Problem 2 The AB manufacturing company has two kinds of products: A and B. They earn$180 profit for each Product A and $90 profit for each Product B. To produce Products A and B,the company needs three resources: Q, R, S. Each unit of Product A consumes 6 units of ResourceQ and 2 units of Resource R. Each unit of Product B consumes 8 units of Resource Q and 3 unitsof Resource S. The AB manufacturing company only can use 48 units of Q, 12 units of R and 12units of S per day.ResourceᎡSProfit per unitProduct A620180Product B Resource Available∞03904812121. Formulate a linear program and use the simplex algorithm to solve it.2. Write down a linear programming model to determine the minimum total amount the insur-ance company should pay to AB manufacturing when they lose all the available resource.(Hint: consider the relationship between this LP for the insurance company and the originalLP for the AB company).3. What are the shadow prices for each resource? Which is the most "scarce" resource for ABmanufacturing?

Objective Functions and Constraints: Based on the given details, the objective…

Answered: Problem 2 The AB manufacturing company…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 73P

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Problem: A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50, and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and a tiller are rented. The tractor is available for 150 hours, and the tiller is available for 200 hours. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. How many acres of each crop should be planted to maximize the farmer’s profit? (Let W be the number of acres of wheat to be planted, B the number of acres of barley to be planted and P the profit) (a) What is the objective function for the problem? (b) Excluding the non-negative constraint, how many constraints does the problem have? (c) What is the linear programming model of the problem?
Problem 2 . A hospital outpatient clinic performs four types of operations. The profit per operation, as well as the minutes of X-ray time and laboratory time used, are given in the table below. The clinic has 500 private rooms and 500 intensive care rooms. Type 1 and type 2 operations require a patient to stay in an intensive care room for one day, whereas type 3 and type 4 operations require a patient to stay in a private room for one day. Each day, the hospital is required to perform at least 100 operations of each type. The hospital has set the following goals (listed in order of priority): ■ Goal 1: Earn a daily profit of at least$100,000.■ Goal 2: Use at most 50 hours daily of X-ray time.■ Goal 3: Use at most 40 hours daily of laboratory time. Use goal programming to determine the types of operations that should be performed.
Problem There are three factories on the Momiss River. Each emits two types of pollutants, labeled P1 and P2, into the river. If the waste from each factory is processed, the pollution in the river can be reduced. It costs $1500 to process a ton of factory 1 waste, and each ton processed reduces the amount of P1 by 0.10 ton and the amount of P2 by 0.45 ton. It costs $1000 to process a ton of factory 2 waste, and each ton processed reduces the amount of P1 by 0.20 ton and the amount of P2 by 0.25 ton. It costs $2000 to process a ton of factory 3 waste, and each ton processed reduces the amount of P1 by 0.40 ton and the amount of P2 by 0.30 ton. The state wants to reduce the amount of P 1in the river by at least 30 tons and the amount of P2 by at least 40 tons. Use Solver to determine how to minimize the cost of reducing pollution by the desired amounts. Use SolverTable to investigate the effects of increases in the minimal reductions required by the state. Specifically, see what…
The Midland Tool Shop has four heavy presses it uses to stamp out prefabricated metal covers and housings for electric consumer products. All four presses operate differently and are of different sizes. Currently the firm has a contract to produce three products. The contract calls for 400 units of product A, 570 units of product B, and 320 units of product C. The time (in minutes) required for each product to be produced on each machine is as follows: Machine Product 1 2 3 4 A 35 41 34 39 B 40 36 32 43 C 38 37 33 40 Machine 1 is available for 150 hours, machine 2 for 240 hours, machine 3 for 200 hours, and machine 4 for 250 hours. The products also result in different profits, according to the machine they are produced on, because of time, waste, and operating cost. The profit per unit per machine for each product is summarized as follows: Machine Product 1 2 3…
Instruction: Formulate the dual problems for X2 the following linear programming model No need to compute/doing a tableau 1. Maximize Z = 2X1 + X2 Subject to: X2 < 10 2X1 + 5X2 < 60 X1 + X2 <18 3X1 + X2 < 44 2. Maximize Z = 2X1 + X2 +3X3 Subject X1 + X2 + 2X3 < 400 2X1 + X2 + X3 < 500 3. Maximize Z = 80X1 + 60X2 Subject X1 + X2 = 200 X1 < 50 X2 > 80
Problem 3 Carrington Oil produces gas 1 and gas 2 from two types of crude oil: crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for $72 per barrel, whereas gas 2 sells for $84 per barrel. Up to 4200 barrels of gas 1 and up to 4300 barrels of gas 2 can be sold. The cost per barrel of each crude, their availability, and the level of impurities in each crude are listed in the file. Before blending the crude oil into gas, any amount of each crude can be "purified" for a cost of $3.50 per barrel. Purification eliminates half of the impurities in the crude oil. Determine how to maximize profit. Use SolverTable to determine how an increase in the availability of crude 1 affects the optimal profit. Use SolverTable to determine how an increase in the availability of crude 2 affects the optimal profit. Use SolverTable to determine how a change in the price of gas 2 changes the optimal profit and the types of…
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Due to the unavailability of desired quality of raw materials, ABC Ltd. can manufacture maximum 80 units of product A & 60 units of product B. A consumes 5 units and B consumes 6 units of raw materials in the manufacture respectively and their respective profit margins one RO 50 & RO 80. Further, A requires 1 man-day of labour per unit & B requires 2 man-day of labour per unit. The constraints operating are: Supply of raw material - maximum 600 units Supply of labour - maximum 160 man-days. Formulate as a LP model & solve graphically.
The following information relates to Questions 10-11: The production planner for Rienzo, Inc. produces two chemicals: Alpha (A) and Delta (D). Two of his resources are constrained: Jaya, of which he can get at most 8,400 litres per week; and Sanga, of which he can get at most 4,600 litres per week. Each unit of Alpha chemical requires 16 litres of Jaya and 7 litres of Sanga, while a unit of Delta chemical uses 8 litres of each type of raw material. Profits for the Alpha blend are £3.00 per unit, and profits for the Delta blend are £5.00 per unit. Question 10 What is the objective function? A £1 A + £5 D = Z B £8 A + £16 D = Z C I do not wish to answer this question. D £3 A + £5 D = Z E £16 A + £8 D = Z F £7 A + £8 D = Z
Philadelphia fun-cycles (PFC) is buying new cycles for rentals during the next summer season for tourists visiting Fairmount park. It has a budget of has $180,000 that can be used to purchase new rental cycles that can have 1, 2, or 4 seats. The cycles can be purchased from two different manufacturers. PFC would like to purchase at least 60 cycles in total. These cycles can be any mix of 1, 2, or 4 seaters. PFC wants a total seating capacity of at least 150. PFC also wants to buy at least 45 one-seaters, 45 two-seaters and 45 four-seaters They would like to purchase the same number of cycles from both manufacturers to maintain goodwill. Cycle Builder Cost Seating Capacity Expected Daily Profit The Roadie Mighty Cycles $500 1 $100 The Doubler Mighty Cycles $900 2 $140 The Rickshaw Mighty Cycles $1800 4 $200 The Speedy S&K Cycles $800 1 $115 The Cozy S&K Cycles $1000 2 $150 The Caravan S&K Cycles…
Philadelphia fun-cycles (PFC) is buying new cycles for rentals during the next summer season for tourists visiting Fairmount park. It has a budget of has $180,000 that can be used to purchase new rental cycles that can have 1, 2, or 4 seats. The cycles can be purchased from two different manufacturers. PFC would like to purchase at least 60 cycles in total. These cycles can be any mix of 1, 2, or 4 seaters. PFC wants a total seating capacity of at least 150. PFC also wants to buy at least 45 one-seaters, 45 two-seaters and 45 four-seaters They would like to purchase the same number of cycles from both manufacturers to maintain goodwill. Cycle Builder Cost Seating Capacity Expected Daily Profit The Roadie Mighty Cycles $500 1 $100 The Doubler Mighty Cycles $900 2 $140 The Rickshaw Mighty Cycles $1800 4 $200 The Speedy S&K Cycles $800 1 $115 The Cozy S&K Cycles $1000 2 $150 The Caravan S&K Cycles…

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