Solve the following:
Q1: A storage company has identified a new location to build climate-controlled storage units and rent them. Storage units will be built in three sizes. These are 10x10 ft, 10x15 ft, and 10x20 ft. The monthly operating cost for each type of storage unit, and the monthly amount they will be rented for is provided in the below table. The maximum space available for building storage units at the new location is 16,000 square ft. Based on experience from storage facilities at other locations, the management has decided that for the new storage facility:
- They should build at least 40 units of 10x10
- They should build at least 20 units of 10x20
- At most, 50 units of 10x20 are anticipated to be rented, i.e. maximum demand for 10x20 units is forecast to be 50.
- Given the popularity of 10x15 units, they should build at least twice as many 10x15 units as they will build 10x20 units.
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Monthly Operating Cost, ($)
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Monthly Rent, ($)
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10x10
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30
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120
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10x15
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45
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160
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10x20
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60
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200
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a) Formulate the problem.
b) Solve the problem using LINDO.
Q2: A company makes 3 products. The selling price for each unit of products A, B, and C is $60, $75, and $100, respectively. Labor cost for producing a single unit of each of the three products is given in the below table.
The three products are made from Material-1 and Material-2. Material-1 costs $5 per pound and Material-2 costs $10 per pound. The amount of Material-1 and Material 2 required to produce a single unit of each of the three products is given in the below table also. On any given day, 400 pounds of Material-1 and 300 pounds of Material-2 is available.
The 3 products are manufactured using two types of machines. The amount of machine-time each product requires on each of the two machine types is also given in the below table. On any given day, the available capacity for machine type-1 100 hours for machine type-2 is 130 hours.
Formulate the problem to maximize the daily profits.
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Labor Cost, ($)
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Material-1 Needed, (lb)
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Material-2 Needed, (lb)
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# of Hours Required on Machine-1
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# of Hours Required on Machine-2
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Product-A
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10
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1
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1
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1
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2
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Product-B
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15
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1.2
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1.5
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2
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3
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Product-C
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20
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1.3
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1.8
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4
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4
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a) Formulate the problem.
b) Solve the problem using LINDO.
c) Which limitations does the company need to improve on, and by how much, in order to increase their profits by 10%?
Q3: Wivco produces two types of perfumes, Perfume-1 and Perfume-2, by processing raw material. Up to 90 ounces of raw material may be purchased at a cost of $10 per ounce. One ounce of raw material can be used to produce either 1.0 ounce of Perfume-1 or 0.33 ounce of Perfume-2. Producing one ounce of Perfume-1 requires two hours of labor, or 3 hours to produce 0.33 ounce of Perfume-2. A total of 200 hours of labor are available, and at most 40 ounce of Perfume-2 can be sold. Perfume-1 sells for $13 per ounce and Perfume-2 sells for $40 per ounce.
a) Formulate the problem.
b) Solve the problem using LINDO
Q4: Sales at BB Toy Company have significantly increased in recent years. A major contributor to the increase has been catalog sales. Steadily, more and more of their catalog business is handled on-line. However, to accommodate customers who do not have access to the internet, they have assigned Jim to take orders by phone. If Jim is occupied on one line, the incoming calls are answered automatically by a recording and asked to wait. As soon as Jim is free, the customer that has waited the longest is transferred and answered first. Incoming calls follow Poisson distribution and arrive at a rate of about 12 per hour. Jim is able to take an order at an average rate of 4 minutes; with exponentially distributed service times. Jim is paid $10 per hour; but because of lost goodwill and future sales, BB Toys loses about $50 per hour of customer time spent waiting for the clerk to take an order
(A) What is the average time that customers must wait before their calls are transferred to Jim?
(B) What is the average number of customers waiting to place an order?
(C) Explain if it makes sense to add another clerk, also paid $10 per hour, to take phone orders?
Q5: Innovent Computers' shipping and receiving department (S&R) has four full-time clerks dedicated to packaging and shipping customer orders. Following Poisson distribution, orders arrive in S&R at an average rate of 12 per hour; and wait for the next available clerk. It takes an average of 15 minutes to package and ship out an order. Service times are exponentially distributed.
Determine the following:
(A) What is the probability that there are no orders in S&R to be processed?
(B) What is the average number of orders in S&R?
(C) What is the average time between when an order arrives in S&R and when it is shipped out?
(D) What is the average time an order is waiting for the next available clerk?
(E) What is the average number of orders waiting to be processed?
(F) What is the S&R utilization factor?
(G) In November and December, with higher Holiday sales, Innovent increases the number of S&R clerks to five. How does this impact S&R utilization factor in November/December?