Problems for Mathematical Formulation and Solution Using LINGO 1. Media Selection

Transcription

1 Problems for Mathematical Formulation and Solution Using LINGO 1. Media Selection A company has budgeted up to $8,000 per week for local advertising. The money is to be allocated among four promotional media: TV spots, newspaper ads, and two types of radio advertisements. The company s goal is to reach the largest possible high-potential audience through the various media. The following table presents the number of potential customers reached by making use of an advertisement in each of the four media. It also provides the cost per advertisement placed and the maximum number of ads that can be purchased per week. Table: Advertisement Audience reached and cost MEDIUM AUDIENCE REACHED PER AD COST PER AD ($) MAXIMUM ADS PER WEEK TV spot (one minute) 5, Daily newspaper (full-page ad) 8, Radio spot (30 seconds, prime time) 2, Radio spot (one minute, afternoon) 2, The company s contractual arrangements require that at least 5 radio spots be placed each week. To ensure a broad-scoped promotional campaign, management also insists that no more than $1,800 be spent on radio advertising every week. Various media has informed that there is a revision in cost in near future and the estimate of the revised costs are as follows: MEDIUM COST PER AD ($) TV spot (one minute) 1000 Daily newspaper (full-page ad) 1125 Radio spot (30 seconds, prime time) 430 Radio spot (one minute, afternoon) 400 Determine the new media mix and show the sensitivity of the result. 12

2 2. Production Mix Fifth Avenue Industries, a manufacturer of menswear, produces four varieties of ties. One is an expensive, all-silk tie, one is an all-polyester tie, and two are blends of polyester and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process. MATERIAL COST PER MATERIAL AVAILABLE YARD ($) PER MONTH (YARDS) Silk Polyester 6 3,000 Cotton 9 1,600 The firm has fixed contracts with several major department store chains to supply ties. The contracts require that Fifth Avenue Industries supply a minimum quantity of each tie, but allow for a larger demand if Fifth Avenue chooses to meet that demand. (Most of the ties are not shipped with the name Fifth Avenue on their label, incidentally, but with Private Stock labels supplied by the stores). Table below summarizes the contract demand for each of the four styles of ties, the selling price per tie, and the fabric requirements of each variety. Table: DATA FOR FIFTH AVENUE INDUSTRIES VARIETY OF TIE SELLING PRICE PER TIE ($) MONTHLY CONTRACT MINIMUM MONTHLY DEMAND MATERIAL PER TIE (YARDS) MATERIAL REQUIREMENTS All Silk ,000 7, % silk All polyester ,000 14, % polyester Poly-cotton Blend # ,000 16, % polyester 50% cotton Poly-cotton blend # ,000 8, % polyester 70% cotton There is a revision in monthly contract and monthly demand, and the revised estimates are as follows. VARIETY OF TIE MONTHLY CONTRACT MINIMUM MONTHLY DEMAND All Silk 4,000 5,000 All polyester 12,000 15,000 Poly-cotton Blend #1 11,000 15,000 Poly-cotton blend # 2 8,000 10,500 Show the effect of these changes in the solution and also show the sensitivity of new solution. 13

3 3. Production Scheduling Greenberg Motors, Inc., manufactures two different electrical motors for sale under contract to Drexel Corp., a well-known producer of small kitchen appliances. Its model GM3A is found in many Drexel food processors and its model GM3B is used in the assembly of blenders. Three times each year, the procurement office at Drexel contacts Irwin Greenberg, the founder of Greenberg Motors, to place a monthly order for each of the coming four months. Drexel s demand for motors varies each month based on its own sales forecasts, production capacity, and financial position. Greenberg has just received the January- April order and must begin his own four-month production plan. The demand for motors is shown in table below. Table: Four-Month Order Schedule for Electrical Motors MODEL JANUARY FEBRUARY MARCH APRIL GM3A ,000 1,100 GM3B 1,000 1,200 1,400 1,400 The production costs are currently $10 per GM3A motor produced and $6 per GM3B unit. A labour agreement going into effect on March 1 will raise each figure by 10%, however. Each GM3A motor held in stock costs $0.18 per month, while each Gm3B has a carrying cost of $0.13 per month. Greenberg s accountants allow monthly ending inventories as an acceptable approximation to the average inventory levels during the month. The inventory at the end of the planning period for GM3A and GM3B are 450 and 300 respectively. The storage area for Greenberg motors can hold a maximum of 3300 motors of either type at any one time. Greenberg has a base employment level of 2240 labor hours per month. In a busy period, the company can bring two skilled former employees on board to increase capacity to 2,560 hours per month. Each GM3A motor produced requires 1.3 hours of labor, while each GM3B takes a worker 0.9 hours to assemble Show the effect of labour agreement on decision. 14

4 4. Labour Planning XYZ Bank of Commerce and Industry is a busy bank that has requirements for between 10 and 18 tellers depending on the time of day. The lunch time, from noon to 2 P.M., is usually heaviest. Table below indicates the workers needed at various hours that the bank is open. TABLE: XYZ Bank of Commerce and Industry TIME PERIOD NUMBER OF TELLERS 9 A.M. 10 A. M A. M A. M A.M. Noon 14 Noon 1 P.M P.M. 2 P.M P.M. 3 P.M P.M. 4 P.M P.M. 5 P.M. 10 The bank now employs 12 full-time tellers, but many people are on its roster of available part-time employees. A part-time employee must put in exactly four hours per day, but can start anytime between 9 A.M. and 1 P.M. Part-timers are a fairly inexpensive labor pool, since no retirement or lunch benefits are provided for them. Full-timers, on the other hand, work from 9 A.M. 5. P.M., but are allowed one hour for lunch. (Half of the full-timers eat at 11 A.M., the other half at noon) Full-timers thus provide 35 hours per week of productive labor time. By corporate policy, the bank limits part-time hours to a maximum of 50% of the day s total requirement. Part-timers earn $4 per hour (or $ 16 per day) on the average, while full-timers earn $50 per day in salary and benefits, on the average. The bank would like to set a schedule that would minimize its total personnel costs. It will release one or more of its full-time tellers if it is profitable to do so. 15

5 5. Portfolio Selection ABC Trust invests in short-term trade credits, corporate bonds, gold stocks, and construction loans. The board of directors has placed limits on the amount that can be committed to any one type of investment in order to encourage a diversified portfolio. ABC Trust has $5 million available for immediate investment and wishes to do two things: (1) maximize the interest earned on the investments made over the next six months, and (2) satisfy the diversification requirements as set by the board of directors. The specifics of the investment possibilities are: INVESTMENT INTEREST EARNED (%) MAXIMUM INVESTMENT ($ MILLION) Trade Credit Corporate bonds Gold stocks Construction loans In addition, the board specifies that at least 55% of the funds invested must be in gold stocks and construction loans, and that no less than 15% be invested in trade credit. The board also want to know that how the interest earned changes when the investment restrictions are at least 40 % of the funds invested must be in gold stocks and construction loans, and that no less than 20% be invested in trade credit. 16

6 6. Shipping Problem A Bicycle Company manufactures and markets a line of ten-speed bicycles nationwide. The firm has final assembly plants in two cities (A and B) in which labor costs are low. Its three major warehouses are located near the large market areas namely X, Y, and Z. The sales requirements for the next year at the warehouse of the city X are 10,000 bicycles, at the warehouse of the city Y are 8,000 bicycles, and at the warehouse of the city Z are 15,000 bicycles. The factory capacity at each location is limited. The plant at city A can assemble and ship 20,000 bicycles, while the plant at city B can produce 15,000 bicycles per year. The cost of shipping one bicycle from each factory to each warehouse differs, and these unit-shipping costs are: FROM TO X Y Z A $2 $3 $5 B $3 $1 $4 The company wishes to develop a shipping schedule that will minimize its total annual transportation costs. A revision of shipping cost is expected soon and the revised cost is as follows: FROM TO X Y Z A $2.5 $4 $5.5 B $4 $2 $4.5 Determine the effect of revision of cost. 17

7 7. Diet Problem The Whole Food Nutrition Centre uses three bulk grains to blend a natural cereal that it sells by the pound. The store advertises that each two-ounce serving of the cereal, when taken with ½ cup of whole milk, meets an average adult s minimum daily requirement for protein, riboflavin, phosphorus, and magnesium. The cost of each bulk grain and the protein, riboflavin, phosphorus, and magnesium units per pound of each are shown in Table below. TABLE: Whole Food s Natural Cereal Requirements GRAIN COST PER POUND (Rs) PROTEIN (UNITS/LB) FIBOFLAVIN (UNITS/LB) PHOSPHORUS (UNITS/LB) MAGNESIUM (UNITS/LB) A B C The minimum adult daily requirement (called the Recommended Daily Allowance, or RDA) for protein is 3 units; for riboflavin, 2 units; for phosphorus, 1 unit; and for magnesium, units. Whole Food wants to select the blend of grains that will meet the RDA at a minimum cost. Whole Food is considering another grain D about which the following is known. GRAIN COST PER PROTEIN FIBOFLAVIN PHOSPHORUS MAGNESIUM POUND (Rs) (UNITS/LB) (UNITS/LB) (UNITS/LB) (UNITS/LB) D What may be the effect of considering this grain? 18

8 8. Ingredient Mix and Blending Problem The Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution. The grades, regular and economy, are produced by refining a blend of two types of crude oil, Type X100 and Type X220. Each crude oil differs not only in cost per barrel, but in composition as well. The accompanying table indicates the percentage of crucial ingredients found in each of the crude oils and the cost per barrel for each. CRUDE OIL TYPE INGREDIENT A (%) INGREDIENT B (%) COST/BARREL ($) X X Weekly demand for the regular grade of Low Knock gasoline is at least 25,000 barrels, while demand for the economy is at least 32,000 barrels per week. At least 45% of each barrel of regular must be ingredient A. At most 50% of each barrel of economy should contain ingredient B. The Low Knock management must decide how many barrels of each type of crude oil to buy each week for blending in order to satisfy demand at minimum cost. If the proportion is changed as follows, what is the effect on the solution? At least 40% of each barrel of regular must be ingredient A. At most 55% of each barrel of economy should contain ingredient B. 19

9 9. Product Mix Problem The flair Furniture Company produces inexpensive tables and chairs. The production process for each is similar in that both require a certain number of hours of carpentry work and a certain number of labour hours in the painting and varnishing department. Each table takes 4 hours of carpentry and 2 hours in the painting and varnishing shop. Each chair requires 3 hours in carpentry and 1 hour in painting and varnishing. During the current production period, 240 hours of carpentry time are available and 100 hours in painting and varnishing time are available. Each table and chair must be inspected and may need rework. The amount of inspection and rework required for each table and chair are 0.5 hours and 0.6 hours respectively. Number of labour hours available for this purpose is 36. The flair faces a resource constraint relating to the lumber needed for each table or chair and the amount available each day is 1248 linear feet of lumber. The required quantity for table and chair is 32 and 10 linear feet of lumbers respectively. The demand for tables is found to be a maximum of 40 daily and there are no similar constraints regarding chairs. Each table sold yields a profit of Rs 350; each chair produced is sold for Rs 250 profit. How many tables and chairs should flair furniture produced daily? What is the profit generated by this solution? Will Flair use all of its resources to their limits each day? Should flair purchase more lumber if it is available at Rs 4 per linear foot? Should it hire more carpenters at Rs. 30 per hour? What is the range within which the carpentry hours, painting hours, and inspection/rework hours can fluctuate before the optimal solution changes? Fair s owner has been approached by a friend whose company would like to use several hours in the painting facility every day. Should flair sell time to the other firm? If so, how much? Explain. Within what range for the current solution can the profit contribution of tables and chairs change? As a result of improvement in production process, the inspection/rework time required for tables and chairs is reduced to 0.2 and 0.4 hours. Flair want to earmark 24 hours daily for this purpose instead of 36 hours. What is the effect of this improvement, in solution? There is a saving of 12 hours of inspection/rework due to the improvement and where it can be used? Assume that the workers are skilled for all kind work in the production shop of the company. 20

10 10. Product Mix Problem A Chicago manufacturer of office equipment is desperately attempting to control its profit and loss statement. The company currently manufactures 15 different products, each coded with a one-letter and three-digit designation. PRODUCT STEEL ALLOY (LBS.) PLASTIC (SQ. FT.) WOOD (BD. FT.) ALUMINUM (LBS.) FORMICA (BD. FT.) LABOR (HOURS) MINIMUM MONTHLY DEMAND (UNITS) CONTRI- BUTION TO PROFIT A $ B C D E F G H I J K L M N P Availability Per Month ,500 1,800 1,000 a. How many of each of the 15 products should be produced each month? b. Clearly, explain the meaning of each shadow price. c. A number of workers interested in saving money for the holidays have offered to work overtime next month at a rate of $12.50 per hour. What should the response of management be? d. Two tons of steel alloy are available from an overstocked supplier at a total cost of $8,000. Should the steel be purchased? All or part of the supply? e. The accountants have just discovered that an error was made in the contribution to profit for product N150. The correct value is actually $8.88. What are the implications of this error? f. Management is considering the abandonment of five product lines (those beginning with the letters A through E). If no minimum monthly demand is established, what are the implications? Note that there already is no minimum for two of these products. Use the corrected value for N