) 5 $906.36. So H&J wants your assistance in determining the best way to convert all of its non-U.S. currency holdings back into U.S. dollars.

1. Draw a network flow diagram for this problem.

2. Create a spreadsheet model for this problem and solve it.

3. What is the optimal solution?

4. If H&J converted each non-U.S. currency it owns directly into U.S. dollars, how many U.S. dollars would it have?

5. Suppose H&J wants to perform the same conversion but also leave $5 million in Australian dollars. What is the optimal solution in this case?

Part -3: Removing Snow in Montreal

Based on: Campbell, J., and A. Langevin. "The Snow Disposal Assignment Problem." Journal of the Operational Research Society, 1995, pp. 919-929.

Snow removal and disposal are important and expensive activities in Montreal and many northern cities. While snow can be cleared from streets and sidewalks by plowing and shoveling, in prolonged subfreezing temperatures, the resulting banks of accumu- lated snow can impede pedestrian and vehicular traffic and must be removed.

To allow timely removal and disposal of snow, a city is divided up into several sectors, and snow removal operations are carried out concurrently in each sector. In Montreal, accumulated snow is loaded onto trucks and hauled away to disposal sites (for example, rivers, quarries, sewer chutes, and surface holding areas). For contrac- tual reasons, each sector may be assigned to only a single disposal site. (However, each disposal site may receive snow from multiple sectors.) The different types of disposal sites can accommodate different amounts of snow due to either the physical size of the disposal facility or environmental restrictions on the amount of snow (often con- taminated by salt and de-icing chemicals) that can be dumped into rivers. The annual capacities for five different snow disposal sites are given in the following table (in 1,000s of cubic meters).

			Disposal Site
	1	2	3	4	5
Capacity	350	250	500	400	200

The cost of removing and disposing of snow depends mainly on the distance it must be trucked. For planning purposes, the city of Montreal uses the straight-line distance between the center of each sector to each of the various disposal sites as an approxima- tion of the cost involved in transporting snow between these locations. The following table summarizes these distances (in kilometers) for 10 sectors in the city.

Disposal Site

Sector	1	2	3	4	5
1	3.4	1.4	4.9	7.4	9.3
2	2.4	2.1	8.3	9.1	8.8
3	1.4	2.9	3.7	9.4	8.6
4	2.6	3.6	4.5	8.2	8.9
5	1.5	3.1	2.1	7.9	8.8
6	4.2	4.9	6.5	7.7	6.1
7	4.8	6.2	9.9	6.2	5.7
8	5.4	6.0	5.2	7.6	4.9
9	3.1	4.1	6.6	7.5	7.2
10	3.2	6.5	7.1	6.0	8.3

Using historical snowfall data, the city is able to estimate the annual volume of snow requiring removal in each sector as four times the length of streets in the sectors in meters (that is, it is assumed each linear meter of street generates four cubic meters of snow to remove over an entire year). The following table estimates the snow removal requirements (in 1,000s of cubic meters) for each sector in the coming year.

Estimated Annual Snow Removal Requirements

1	2	3	4	5	6	7	8	9	10
153	152	154	138	127	129	111	110	130	135

1. Create a spreadsheet that Montreal could use to determine the most efficient snow removal plan for the coming year. Assume it costs $0.10 to transport 1 cubic meter of snow 1 kilometer.

2. What is the optimal solution?

3. How much will it cost Montreal to implement your snow disposal plan?

4. Ignoring the capacity restrictions at the disposal sites, how many different assign- ments of sectors to disposal sites are possible?

5. Suppose Montreal can increase the capacity of a single disposal site by 100,000 cubic meters. Which disposal site's capacity (if any) should be increased, and how much should the city be willing to pay to obtain this extra disposal capacity?

Part -4: Planning Diets for the Food Stamp Program

Based on: S. Taj, "A Mathematical Model for Planning Policies for Food Stamps," Applications of Management Science, vol. 7, 1993, pp. 25248.

The United States Department of Agriculture (USDA) is responsible for managing and administering the national food stamp program. This program provides vouchers to low income families that can be used in place of cash to purchase food at grocery stores. In determining the cash value of the vouchers issued, the USDA must consider how much it costs to obtain a nutritional, well-balanced diet for men and women in vari- ous age groups. As a first step in this process, the USDA identified and analyzed 31 different food groups and determined the contributions a serving from each group makes to 24 different nutritional categories. A partial listing of this information is given in Figure 7.20 (and in the file Fig7-20.xlsm that accompanies this book).

The last two rows in this spreadsheet indicate the minimum and/or maximum nutrients required per week for men between the ages of 20 and 50. (Maximum values of 9999 indicate that no maximum value applies to that particular nutritional requirement.) The USDA uses this information to design a diet (or weekly consumption plan) that meets the indicated nutritional requirements. The last two columns in Figure 7.20 represent two different objectives that can be pursued in creating a diet. First, we may want to identify the diet that meets the nutritional requirements at a minimum cost. Although such a diet might be very economical, it might also be very unsatisfactory to the tastes of the people who are expected to eat it. To help address this issue, the USDA conducted a survey to assess people's preferences for different food groups.

Summarizes these preference ratings, with higher scores indicating more desirable foods, and lower scores indicating less desirable foods. Thus, another objective that could be pursued is determining the diet that meets the nutritional requirements and produces the highest total preference rating. However, this solution is likely to be quite expensive. Assume that the USDA has asked you to help them analyze this situation using MOLP.

1. Find the weekly diet that meets the nutritional requirements in the least costly manner. What is the lowest possible minimum cost? What preference rating does this solution have?

2. Find the weekly diet that meets the nutritional requirements with the highest preference rating. What preference rating does this solution have? What cost is associated with this solution?

3. Find the solution that minimizes the maximum percentage deviation from the optimum values for each individual objective. What cost and preference rating is associated with this solution?

4. Suppose that deviations from the optimal cost value are weighted twice as heavily as those from the optimal preference value. Find the solution that minimizes the maximum weighted percentage deviations. What cost and preference rating is associated with this solution?