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Feb 20, 2024
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7
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1 Formulating Linear Simultaneous-Decision Problems Notes by Shan Li 7&12&14-Sep-2020 (with annotation) Recorded zoom lecture (9/12): https://baruch.zoom.us/rec/share/MVvxOHoNP6Hzl9Lwm2vPjxgzor72iT6fpX2K8eFc
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There is a wide range of management problems where multiple decisions must be made simultaneously. 1. Resource allocation problem Example 1 – A Product-mix problem: 1) BlubberMaid manufacturer’s three rubber-based products: Airtex, Extendex, and Resistex. All three products require the same three chemical polymers and a base. The available supplies and amount of each ingredient used per pound of final product are given in the following table, together with the profits for each of the products. Determine an optimal production plan. A B C Base Profit (oz/lb of product) ($/lb) Airtex 4 2 4 6 7 Extendex 3 2 2 9 7 Resistex 6 3 5 2 6 Supply (lbs) 500 425 650 1100 Decision variables: X
1
: production amount of Airtex (in lb) X
2
: production amount of Extendex (in lb) X
3
: production amount of Resistex (in lb) Max 7X
1
+7X
2
+6X
3
4X
1
+3X
2
+6X
3
<=500*16 2X
1
+2X
2
+3X
3
<=425*16 4X
1
+2X
2
+5X
3
<=650*16 6X
1
+9X
2
+2X
3
<=1100*16 X
1
,X
2
,X
3
>=0
2 2) Management points out that the optimal BlubberMaid production plan ignores demand. To satisfy their customers, they wish to produce at least 1000 lbs Airtex, 500 lbs Extendex, and 400 lbs Resistex. Revise your LP formulation accordingly. Add three more constraints: X
1
>=1000 X
2
>=500 X
3
>=400 Example 2 – A capital budgeting problem: The Think-Big Development Co. is a major investor in commercial real-
estate development projects. It currently has the opportunity to share in three large construction projects: Project 1: Construct a high-rise office building. Project 2: Construct a hotel. Project 3: Construct a shopping center. Each project requires each partner to make investments at four different points in time: a down payment now, and additional capital after one, two and three years. The table below shows for each project the total amount of investment capital required from all the partners at these four points in time. Investment Capital Requirements (in Million Dollars) Year Office Building Hotel Shopping Center 0 40 80 90 1 60 80 50 2 90 80 20 3 10 70 60 Net present value 45 70 50 Based on current estimates of future cash flows, the estimated net present value for each project is shown in the bottom row of the table. For each project, participation shares are being sold to major investors, such as Think-Big, who become the partners for the project
3 by investing their proportional shares at the four specified points in time. The company currently has $25 million available for capital investment. Projections are that another $20 million will become available after one year, $20 million more after two years, and another $15 million after three years. What share should Think-Big take in the respective projects to maximize the total net present value of these investments?
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4 2. Cost-benefit trade-off problem Example 1- Diet Problem The kitchen manager for Sing Sing Prison is trying to decide what to feed its prisoners. She would like to offer some combination of milk, beans, and oranges. The goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional content of each food, along with the minimum nutritional requirements, are shown below. What diet should be fed to each prisoner?
5 3. Mixed Problem Example -The blending problem of Agri-Pro Agri-Pro mixes four feeds for farm animals to meet customer-specified corn, grain, and mineral requirements. The percentage of corn, grain, and minerals, and the cost per pound for each feed are given in the following table. Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Corn 30% 5% 20% 10% Grain 10% 30% 15% 10% Minerals 20% 20% 20% 30% Cost ($/lb) 0.25 0.30 0.32 0.15 Determine the least-cost amount of each feed to use so that the resulting mixture of 8000 pounds has at least 20% corn, 15% grain, and 15% minerals.
6 4. Production scheduling problem (multi-period problem)
Example The TE Appliance Company is thinking of manufacturing and selling trash 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. Table below gives these forecast costs and prices. The firm can sell as many as 300 units per month, but i ts operation is limited by the size of its warehouse, which holds a maximum of 100 compactors. TE’s operations manager, Richard Anderson, needs to determine the number of compactors to manufacture and sell each month to maximize the firm’s profit. TE 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.
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7 5. Transportation problem Example – The Big M Company Transportation Problem The Big M Company produces a variety of heavy duty machines at two factories. One of its products is a large turret lathe. Orders have been received from three customers to purchase some of these turret lathes next month. These lathes will be shipped individually and the table below shows what the cost will be for shipping each lathe from each factory to each customer. This table also shows how many lathes have been ordered by each customer and how many will be produced by each factory. The company’s distribution manager now wants to determine how many machines to ship from each factory to each customer to minimize the total shipping cost. Shipping Cost for each lathe To Customer 1 Customer 2 Customer 3 Output From Factory 1 $700 $900 $800 12 lathes Factory 2 800 900 700 15 lathes Order Size 10 lathes 8 lathes 9 lathes