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Carl's Custom Cans produces small containers which are purchased by candy and snack food producers. The production facility operates 250 days per year and has annual demand of 10,400 units for one of its custom cans. They can produce up to 200 of these cans each day. It costs $20.61 to set up one of their production lines to run this can. (Carl pays $17 per hour for setup labor.) The cost of each can is $3.40 and annual holding costs are $2.60 per can.What is the optimal size of the production run for this can? (Display your answer to the nearest whole number).   Given your answer to the previous question, how many production runs will be required each year in order to meet the annual demand? (Round your answer UP to the next whole number.)   Suppose the customer for this custom can wants to purchase in quantities of 400 units. What is the required setup cost to make this order quantity an optimal production run quantity for Carl's Custom Cans? (Display your answer to two decimal places.)   Based on your answer to the previous question (reduced setup cost), how long (in minutes) should it take to set up this production line? (Display your answer to the nearest whole number.)

Question

Carl's Custom Cans produces small containers which are purchased by candy and snack food producers. The production facility operates 250 days per year and has annual demand of 10,400 units for one of its custom cans. They can produce up to 200 of these cans each day. It costs 20.61 tosetuponeoftheirproductionlinestorunthiscan.(Carlpays20.61 to set up one of their production lines to run this can. (Carl pays 17 per hour for setup labor.) The cost of each can is 3.40 andannualholdingcostsare3.40 and annual holding costs are 2.60 per can.What is the optimal size of the production run for this can? (Display your answer to the nearest whole number).   Given your answer to the previous question, how many production runs will be required each year in order to meet the annual demand? (Round your answer UP to the next whole number.)   Suppose the customer for this custom can wants to purchase in quantities of 400 units. What is the required setup cost to make this order quantity an optimal production run quantity for Carl's Custom Cans? (Display your answer to two decimal places.)   Based on your answer to the previous question (reduced setup cost), how long (in minutes) should it take to set up this production line? (Display your answer to the nearest whole number.)

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Solution

Sure, let's go through the steps to solve each part of the problem.

Part 1: Optimal Size of the Production Run

To find the optimal size of the production run, we use the Economic Production Quantity (EPQ) formula:

EPQ=2DSH⋅PP−D EPQ = \sqrt{\frac{2DS}{H} \cdot \frac{P}{P - D}}

Where:

  • D D is the annual demand (10,400 units)
  • S S is the setup cost per production run ($20.61)
  • H H is the annual holding cost per unit ($2.60)
  • P P is the production rate (200 units per day)
  • d d is the daily demand rate (10,400250=41.6\frac{10,400}{250} = 41.6 units per day)

Plugging in the values:

EPQ=2×10,400×20.612.60⋅200200−41.6 EPQ = \sqrt{\frac{2 \times 10,400 \times 20.61}{2.60} \cdot \frac{200}{200 - 41.6}}

First, calculate the term inside the square root:

2×10,400×20.612.60=428,6882.60=164,880 \frac{2 \times 10,400 \times 20.61}{2.60} = \frac{428,688}{2.60} = 164,880

Next, calculate the fraction:

200200−41.6=200158.4≈1.263 \frac{200}{200 - 41.6} = \frac{200}{158.4} \approx 1.263

Now, multiply these results:

164,880×1.263≈208,200 164,880 \times 1.263 \approx 208,200

Finally, take the square root:

EPQ=208,200≈456 EPQ = \sqrt{208,200} \approx 456

So, the optimal size of the production run is approximately 456 units.

Part 2: Number of Production Runs Required Each Year

To find the number of production runs required each year, divide the annual demand by the optimal production run size:

Number of runs=10,400456≈22.81 \text{Number of runs} = \frac{10,400}{456} \approx 22.81

Since we need to round up to the next whole number:

Number of runs=23 \text{Number of runs} = 23

Part 3: Required Setup Cost for an Order Quantity of 400 Units

To find the required setup cost for an order quantity of 400 units to be optimal, we use the EPQ formula and solve for S S :

400=2DSH⋅PP−D 400 = \sqrt{\frac{2DS}{H} \cdot \frac{P}{P - D}}

Square both sides:

160,000=2DSH⋅PP−D 160,000 = \frac{2DS}{H} \cdot \frac{P}{P - D}

Rearrange to solve for S S :

S=160,000⋅H2D⋅P−DP S = \frac{160,000 \cdot H}{2D} \cdot \frac{P - D}{P}

Plug in the values:

S=160,000⋅2.602×10,400⋅200−41.6200 S = \frac{160,000 \cdot 2.60}{2 \times 10,400} \cdot \frac{200 - 41.6}{200}

Calculate the first fraction:

160,000⋅2.6020,800=416,00020,800=20 \frac{160,000 \cdot 2.60}{20,800} = \frac{416,000}{20,800} = 20

Calculate the second fraction:

200−41.6200=158.4200=0.792 \frac{200 - 41.6}{200} = \frac{158.4}{200} = 0.792

Multiply these results:

S=20×0.792=15.84 S = 20 \times 0.792 = 15.84

So, the required setup cost is $15.84.

Part 4: Setup Time in Minutes

Given the setup cost is 15.84andthelaborcostis15.84 and the labor cost is 17 per hour, we can find the setup time:

Setup time (hours)=15.8417≈0.932 hours \text{Setup time (hours)} = \frac{15.84}{17} \approx 0.932 \text{ hours}

Convert hours to minutes:

0.932×60≈56 minutes 0.932 \times 60 \approx 56 \text{ minutes}

So, the setup time should be approximately 56 minutes.

This problem has been solved

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