MS987 – Optimisation in Data Analytics Question bank in Linear programming modelling 1. Red Brand Canners used to be a medium-size company canning and distributing a variety of fruit and vegetable products in the western part of USA in the 1960s. In one year, they signed an agreement at planting time, to purchase the entire tomato crop in a large field at an average delivered price of 6 cents per lb. At harvest time, Produce Inspection estimates that the total crop will be 3 ∗ 106 lb, of which 20% is expected to be Grade A and the remaining portion expected to be Grade B. The company makes three different tomato products, and they set the selling prices of these products in light of the long-term marketing strategy of the company. Moreover, the company uses a numerical scale to record the quality of both the raw produce and prepared products. The scale runs from 0 to 10 points, higher numbers representing better quality. On this scale, Grade A tomatoes were valued at 9 points/lb, and Grade B tomatoes 5 points/lb. Product Selling price (£/case) Tomatoes Used (lb/case) 18 Cost of production (£/case) 2.52 Demand Forecast (cases) Unlimited MAIQR Quality requirement 8 Whole tomatoes Tomato Juice Tomato paste 4 4.5 3.8 20 25 3.18 1.95 50,000 80,000 6 5 Write an LP to help the company to maximize their profits. 2. A company makes 3 products P1, P2, and P3 using limestone, electricity, water, fuel, and Labour as inputs. Labour is measured in hours and other inputs have suitable units. Each input is available from one or more sources. The company has its own quarry for limestone, which can supply up to 250 units/day at a cost of £20/unit. Beyond that limestone can be purchased from an outside supplier at £50/unit. Electricity is only available from local utility. They charge £30/unit for first 1000 units/day, £45/unit for upto an additional 500 units/day beyond the initial 1000 units/day, £75/unit for amounts beyond 1500 units/day. Water is available up to 800 units/day from the local utility at £6/unit. Beyond that, they charge £7/unit. There is a single supplier for fuel, who can supply at most 3000 units/day at £40/unit. There is no supply beyond that amount for fuel. Their regular workforce have 640 hours/day at £10/hour. Beyond that, they can get up to 160 hours/day at £17/ hour from a pool of freelance workers. They can sell up to 50 units/day of P1 at £3000/units in upscale market. Beyond that they can sell up to 50 more units/day of P1 to wholesaler at £250/unit. They can sell up to 100 units/day of P2 at £3500/unit in upscale market. They can sell any quantity of P3 produced at a constant rate of £4500/unit. Data on the inputs needed to make the various products are given in Table below. Formulate a LP for the product mix problem in order to maximise the profit/day at this company. Product P1 P2 P3 Limestone 1/2 1 3/2 Units/product requirement Electricity Water Fuel 3 1 1 2 1/4 1 5 2 3 Labour 2 1 1 3. Consider an investment allocation problem, where the investor has £300,000 that can be invested. In addition to the money at hand, it is possible to borrow up to £100,000 at 12% interest. This money can be used for leveraging (borrowing to invest). The investor has narrowed down the choices to six alternatives, shown in Table below. Investment type Real estate Silver Savings account Blue chip stocks Bonds Hi-tech stocks Expected annual interest or dividend 0% 0% 2% 3% 4% 0% Expected annual increase in value 18% 10% 0% 6% 0% 20% Average risk per dollar 20 12 1 7 3 30 The table also shows the expected annual interest or dividend for the investment alternatives, the expected annual increase of the value of the investment, and an indication of the risk of the investment (per pound). Maximize the expected value of the assets at the end of the planning period (1 year). The value of the assets after 1 year equals today’s value of the investment plus the expected interest or dividend plus the expected change in value within a year minus the amount of money that was borrowed (principal and interest). In addition, the decision maker faces the following constraints: • The expected value of assets (exclusive interest) at the end of the planning period should be at least 7% higher than at the beginning. • Invest at least 50% of all the money actually invested in stocks and bonds combined. • Invest no more than 20% of total amount available (excluding the amount borrowed) in real estate and silver combined, and. • The average risk of the portfolio should not exceed 10 4. We are to blend two table wines from the Moselle region in Germany. The two blends are the Filzener Hexenhammer and the Leiwener Hosenscheisser. Both products are blends of wines from three white grapes, viz., Riesling, Müller-Thurgau, and Silvaner. The original wines are available in quantities of 10,000, 5000, and 6000 gallons at a cost of $8, $6, and $5 per gallon. The estimated demands for the two blends are 7000 and 8000 gallons (our customers will purchase these or any higher amounts from us) and the estimated sales prices are $16 and $18 per gallon, respectively. The blending rules provide the minimum and maximum percentage of individual basic wines that are allowed in the two blends. The blending rules are given in the table below: Basic wines Filzener Hexenhammer Leiwener Hosenscheisse Riesling [45%,55%] [20%,50%] Müller-Thurgau [10%,15%] [10%,60%] Silvaner [35%,35%] [30%,40%] Formulate a LP to maximise the profits of the wine company. 5. A newly opened supermarket location is hiring staff. Based on their previous experience, they have forecasted the number of staff required in a 4-hour timeslot throughout the day: Time slots Required number of employees 06.0010.00 17 10.0014.00 9 14.0018.00 19 18.0022.00 12 22.00-2.00 2.00-6.00 5 8 The supermarket has to hire staff to work on an 8hr shift. Formulate a LP to minimize the number of staff they need to hire in order to satisfy the time slot requirement. For simplicity, you can ignore integer requirements. 6. An electronics company produces three types of parts for automatic washing machine. It purchases castings of the parts from a local foundry and then finishes the part on drilling, shaping and polishing machines. The selling prices of part A, B and C respectively are £8, £10 and £14. All parts made can be sold. Castings for parts A, B and C respectively cost £5, £6 and £10. The shop possesses only one of each type of machine. Costs per hour to run each of the three machines are £20 for drilling, £30 for shaping and £30 for polishing. The capacities (parts per hour) for each part on each machines are shown in the following table: Machine Drilling Shaping Polishing Part A 25 25 40 Capacity (parts per hour) Part B Part C 40 25 20 20 30 40 The management of the shop wants to know how many parts of each type it should produce per hour in order to maximise profit for an hour's run. Formulate this problem as an LP model. 7. A co-operative farm owns 100 acres of land and has £25,000 in funds available for investment. The farm members can produce a total of 3500 hours worth of labour during the months of September-May and 4,000 hours during June-August. If any of these hours are not needed, some members of the farm will use them to work on a neighbouring farm for £2 per hour during September-May and £3 per hour during June-August. Cash income can be obtained from the three main crops and two types of livestock, dairy cows and poultry. No investment funds are needed for the crops. However, each cow will require an investment of £3, 200 and each hen will require £15. In addition, each cow will require 1.5 acres of land, 100 hours of labour during Sep-May and 50 hours during Jun-Aug. Each cow will produce a net annual cash income of £3,500 for the farm. The corresponding figures for each hen are no acreage, 0.6 manhours during Sep-May and 0.4 manhours during Jun-Aug and an annual net income of £200. The chicken house can accommodate a maximum of 4,000 hens and the size of cattle shed has a maximum capacity of 32 cows. Estimated labour hours and income per acre planned in each of the three crops are: Labour (hrs) Sep – May Jun-Aug Net annual income (£) Crop1 40 Crop2 20 Crop3 25 50 1,200 35 800 40 850 The co-operative farm wishes to determine how much acreage should be planted in each of the crops and how many cows and hens should be kept in order to maximise its net cash income. 8. An investor has money making activities A1, A2, A3 and A4. He has only £100,000 to invest. In order to avoid excessive investment, no more than 50% of the total investment can be placed in activity A1 and/or activity A3. Activity A1 is very conservative while activity A4 is speculative. To avoid excessive speculation, at least £1 must be invested in activity A1 for every £3 invested in activity A4. Return on investment on the 4 activities are 10%,12%,14% and 16% respectively. The investor wishes to know how much to invest in each activity to maximise the total return on the investment.
