Tutorial Sheet ( ) Formulation of Optimization Problems Dr. Nidhal M. Al-Azzawi

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1 Tutorial Sheet ( ) Formulation of Optimization Problems Dr. Nidhal M. Al-Azzawi Q1) A company must ship coal from four mines, labeled A, B, C and D to supply three factories, labeled a, b and c. The per unit transport cost, the outputs of the mines, and the requirements of the factories are given below. Factories Transport Cost unit of money Max need A B C D a b c Max Production For instance, mine B can produce 32 units of coal, factory a requires 34 units of coal, and it costs 3 units of money to ship one unit of coal from B to a. What is the minimal cost of transporting coal from the mines to the factories? Q2) A factory produces three types of sugar, types X, Y, and Z, from three types of syrup, labeled A, B, and C. The following table contains the number of liters of syrup necessary to make each kilogram of sugar. Liter of Syrup kg of Sugar X Y Z A B C For instance, one kilogram of type X sugar requires 3 liters of A, 2 liters of B, and 4 liters of C. The factory can sell each type of sugar for one pound per kilogram. Assume that the factory owner can use no more than 44 liters of A and 51 liters of B, but is required by law to use at least 12 liters of C. If the goal is to maximize profit, how many kilograms of each type of sugar should the factory produce? Q3) An airline is reducing the service from its hub because of low demand. It wants to reduce the number of agents working at the hub but doesn t know what is the right number of agents. The primary goal of reducing work is of course cost cutting. The airline does want to maintain a satisfactory service level. Based on an OR study, a minimum number of agents needed at different times of the day was determined. Time Period Time Period Covered Shift Minimum Number of Agents Needed 6 am 8 am # 48 8 am 10 am # # am noon # # 65 noon 2 pm # # # 87 2 pm 4 pm # # 64 4 pm 6 pm # # 73 6 pm 8 pm # # 82 8 pm 10 pm # pm midnight # # 52 midnight - 6 am # 15 Let c 1, c 2, c 3, c 4, c 5 be the costs of hiring an agent in the five shifts respectively. The airline wants to find out how many people should be hired in each shift to meet the service level at the minimum possible cost.

2 Q4) A problem is to determine what mix of four products we should produce during the upcoming week using three machines. Objective is to maximize total profit. Machine Production Time per Unit Product hrs. Production Time Available P1 P2 P3 P4 per week, hrs Roll Cut Weld Profit/unit Q5) An open-topped rectangular tank is to be constructed so that the sum of the height and the perimeter of the base is 30 meters. Find the dimensions which maximize the surface area of the tank. What is the maximum value of the surface area? [You may assume that the maximum exists, and that the corresponding dimensions of the tank are strictly positive.] Q6) 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? Q7) A gold processor has two sources of gold ore, source A and source B. In order to keep 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? Q8) A publisher has orders for 600 copies of a certain text from San Francisco and 400 copies from Sacramento. The company has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs $5 to ship a text from Novato to San Francisco, but it costs $10 to ship it to Sacramento. It costs $15 to ship a text from Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento. How many copies should the company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost? Q9). A vitamin manufacturer wants to produce a vitamin A elixir. There are two sources of vitamin A, xylocol and yellow gel. The manufacturer wants to produce 10 ml. bottles of the elixir and wants each bottle to cost less than $12. Xylocol costs $2 per ml. and yellow gel costs $1 per ml. The process used requires more yellow gel than xylocol. If xylocol produces 3 mg. of vitamin A per ml. and yellow gel produces 1 mg. of vitamin A per ml., how many ml. of each should be used to maximize the output of vitamin A? Q10). A manufacturer produces two items, bookcases and library tables. Each item requires processing in each of two departments. Department 1 has 40 hours available and department 2 has 36 hours available each week for production. To manufacture a bookcase requires 2 hours in department 1 and 4 hours in department 2 while a library table requires 3 hours in department 1 and 2 hours in department 2. Profits on the items are $6 and $7 respectively. If all the units can be sold, how many of each should be made to maximize profits? Q11) 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? Q12) 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?

3 Q13) 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 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 Q14) You have $12,000 to invest, and three different funds from which to choose. The municipal bond fund 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? Solve it with two variables. Q15) A building supply has two locations in town. The office receives orders from two customers, each requiring 3/4-inch plywood. Customer A needs fifty sheets and Customer B needs seventy sheets. The warehouse on the east side of town has eighty sheets in stock; the west-side warehouse has forty-five sheets in stock. Delivery costs per sheet are as follows: $0.50 from the eastern warehouse to Customer A, $0.60 from the eastern warehouse to Customer B, $0.40 from the western warehouse to Customer A, and $0.55 from the western warehouse to Customer B. Find the shipping arrangement which minimizes costs. Q16) A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy Q17) A potter is making cups and plates. It takes her 6 minutes to make a cup and 3 minutes to make a plate. Each cup uses 3/4 lb. of clay and each plate uses one lb. of clay. She has 20 hours available for making the cups and plates and has 250 lbs. of clay on hand. She makes a profit of $2 on each cup and $1.50 on each plate. How many cups and how many plates should she make in order to maximize her profit? Q18) A plant makes aluminum and copper wire. Each pound of aluminum wire requires 5 kwh of electricity and 1/4 hr. of labor. Each pound of copper wire requires 2 kwh of electricity and 1/2 hr. of labor. Production of copper wire is restricted by the fact that raw materials are available to produce at most 60 lbs./day. Electricity is limited to 500 kwh/day and labor to 40 person hrs./day. If the profit from aluminum wire is $.25/lb. and the profit from copper is $.40/lb., how much of each should be produced to maximize profit and what is the maximum profit? Q١٩) A company makes two types of sofas, regular and long, at two locations, one in Hickory and one in Lenoir. The plant in Hickory has a daily operating budget of $45,000 and can produce at most 300 sofas daily in any combination. It costs $150 to make a regular sofa and $200 to make a long sofa at the Hickory plant. The Lenoir plant has a daily operating budget of $36,000, can produce at most 250 sofas daily in any combination and makes a regular sofa for $135 and a long sofa for $180. The company wants to limit production to a maximum of 250 regular sofas and 350 long sofas each day. If the company makes a profit of $50 on each regular sofa and $70 on each long sofa, how many of each type should be made at each plant in order to maximize profit? What is the maximum profit? Q2٠) 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 maximize 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.

4 Q2١) 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 craftsman time for hand finishing. The table below gives the number of minutes required for each item: Machine time Craftsman time X 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 cost 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. Q2٢) A company manufactures two products (A and B) and the profit per unit sold is 3 and 5 respectively. Each product has to be assembled on a particular machine, each unit of product A taking 12 minutes of assembly time and each unit of product B 25 minutes of assembly time. The company estimates that the machine used for assembly has an effective working week of only 30 hours (due to maintenance/breakdown). Technological constraints mean that for every five units of product A produced at least two units of product B must be produced. Formulate the problem of how much of each product to produce. The company has been offered the chance to hire an extra machine, thereby doubling the effective assembly time available. What is the maximum amount you would be prepared to pay (per week) for the hire of this machine and why? Q2٣) 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 to maximize the profit. Q2٤) A plant produces two types of refrigerators, A and B. There are two production lines, one dedicated to producing refrigerators of Type A, the other to producing refrigerators of Type B. The capacity of the production line for A is 60 units per day, the capacity of the production line for B is 50 units per day. A requires 20 minutes of labor whereas B requires 40 minutes of labor. Presently, there is a maximum of 40 hours of labor per day which can be assigned to either production line. Profit contributions are $20 per refrigerator of Type A produced and $30 per Type B produced. What should the daily production be? Q2٥) ChemCo produces two chemicals: A and B. These chemicals are produced via two manufacturing processes. Process 1 requires 2 hours of labor and 1 lb. of raw material to produce 2 oz. of A and 1 oz. of B. Process 2 requires 3 hours of labor and 2 lb. of raw material to produce 3 oz. of A and 2 oz. of B. Sixty hours of labor and 40 lb. of raw material are available. Chemical A sells for $16 per oz. and B sells for $14 per oz. Formulate a linear program that maximizes ChemCo s revenue. Hint: Define the amounts of Process 1 and Process 2 used, as your decision variables Q2٦) An electronics firm manufactures radio models A, B and C, which have profit contributions of 16, 30 and 50 respectively. Minimum production requirements are 20, 120 and 60 for the three models, respectively. A dozen units of Model A requires 3 hours for manufacturing of components, 4 for assembling, and 1 for packaging. The corresponding figures for a dozen units of Model B are 3.5, 5, and 1.5, and for Model C are 5, 8, and 3. During the forthcoming week the company has available 120 plant-hours for manufacturing components, 160 for assembly, and 48 for packaging. Formulate this production planning to maximize the profit. Q2٧) A company manufactures checker sets and chess sets. Suppose each day the company has available 1900 boards (which can be used for both games) and 80,000 units of wood for making pieces. Each checker set uses 20 units of wood and each chess set uses 80 units of wood. The distributors the company sells to can take up to 1250 checker sets per day and up to 750 chess sets per day. The company makes a profit of $1 on each checker set and $1.25 on each chess set. How many checker sets and how many chess sets should the company make each day in order to maximize its profits? What is the profit per day using this strategy?

5 Q2٨) ChemCo produces two fertilizers, FastGro and ReallyFastGro. Each is made of a mix of two growth agents. FastGro is a 50/50 mix of the two, and it sells for $13 per pound. ReallyFastGro is a 25/75 mix (i.e., its 25% Agent 1 and 75% Agent 2), and sells for $15/pound. Agent 1 costs $2/pound and Agent 2 costs $3/pound. ChemCo can purchase up to 250 pounds of Agent 1 and up to 350 pounds of Agent 2 each day. What is ChemCo's optimal production strategy? i.e., How much of each product should it produce in order to maximize its profits? Q٢٩) We need to blend iron ore from 4 mines in order to produce tire tread. To assure proper quality minimum requirements of aluminum, boron and carbon must be present in the final blend. In particular, there must be 5,100 and 30 pounds, respectively, in each ton of ore. These elements exist at different levels in the mines. In addition the cost of mining differs for each mine. The data are shown in table below. How much should be mined from each mine in order to attain a proper blend at minimum cost? Mine Aluminum (lbs/ton) Boron (lbs/ton Carbon (lbs/ton) Cost $800 $400 $600 $500 Q3٠) A company produces two varieties of a product. Variety A has a profit per unit of 2.00 and variety B has a profit per unit of Demand for variety A is at most four units per day. Production constraints are such that at most 10 hours can be worked per day. One unit of variety A takes one hour to produce but one unit of variety B takes two hours to produce. Ten square meters of space is available to store one day's production and one unit of variety A requires two square meters whilst one unit of variety B requires one square meter. Formulate the problem of deciding how much to produce per day to maximize the profit. Q3١) A cement manufacturer produces two types of cement, namely granules and powder. He cannot make more than 1600 bags a day due to a shortage of vehicles to transport the cement out of the plant. A sales contract means that he must produce at least 500 bags of powdered cement per day. He is further restricted by a shortage of time - the granulated cement requires twice as much time to make as the powdered cement. A bag of powdered cement requires 0.24 minutes to make and the plant operates an 8 hour day. His profit is 4 per bag for granulated cement and 3 per bag for the powdered cement. Formulate the problem of deciding how much he should produce to maximize profit.

6 Q3٢) A soft drink company has two bottling plants, one located at P and the other at Q. Each plant produces three different soft drinks A, B, and C. The capacities of the two plants in number of bottles per day, are as follows: Plants Products P Q A B C A market survey indicates that during the month of May, there will be a demand for bottles of A, bottles of B and bottles of C. The operating cost per day of running plants P and Q are respectively $1000 and $800. How many days should the firm run each plant in the month of May so that the production cost is minimized while still meeting the market demand? Q33) A patient just had a surgery and is required to have at least 60 mg of drug D1 and 12 mg of drug D2, also he had to take 10 mg of drug D3 and 84 mg of drug D4. These drugs have to be consumed by taking a certain amount of substances M and N, both of which contain unwanted drug D5. Each gm of M contains 20 mg of D1, 8 mg of D2, 1.4 mg of D3, 14 mg of D4 and 23 mg of D5. Each gm of N contains 15 mg of D1, 1.5 mg of D2, 1 mg of D3, 6 mg of D4 and 12 mg of D5. How many grams of M and N should be taken to fulfill the requirement of the four drugs also to minimize the intake of unwanted D5? Q34) An open rectangular box with square base is to be made from 48 ft 2 of material, what dimensions will result in a box with the largest possible volume? Q35) ) A refinery has available Three crude oils that have the yields shown below; Product Volume % yield Sale Price Max. Allowable Crude 1 Crude 2 Crude 3 $/bbl product rate 1000 bbl/day Gasoline Kerosene Diesel oil Fuel oil Residue No limit Processing cost $/bbl ٢ ٢ Raw material cost $/bbl Formulate this optimization problem to maximize the profit of the refinery.

7 Q36) A cylindrical can is to hold 20 m. 3 The material for the top and bottom costs $10/m. 2 and material for the side costs $8/m. 2 Find the radius r and height h of the most economical can. Q37) A sheet of cardboard 3. by 4. will be made into a box by cung equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume? Q38) Oil is to be shipped in barrels having an undetermined height h and radius r. Each barrel must have a fixed volume V. We want to pick r and h so as to minimize the surface area A of each barrel, and thereby minimize the cost of making the barrels. Find the radius at which A(r) has its globall minimum. What is the ratio h/r when r has the optimal value you just derived? Q39) Suppose you are renting a car at $1.2/hr, the gasoline to be supplied by you costs $0.35/gallon. You propose to drive from Baghdad to Amman, a distance of 900 miles. The speed must be between 40 to 80 miles/hr, assume that the following relation describes the mileage; m = 26 s/5 m = number of miles per gallon. and s = speed in miles/hr. How can you make this tripe as inexpensively as possible by good choice of s. Q40) Universal Electric manufactures and sells two models of lamps L 1 and L 2 the profit being $15 and $10, respectively. The process involves two workers W 1 and W 2 who are available for this kind of work 100 and 80 hours per month respectively.w 1 assembles L 1 in 20 minutes and L 2 in 30 minutes. W 2 paints L 1 in 20 minutes and L 2 in 10 minutes. Assuming that all the lamps made can be sold without difficulty, determine production of L 1 and L 2 that maximize the profit. Solve graphically and simplex method with constraints.