1. A manuafactuing shop is designed to operate most efficiently at an output of 550 units per day. In the past month the plant produced 490 units. What was its capacity utilation rate last month? 89% 2. A company has a factory that is designed so that it is most efficient (avererage unit cost is minimized) when producing 15,000 units of output each month. However, it has an absolute maximum output capability 17,250 units per month, and can produce as little as 7,000 units per month without corporate headquarters shifting production to another plant. If the factory produces 10,925 units in October, what is the capacity utilization rate in October for this factory? 73% Hoosier Manuafacturing operates a production shop that is designed to have the lowest unit production cost at an output rate of 100 units per hour. In the month of July, the company operated the production line for a total of 175 hours and produced 16,900 units of output. What was its capacity utiliztion rate for the month? 175% AlwaysRain Irrigation, Inc., would like to to determine capacity requirements for the next four years. Currently two production lines are in place for making bronze and plastic sprinklers. Three types of sprinklers are availabe in both bronze and plastic: 90-degree nozzle sprinklers, 180-degree nozzle sprinklers, and 360-degree nozzle sprinklers. Management has forecast demand for the next four years as follows: 1(in 000's) Plastic 90 Plastic 180 Plastic 360 Bronze 90 Bronze 180 Bronze 360 Yearly 2(in 000s) 32 15 50 7 3 11 Demand 3 (in 000s) 4 (in 000s) 44 55 56 16 17 18 55 64 67 8 9 10 4 5 6 12 15 18 Both production lines can produce all the different types of nozzles. The bronze machines needed for the bronze sprinklers require two operators and can produce up to 12,000 sprinklers. The plastic injection molding machine needed for the plastic sprinklers requires four operators and can produce up to 200,000 sprinklers. Three bronze machines and only one injection molding machine are available. What are the capcity requirements for the next four years? (Asuume that there is no learning.) Year 1 97 21 Plastic (000s) Bronze (000s) Total Available Capacity Plastic Bronze 2 115 24 3 136 29 4 141 34 200000 36000 1 48.5% 0.485 1.94 58.33% 0.583 3.50 Plastic % Plastic (000s) Labor Requirement Bronze % Bronze (000s) Labor Requirement Capacity Requirements 2 3 4 57.5% 68.0% 70.5% 0.575 0.680 0.705 2.30 2.72 2.82 66.67% 80.56% 94.44% 0.667 0.806 0.944 4.00 4.83 5.67 5. Suppose that AlwaysRain Irrigation's marketing department will undertake an intense ad campaign for the bronze sprilnklers, which are more expensive but also more durable than the plastic ones. Forcast demand for the next four years is 1(in 000's) Plastic 90 Plastic 180 Plastic 360 Bronze 90 Bronze 180 Bronze 360 Yearly 2(in 000s) 32 15 50 11 6 15 Demand 3 (in 000s) 4 (in 000s) 44 55 56 16 17 18 55 64 67 15 18 23 5 6 9 16 17 20 Year 1 97 32 3 136 41 4 141 52 Capacity Requirements 48.5% 57.5% 68.0% 0.485 0.575 0.680 1.94 2.30 2.72 88.89% 100.00% 113.89% 0.889 1.000 1.139 5.33 6.00 6.83 70.5% 0.705 2.82 144.44% 1.444 8.67 Plastic (000s) Bronze (000s) Plastic % Plastic (000s) Labor Requirement Bronze% Bronze (000s) Labor Requirement 2 115 36 Based on the new ad campaign for bronze, the company will not have enough machine production or labor resource after year 2. 6. In anticipation of the ad campaign, AlwaysRain bought an additional bronze machine. Will this be enough to ensure that enough capacity is available? 1(in 000's) Plastic 90 Plastic 180 Plastic 360 Bronze 90 Bronze 180 32 15 50 11 6 Yearly 2(in 000s) Demand 3 (in 000s) 4 (in 000s) 44 55 56 16 17 18 55 64 67 15 18 23 5 6 9 Bronze 360 15 16 17 20 Year 1 2 3 4 97 32 115 36 136 41 141 52 Capacity Requirements 48.5% 57.5% 68.0% 0.485 0.575 0.680 1.94 2.30 2.72 66.7% 75.0% 85.4% 0.667 0.750 0.854 5.33 6.00 6.83 70.5% 0.705 2.82 108.3% 1.083 8.67 Plastic (000s) Bronze (000s) Plastic % Plastic (000s) Labor Requirements Bronze % Bronze (000s) Labor Requirements Based on the new ad campaign and one new machine they will still not have enough capacity to meet the requirements because they would have to hire 2 new people. 7. Suppose that operaters have enough training to operate both the bronze machines and the injection molding machines for the plastic sprinklers. Currently AlwaysRain has 10 such employees. In anticipation of the ad campaign described in problem 5, management approved the purchase of two additional bronze machines. What are the labor requirement implications? 1(in 000's) Plastic 90 Plastic 180 Plastic 360 Bronze 90 Bronze 180 Bronze 360 32 15 50 11 6 15 Yearly 2(in 000s) Demand 3 (in 000s) 4 (in 000s) 44 55 56 16 17 18 55 64 67 15 18 23 5 6 9 16 17 20 Year 1 Plastic (000s) Bronze (000s) Plastic % Plastic (000s) Labor Requirements Bronze % Bronze (000s) Labor Requirements 2 3 4 97 32 115 36 136 41 141 52 Capacity Requirements 48.5% 57.5% 68.0% 0.485 0.575 0.68 1.94 2.3 2.72 53.3% 60.0% 68.3% 0.533 0.600 0.683 5.33 18.00 20.50 70.5% 0.705 2.82 86.7% 0.867 26.00 8. Expando, Inc., is considering the possiblity of building an additional factory that would produce a new addition to it product line. The company is currently considering two options. The first is a small fcility that it could build at cost of $6 million. If demand for new products is low, the company expects to receive $10 million in discounted revenues (present value of future revenues) with the small facility. On the other hand, if demand is high, it expects $12 million in discounted revenues using the small facility. The second options is to build a large factory at a cost of $9 million. Were demand to be low, the company would expect $10 million in discounted revenues with the large plant. If demand is high, the company estimates the discounted revenues would be $14 million. In either case, the probability of demand being high is .40, and the probability of it being low is .60. Not constructing a new factory would result in no additional revenue being genterareted because the curren factories cannot produce these new products. Construct a decision tre to help Expando make the best decision. Choices Revenue Cost Value Sm Facility, Low Demand $10,000,000 $6,000,000 Err:509 Sm Facility, High Demand $12,000,000 $6,000,000 Err:509 Lg Factory, Low Demand $10,000,000 $9,000,000 Err:509 Large Factory, High Demand $14,000,000 $9,000,000 Err:509 Expected value of the alternative building Small Factory $4,800,000 Large Factory $2,600,000 According to the decision tree analysis, it's best to build the small factory. 9. a builder has located a piece of property that she would like to buy and eventually build on. The land is currently zoned for four homes per acre, but she is planning to request new zoning. What she builds on approval of zoning requests and your analysis of this problem to advise her. With her input and your help, the decision process has been reduced to the followign costs, alternatives, and probabilities: Cost of land: $2 million. Probability of rezoning: .60. If the land is rezoned, there will be additional costs for new roads, lightning , and so on, of $1 million. If the land is rezoned, the contractor must decide whether to build a shopping center or 1,500 apartments that the tentative plan shows would be possible. If she builds a shopping center, there is a 70 percent chance that she can sell the shopping center to a large dpartment store chain for $4 million over construction cost, which excludes the land; and there is a 30 percent chance that she can sell it to an insurance company for $5 million over her construction cost (also excluding the land). If, instead of the shopping center, she decides to build the 1,500 apartments, she places probabilities on the profits as follows: There is a 60 percent chance that she can sell the apartments to a real estate investment corporation for $3,000 each over her construction cost. (Both exclude the land cost.) If the land is not rezoned, she will comply with the existing zoning restrictions and simply build 600 homes, on which she expects to make $4,000 over the construction cost on each one(exclluding the cost of land). Draw a decision tree of the problem and determin the best solution and the expected net profit. Choices Revenue Cost Value Rezone, Shopping Center, Lg dept chain $4,000,000 $3,000,000 $1,000,000 Rezone, Shopping Center,Insurance Company $5,000,000 $3,000,000 $2,000,000 Rezone, 1500 Apts., $3,000 each $4,500,000 $3,000,000 $1,500,000 Rezone, 1500 Apts., $2,000 each $3,000,000 $3,000,000 $0 Not rezone, 600 homes, $4,000 each $2,400,000 $2,000,000 $400,000 Expected Value of alternatives Rezoned Shopping Center Rezoned Apartments $1,300,000 $900,000 Owners of a local restaurant are concerned about their ability to provide quality service as they continue to grow and attract more customers. They have collected date from Friday and Saturday nights, their busiest times of the week. During these time periods about 75 customers arrive per hour for service. Given the number of tables and chairs, and they typical time time it takes to serve a customer, the owners estimate they can serve on average about 100 customers per hour. During these nights, are they in the zone of service, the critical zone, or the zone of the nonservice? Capacity Utilization 0.75 They are in the critical zone, because they are at 75% of service. Anything between 70% and 100% is in the critical range. Owners of the restaurant in the prior problem anticipate that in one year their demand will double as long as they can provide good service to their customers. How much will they have to increase their service capacity to stay out of their critical zone? Capacity Utilization 214 If the restaurant increases their service capacity to 214 customers per hour to stay out of the critical zone