Q1- Best Seats produces two types of electric golf cart seats: a six-seat version and a ten-seat version. The company has received a large order for 30,000 six-seaters and 15,000 ten-seaters. The production infrastructure of the company consists of three departments: Production, Assembly and Packaging. The table below summarizes the hours of processing time available and the processing time required by each department for both types of seats.
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Hours required per trimmer
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Electric
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Gas
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Hours available
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Production
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0.20
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0.40
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10,000
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Assembly
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0.30
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0.50
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15,000,
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Packaging
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0.10
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0.10
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5,000
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The company makes its in-house six-seaters for $55 and ten-seaters $85 for the ten-seaters. A new vendor is willing to manufacture and supply the company with unbranded seats according to Best Seats' specifications for $67 and $95 respectively. How many six and ten-seaters should Best Seats make and how many should they buy in order to meet the large order received, at the least cost?
a. Formulate a linear programming model for this problem.
b. Create a spreadsheet model for this problem and solve it using Solver.
c. What is the optimal solution?
Q2- Howard Jones, the owner of a bath tub manufacturing company is facing a new problem. Although sales of the two types of bath tubs manufactured by his company are experiencing brisk sales, the company is not earning the level of profits that Howard wants to achieve. Since the bath tubs enjoy a reputation for high quality and reliability, Howard believes that the price of bath tubs can be increased. He engages a market research firm which finds that the ideal price for the bath tubs should be between $1000 and $1,500 and the demand for the tubs as follows:
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Aqua-Spa
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Hydro-Luxe
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Raw material inventory
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Pumps (units)
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1
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1
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200
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Labor (Hours)
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9
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6
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1566
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Tubing (meters)
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12
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16
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2800
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Demand for Aqua-Spas: 300-0.175 x (the price of Aqua-Spas)
Demand for Hydro-Luxes: 325-0.15 x (the price of Hydro-Luxes)
The manufacturing costs have been determined to be $850 and $700 per unit for the Aqua-Spas and Hydro-Luxes respectively.
Ideally, Howard would like to produce enough to meet the demand and hold no excess inventory. The requirements for manufacturing each Aqua-Spa and Hydro-Luxe are as follows:
Howard would like to know how much to charge for each type of product and, how many units to produce.
a. Formulate a non-linear programming model in algebraic form.
b. Implement your model in a spreadsheet and solve it.
c. What is the optimum profit value and the number of units of Aqua-Spas and Hydro-Luxes that will need to be produced for achieving the optimum profit?
Q3- A car dealership needs to determine how to allocate its $20,000 advertising budget. Senior managementhas decided to spend at least $500 on each medium of advertising during the first month. The company has estimated the expected profit from each dollar (X) spent in four different advertising media as follows:
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Medium
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Expected profit
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Newspaper
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100X0.7
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Radio
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125X0.65
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TV
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180X0.6
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Direct mail
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250X0.5
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a. Formulate a non-linear programming model in algebraic form.
b. If the company spends at least $500 on each medium, how should it allocate its advertising budget in order to maximize its profit?
c. What is the expected profit value with the $500 spent on each medium?
Q4- South Bend, Indiana has a large number of motorcycle enthusiasts who buy motorcycles and enjoy riding them during the lovely, warm summer time. Sensing the demand for motorcycles, an entrepreneur establishes a motorcycle retailer ship in Granger, IN to sell and service several brands of motorcycles. The demand, unit costs and storage requirements for new motorcycles are given in the table below:
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Model 1
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Model 2
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Model 3
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Annual Demand
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800
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500
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1,500
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Unit Cost
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$300
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$1,100
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$600
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Storage Space required
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9 Sq. ft.
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25 Sq. ft.
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16 Sq. ft.
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In addition to the costs above, it costs the dealership $60 to receive and process new motorcycle orders and a 25% annual carrying cost for all items it holds in inventory. The dealership has 3,000 Sq. ft. of space and $45,000 invested in inventory. The manager of the dealership learns that IUSB students are good at solving problems to determine the optimum solution for product mixes, to maximize returns. Impress the dealer by answering the questions below:
a. Formulate the problem in algebraic form, specifying all constraints.
b. Implement your problem in a spreadsheet and solve it.
c. What are the optimum quantities of motorcycle models to be purchased?
Q5- An oil company that operates the pipeline network shown below, where eachpipeline is labeled with its maximum flow rate in million cubic feet (MMcf) per day. A new oil well has been constructed near A. They would like to transport oil from the well near A to their refinery at G.
a) Formulate and solve a network optimization model to determine the maximum flow rate From A to G.
b) Draw the optimized network diagram for your solution
Q6- A company produces Snow blowers at its two different plants, in Chicago, IL and LA Porte, IN. The cost of production (per unit) in Chicago and LA Porte are $400 and $ 360 respectively. Each plant can produce a maximum of 300 units per month. Inventory holding costs are assessed at $30 per unit in beginning inventory each month. The company estimates the demand for its product to be 300, 400 and 500 units respectively over the next three months and wants to be able to meet this demand at minimum cost.
Assume starting inventory as 0 units.
a. Formulate an LP model for this problem.
b. Implement your model in a spreadsheet and solve it.
c. What is the optimal solution?
d. How does the solution change if the each plant is required to produce at least 50 units per month in order to fulfil labor union agreements?