Chapter 11, Problem 7P

SummerFun. Inc., produces a variety of recreation and leisure products. The production manager has developed an aggregate forecast:

Use the following information to develop aggregate plans.

Develop an aggregate plan using each of the following guidelines and compute the total cost for each plan. Hint: You will need extra output in April and August to accommodate demand in the following months.

a. Use regular production. Supplement using inventory, overtime, and subcontracting as needed. No backlogs allowed.

b. Use a level strategy. Use a combination of backlogs, subcontracting, and inventory to handle variations in demand. There should not be a backlog in the final period.

Summary Introduction

To determine: The total cost using an aggregate plan.

Introduction:The aggregate plan is the output of sales and operations planning. The major concern of aggregate planning is the production time and quantity for the intermediate future. Aggregate planning would encompass a time prospect of approximately 3 to 18 months.

Answer to Problem 7P

The total cost of the aggregate plan is $31,250.

Explanation of Solution

Given information:

Regular production cost is $80, overtime production cost is $120, subcontracting cost is $140, backorder cost is $20, holding cost is $10, regular capacity is 40 units, overtime cost is 8 units, and subcontracting capacity is 12 units. Beginning inventory is given as 0 units. In addition to this forecast for 7 months is given as follows:

Month	March	April	May	June	July	August	September	Total
Forecast	50	44	55	60	50	40	51	350

Determine the total cost of the plan:

It is given that regular productions should be used. No backlogs are allowed. Supplements can be satisfied using overtime, subcontracting, and inventory.

Supporting explanation:

Forecast and regular time capacity are given. Regular time capacity is given as 40 units, remaining units should be produced using the overtime capacity. Maximum overtime capacity is given as 8 units. Thus, if the forecast is not satisfied, it can be produced using the subcontract:

Calculate the difference for the month of March:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is 0 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overitme}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+2\right)-50\\ =0\end{array}$

Calculate the difference for the month of April:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is 4 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overitme}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+0\right)-44\\ =4\end{array}$

Calculate the difference for the month of May:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is -4 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overitme}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+3\right)-55\\ =-4\end{array}$

Note: The calculation repeats for all the months.

Beginning inventory:

The initial inventory is given as 0. For the remaining months, ending inventory of previous month would be the beginning inventory of present month.

Ending inventory for the month of March:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+0\\ =0\end{array}$

Ending inventory for the month of April:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 4 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+4\\ =4\end{array}$

Ending inventory for the month of May:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =4-4\\ =0\end{array}$

Note: The calculation repeats for all the months.

Average inventory for the month of March:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 0 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+0}{2}\\ =0\end{array}$

Average inventory for the month of April:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 2 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+4}{2}\\ =2\end{array}$

Average inventory for the month of May:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 2 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{4+0}{2}\\ =2\end{array}$

Note: The calculation repeats for all the months.

Calculate the regular time cost for the month of March:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of April:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of May:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Note: The calculation repeats for all the months.

Calculate the total regular time cost:

It is calculated by adding the regular time cost of all the months.

$\begin{array}{c}\text{Total regular time cost}=\text{Regular time of all the months}\\ =\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200\\ =\$22,400\end{array}$

Hence, the total regular time cost is $22,400.

Calculate the overtime cost for the month of March:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of April:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of May:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Note: The calculation repeats for all the months.

Calculate the total overtime cost:

It is calculated by adding the overtime cost of all the months.

$\begin{array}{c}\text{Total overtime cost}=\text{Overtime of all the months}\\ =\$960+\$960+\$960+\$960+\$960+\$360+\$960\\ =\$6,120\end{array}$

Hence, the total overtime cost is 6,120.

Calculate the subcontract cost for the month of March:

Subcontract cost per unit is given as $140 and subcontract unit is given as 2. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $280.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 2\\ =\$280\end{array}$

Calculate the subcontract cost for the month of April:

Subcontract cost per unit is given as $140 and subcontract unit is given as 0. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $0.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 0\\ =\$0\end{array}$

Calculate the subcontract cost for the month of May:

Subcontract cost per unit is given as $140 and subcontract unit is given as 3. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $420.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 3\\ =\$420\end{array}$

Note: The calculation repeats for all the months.

Calculate the total subcontract cost:

It is calculated by adding the subcontract cost of all the months.

$\begin{array}{c}\text{Total subcontract cost}=\text{Subcontract of all the months}\\ =\$280+\$0+\$420+\$1,680+\$280+\$0+\$0\\ =\$2,660\end{array}$

Hence, the total subcontract cost is $2,660.

Calculate the inventory cost for the month of March:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $0.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 0\\ =\$0\end{array}$

Calculate the inventory cost for the month of April:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $20.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 2\\ =\$20\end{array}$

Calculate the inventory cost for the month of May:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $20.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 2\\ =\$20\end{array}$

Note: The calculation repeats for all the months.

Calculate the total inventory cost:

It is calculated by adding the inventory cost of all the months.

$\begin{array}{c}\text{Total inventory cost}=\text{Inventory cost of all the months}\\ =\$0+\$20+\$20+\$0+\$0+\$15+\$15\\ =\$70\end{array}$

Hence, the total inventory cost is $70.

Calculate the backorder cost for the month of March:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Calculate the backorder cost for the month of April:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Calculate the backorder cost for the month of May:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Note: The calculation repeats for all the months.

Calculate the total cost of the plan:

It is calculated by adding the total regular time cost, overtime cost, subcontract cost, and inventory cost.

$\begin{array}{c}\text{Total cost of the plan}=\left(\begin{array}{l}\text{Total regular time cost}+\text{Total overtime cost}+\\ \text{Total subcontract cost}+\text{Total inventory cost}+\\ \text{Total backorder cost}\end{array}\right)\\ =\$22,400+\$6,120+\$2,660+\$70+\$0\\ =\$31,250\end{array}$

Hence, the total cost of the plan is $31,250.

Summary Introduction

To determine: The total cost using a level strategy of aggregate planning.

Introduction:Level production strategy is a production strategy used to produce at a constant rate. This strategy keeps constant level of workforce and backlog of demand.

Answer to Problem 7P

The total cost of the aggregate plan is $30,500.

Explanation of Solution

Given information:

Regular production cost is $80, overtime production cost is $120, subcontracting cost is $140, backorder cost is $20, holding cost is $10, regular capacity is 40 units, overtime cost is 8 units, and subcontracting capacity is 12 units. Beginning inventory is given as 0 units. In addition to this forecast for 7 months is given as follows:

Month	March	April	May	June	July	August	September	Total
Forecast	50	44	55	60	50	40	51	350

Determine the total cost of the plan:

Subcontracting, inventory, and backlogs can be used to handle the fluctuations in the demand. Initial solution using regular time and overtime without using subcontracting is as follows:

Supporting explanation:

Determine the regular time productivity:

It is calculated by taking an average of the given forecast.

$\begin{array}{c}\text{Regular time productivity}=\frac{\text{Sum of forecast values}}{\text{Number of months}}\\ =\frac{50+44+55+60+50+40+51}{7}\\ =\frac{350}{7}\\ =40\end{array}$

Calculate the difference for the month of March:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is -2 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overtime}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+0\right)-50\\ =-2\end{array}$

Calculate the difference for the month of April:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is 4 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overtime}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+0\right)-44\\ =4\end{array}$

Calculate the difference for the month of May:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is --7 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overitme}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+0\right)-55\\ =-7\end{array}$

Note: The calculation repeats for all the months.

Beginning inventory:

The initial inventory is given as 0. For the remaining months, ending inventory of previous month would be the beginning inventory of present month.

Ending inventory for the month of March:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+0\\ =0\end{array}$

Ending inventory for the month of April:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Difference between output and forecast is 2 (4-2). Hence, the ending inventory is 2 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+2\\ =2\end{array}$

Ending inventory for the month of May:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =4-4\\ =0\end{array}$

Note: The calculation repeats for all the months.

Average inventory for the month of March:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 0 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+0}{2}\\ =0\end{array}$

Average inventory for the month of April:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 1unit.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+2}{2}\\ =1\end{array}$

Average inventory for the month of May:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 1unit.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{2+0}{2}\\ =1\end{array}$

Note: The calculation repeats for all the months.

Backlogs for the month of 1:

It is number of units required in the month. As there is no previous month for 1^st month, the difference would be the backlog. Hence, the backlog for 1^st is 2 units.

Backlogs for the month of 2:

As the difference is positive, there would not be backlog.

Backlogs for the month of 3:

It is number of units required in the month. It is calculated by adding the backlog of previous month and the difference between output and forecast of current month (without considering the negative sign). Hence, the backlog is 5 units.

$\begin{array}{c}\text{Backlogs}=\text{Backlog of previous month}+\text{Difference between output and forecast}\\ =2-7\\ =5\end{array}$

Note: The calculation repeats for all the months.

Calculate the regular time cost for the month of March:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of April:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of May:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Note: The calculation repeats for all the months.

Calculate the total regular time cost:

It is calculated by adding the regular time cost of all the months.

$\begin{array}{c}\text{Total regular time cost}=\text{Regular time of all the months}\\ =\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200\\ =\$22,400\end{array}$

Hence, the total regular time cost is $22,400.

Calculate the overtime cost for the month of March:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of April:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of May:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Note: The calculation repeats for all the months.

Calculate the total overtime cost:

It is calculated by adding the overtime cost of all the months.

$\begin{array}{c}\text{Total overtime cost}=\text{Overtime of all the months}\\ =\$960+\$960+\$960+\$960+\$960+\$960+\$960\\ =\$6,720\end{array}$

Hence, the total overtime cost is 6,120.

Calculate the subcontract cost for the month of March:

Subcontract cost per unit is given as $140 and subcontract unit is given as 0. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $140.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 0\\ =\$140\end{array}$

Calculate the subcontract cost for the month of April:

Subcontract cost per unit is given as $140 and subcontract unit is given as 0. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $0.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 0\\ =\$0\end{array}$

Calculate the subcontract cost for the month of May:

Subcontract cost per unit is given as $140 and subcontract unit is given as 0. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $0.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 0\\ =\$0\end{array}$

Note: The calculation repeats for all the months.

Calculate the total subcontract cost:

It is calculated by adding the subcontract cost of all the months.

$\begin{array}{c}\text{Total subcontract cost}=\text{Subcontract of all the months}\\ =\$0+\$0+\$0+\$0+\$0+\$0+\$0\\ =\$0\end{array}$

Hence, the total subcontract cost is $0.

Calculate the inventory cost for the month of March:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $0.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 0\\ =\$0\end{array}$

Calculate the inventory cost for the month of April:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $10.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 1\\ =\$10\end{array}$

Calculate the inventory cost for the month of May:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $10.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 1\\ =\$10\end{array}$

Note: The calculation repeats for all the months.

Calculate the total inventory cost:

It is calculated by adding the inventory cost of all the months.

$\begin{array}{c}\text{Total inventory cost}=\text{Inventory cost of all the months}\\ =\$0+\$10+\$10+\$0+\$0+\$0+\$0\\ =\$20\end{array}$

Hence, the total inventory cost is $70.

Calculate the backorder cost for the month of March:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $40.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 2\\ =\$40\end{array}$

Calculate the backorder cost for the month of April:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Calculate the backorder cost for the month of May:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $100.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 5\\ =\$100\end{array}$

Note: The calculation repeats for all the months.

Calculate the total backorder cost:

It is calculated by adding the backorder cost of all the months.

$\begin{array}{c}\text{Total backorder cost}=\text{Backorder cost of all the months}\\ =\$40+\$0+\$100+\$340+\$380+\$220+\$280\\ =\$1,360\end{array}$

Hence, the total backorder cost is $1,360.

Calculate the total cost of the plan:

It is calculated by adding the total regular time cost, overtime cost, subcontract cost, and inventory cost.

$\begin{array}{c}\text{Total cost of the plan}=\left(\begin{array}{l}\text{Total regular time cost}+\text{Total overtime cost}+\\ \text{Total subcontract cost}+\text{Total inventory cost}+\\ \text{Total backorder cost}\end{array}\right)\\ =\$22,400+\$6,720+\$0+\$20+\$1,360\\ =\$30,500\end{array}$

Hence, the total cost of the plan is $30,500.

Determine the total cost of the plan:

Subcontracting, inventory, and backlogs can be used to handle the fluctuations in the demand. Final solution using regular time, overtime, and subcontracting is as follows:

Supporting explanation:

Determine the regular time productivity:

It is calculated by taking an average of the given forecast.

$\begin{array}{c}\text{Regular time productivity}=\frac{\text{Sum of forecast values}}{\text{Number of months}}\\ =\frac{50+44+55+60+50+40+51}{7}\\ =\frac{350}{7}\\ =40\end{array}$

Calculate the difference for the month of March:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is 0 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overtime}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+2\right)-50\\ =0\end{array}$

Calculate the difference for the month of April:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is 4 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overtime}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+0\right)-44\\ =4\end{array}$

Calculate the difference for the month of May:

It is the calculation of difference between forecast and output. Hence, it can be calculated by subtracting the forecast from the output. Hence, the difference is -4 units.

$\begin{array}{c}\text{Difference}=\text{Output}-\text{Forecast}\\ \text{=}\left(\text{Regular}+\text{Overitme}+\text{Subcontract}\right)-\text{Forecast}\\ \text{=}\left(40+8+3\right)-55\\ =-4\end{array}$

Note: The calculation repeats for all the months.

Beginning inventory:

The initial inventory is given as 0. For the remaining months, ending inventory of previous month would be the beginning inventory of present month.

Ending inventory for the month of March:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+0\\ =0\end{array}$

Ending inventory for the month of April:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 4 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =0+4\\ =4\end{array}$

Ending inventory for the month of May:

Ending inventory can be determined by adding the beginning inventory and difference between output and forecast. Hence, the ending inventory is 0 units.

$\begin{array}{c}\text{Ending inventory}=\text{Beginning inventory}+\text{Difference between output and forecast}\\ =4-4\\ =0\end{array}$

Note: The calculation repeats for all the months.

Average inventory for the month of March:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 0 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+0}{2}\\ =0\end{array}$

Average inventory for the month of April:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 2 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{0+4}{2}\\ =2\end{array}$

Average inventory for the month of May:

It is calculated by taking an average of beginning inventory and ending inventory. Hence, the average inventory is 2 units.

$\begin{array}{c}\text{Average inventory}=\frac{\text{Beginning inventory}+\text{Ending inventory}}{2}\\ =\frac{4+0}{2}\\ =2\end{array}$

Note: The calculation repeats for all the months.

Backlogs for the month of June:

It is number of units required in the month. It is calculated by adding the backlog of previous month and the difference between output and forecast of current month (without considering the negative sign). Hence, the backlog is 3 units.

$\begin{array}{c}\text{Backlogs}=\text{Backlog of previous month}+\text{Difference between output and forecast}\\ =0+3\\ =3\end{array}$

Backlogs for the month of July:

It is number of units required in the month. It is calculated by adding the backlog of previous month and the difference between output and forecast of current month (without considering the negative sign). Hence, the backlog is 5 units.

$\begin{array}{c}\text{Backlogs}=\text{Backlog of previous month}+\text{Difference between output and forecast}\\ =3+2\\ =5\end{array}$

Note: The calculation repeats for all the months.

Calculate the regular time cost for the month of March:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of April:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Calculate the regular time cost for the month of May:

Regular time cost per unit is given as $80 and regular time unit is given as 40. Regular time cost is calculated by multiplying regular time unit and regular time cost per unit. Hence, the regular time cost is $3,200.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time cost per unit}\times \text{Regular time units}\\ =\$80\times 40\\ =\$3,200\end{array}$

Note: The calculation repeats for all the months.

Calculate the total regular time cost:

It is calculated by adding the regular time cost of all the months.

$\begin{array}{c}\text{Total regular time cost}=\text{Regular time of all the months}\\ =\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200+\$3,200\\ =\$22,400\end{array}$

Hence, the total regular time cost is $22,400.

Calculate the overtime cost for the month of March:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of April:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Calculate the overtime cost for the month of May:

Overtime cost per unit is given as $120 and overtime unit is given as 8. Overtime cost is calculated by multiplying overtime unit and overtime cost per unit. Hence, the overtime cost is $960.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime cost per unit}\times \text{Overtime units}\\ =\$120\times 8\\ =\$960\end{array}$

Note: The calculation repeats for all the months.

Calculate the total overtime cost:

It is calculated by adding the overtime cost of all the months.

$\begin{array}{c}\text{Total overtime cost}=\text{Overtime of all the months}\\ =\$960+\$960+\$960+\$960+\$960+\$960+\$960\\ =\$6,720\end{array}$

Hence, the total overtime cost is 6,120.

Calculate the subcontract cost for the month of March:

Subcontract cost per unit is given as $140 and subcontract unit is given as 2. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $280.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 2\\ =\$280\end{array}$

Calculate the subcontract cost for the month of April:

Subcontract cost per unit is given as $140 and subcontract unit is given as 0. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $0.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 0\\ =\$0\end{array}$

Calculate the subcontract cost for the month of May:

Subcontract cost per unit is given as $140 and subcontract unit is given as 3. Subcontract cost is calculated by multiplying subcontract unit and subcontract cost per unit. Hence, the subcontract cost is $420.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract cost per unit}\times \text{Subcontract units}\\ =\$140\times 3\\ =\$420\end{array}$

Note: The calculation repeats for all the months.

Calculate the total subcontract cost:

It is calculated by adding the subcontract cost of all the months.

$\begin{array}{c}\text{Total subcontract cost}=\text{Subcontract of all the months}\\ =\$280+\$0+\$420+\$1,260+\$0+\$0+\$0\\ =\$1,960\end{array}$

Hence, the total subcontract cost is $1,960.

Calculate the inventory cost for the month of March:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $0.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 0\\ =\$0\end{array}$

Calculate the inventory cost for the month of April:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $20.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 2\\ =\$20\end{array}$

Calculate the inventory cost for the month of May:

It is calculated by average balance inventory cost and the average inventory units. Hence, the inventory cost is $20.

$\begin{array}{c}\text{Inventory cost}=\text{Inventory cost per unit}\times \text{Average inventory}\\ =\$10\times 2\\ =\$20\end{array}$

Note: The calculation repeats for all the months.

Calculate the total inventory cost:

It is calculated by adding the inventory cost of all the months.

$\begin{array}{c}\text{Total inventory cost}=\text{Inventory cost of all the months}\\ =\$0+\$20+\$20+\$0+\$0+\$15+\$15\\ =\$70\end{array}$

Hence, the total inventory cost is $70.

Calculate the backorder cost for the month of March:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$20\end{array}$

Calculate the backorder cost for the month of April:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Calculate the backorder cost for the month of May:

It is calculated by multiplying the backorder cost and the backlog. Hence, the backorder cost is $0.

$\begin{array}{c}\text{Backorder cost}=\text{Backorder cost per unit}\times \text{Backlogs}\\ =\$20\times 0\\ =\$0\end{array}$

Note: The calculation repeats for all the months.

Calculate the total backorder cost:

It is calculated by adding the backorder cost of all the months.

$\begin{array}{c}\text{Total backorder cost}=\text{Backorder cost of all the months}\\ =\$0+\$0+\$0+\$60+\$100+\$0+\$0\\ =\$160\end{array}$

Hence, the total backorder cost is $160.

Calculate the total cost of the plan:

It is calculated by adding the total regular time cost, overtime cost, subcontract cost, and inventory cost.

$\begin{array}{c}\text{Total cost of the plan}=\left(\begin{array}{l}\text{Total regular time cost}+\text{Total overtime cost}+\\ \text{Total subcontract cost}+\text{Total inventory cost}+\\ \text{Total backorder cost}\end{array}\right)\\ =\$22,400+\$6,720+\$0+\$70+\$160\\ =\$31,310\end{array}$

Hence, the total cost of the plan is $31,310.