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Question Three

You have been asked to put together a capacity plan for a critical operation at the W/MM company. Your capacity measure is the number of machines. Three products (Aram cola, mango juice, and Uhai drinking water) are manufactured. The time standards (processing and setup), lot size, and demand forecasts are given in the following table. The firm operates two 8-hour shifts, 5 days per week, 50 weeks per year. Experience shows that a capacity cushion of 10 percent is sufficient.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Product & \multicolumn{2}{|c|}{Time Standard} & Uptime & Lot Size & Demand Forecast \\
\hline
& Processing (hr/item) & Setup (hr/item) & & & \\
\hline
Aram cola & 0.5 & 0.5 & & 240 & 80,000 \\
\hline
Mango juice & 0.10 & 0.25 & & 180 & 60,000 \\
\hline
Uhai drinking water & 0.02 & 4 & & 100 & 120,000 \\
\hline
\end{tabular}

c. How many machines are needed?

d. If the current operation has two machines, what is the capacity gap?



Answer :

GinnyAnswer
Sure, let's go through the solution step-by-step for each part of the question.

### Given Data:
1. Processing Times (in hours/item):
- Aram cola: 0.5 hours/item
- Mango juice: 0.10 hours/item
- Uhai drinking water: 0.02 hours/item

2. Setup Times (in hours):
- Aram cola: 0.05 hours
- Mango juice: 0.25 hours
- Uhai drinking water: 4 hours

3. Lot Sizes (in units):
- Aram cola: 240 units
- Mango juice: 180 units
- Uhai drinking water: 100 units

4. Demand Forecasts (units per year):
- Aram cola: 80000 units/year
- Mango juice: 60000 units/year
- Uhai drinking water: 120000 units/year

5. Operational Parameters:
- Shifts per day: 2
- Hours per shift: 8
- Days per week: 5
- Weeks per year: 50
- Capacity cushion: 10%

### a. Calculate Total Available Hours Per Year:
First, let’s calculate the total available hours per year:
[tex]\[ \text{Total available hours/year} = \text{shifts/day} \times \text{hours/shift} \times \text{days/week} \times \text{weeks/year} \][/tex]
[tex]\[ \text{Total available hours/year} = 2 \times 8 \times 5 \times 50 = 4000 \text{ hours/year} \][/tex]

### b. Calculate the Total Time Required for Each Product:
To calculate the total time required for each product, we need to find the time to produce one unit:
[tex]\[ \text{Time per unit} = \text{Processing time} + \frac{\text{Setup time}}{\text{Lot size}} \][/tex]

Now, calculate this for each product:

1. Aram Cola:
[tex]\[ \text{Time per unit} = 0.5 + \frac{0.05}{240} = 0.5 + 0.0002083333 = 0.5002083333 \text{ hours/unit} \][/tex]
[tex]\[ \text{Total time required} = 0.5002083333 \times 80000 = 40016.66666667 \text{ hours} \][/tex]

2. Mango Juice:
[tex]\[ \text{Time per unit} = 0.10 + \frac{0.25}{180} = 0.10 + 0.0013888889 = 0.1013888889 \text{ hours/unit} \][/tex]
[tex]\[ \text{Total time required} = 0.1013888889 \times 60000 = 6083.33333333 \text{ hours} \][/tex]

3. Uhai Drinking Water:
[tex]\[ \text{Time per unit} = 0.02 + \frac{4}{100} = 0.02 + 0.04 = 0.06 \text{ hours/unit} \][/tex]
[tex]\[ \text{Total time required} = 0.06 \times 120000 = 7200 \text{ hours} \][/tex]

### c. Calculate the Total Time Required for All Products:
[tex]\[ \text{Total time for all products} = 40016.66666667 + 6083.33333333 + 7200 = 53300 \text{ hours} \][/tex]

### d. Adjust for Capacity Cushion:
Considering a capacity cushion of 10%, we adjust the total time:
[tex]\[ \text{Total time with capacity cushion} = \frac{\text{Total time for all products}}{1 - 0.10} = \frac{53300}{0.90} = 59222.2222222 \text{ hours} \][/tex]

### e. Calculate the Number of Machines Needed:
To find the number of machines needed:
[tex]\[ \text{Number of machines needed} = \frac{\text{Total time with capacity cushion}}{\text{Total available hours per year}} \][/tex]
[tex]\[ \text{Number of machines needed} = \frac{59222.2222222}{4000} = 14.8055555556 \][/tex]
We need approximately 14.81 machines. Since you cannot have a fraction of a machine, you would need 15 machines.

### f. Calculate the Capacity Gap:
Finally, if the current operation has two machines, the capacity gap is:
[tex]\[ \text{Capacity gap} = \text{Current number of machines} - \text{Number of machines needed} \][/tex]
[tex]\[ \text{Capacity gap} = 2 - 14.8055555556 = -12.8055555556 \][/tex]
This means there is a shortage of approximately 12.81 machines.

### Conclusion:
- Total Time Required for All Products: 53300 hours
- Total Time with Capacity Cushion: 59222.22 hours
- Number of Machines Needed: 14.81 (needs to be rounded up to 15)
- Capacity Gap: -12.81 machines (indicating the need for 13 more machines)

These calculations provide a complete capacity plan for the critical operation in the company.

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