Formulation problems
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(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:
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: Municipal bonds should constitute at least 20% of the investment At least 40% of the funds should be placed in a combination of electronics firms, aerospace firms and drug manufacturers 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.
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:
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: Current month’s normal (regular) time production Current month’s overtime production 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:
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:
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