Formulation problems
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FORMULATION PROBLEMS (Product mix) MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and Beta 5. The firm employs five technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (ie. All 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate a $1,200 profit per unit, and Beta 5s yield $1,800 each. (Product mix) The Marriott Tub Company manufactures two lines of bathtubs, called model A and model B. Every tub requires blending a certain amount of steel and zinc; the company has available a total of 25,000 pounds of steel and 6,000 pounds of zinc. Each model A bathtub requires a mixture of 125 pounds of steel and 20 pounds of zinc, and each yields a profit to the firm of $90. Each model B tub produced can be sold for a profit of $70; it in turn requires 100 pounds of steel and 30 pounds of zinc. (Product mix).The Valley Wine Company produces two kinds of wine-Valley Nectar and Valley Red. The wines are produced from 64 tons of grapes the company has acquired this season. A 1,000 gallon batch of nectar requires 4 tons of grapes and a 1,000 gallon batch of red requires 8 tons. However, production is limited by the availability of only 50 cubic yards of storage space for aging and 120 hours of processing time. A batch of each type of wine requires 5 yd3 of storage space. The processing time for a batch of Nectar is 15 hours and the processing time for a batch of red is 8 hours. Demand for each type of wine is at most 7 batches. The profit for a batch of nectar is $9,000 and the profit for a batch of red is $12,000 Define the decision variables and formulate the linear programming model for this problem. (Product mix) A company produces two brands of perfumed bath oils. Each brand contains a combination of three of four perfume essences, the quantities required per bottle being those set out in the table below:
While the remaining ingredients are readily available, quantities of these essences are in limited supply for the coming week. Supplies are shown as follows:
Each bottle of bath oil A produced yields a profit of $1.20 and each bottle of bath oil B a profit of $1.50. Define the decision variables and formulate the problem. (Product mix) A California grower has a 50-acre farm on which to plant strawberries and tomatoes. The grower has available 300 hours of labor per week and 800 tons of fertilizer, and has contracted for shipping space for a maximum of 26 acres’ worth of strawberries and 37 acres’ worth of tomatoes. An acre of strawberries requires 10 hours of labor and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labor and 20 tons of fertilizer. The profit from an acre of strawberries is $400 and the profit from an acre of tomatoes is $300. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit. (Product mix) Angela and Bob Ray keep a large garden in which they grow cabbage, tomatoes and onions to make two kinds of relish- chow-chow and tomato. The chow-chow is made primarily of cabbage, whereas the tomato relish is made mostly from tomatoes Both relishes include onions, bell peppers and spices. A jar of chow-chow contains 8 ounces of cabbage, 3 ounces of tomatoes and 3 ounces of onions, whereas a jar of tomato relish contains 6 ounces of tomatoes, 6 ounces of cabbage and 2 ounces of onions. The Rays grow 120 pounds of cabbage, 90 pounds of tomatoes and 45 pounds of onions each summer. The Rays can produce no more than 24 dozen jars of relish. They make $2.25 in profit from a jar of chow-chow and $1.95 in profit from a jar of tomato relish. The Rays want to know how many jars of each kind of relish to produce to generate the most profit. Formulate the LP model (Product mix) The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood, each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. (Product mix) A distributor handles three types of television sets- large color, small color and portable black and white- purchased directly from an importer. The distributor has 5.000 cubic feet of storage space. Each large color set requires 8, each small color set requires 5 and each black and white portable requires 4 cubic feet of storage space. Large color sets cost the distributor $360 each, while costs per unit are $300 for small color and $80 for black and white portable sets. The distributor calculates that he can make a $50 profit on each large color, $40 on each small color and $20 on each black and white portable set. Due to the restrictions on marketing possibilities, it is felt that no more than 450 color sets can be purchased for the coming month. The distributor has $32.000 available for the purchase of television sets. Define the decision variables and formulate this linear programming problem. (Feed mix blending) The Feed’n Ship Ranch fattens cattle for local farmers and ships them to meat markets in Kansas City and Omaha. The owners of the ranch seek to determine the amounts of cattle feed to buy so that minimum nutritional standards are satisfied, and at the same time total feed costs are minimized. The feed mix used can be made up of the three grains that contain the following ingredients per pound of feed.
The cost per pound of stocks X and Y and Z are $2, $4, and $2.50, respectively. The minimum requirement per cow per month is 4 pounds of ingredient A, 5 pounds of ingredient B, 1 pound of ingredient C and 8 pounds of ingredient D. The ranch faces one additional restriction: it can only obtain 500 pounds of stock Z per month from the feed supplier regardless of its need. Because there are usually 100 cows at the Feed’n Ship Ranch at any given time, this means that no more than 5 pounds of stock Z can be counted on for use in the feed of each cow per month (Product Mix) Three products are produced through 2 departments:K and L. In order to produce one unit of product A, 7 hours of processing is required in department K and 2 hours of processing is required in department L. One unit of product B requires 3 hours in dept. K and 4 hours in dept L. The processing requirements for product C is 1 hour and 6 hours respectively. Department K has an available weekly capacity of 280 hours and department L has an available weekly capacity of 190 hours. Management wishes to increase the capacity by purchasing new machines. In order to increase the capacity of department K there are two alternatives. Machine K1, which can increase the capacity of this department by 50 hours, can be purchased at a cost of $50,000; or Machine K2, which can increase the capacity by 150 units, can be purchased at a cost of $80,000. A similar opportunity exists for Department L. It is possible to increase the capacity of this department by 120 hours or by 320 hours by purchasing either Machine L1 or L2 respectively. L1 costs $30,000 and L2 costs $90,000. The profit contributions of the three products are $280, $120 and $215 respectively. Formulate the problem by assuming that the investment budget is $150,000 (Agriculture planning) The seasonal yield of olives in a Piraeus, Greece, vineyard is greatly influenced by a process of branch pruning. If olive trees are pruned every two weeks, output is increased. The pruning process, however, requires considerably more labor than permitting the olives to grow on their own that results in a smaller size olive. It also, though, permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labor and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labor hours but takes 2 acres of land. An olive grower has 250 hours of labor available and a total of 150 acres for growing. Because of the olive size difference, a barrel of olives produced on pruned trees makes a profit of $200, whereas a barrel of regular olives makes a profit of $300 per barrel. The grower has determined that because of uncertain demand, no more than 40 barrels of pruned olives should be produced. (Fertilizer Mix) A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are 2 brands of fertilizer to choose from, Super-gro and Crop-quick. Each brand yields a specific amount of nitrogen and phosphate, as follows: Chemical Contribution
The farmer’s field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. Super-gro costs $6 per bag, and Crop-quick costs $3. The farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing. (Workforce scheduling) A company employs telephone operators who work 8-hour shifts, either from 6:00 a.m. to 2:00 p.m; from 10:00 a.m to 6:00 p.m. or from 2:00 p.m. to 10:00 p.m. Those working the first shift are paid $40 per day, those the second $43 per day and those working the third $45 per day. The company has determined that the minimum numbers of operators that must be available at various times of the day are:
Define the decision variables and formulate the problem if the objective is to meet these requirements at the lowest possible cost (Workforce scheduling) Mazy’s Department Store has decided to stay open for business on a 24-hour basis. The store manager has divided the 24-hour day into six 4-hour periods and has determined the following minimum personel requirements for each period
Store personel must report to work at the beginning of one of the above time periods and must work for eight consecutive hours. The store manager wants to know the minimum number of employees to assign to each four-hour segment to minimize the total number of employees. Formulate and solve the problem. (Product Mix) Excellent Doors Company manufactures doors for sale to construction companies. It sells all the doors it manufactures. Each week 20 employees, each working eight-hour shifts, five days a week are assigned to the three processes- wood cutting, manufacturing and finishing. The following table gives the cutting, manufacturing and finishing times per door and the unit profits.
For the upcoming week, Darien, the manager has committed himself to satisfying a contract for at least 280 standard, 120 high-glazed and 100 engraved doors for the Angora Houses. To satisfy the contract, Darien may have to purchase some premanufactured doors from an outside supplier. Darien will only use premanufactured doors in the production of standard and high glazed models but not in the production of engraved doors. Those sold as standard doors require only six minutes of finishing time to meet quality specifications, and will net Darien only $15 profit. Those used for high-glazed doors require only 30 minutes of finishing time and yield Darien a net $50 profit. Clearly define the decision variables and formulate the problem. Sinking Fund. An investor seeks to establish an investment portfolio using the least possible initial investment that will generate specific amounts of capital at specific time periods in the future. Consider Lary Frendentall, who is trying to plan for his daughter Susan’s college expenses. Based on current projections (it is now the start of year 1), Larry anticipates that his financial needs at the start of each of the following years is as follows:
Larry has several investment choices to choose from at the present time. Each choice has a fixed return on investment and a specified maturity date. Assume that each choice is available for investment at the start of every year Since choices C and D are relatively risky choices, Larry wants no more than 30% of his total investment in those two choices at any point in time.
Larry wants to establish a sinking fund to meet his requirements. Note that at the start of year 1, the entire investment is available for investing in the choices. However, in subsequent years, only the amount maturing from a prior investment is available for investment. (Course Scheduling) Brenda Last, an undergraduate business major at State University, is attempting to determine her course schedule for Spring semester. She is considering seven 3-credit hour courses, which are shown in the following table. Also included are the average number of hours she expects to have to devote to each course each week (based on information from other students) and her minimim expected grade in each course based on an analysis of the grading records of the teachers in each course
An A in a couse earns 4 quality credits per hour, a B earns 3 quality credits, a C earns 2 quality credits, a D earns 1 quality credit, and an F earns no quality credits per hour. Brenda wants to select a schedule that will provide at least 2.0 grade point average. In order to remain a full-time student, which she must do to continue receiving financial aid, she must take at least 12 credit hours. Principles of Accounting, Corporate Finance, Quantitative methods and C-Programming all require a lot of computing and mathematics, and Brenda would like to take no more than two of these courses. To remain on schedule and meet prerequisites, she needs to take at least three of the following courses: Management I, Principles of Accounting, C-programming, and English. Brenda wants to develop a course schedule that will minimize the number of hours she has to work each week. Define the decision variables and formulate the model Solve the problem using computer. Indicate how many total hours Brenda should expect to work on these courses each week and her minimum grade point average (Plane Loading) Turkish airlines aims to load the cargo plane (flying on the Ankara-Hamburg route) in the best way. There are four sections of the plane: one main cabin, two wing cabins and one bottom cabin. The freight costs of each of the cabins differ according to the properties of cabins. A kg. of load costs $800 in the main cabin, $600 in the wing cabins and $500 in the bottom cabin. The loading capacity of the plane is 50 tons. In order to maintain flight safety, the loads of the wing cabins should be equal to each other. The load of the main cabin should be at least equal to the total of the loads of the wing cabins; and should be at least 2 times of the load of the bottom cabin. The plane should fly fully loaded. Define the decision variables and formulate the problem. (* MS-WS-2’de ve 347-WS-2’de var) (Course Scheduling) Three professors must be assigned to teach six sections of finance. Each professor must teach two sections of finance, and each has ranked the six time periods during which finance is taught. Professors are paid differing rates according to the time periods. The costs of each assignment is as shown on the Table below. (professor 1 is not available at 9 a.m., but is available from then on. Formulate the problem so as to assign professors to sections that will minimize the costs
(Product mix) Modem Corporation of Americia (MCA) is the world’s largest producer of modem communication devices for microcomputers. MCA sold 9,000 of the regular model and 10,400 of the smart (intelligent) model this September. Its income statement for the month is shown below. Costs presented are typical of prior months and are expected to remain at the same levels in the near future. The firm is facing several constraints as it prepares its November production plan. First, it has experienced a tremendous demand and has been unable to keep any significant inventory in stock. This situation is not expected to change. Second, the firm is located in a small Iowa town from which additional labor is not readily available. Workers can be shifted from production of one modem to another, however. To produce the 9,000 regular modems in September required 5,000 direct labor hours. The 10,400 intelligent modem absorbed 10,400 direct labor hours. Third, MCA is experiencing a problem affecting the intelligent modem model. Its component supplier is able to guarantee only 8,000 microprocessors for November delivery. Each intelligent modem requires one of these specially made microprocessors. Alternative suppliers are not available on short notice. MCA wants to plan the optimal mix of the two modem models to produce in November to maximize profits for MCA
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