2. Find solution of following LPPs and solution of their dual problem too: a.

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1 1. Find solution of following LPPs with using Simplex method: x + 2x x + x max;10x 4x + x = 25; 2 x + 3x + x = 26; x ,...,4 b. x1 + 2x2 max;10x1 4x2 25; 2x1 + 3x2 6; x1 0; x2 0 3x x + x + 2x min; 2x x x = 0; x + x = 10; x + 3x = 4; x 0 c ,...,4 x + 3x min; 2x + x 6; 2x + 2x 6; 4x x 15; x 0 d ,2 7x 4x max; 3x + 5x 15; x + x 6; x 0 e ,2 2. Find solution of following LPPs and solution of their dual problem too: x 3x max; 2x + x 10; x + 3x 9; x + x b. x1 + 2x2 max;10x1 4x2 25; 2x1 + 3x2 6; x1 0; x2 0 c. x1 + 2x2 min; x1 + x2 1; 2x1 + 3x2 4; 2x1 10x2 5; x1 0; x2 0 d. x1 x2 max; x1 + 3x2 9; x1 + 2x2 14; x1 + 2x2 3; x1 6 e. x1 x2 x1 x2 x1 x2 x1 x2 x1 x2 f. 7x1 4x2 max; x1 2x2 11; 3x1 + 5x2 15; x1 + 2x2 4; x1 + x min; ; ; ; x x max; 3x + 3 x ; 5x x 6; 3x + x 1; 2x + x 3; x 0 g ,2 3. Find solution of dual problems of following LPPs with using primary problem solving. x 4x x max; 4x + 2x 2x 3; 3x x x 1; x ,2,3 12x + 8x min; 2x + x 4; 2x + 3x 8; x + 6x 6; x 0 b ,2 4. Solve given ILPPs: x + x max;10x + 7x 35; 2x + x 2; x 0; x Z ,2 1,2 x + x + x max; 2x x 3; x 4x + 3x 5; x 0; x Z b ,2,3 1,2,3 3x + 2x + 4x max; x + x + 2x 4; 2x + x 5; 2x + x + 3x 7; x 0; x Z c ,2,3 1,2,3 x min; 3x + 2x 6; 3x + 2x 0; x 0; x Z d ,2 1,2 5. You have $12,000 to invest, and three different funds from which to choose. The municipal bond fund (MBF) has a 7% return, the local bank's CDs have an 8% return, and the high-risk account has an expected (hoped-for) 12% return. To minimize risk, you decide not to invest any more than $2,000 in the high-risk account. For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the year-end yields are as expected, what are the optimal investment amounts? [MBF: 7500$; CD: 2500$; H-RF: 2000$] 6. A carpenter makes tables and chairs. Each table can be sold for a profit of 30 and each chair for a profit of 10. The carpenter can afford to spend up to 40 hours per week working and takes six hours to make a table and three hours to make a chair. Customer demand requires that he makes at least three times as many chairs as tables. Tables take up four times as much storage space as chairs and there is room for at most four tables each week. Formulate this problem as a linear programming problem and solve. [2 tables, 8 chairs] 7. In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of

2 carbohydrates, and 4 g of protein. But the rabbits should be fed no more than five ounces of food a day. Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0.20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0.30 per ounce. What is the optimal blend? [2/3 ounces of X, 8/3 ounces of Y] b. Solve this problem, if food X contains 8 g of fat, 6 g of carbohydrates, and 2 g of protein per ounce. [2/3 ounces of X, 8/3 ounces of Y] 8. At a certain refinery, the refining process requires the production of at least two gallons of gasoline for each gallon of fuel oil. To meet the anticipated demands of winter, at least three million gallons of fuel oil a day will need to be produced. The demand for gasoline, on the other hand, is not more than 6.4 million gallons a day. If gasoline is selling for $1.50 per gallon and fuel oil sells for $1.90/gal, how much of each should be produced in order to maximize revenue? [unfeasible problem] 9. Solve previous problem, if the demand for gasoline is not more than 16.4 (no 6,4) million gallons a day. [oil: 82/15 mil; gasoline: 16,4 mil] 10. A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? [SC: 100; GC: 170] 11. You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume? 12. A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B. At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours. The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week. Formulate the problem of deciding how much of each product to make in the current week as a linear program. Solve this linear program. [X: 45; Y: 6] 13. A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic processing and

3 craftsman time for hand finishing. The table below gives the number of minutes required for each item: Machine time Craftsman time Item X Item Y The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time is costed at 10 per hour worked and craftsman time is costed at 2 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is 20 for X and 30 for Y. The company has a specific contract to produce 10 items of X per week for a particular customer. Formulate the problem of deciding how much to produce per week as a linear program and solve it. [X: 105; Y: 0] 14. A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye how many acres of each should be planted to maximize profits? [plant: 4 acres; rye: 4 acres] 15. A gold processor has two sources of gold ore, source A and source B. In order to kep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints? [source A: 2 tons; source B: 4 tons] 16. A publisher has orders for 600 copies of a certain text from San Francisco, 400 copies from Sacramento and 300 copies from Dallas. The company has 700 copies in a warehouse in Novato, 800 copies in a warehouse in Lodi and 400 copies in a warehouse in Souris. It costs $5 to ship a text from Novato to San Francisco, but it costs $10 to ship it to Sacramento and it costs $10 to ship it to Dallas. It costs $15 to ship a text from Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento and it costs $11 to ship it from Lodi to Dallas. It costs $9 to ship a text from Souris to San Francisco, but it costs $7 to ship it to Sacramento and it costs $6 to ship it to Dallas. How many copies should the company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost? [6400 $] 17. Two students Ann and Carl work x and y hours per week, respectively. Together they can work at most 40 hours per week. According to the rules for part timers Ann can work at most 8 hours more that Carl. But Carl can work at most 6 hours more than Ann. There is an extra constraint 18 2y + x. Find their maximum combined income per week for next cases. If Ann and Carl earn $15 and $17 per hour, respectively. b. If Ann and Carl earn $17 and $15 per hour, respectively. c. If Ann and Carl both earn $16 per hour, respectively.

4 18. We have 15 bar pieces each with the length of 10 meters. We need to cut 12 bar pieces with the length of 3 meters and 15 bar pieces with the length of 4 meters. Suggest the optimal solution by minimizing the scrap. 19. We have 10 bar pieces each with the length of 10 meters. We need to cut 12 bar pieces with the length of 3 meters and 15 bar pieces with the length of 4 meters. Suggest the optimal solution by minimizing the scrap. 20. We have 10 bar pieces each with the length of 10 meters. We need to cut 15 bar pieces with the length of 3 meters and 12 bar pieces with the length of 4 meters. Suggest the optimal solution by minimizing the scrap. 21. There is a cutting machine available for a cutting line which is able to cut standardized bales with the width of 2,5 meters. From these standardized bales we have to cut the required amount of bales with following widths: 862 pc by 112 cm, 341 pc by 90 cm, 87 pc by 77 cm and 216 pc by 35 cm. Let s assume that we have sufficient number of the standardized bales and we cut only the required widths (of course scrap will be caused by this). Suggest the setting of the cutting tools in the cutting machine (and also their presetting) so that the scrap would be minimized 22. There is a cutting machine available for a cutting line which is able to cut standardized bales with the width of 2 meters. From these standardized bales we have to cut the required amount of bales with following widths: 862 pc by 112 cm, 341 pc by 77 cm and 216 pc by 35 cm. Let s assume that we have sufficient number of the standardized bales and we cut only the required widths (of course scrap will be caused by this). Suggest the setting of the cutting tools in the cutting machine (and also their presetting) so that the scrap would be minimized 23. A chain of hypermarkets has its central stores in BA, LM and KE. These central stores dispose of the amounts 40,20 and 40 units of the same item. The individual hypermarkets need these amounts of this item: TN 25, ZA 10, RV 20, BB 30, PP 15. Transport costs of 1 unit of this item from central stores into hypermarkets are listed in the following table. Design the supply with this item in order to minimize the transport costs. TN ZA RV BB PP KE BA LM A chain of hypermarkets has its central stores in BA, LM and KE. These central stores dispose of the amounts 60,20 and 40 units of the same item. The individual hypermarkets need these amounts of this item: TN 25, ZA 10, RV 20, BB 30, PP 15. Transport costs of 1 unit of this item from central stores into hypermarkets are listed in the following table. Design the supply with this item in order to minimize the transport costs. TN ZA RV BB PP KE BA LM

5 25. Find the optimal transport plan of the building material from 4 stone pits into 4 building sites warehouses, if the transport costs of 1 metric tonne of building material, stone pits capacity and demands of the building sites warehouses are listed in the following table: S 1 S 2 S 3 S 4 a i K K K K b j Find the optimal transport plan of the building material from 3 stone pits into 4 building sites warehouses, if the transport costs of 1 metric tone of building material, stone pits capacity and demands of the building sites warehouses are listed in the following table: S 1 S 2 S 3 S 4 a i K K K b j There is a sugar supply supposed to be delivered from three sugar refineries into 4 wholesales according to the following table. Design the supply of the wholesales with the sugar in order to minimize the transport costs. P 1 P 2 P 3 P 4 a i C C C b j Find the optimal solution of transport problem, which is determined by the following table (quantity of goods is given in tonnes and the price of transportation is in the hundreds of euros) O 1 O 2 O 3 O 4 a i D D D b j Find the optimal solution of transport problem, which is determined by the following table (quantity of goods is given in tonnes and the price of transportation is in the hundreds of euros) O 1 O 2 O 3 a i D D D b j

6 30. Find the optimal solution of transport problem, which is determined by the following table (quantity of goods is given in tonnes and the price of transportation is in the hundreds of dolars) O 1 O 2 O 3 a i D D D D b j