Question

set-up time reductiondemand: 4,800 pints/week for each of 5 flavorsproduction rate: 6 pints/minute = 28,800 pints/weeksetup time: 0.2 weeks, regardless of flavorsetup time | cycle time | batch size (pints) | avg. inv. per flavor (pints) | total avg. inv. (pints)16 hrs | 6 wks | 28,800 | 12,000 | 60,0008 hrs | | | | 4 hrs | | | | 2 hrs | | | | 1 hr | | | | 30 min | | | | 15 min | | | | note: recall that the line operates 80 hours per week!

Explanation:

First, we define key conversions and formulas using the given data:

• Weekly operating hours = 80 hrs/week, so 1 week = 80 hrs, meaning setup time in weeks = $\frac{\text{Setup Time (hrs)}}{80}$
• Cycle Time (weeks) = Setup Time (weeks) + $\frac{\text{Batch Size}}{\text{Production Rate}}$
• Batch Size = Demand per flavor per week × Cycle Time
• Average Inventory per flavor = $\frac{\text{Batch Size}}{2}$
• Total Average Inventory = Average Inventory per flavor × 5 (number of flavors)

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Step1: Convert setup time to weeks

For each setup time, calculate:
$\text{Setup Time (weeks)} = \frac{\text{Setup Time (hrs)}}{80}$
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Step2: Calculate cycle time

Use the relationship:
$\text{Cycle Time } T = \frac{\text{Setup Time (weeks)}}{1 - \frac{\text{Demand per flavor}}{\text{Production Rate}}}$
Where demand per flavor = 4800 pints/week, production rate = 28800 pints/week, so $\frac{4800}{28800} = \frac{1}{6}$
Simplified: $T = \frac{\text{Setup Time (weeks)}}{1 - \frac{1}{6}} = \frac{6 \times \text{Setup Time (weeks)}}{5}$
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Step3: Calculate batch size

$\text{Batch Size} = 4800 \times T$
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Step4: Calculate avg inv per flavor

$\text{Avg. Inv. per Flavor} = \frac{\text{Batch Size}}{2}$
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Step5: Calculate total avg inv

$\text{Total Avg. Inv.} = \text{Avg. Inv. per Flavor} \times 5$

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For 8 hrs Setup Time:

Step1: Convert setup time to weeks

$\frac{8}{80} = 0.1$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.1}{5} = 0.12$ weeks

Step3: Calculate batch size

$4800 \times 0.12 = 5760$ pints

Step4: Calculate avg inv per flavor

$\frac{5760}{2} = 2880$ pints

Step5: Calculate total avg inv

$2880 \times 5 = 14400$ pints

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For 4 hrs Setup Time:

Step1: Convert setup time to weeks

$\frac{4}{80} = 0.05$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.05}{5} = 0.06$ weeks

Step3: Calculate batch size

$4800 \times 0.06 = 2880$ pints

Step4: Calculate avg inv per flavor

$\frac{2880}{2} = 1440$ pints

Step5: Calculate total avg inv

$1440 \times 5 = 7200$ pints

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For 2 hrs Setup Time:

Step1: Convert setup time to weeks

$\frac{2}{80} = 0.025$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.025}{5} = 0.03$ weeks

Step3: Calculate batch size

$4800 \times 0.03 = 1440$ pints

Step4: Calculate avg inv per flavor

$\frac{1440}{2} = 720$ pints

Step5: Calculate total avg inv

$720 \times 5 = 3600$ pints

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For 1 hr Setup Time:

Step1: Convert setup time to weeks

$\frac{1}{80} = 0.0125$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.0125}{5} = 0.015$ weeks

Step3: Calculate batch size

$4800 \times 0.015 = 720$ pints

Step4: Calculate avg inv per flavor

$\frac{720}{2} = 360$ pints

Step5: Calculate total avg inv

$360 \times 5 = 1800$ pints

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For 30 min (0.5 hr) Setup Time:

Step1: Convert setup time to weeks

$\frac{0.5}{80} = 0.00625$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.00625}{5} = 0.0075$ weeks

Step3: Calculate batch size

$4800 \times 0.0075 = 360$ pints

Step4: Calculate avg inv per flavor

$\frac{360}{2} = 180$ pints

Step5: Calculate total avg inv

$180 \times 5 = 900$ pints

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For 15 min (0.25 hr) Setup Time:

Step1: Convert setup time to weeks

$\frac{0.25}{80} = 0.003125$ weeks

Step2: Calculate cycle time

$T = \frac{6 \times 0.003125}{5} = 0.00375$ weeks

Step3: Calculate batch size

$4800 \times 0.00375 = 180$ pints

Step4: Calculate avg inv per flavor

$\frac{180}{2} = 90$ pints

Step5: Calculate total avg inv

$90 \times 5 = 450$ pints

Answer:

Setup time	Cycle Time	Batch Size (pints)	Avg. Inv. per Flavor (pints)	Total Avg. Inv. (pints)
8 hrs	0.12 wks	5,760	2,880	14,400
4 hrs	0.06 wks	2,880	1,440	7,200
2 hrs	0.03 wks	1,440	720	3,600
1 hr	0.015 wks	720	360	1,800
30 min	0.0075 wks	360	180	900
15 min	0.00375 wks	180	90	450