UGBS 202 – BUSINESS MATHEMATICS
TUTORIAL SET 4 (MATHEMATICAL MODELLING AND INDICES) MARCH 2016
Not to be submitted
1. Mikey W. Smitty, an emerging rapper, is getting ready to cut his first CD, called “Western Rap.” The cost of recording the CD is $5000 but copies are $5 apiece. If the CDs can be sold for $15 each, how many CDs must be sold to break even?
2. Esi, a doll maker from Takoradi, is interested in the mass marketing and production of a ceramic doll of her own design called Oman. The initial investment required for plant and equipment is estimated at GHS25000. Labour and material costs are approximately GHS10 per doll. If the dolls can be sold for GHS50 each, what volume of demand is necessary for the Oman doll to break even?
3. The school cafeteria can make pizza for approximately $0.30 a slice. The cost of kitchen use and cafeteria staff runs about $200 per day. The Pizza Den nearby will deliver whole pizzas for $9.00 each. The cafeteria staff cuts the pizza into eight pieces and serves them in the usual cafeteria line. With no cooking duties, the staff can be reduced by half, for a fixed cost of $75 per day. Should the school cafeteria make or buy its pizzas?
4. David Austin recently purchased a chain of dry cleaners in northern Wisconsin. Although the business is making a modest profit now, David suspects that if he invests in a new Tutorial press, he could recognize a substantial increase in profits. The new press costs $15,400 to purchase and install and can press 40 shirts an hour (or 320 per day). David estimates that with the new press, it will cost $0.25 to launder and press each shirt. Customers are charged $1.10 per shirt.
a. How many shirts will David have to press to break even?
b. So far, David’s workload has varied from 50 to 200 shirts a day. How long would it take to break even on the new press at the low-demand estimate? At the high demand estimate?
c. If David cuts his price to $0.99 a shirt, he expects to be able to stabilize his customer base at 250 shirts per day. How long would it take to break even at the reduced price of $0.99? Should David cut his price and buy the new press?
5. Koko King Company delivers fresh breakfasts each morning to vending stations throughout the city. The company makes three kinds of breakfasts—Hausa Koko, wheat meal, and oatmeal. A Hausa Koko breakfast requires a worker 0.45 minutes to package, a wheat meal breakfast requires 0.41 minutes, and an oatmeal breakfast requires 0.50 minutes to package. The company has 16 available hours each night for breakfast package. Vending stations’ capacity is available for 2,000 packs of breakfast each day. The company knows from past sales records that its customers buy as many Hausa Koko breakfasts as the other two breakfasts combined, if not more so, but customers need a variety of breakfasts available, so Koko King stocks at least 200 of each. Koko King’s management wants to know how many of each breakfast it should stock to maximize profit.
6. Mr. Mensah owns Menstech, a computer shop. He stocks three types of computers—dell, Hp, and Toshiba. A dell machine costs GH 1,200, an Hp machine costs GH 1,700, and a Toshiba machine costs GH 900. He has GH 12,000 available this month to purchase machines. Each machine must be assembled; a dell machine requires 8 hours to assemble, an Hp machine requires 12 hours, and a Toshiba machine requires 16 hours. He estimates that he and his employees have 120 hours available to assemble machines. He has enough space in his store to order 20 machines this month. Based on past sales, Mr. Mensah wants to stock at least twice as many Toshiba machines as the other two combined because Toshiba machines sell better. Formulate a mathematical model for this problem.
7. The Fuga and Turkson Cabinet and Furniture Company (FTFC) produces sofas, tables and chairs at its plant at Kade. The plant uses three main resources to make furniture: - wood, upholstery and labour. The resource requirements for each piece of furniture and the total resources available weekly are given in the table below:
Product
Resource Requirement
Wood (lb) Upholstery (yds.) Labour (hr.)
Sofa 7 12 6
Table 5 - 9
Chair 4 7 5
Available 2250 1000 240
The furniture is produced on weekly basis and stored in a warehouse until the end of the week, when it is shipped out. The warehouse has a total capacity of 650 pieces of furniture. The company wants to know how many pieces of each type of furniture to make per week to maximise profit.
(a) Define the decision variables for this problem. (b) Formulate a mathematical model for this problem
8. The artisans at Jewellery Palace, Nungua, are preparing to make gold jewellery during 2-month period for the Christmas season. They can make bracelets, necklaces, and pins. Each bracelet requires 6.3 ounces of gold and 17 hours of labour, each necklace requires 3.9 ounces of gold and 10 hours of labour, and each pin requires 3.1 ounces of gold and 7 hours of labour. Jewellery Palace has available 125 ounces of gold and 320 hours of labour. How many pieces of bracelets, necklaces and pins can be produced to meet these resource availabilities?
Making this decision requires a consideration of the following key factors:
Each product is put through Kpogas’ machining operations in both department A and department B. For next month’s production, the two departments have 150 and 160 hours available time, respectively. Each chair produced uses 10 hours of machining in department A and 20 hours of machining in department B, whereas each table produced uses 15 hours of machining in department A and 10 hours of machining in department B.
In order for management to honour an agreement with the union, the total labour hours used in the next month’s testing of finished products cannot fall more than 10% below an arbitrated goal of 150 hours. Since 10% of 150 is 15, the total hours devoted to testing cannot fall below 135. The testing is performed in the third department and has nothing to do with department A and B. Each chair is given 30 hours of testing and each table is given 10.
To maintain current market position, management has decided that it is necessary to build at least one chair for every three tables produced.
A major dealer has ordered a total of at least five chairs and tables (in any combination) for next month, and so at least that number must be produced.
Provide the necessary mathematical model for this problem
10. The RMC, Inc., is a small firm that produces a variety of chemical-based products. In a particular production process, three raw materials are used to produce three products: a fuel additive, a solvent base, and a carpet cleaning fluid. Each ton of fuel additive is a blend of 0.4 tons of material 1 and 0.6 tons of material 3. Each ton of solvent base requires 0.5 tons of material 1, 0.2 tons of material 2, and 0.3 tons of material 3. Each ton of carpet cleaning fluid is a blend of 0.6 tons of material 1, 0.1 tons of material 2, and 0.3 tons of material 3. RMC has 20 tons of material 1, 5 tons of material 2, and 21 tons of material 3 available. The firm also establishes the following maximum production capacities: 50 tons of fuel additive, 25 tons of solvent base, and 40 tons of carpet cleaning fluid.
a) Define ALL the decision variables for this mixed integer problem b) Formulate a mathematical model for this problem
11. The UGBS endowment fund has available GH¢1 million that can be invested in utility stocks, tech stocks, and gold stocks. The annual rates of return for the three types of stocks are 9% for utility stock, 10% for tech stock, and 12% for gold stock. Management of the endowment fund has decided that at least 45% of the available fund must be invested in utility stocks. In addition, management has specified that the ratio of the amount invested in tech stocks to that of gold stocks should not exceed 2. Each GH¢1 million invested in utility stocks, tech stocks, and gold stock has a risk index of 0.006, 0.004, and 0.004 respectively. Management of the endowment fund can only accept a total risk index of 4500 from the three stocks.
12. Cal Brokers Ltd was instructed by one of its clients, Asare, to invest GH¢250,000 of his money. Asare has a good deal of trust in the investment house, but he also has his own ideas about the distribution of the funds being invested. In particular, he requests that the firm selects whatever stocks and bonds they believe are well rated, but within the following guidelines:
Municipal bonds should constitute at least 20% of the investment.
At least 40% of the funds should be placed in a combination of Aluworks stock, Cal stock, AYRTON stock.
No more than 50% of the amount invested in municipal bonds should be placed in a risk, high-yield GOIL stock.
a. Identify the decision variable in portfolio selection problem b. Formulate a mathematical model for the problem
13. The quarterly advertised prices for a car is given by the following:
Qtr one Qtr two Qtr three Qtr four
10450 10800 11450 9999
a. How will you describe the price changes?
14. The information given below are the monthly price indices for a product for a particular year.
1 2 3 4 5 6 7 8 9 10 11 12
121 112 98 81 63 57 89 109 131 147 132 120 a. If sales in month three is 240, what is the value of sales in month 8?
b. If sales in month 10 is 1200, what is the value of sales in month 2?
15. Why is it necessary to change the base period in index number calculations?
16. What is the difference between a 10% change and a 10 percentage point’s change?
17. Due to changes in market behaviours, the management of Ebitiyie Company decided to change their price base periods in computing their price indices. The price indices for the company is given as follows for the two base periods.
1 2 3 4 5 6 7 8
100 138 162 196 200
100 125 145 165
a. What are the base periods
b. What would the index values be from year 6 to year 8?
18. The table below gives the price of four commodities in a four year period.
Commodity 2010 2011 2012 2013
Wheat 23 23 24 26
Rice 24 40 56 72
Sugar 25 28 28 29
Beans 81 84 90 99
Using 2010 as base, find the mean price relative and the simple aggregate indices of 2011, 2012 and 2013.
19. A company buys four products with the following features
Number of units Price paid per unit
Year one Year two Year one Year two
A 20 24 10 11
B 55 51 23 25
C 63 84 17 17
D 28 34 19 20
a. What are the price indices for each product in year 2 using year one as base year b. Calculate the base-weighted index for the products
c. Calculate the current weighted index
20. The table below gives the food basket of an economy containing only 2 items, wine and cheese. The quantities consumed in 2006 and 2007 are as follows
Wine Cheese
2006 50 bottles 100 wheels
2007 45 bottles 150 eels
