# Anderson Electronics is considering the production of four ...

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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:

| |Quantity of Perfume Essence (in ounces) |

|Bath Oil |1 |2 |3 |4 |

|A |0.2 |0.4 |0 |0.2 |

|B |0.5 |0 |0.2 |0.1 |

While the remaining ingredients are readily available, quantities of these essences are in limited supply for the coming week. Supplies are shown as follows:

|Availability of Essence (in ounces) |

|1 |2 |3 |4 |

|200 |200 |150 |150 |

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.

| |Feed (oz) |

|Ingredient |Stock X |Stock Y |Stock Z |

|A |3 |2 |4 |

|B |2 |3 |1 |

|C |1 |0 |2 |

|D |6 |8 |4 |

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

| |Nitrogen (lb/bag) |Phosphate |

|Product | |(lb/bag) |

|Super-gro |2 |4 |

|Crop-quick |4 |3 |

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:

|Time |Minimum Number of Operators |

|8 a.m. – 10 a.m |3 |

|10 a.m.- 2 p.m. |4 |

|2 p.m.- 4 p.m. |12 |

|4 p.m.- 6 p.m. |5 |

|8 p.m. – 10 p.m. |2 |

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

|Time |Personel Needed |

|Midnight-4:00 a.m. |90 |

|4:00-8:00 a.m. |215 |

|8:00 a.m.- Noon |250 |

|Noon-4:00 p.m. |65 |

|4:00- 8:00 p.m. |300 |

|8:00 p.m.- Midnight |125 |

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.

| | | |Finishing Time | |

| |Cutting Time (minutes) |Manufacturing Time (minutes) |(minutes) |Unit Profit (\$) |

|Door | | | | |

|Standard |45 |30 |15 |\$45 |

|High glazed |60 |30 |30 |\$90 |

|Engraved |30 |60 |30 |\$120 |

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:

|Year 3 |\$20,000 |

|Year 4 |\$22,000 |

|Year 5 |\$24,000 |

|Year 6 |\$26,000 |

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.

|Choice |ROI |Maturity |

|A |5% |1 year |

|B |13% |2 years |

|C |28% |3 years |

|D |40 % |4 years |

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

|Course |Average Hours per Week |Minimum Grade |

|1. Management I |5 |B |

|2. Principles of Accounting |10 |C |

|3.Corporate Finance |8 |C |

|4. Quantitative Methods |12 |D |

|5. Marketing Management |7 |C |

|6 C-programming |10 |D |

|7. English Literature |8 |B |

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.

a. Define the decision variables and formulate the model

b. 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

| |9 a.m |10 a.m. |11 a.m. |1 p.m. |2 p.m. |3 p.m. |

|Professor 1 |8 |7 |6 |5 |7 |6 |

|Professor 2 |9 |9 |8 |8 |4 |4 |

|Professor 3 |3 |7 |9 |6 |9 |9 |

(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

| |Regular Modems |Intelligent Modems |

|Sales |\$ 450,000 | 640,000 |

|Less: Discounts | 10,000 | 15,000 |

|Returns |12,000 |9,500 |

|Warranty replacements |4,000 |2,500 |

|Net Sales | 424,000 | 613,000 |

|Sales Costs | | |

| Direct labor | 60,000 | 76,800 |

|Indirect labor |9,000 |11,520 |

|Materials cost |90,000 |128,000 |

|Depreciation |40,000 |50,800 |

|Cost of sales |199,000 |267,120 |

|Gross profit | 225,000 | 345,880 |

|Selling and general expenses | | |

| General expenses- variable | 30,000 | 35,000 |

|General expenses-fixed |36,000 |40,000 |

|Sales commissions |31,000 |60,000 |

|Total operating cost |125,000 |160,000 |

|Pretax income | 100,000 | 185,880 |

|Income taxes (25%) | 25,000 | 46,470 |

|Net income | 75,000 | 139,410 |

(Production planning) Ford produces model A automobiles. The demand for the next three months are: 750, 850 and 900 units of A model. Production is realized by using normal time and overtime. The normal time production capacity of the plant is 700 units per month. Overtime hours cannot exceed 30% of normal time. Normal time costs \$5/hour, overtime costs 20% more. It is possible to stock automobiles from one month to the next. Inventory holding cost is \$25/automobile/month. It takes 10 hours to produce an automobile. There is no inventory at the beginning of the 3-month period, and no inventory accumulation is required at the end of the third month. Formulate the problem assuming that the demand is fully met.

(Production planning) The T.E. Callarman Appliance Company is thinking of manufacturing and selling compactors on an experimental basis over the next 6 months. The manufacturing costs and selling prices of the compactors are projected to vary from month to month. These are given below:

|Month |Manufacturing Cost |Selling Price |

|July |\$60 |\$90 |

|August |\$60 |\$80 |

|September |\$50 |\$60 |

|October |\$60 |\$70 |

|November |\$70 |\$80 |

|December |\$60 |\$90 |

Shipments are made in one large load at the end of that month. The firm can sell as many as 300 units per month, but its operation is limited by the size of its warehouse, which holds a maximum of 100 compactors.

Callarman’s operations manager needs to determine the number of compactors to manufacture and sell each month in order to maximize the firm’s profit. Callarman has no compactors on hand at the beginning of July and wishes to have no compactors on hand at the end of the test period in December. Define the decision variables and formulate the problem assuming all demand is to be met and also assuming that no inventory holding costs are incurred.

(Portfolio selection) Heinlein and Krampf Brokerage firm has just been instructed by one of its clients to invest \$250,000 for her. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever bonds they believe are well rated, but within the following guidelines:

a. Municipal bonds should constitute at least 20% of the investment

b. At least 40% of the funds should be placed in a combination of electronics firms, aerospace firms and drug manufacturers

c. No more than 50% of the amount invested in municipal bonds should be placed in nursing home bonds.

Subject to these restraints, the client’s goal is to maximize projected return on investments. The list of high-quality stoks and bonds and their corresponding rates of return are shown below.

|Investment |Projected Rate of Return (%) |

|Los Angeles municipal bonds |5.3 |

|Thompson Electronics, Inc. |6.8 |

|United Aerospace Corp. |4.9 |

|Palmer Drugs |8.4 |

|Happy Days Nursing Homes |11.8 |

Define the decision variables and formulate the problem using LP (20 points)

(Portfolio selection) The stock brokerage firm of Blank, Leibowitz and Weinberger has analyzed and recommended two stocks to an investors’ club of college professors. The professors were interested in factors such as short-term growth, intermediate-term growth and dividend rates. The data on each stock are as follows:

| |Stock (\$) |

|Factor |Louisiana Gas and Power |Trimex Insulation Company |

|Short-term growth potential per dollar invested| | |

| |.36 |.24 |

|Intermediate-term growth potential (over the | | |

|next three years), per dollar invested |1.67 |1.50 |

|Dividend rate potential |4% |8% |

Each member of the club has an investment goal of (1) an appreciation of no less than \$720 in the short term, (2) an appreciation of at least \$5,000 in the next three years, and (3) a dividend income of at least \$200 per year. What is the smallest investment that a professor can make to meet these three goals?

(Production planning) Boralis manufactures backpacks for serious hikers. The demand for the product occurs during March to May of each year. Boralis estimates the demand for the 3 months to be 100, 200 and 180 units respectively. The monthly demand must be met on time.

It is estimated that the normal time capacity of the facility is 180 units and overtime capacity is 20 units per month from March to May. Because the production capacity and demand for the different months do not match, the current month’s demand may be satisfied in one of three ways:

a. Current month’s normal (regular) time production

b. Current month’s overtime production

c. Surplus production in an earlier month

In the first case, the (regular time) production cost per backpack is \$40. In the second case, the overtime production cost is \$60. The third case incurs an additional holding cost of \$2 per backpack per month. There is no inventory at the beginning of the first month and it is not desired to have inventory at the end of the third month. Boralis wishes to determine the optimal production schedule for the 3 months. Define the decision variables and formulate the problem.

(Product mix) Consider an economy with just 3 industries (steel, coal and electricity) To produce one unit of steel, 0.05 units of steel, 0.40 units of coal and 0.25 units of electricity are used. To produce one unit of coal, 0.10 units of steel, 0.15 units of coal and 0.30 units of electricity are used. To produce one unit of electricity,0.20 units of steel, 0.60 units of coal and 0.20 units of electricity are used. The economy needs 200 units of steel, 150 units of coal and 180 units of electricity as final demands for the next quarter. Define the decision variables and formulate the constraints of this problem.

(Product mix-production planning) A certain corporation has three branch plants with excess production capacity. All three plants have the capability for producing a certain product and management has decided to use some of the excess production capacity in this way. This product can be made in three sizes- large, medium and small- that yield a net unit profit of \$140, \$120 and \$100 respectively. Plants 1, 2 and 3 have the excess manpower and equipment capacity to produce 750, 900 and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved. However, the amount of available in-process storage space also imposes a limitation on the production rates. Plants 1, 2 and 3 have 13.000, 12.000 and 5.000 square feet of in-process storage space available for a day’s production of this product. Each unit of the large, medium and small sizes produced per day requires 20, 15 and 12 square feet of storage space respectively.

Sales forecasts indicate that at most 900, 1200 and 750 units of the large, medium and small sizes respectively can be sold per day.

To maintain a uniform work load among the plants and to retain some flexibility, management has decided that the additional production assigned to each plant must use the same percentage of the excess manpower and equipment capacity.

Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profits. Define the decision variables and formulate the problem.

(Central planning) One of the interesting social experiments in the Mediterranean region is the system of kibbutzim on communal farming communities, in Israel. It is common for groups of kibbutzim to join together to share common technical services and to coordinate their production. Our example concerns one such group of three kibbutzim, which we call the Southern Confederation of Kibbutzim (SCK). Overall planning for the SCK is done in its Coordinating Technical Office. This office currently is planning agricultural production for the coming year.

The agricultural output of each kibbutz is limited by both the amount of available irrigable land and by the quantity of water allocated for irrigation by the water commissioner (a national government official). These data are given below:

|Kibbutz |Usable land (acres) |Water allocation (acre feet) |

|1 |400 |600 |

|2 |600 |800 |

|3 |300 |375 |

The crops suited for this region include sugar beets, cotton and sorghum and are the only ones being considered for the upcoming season. These crops differ primarily in their expected net return per acre and their consumption of water. In addition, the Ministry of Agriculture has set a maximum quota for the total acreage that can be devoted to each of these crops by the SCK as shown in the table below:

| |Maximum quota (acres) |Water consumption (acre feet/acre) |Net return (dollars/acre)|

|Crop | | | |

|Sugar beets |600 |3 |400 |

|Cotton |500 |2 |300 |

|Sorghum |325 |1 |100 |

The three kibbutzim belonging to the Southern Confederation have agreed that every kibbutz will plant the same proportion of its available irrigable land. However, any combination of the crops may be grown at any of the kibbutzim. The job facing the Coordinating Technical Office is to plan how many acres to devote to each crop at the respective kibbutzim while satisfying the above restrictions. The objective is to maximize the total net return to the Southern Confederation as a whole. Define the decision variables and formulate the problem.

(Transportation) A corporation manufactures fertilizer and currently has quantities available at three depots. There are 600 bags available at depot 1, 500 bags available at depot 2 and 400 bags at depot 3. Orders have been placed by three distribution centers. 300 bags are required at center A; 700 bags at center B and 500 bags at center C. The table below shows the cost per bag, in dollars, of shipping fertilizer from each depot to each distribution center.

|  |Distribution Center |

|Depot |A |B |C |

|1 |10 |7 |5 |

|2 |8 |6 |9 |

|3 |6 |7 |4 |

Define the decision variables and formulate the problem.

(Transportation) The Easy Drive Car Rental Agency needs 500 new cars in its Nashville operation and 300 new cars in Jacksonville and it currently has 400 new cars in both Atlanta and Birmingham. It costs \$30 to move a car from Atlanta to Nashville, \$70 to move a car from Atlanta to Jacksonville, \$40 from Birmingham to Nashville and \$60 from Birmingham to Jacksonville. The agency wants to determine how many cars should be transported from the agencies in Atlanta and Birmingham to the agencies in Nashville and Jacksonville in order to meet demand while minimizing the transport costs.

(Transportation) A manufacturer of television sets has two plants and two distribution centers. For the coming week, distribution center A requires 300 sets, and center B 250 sets. At most, 275 sets will be available at plant 1 and at most 325 sets will be available at plant 2. Television sets can be shipped from either plant to either distribution center. However, unit shipping costs differ along the four routes. It costs \$10 per set for shipments from plant 1 to center A, \$12 per set for shipments from plant 1 to center B, \$14 per set for shipments from plant 2 to center A and \$11 per set for shipments from plant 2 to center B. Demand at the two centers is to be fully met. Define the decision variables and formulate the problem (* MS-WS-2’de ve 347-WS-2’de var)

(Transportation) Don Yale, president of Hardrock Concrete Company has plants in three locations and is currently wondering on three major construction projects, located ad different sites. The shipping cost per truckload of concrete, plant capacities and project requirements are provided in the accompanying table

| To |Const. |Cons. |Cons. |Plant Capacities |

|From |Project |Project |Project | |

|Plant 1 |\$10 |4 |11 |70 |

|Plant 2 |12 |5 |8 |50 |

|Plant 3 |9 |7 |6 |30 |

|Project requirements |40 |50 |50 |150 |

| | | | |140 |

(Transportation). Oranges are grown, picked and then stored in warehouses in Tampa, Miami and Fresno. These warehouses supply oranges to markets in New York, Phildelphia, Chicago and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply and demand. Because of an agreement between distributors, shipments are prohibited from Miami to Chicago:

| To | | | | | |

|From |New York |Philadelphia |Chicago |Boston |Supply |

|Tampa |\$9 |14 |12 |17 |200 |

|Miami |11 |10 |6 |10 |200 |

|Fresno |12 |8 |15 |7 |200 |

|Demand |130 |170 |100 |150 | |

Formulate this problem as a linear programming model

(Transportation) During the war in Iraq, large amounts of military material and supplies had to be shipped daily from supply depots in the US to bases in the Middle East. The critical factor in the movement of these supplies was speed. The following table shows the number of plane loads of supplies available each day from each of six supply depots and the number of daily loads demanded at each of five bases. (Each planeload is approximately equal in tonnage.) Also included are the transprort hours per plane, including loading and fueling, actual flight time and unloading and refueling.

| | Military | |Base | |

|Supply Depot | | | | | | |

| |A |B |C |D |E |Supply |

|1 |36 |40 |32 |43 |29 |7 |

|2 |28 |27 |29 |40 |38 |10 |

|3 |34 |35 |41 |29 |31 |8 |

|4 |41 |42 |35 |27 |36 |8 |

|5 |25 |28 |40 |34 |38 |9 |

|6 |31 |30 |43 |38 |40 |6 |

|Demand |9 |6 |12 |8 |10 | |

Determine the optimal daily flight Schedule that will minimize total transport time

(Transshipment). Walsh’s Fruit Company contracts with growers in Ohio, Pennsylvania and New York to produce grapes. The grapes are processed into juice at the farms and stored in refrigerated vats. Then the juice is shipped to two plants, where it is processed into bottled grape juice and frozen concentrate. The juice and concentrate are then transported to three food warehouses/ distribution centers. The transportation costs per ton from the farms to the plants and from the plants to the distributors and the supply at the farms and demand at the distribution centers are summarized in the following tables

Plant

|Farm |4. Indiana |5. Georgia |Supply (1,000 tons) |

|1. Ohio |16 |21 |72 |

|2. Pennysylvania |18 |16 |105 |

|3. New York |22 |25 |83 |

Distribution Center

|Plant |6. Virginia |7. Kentucky |8. Lousiana |

|4. Indiana |23 |15 |29 |

|5. Georgia |20 |17 |24 |

|Demand (1,000 t) |90 |80 |120 |

Formulate the problem

(Transportation) A sports apparel company has received an order for a college basketball team’s national championship T-shirt. T-shirts are produced in 3 plants. The shirts are shipped fromthe factories to the 3 distribution centers. Following are the production and transportation costs (\$/shirt) from the T-shirt factories to the distribution centers, plus the supply of T-shirts at the factories and demand for the shirts at the distribution centers.

Production costs:

|Plant 1 |\$100 |

|Plant 2 |\$105 |

|Plant 3 |\$90 |

Distribution costs:

| |Dist. Cen. |DC |DC |Plant Capacities |

|To | | | | |

|From | | | | |

|Plant 1 |\$10 |4 |11 |170 |

|Plant 2 |12 |5 |8 |250 |

|Plant 3 |9 |7 |6 |130 |

| | | | |550 |

|Demand |140 |250 |150 |540 |

(Transportation) Bayville has built a new elementary school so that the town now has a total of four schools- Addison, Beeks, Canfield and Daley. They have capacities (the maximum number of students that can be registered) of 300, 300, 600 and 500 respectively. The school wants to assign children to schools so that their travel time by bus is as short as possible. The school has partitioned the town into three districts conforming to population density- north, south and east. The average bus travel time from each district to each school (from a source to a destination) is shown as follows. Travel time (cost) matrix

|District | | |

|1. Denver |2 |3 |

|2. Atlanta |3 |1 |

| From | | | | |

|To |5. Detroit |6. Miami |7. Dallas |8. New Orleans |

|3. Kansas City |2 |6 |3 |6 |

|4. Louisville |4 |4 |6 |5 |

* The capacities of plants are 600 units at Denver and 400 units in Atlanta.

* The demands at retail outlets are estimated to be 200 units at Detroit, 150 units at Miami, 350 units at Dallas and 250 units at New Orleans.

Draw the network, define the decision variables and formulate this transshipment problem assuming that Ryan does not want to ship any items From Kansas City to Miami.

(Course Assignment) A university department head has five instructors to be assigned to four different courses all the instructors have thaught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table (a perfect sore is 100)

Course

|Instructor |A |B |C |D |

|1 |80 |75 |90 |85 |

|2 |95 |90 |90 |97 |

|3 |85 |95 |88 |91 |

|4 |93 |91 |80 |84 |

|5 |91 |92 |93 |88 |

The department head wants to know the optimal assignment of instructors to courses to maximize the overall average evaluation. The instructor who is not assigned to teach a course will be assigned to grade exams. Formulate the problem.

(Assignment) . There are five professors and four sections of finance. Professors are paid differing rates according to the time periods. The costs of each assignment is as shown on the Table below. Because of personal reasons Professor 4 should not be assigned to the section whose class start time is 1 p.m. The professor who is not assigned to teach a course will be assigned to grade exams. Formulate the problem so as to assign professors to sections that will minimize the costs. What will be the impact of the imbalance of the problem on the optimal solution. (10 points)

| |9 a.m |10 a.m. |11 a.m. |1 p.m. |

|Professor 1 |8 |7 |6 |5 |

|Professor 2 |9 |9 |8 |8 |

|Professor 3 |3 |7 |9 |6 |

|Professor 4 |6 |5 |4 |5 |

|Professor 5 |7 |6 |8 |5 |

(Assignment). 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. Formulate the problem so as to assign professors to sections that will minimize the costs by assuming that it is impossible to assign Professor 1 to the courses that begin at 9 a.m.

State the impact of the imbalance of the problem on the course Schedule. What do you recommend to the department head.

| |9 a.m |10 a.m. |11 a.m. |1 p.m. |2 p.m. |3 p.m. |

|Professor 1 |8 |7 |6 |5 |7 |6 |

|Professor 2 |9 |9 |8 |8 |4 |4 |

|Professor 3 |3 |7 |9 |6 |9 |9 |

(Assignment). A dispatcher for the Citywide Taxi Company has six taxicabs at different locations and five customers who have called for service. The mileage from each taxi’s present location to each customer is shown in the following table. Determine the optimal assignment(s) that will minimize the total mileage traveled.

| | Customer | | |

|Cab |1 |2 |3 |4 |5 |

|A |7 |2 |4 |10 |7 |

|B |5 |1 |5 |6 |6 |

|C |8 |7 |6 |5 |5 |

|D |2 |5 |2 |4 |5 |

|E |3 |3 |5 |8 |4 |

|F |6 |2 |4 |3 |4 |

(Assignment) Given the following cost table for an assignment problem. Determine the optimal assignment and compute the minimum cost. Identify all alternative solutions if there are multiple optimal solutions. (8)

Machine

|Operator |A |B |C |D |

|1 |\$10 |2 |8 |6 |

|2 |9 |5 |11 |9 |

|3 |12 |7 |14 |14 |

|4 |3 |1 |4 |2 |

(Assignment) Prentice-Hall wants to assign recently hired colloge graduates: Jones, Smith, Andy and Wilson to regional sales districts in Omaha, Dallas and Miami. But the firm also has an opening in New York and would send one of the three there if it were more economical than a move to Omaha, Miami, or Dallas. It will cost \$10 to relocate Jones to New York, \$8 to relocate Smith there and \$15 to move Wilson.What is the optimal assignment of personel to offices?

|Office |Omaha |Miami |Dallas |

|Hiree | | | |

|Jones | | | |

|Smith | | | |

|Wilson | | | |

(Integer Programming) Consider a minimization Integer Linear Programming problem. Does the optimal value for the LP relaxation provide an upper or a lower bound for the optimal value of the ILP? Explain your answer

(Binary Programming) Melissa Jacobson, an undergraduate business major at State University, is attempting to determine her course schedule for the fall semester. She is considering nine 3-credit-hour courses, which are shown in the following table. Of these courses Production Management, Organizational Theory, Corporate Finance and Marketing Management are must courses. 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 minimum expected grade in each course based on an analysis of the grading records of the teachers in each course.

| |Average hours per week | |

|Production Management |11 |C |

|Organizational Theory |6 |B |

|Entrepreneurship |6 |B |

|Principles of Accounting |10 |C |

|Corporate Finance |8 |C |

|Quantitative Methods |12 |D |

|Marketing Management |7 |C |

|C-Programming |10 |D |

|English Literature |8 |B |

An A in a course earns 4 quality credits per credit-hour, a B earns 3 quality credits, a C earns 2 credits, a D earns 1 quality credits, an F earns no quality credits per hour. Melissa wants to select a schedule that will provide the highest possible grade point avarage. In order to remain a full-time student, which she must do to continue receiving financial aid, she must take at least 18 credit-hours. Principles of Accounting, Corporate Finance, Quantitative Methods, and C-Programming all require a lot of computing and mathematics, and Melissa would like to take no more than two of these courses. Melissa wants to develop a course which is within her capability of working 58 hours per week. Formulate a 0-1 integer programming model for this problem.

(Binary Programming) 5. A corporation is planning its R&D budget over the next three years the table below shows the costs of each of the three possible projects and the quantity of funds available (both in hundreds of thousands of dollars) in each of the three years.

Project Project Costs

| |Year 1 |Year 2 |Year 3 |

|A |18 |20 |22 |

|B |24 |21 |20 |

|C |21 |23 |47 |

|Available funds |46 |46 |47 |

The present values of expected future profits from the three projects are \$800,000, \$700000 and \$ 850,000 respectively. Formulate the problem. (8)

(Binary Programming) A manufacturing company has decided to expand by building a new factory in either Los Angeles or San Francisco. It is also considering building a new warehouse in whichever city is selected for the new factory. (It may or not may not build a warehouse, but if it decides to build it, it should be built at the city in which the factory is built). The total profitability of each of these alternatives is shown in the Table below. The last column gives the capital required for the respective investments, where the total capital available is \$40,000,000. The objective is to find the feasible combination of alternatives that maximizes the total profitability.

|Alternative |Total annual profitability |Capital requirements |

|Build factory in Los Angeles |\$7 million |\$20 million |

|Build factory in San Fransisco |\$5 million |\$15 million |

|Build warehouse in Los Angeles |\$4 million |\$ 12 million |

|Build warehouse in San Fransisco |\$3 million |\$ 10 million |

Define the decision variables and formulate the problem by taking into account that

Note that:

➢ The company wants to build only one new factory

➢ The company may or may not build a warehouse in a city selected for new factory. But in order to build a warehouse in a city, the decision to build a factory should have to be made, ie. İt cannot build a warehouse in a city in which a factory is not built.

(Fixed Charge) GAP Inc. needs to decide on the locations for two new warehouses. The candidate sites are Philadelphia, Tampa, Denver and Chicago. The following table provides the monthly cpacities and the mothly fixed costs for operating warehouses at each potential site.

|Warehouse |Monthy Capacity |Monthly Fixed Cost |

|Philadelphia (1) |250 units |\$1,000 |

|Tampa (2) |260 units |\$ 800 |

|Chicago (3) |280 units |\$1,200 |

|Denver (4) |270 units |\$ 700 |

The warehouses will need to ship to three marketing areas: North, South, and West. Monthly requirements are 200 units for North, 180 units for South, and 120 units for West. The following Table provides the cost to ship one unit between each location and destination:

|Marketing area |North (N) | South (S) |West (W) |

|Warehouse | | | |

|Philadelphia (1) |\$4 |\$7 |\$9 |

|Tampa (2) |\$6 |\$3 |\$11 |

|Chicago (3) |\$5 |\$6 |\$5 |

|Denver (4) |\$8 |\$10 |\$2 |

In addition the following conditions must be met by the final decision:

• A warehouse must be opened in either Philadelphia or Denver

• If a warehouse is opened in Tampa, then one must also be opened in Chicago

Define the decision variables and formulate the model in order to determine which two sites should be selected for the new warehouses to minimize total fixed and shipping costs

(Fixed Charge and Facility Location) Frijo-Lane Products own farms in Southwest and Midwest where it grows and harvests potatoes. It then ships these potatoes to three processing plants in Atlanta, Baton Rouge and Chicago where different varieties of potato products, including chips, are produced. Recently, the company has experienced a growth in its product demand so it wants to buy one or more new farms to produce more potato products. The company is considering six new farms with the following annual fixed costs and projected harvest.

|Farms |Fixed Annual Costs (\$1,000s) |Projected Annual Harvest |

| | |(thousands of tons) |

|1 |\$405 |11.2 |

|2 |390 |10.5 |

|3 |450 |12.8 |

|4 |368 |9.3 |

|5 |520 |10.8 |

|6 |466 |9.6 |

The company currently has the following additional available production capacity

(tons) at its three plants that it wants to utilize:

|Plant |Available Capacity |

| |(thousands of tons) |

|A |12 |

|B |10 |

|C |14 |

The shipping costs (\$) per ton from the farms being considered for purchase to the plants are as follows:

| |Plant (shipping costs, \$/ton) |

|Farm |A |B |C |

|1 |18 |15 |12 |

|2 |13 |10 |17 |

|3 |16 |14 |18 |

|4 |19 |15 |16 |

|5 |17 |19 |12 |

|6 |14 |16 |12 |

The company wants to know which of the six farms it should purchase that will meet available production capacity at the minimum total cost, including annual fixed costs and shipping costs.

(Fixed Charge). Gandhi Cloth Company is capable of manufacturing three types of clothing: shirts, shorts and pants. The manufacture of each type of clothing requires Gandhi to have the appropriate type of machinery available. There are 3 types of machines among which tha manager can cohhose: the first one costs \$200, the second one has a fixed cost of \$150 per week had the third one costs \$100. The machinery needed to manufacture each type of clothing must be rented at the following rates: shirt machinery, \$200 per week; shorts machinery, \$150 per week; pants machinery, \$100 per week. The first has a capacity of 400 machine hours per week, the second 350 hours qer week and thethird has a production capacity of 350 hours. The manufacture of each type of clothing also requires the amounts of cloth and labor shown below. Each week 150 hours of labor and 160 sq. yd. of cloth are available. The variable unit cost and selling price for each type of clothing are also shown below. (do not forget that Gandhi Cloth need the appropriate machinery if it is going to produce the item that require that machinery)

Define the decision variables and formulate the problem in order to maximize Gandhi’s weekly profits.

Resource Requirements for Gandhi

| |Labor (hours) |Cloth (square yards) |

|Shirt |3 |4 |

|Shorts |2 |3 |

|Pants |6 |4 |

Revenue and Cost Information for Gandhi

| |Sales Price |Variable Cost |

|Shirt |\$12 |\$ 6 |

|Shorts |\$8 |\$4 |

|Pants |\$6 |\$4 |

(Fixed Charge) During the war in Iraq, Terraco Motor Company produced a vehicle for the military. The company is now planning to sell the vehicle to the public. It has four plants that manufacture the vehicle and four regional distributon centers. The company is unsure of public demand for the vehicle, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table. The transportation costs are per thousand vehicles shipped. For example, the cost of shipping 1,000 vehicles from plant 1 to warehouse C is \$32,000.

Transportation costs (\$1,000s)

To Warehouse

| | | | | |Annual Production |Annual Fixed Costs |

|From |A |B |C |D |Capacity | |

|Plant | | | | | | |

|1 |36 |40 |32 |43 |12,000 |2,100,000 |

|2 |28 |27 |29 |40 |18,000 |850,000 |

|3 |34 |35 |41 |29 |14,000 |1,800,000 |

|4 |41 |42 |35 |27 |10,000 |1,100,000 |

|Annual |6,000 |14,000 |8,000 |10,000 | | |

|Demand | | | | | | |

Formulate an integer programming model for this problem that will assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plant to each warehouse to minimize total cost.

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