Chapter 13, Problem 6P

a)

Summary Introduction

To evaluate: Plan D

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 6P

Plan D has been evaluated. The total cost of plan D is $128,000.

Explanation of Solution

Given information:

The production for all the months is 1,600 units. Overtime cost is $50 per unit. The maximum allowable inventor in the warehouse is 400 units or less. The demand requirement from period 0 to 8 is as follows:

Period	Expected demand
0	0
1	1,400
2	1,600
3	1,800
4	1,800
5	2,200
6	2,200
7	1,800
8	1,800

Evaluate the cost of the plan:

Plan D
Period	Demand	Production	Overtime (units)	Ending inventory	Stock-outs (units)	Extra cost
0				200
1	1,400	1,600	-	400		$8,000
2	1,600	1,600	-	400		$8,000
3	1,800	1,600	-	200		$4,000
4	1,800	1,600	-	-		$0
5	2,200	1,600	320	-	280	$44,000
6	2,200	1,600	320	-	280	$44,000
7	1,800	1,600	200	-		$10,000
8	1,800	1,600	200	-		$10,000
Total	14,600		1,040	1,000		$128,000

Note: Production is given as 1,600 units. The ending inventory for December is given as 200 units. If the production is more than or equal to the demand, then there would be inventory. If the demand is more, then there would be stock out. Maximum overtime units allowed is 320 units (20% of production)

Calculate the ending inventory or stock-out for Period 1:

It is calculated by adding the production and ending inventory of the previous month and subtracting the result from the demand. The ending inventory for Period 1 is 400.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory of previous month}\right)-\text{Demand}\\ =\left(1,600+200\right)-1,400\\ =1,800-1,400\\ =400\end{array}$

Calculate the ending inventory or stock-out for Period 2:

It is calculated by adding the production and ending inventory of the previous month and subtracting the result from the demand. The ending inventory for Period 2 is 400.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+400\right)-1,600\\ =2,000-1,600\\ =400\end{array}$

Calculate the ending inventory or stock-out for Period 3:

It is calculated by adding the production and ending inventory of the previous month and subtracting the result from the demand. The ending inventory for Period 3 is 200.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+400\right)-1,800\\ =2,000-1,800\\ =200\end{array}$

Calculate the ending inventory or stock-out for Period 4:

It is calculated by adding the production and ending inventory of the previous month and subtracting the result from the demand. The ending inventory for Period 4 is 0.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+200\right)-1,800\\ =1,800-1,800\\ =0\end{array}$

Calculate the ending inventory or stock-out for Period 5:

As the demand is more than the production, there would be a stock-out.

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the stock-out units for Period 5 is 280 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)-\text{Overtime unit}\\ =\left(2,200-1,600\right)-320\\ =600-320\\ =280\text{ units}\end{array}$

Calculate the ending inventory or stock-out for Period 6:

As the demand is more than the production, there would be a stock-out.

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the stock-out units for Period 6 is 280 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)-\text{Overtime unit}\\ =\left(2,200-1,600\right)-320\\ =600-320\\ =280\text{ units}\end{array}$

Calculate the ending inventory or stock-out for Period 7:

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the stock-out units for Period 7 is 0 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)-\text{Overtime unit}\\ =\left(1,800-1,600\right)-200\\ =200-200\\ =0\text{ units}\end{array}$

Note: Overtime units of 320 units would lead to a negative value. Hence, it was taken as 200 units.

Calculate the ending inventory or stock-out for Period 8:

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the stock-out units for Period 8 is 0 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)-\text{Overtime unit}\\ =\left(1,800-1,600\right)-200\\ =200-200\\ =0\text{ units}\end{array}$

Note: Overtime units of 320 units would lead to negative value. Hence, it was taken as 200 units.

Calculate the extra cost for Period 1:

It is calculated by multiplying the extra inventory units and inventory holding cost. The inventory holding cost is given as $20 per unit. Hence, the extra cost for period 1 is $8,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =400\times \$20\\ =\$8,000\end{array}$

Calculate the extra cost for Period 2:

It is calculated by multiplying the extra inventory units and inventory holding cost. Inventory holding cost is given as $20 per unit. Hence, the extra cost for period 2 is $8,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =400\times \$20\\ =\$8,000\end{array}$

Calculate the extra cost for Period 3:

It is calculated by multiplying the extra inventory units and inventory holding cost. Inventory holding cost is given as $20 per unit. Hence, the extra cost for period 3 is $4,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =200\times \$20\\ =\$4,000\end{array}$

Calculate the extra cost for Period 4:

As there are no overtime units, ending inventory units, or stock-out units, there would not be any extra costs.

Calculate the extra cost for Period 5:

It is calculated by adding the product of overtime units and inventory holding overtime cost per unit, and the product of stock-out units and stock-out cost per unit. Stock-out cost is given as $100 per unit and overtime cost is given as $50 per unit. Hence, the extra cost for period 5 is $44,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Overtime unit}\times \\ \text{Overtime cost per unit}\end{array}\right)+\left(\begin{array}{l}\text{Stock-out units}\times \\ \text{Stock-out cost per unit}\end{array}\right)\\ =\left(320\times \$50\right)+\left(280\times \$100\right)\\ =\$44,000\end{array}$

Calculate the extra cost for Period 6:

It is calculated by adding the product of overtime units and inventory holding overtime cost per unit, and the product of stock-out units and stock-out cost per unit. Stock-out cost is given as $100 per unit and overtime cost is given as $50 per unit. Hence, the extra cost for period 6 is $44,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Overtime unit}\times \\ \text{Overtime cost per unit}\end{array}\right)+\left(\begin{array}{l}\text{Stock-out units}\times \\ \text{Stock-out cost per unit}\end{array}\right)\\ =\left(320\times \$50\right)+\left(280\times \$100\right)\\ =\$44,000\end{array}$

Calculate the extra cost for Period 7:

It is calculated by adding the product of overtime units and inventory holding overtime cost per unit, and the product of stock-out units and stock-out cost per unit. Stock-out cost is given as $100 per unit and overtime cost is given as $50 per unit. Hence, the extra cost for period 7 is $10,000

$\begin{array}{c}\text{Extra cost}=\left(\text{Overtime unit}\times \text{Overtime cost per unit}\right)\\ =\left(200\times \$50\right)\\ =\$10,000\end{array}$

Calculate the extra cost for Period 8:

It is calculated by adding the product of overtime units and inventory holding overtime cost per unit, and the product of stock-out units and stock-out cost per unit. Stock-out cost is given as $100 per unit and overtime cost is given as $50 per unit. Hence, the extra cost for period 8 is $10,000

$\begin{array}{c}\text{Extra cost}=\left(\text{Overtime unit}\times \text{Overtime cost per unit}\right)\\ =\left(200\times \$50\right)\\ =\$10,000\end{array}$

Calculate the total cost of Plan D:

It is calculated by adding the extra costs of all the periods.

$\begin{array}{c}\text{Total cost}=\text{Extra costs of all periods}\\ =\$8,000+\$8,000+\$4,000+\$0+\$44,000+\$44,000+\$10,000+\$10,000\\ =\$128,000\end{array}$

Hence, the total cost for Plan D is $128,000.

b)

Summary Introduction

To determine: Plan E

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 6P

Plan E has been evaluated. The total cost of Plan E is $140,000.

Explanation of Solution

Given information:

Production for all the months is 1,600 units. Overtime cost is $50 per unit. Maximum allowable inventor in the warehouse is 400 units or less. Demand requirement from period 0 to 8 is as follows:

Period	Expected demand
0	0
1	1,400
2	1,600
3	1,800
4	1,800
5	2,200
6	2,200
7	1,800
8	1,800

Evaluate the cost of the plan:

Plan E
Period	Demand	Production	Subcontract (units)	Ending inventory	Extra cost
0				200
1	1,400	1,600	-	400	$8,000
2	1,600	1,600	-	400	$8,000
3	1,800	1,600	-	200	$4,000
4	1,800	1,600	-	-	$0
5	2,200	1,600	600	-	$45,000
6	2,200	1,600	600	-	$45,000
7	1,800	1,600	200	-	$15,000
8	1,800	1,600	200	-	$15,000
Total	14,600		1,600	1,000	$140,000

Note: Production is given as 1,600 units. The ending inventory for December is given as 200 units. If the production is more than or equal to the demand, then there would be inventory. If the demand is more, then there would be subcontract.

Calculate the ending inventory or subcontract for Period 1:

It is calculated by adding the production and ending inventory of previous month and subtracting the demand from the result. The ending inventory for Period 1 is 400.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory of previous month}\right)-\text{Demand}\\ =\left(1,600+200\right)-1,400\\ =1,800-1,400\\ =400\end{array}$

Calculate the ending inventory or subcontract for Period 2:

It is calculated by adding the production and ending inventory of previous month and subtracting the demand from the result. The ending inventory for Period 2 is 400.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+400\right)-1,600\\ =2,000-1,600\\ =400\end{array}$

Calculate the ending inventory or subcontract for Period 3:

It is calculated by adding the production and ending inventory of previous month and subtracting the demand from the result. The ending inventory for Period 3 is 200.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+400\right)-1,800\\ =2,000-1,800\\ =200\end{array}$

Calculate the ending inventory or subcontract for Period 4:

It is calculated by adding the production and ending inventory of previous month and subtracting the demand from the result. The ending inventory for Period 4 is 0.

$\begin{array}{c}\text{Ending inventory}=\left(\text{Production}+\text{Ending inventory}\right)-\text{Demand}\\ =\left(1,600+200\right)-1,800\\ =1,800-1,800\\ =0\end{array}$

Calculate the ending inventory or subcontract for Period 5:

As the demand is more than the production, there would be a subcontract.

It is calculated by subtracting the production from the demand. Hence, the subcontract units for Period 5 is 600 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)\\ =\left(2,200-1,600\right)\\ =600\text{ units}\end{array}$

Calculate the ending inventory or subcontract for Period 6:

As the demand is more than the production, there would be a subcontract.

It is calculated by subtracting the production from the demand. Hence, the subcontract units for Period 6 is 600 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)\\ =\left(2,200-1,600\right)\\ =600\text{ units}\end{array}$

Calculate the ending inventory or subcontract for Period 7:

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the subcontract units for Period 7 is 200 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)\\ =\left(1,800-1,600\right)\\ =200\text{ units}\end{array}$

Calculate the ending inventory or subcontract for Period 8:

It is calculated by subtracting the production from the demand. The result should then be subtracted from the overtime units. Hence, the subcontract units for Period 8 is 0 units.

$\begin{array}{c}\text{Stock-out}=\left(\text{Demand}-\text{Production}\right)\\ =\left(1,800-1,600\right)\\ =200\text{ units}\end{array}$

Calculate the extra cost for Period 1:

It is calculated by multiplying the extra inventory units and inventory holding cost. Inventory holding cost is given as $20 per unit. Hence, the extra cost for period 1 is $8,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =400\times \$20\\ =\$8,000\end{array}$

Calculate the extra cost for Period 2:

It is calculated by multiplying the extra inventory units and inventory holding cost. Inventory holding cost is given as $20 per unit. Hence, the extra cost for period 2 is $8,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =400\times \$20\\ =\$8,000\end{array}$

Calculate the extra cost for Period 3:

It is calculated by multiplying the extra inventory units and inventory holding cost. Inventory holding cost is given as $20 per unit. Hence, the extra cost for period 3 is $4,000

$\begin{array}{c}\text{Extra cost}=\text{Ending inventory}\times \text{Inventory holding cost per unit}\\ =200\times \$20\\ =\$4,000\end{array}$

Calculate the extra cost for Period 4:

As there are no overtime units, ending inventory units, and subcontract units, there would not be extra costs.

Calculate the extra cost for Period 5:

It is calculated by multiplying the subcontract units and subcontract cost per unit. Subcontract cost is given as $75 per unit. Hence, the extra cost for period 5 is $45,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Subcontract units}\times \\ \text{Subcontract cost per unit}\end{array}\right)\\ =\left(600\times \$75\right)\\ =\$45,000\end{array}$

Calculate the extra cost for Period 6:

It is calculated by multiplying the subcontract units and subcontract cost per unit. Subcontract cost is given as $75 per unit. Hence, the extra cost for period 6 is $45,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Subcontract units}\times \\ \text{Subcontract cost per unit}\end{array}\right)\\ =\left(600\times \$75\right)\\ =\$45,000\end{array}$

Calculate the extra cost for Period 7:

It is calculated by multiplying the subcontract units and subcontract cost per unit. Subcontract cost is given as $75 per unit. Hence, the extra cost for period 7 is $15,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Subcontract units}\times \\ \text{Subcontract cost per unit}\end{array}\right)\\ =\left(200\times \$75\right)\\ =\$15,000\end{array}$

Calculate the extra cost for Period 8:

It is calculated by multiplying the subcontract units and subcontract cost per unit. Subcontract cost is given as $75 per unit. Hence, the extra cost for period 8 is $15,000

$\begin{array}{c}\text{Extra cost}=\left(\begin{array}{l}\text{Subcontract units}\times \\ \text{Subcontract cost per unit}\end{array}\right)\\ =\left(200\times \$75\right)\\ =\$15,000\end{array}$

Calculate the total cost of Plan E:

It is calculated by adding the extra costs of all the periods.

$\begin{array}{c}\text{Total cost}=\text{Extra costs of all periods}\\ =\$8,000+\$8,000+\$4,000+\$0+\$45,000+\$45,000+\$15,000+\$15,000\\ =\$140,000\end{array}$

Hence, the total cost for Plan E is $140,000.

Conclusion

The total cost for Plan D is preferable when compared with Plan E. However, Plan C, with the total cost of $86,000 is preferable over other plans.