Chapter 13, Problem 19P

Dwayne Cole, owner of a Florida firm that manufactures display cabinets, develops an 8-month aggregate plan. Demand and capacity (in units) are forecast as follows:

The cost of producing each unit is $1,000 on regular time, $1,300 on overtime, and $1,800 on a subcontract. Inventory carrying cost is $200 per unit per month. There is no beginning or ending inventory in stock, and no backorders are permitted from period to period.

Let the production (workforce) vary by using regular time first, then overtime, and then subcontracting.

a) Set up a production plan that minimizes cost by producing exactly what the demand is each month. This plan allows no backorders or inventory. What is this plan’s cost?

b) Through better planning, regular time production can be set at exactly the same amount, 275 units, per month. If demand cannot be met there is no cost assigned to shortages and they will not be filled. Does this alter the solution?

c) If overtime costs per unit rise from $1,300 to $1,400, will your answer to (a) change? What if overtime costs then fall to $1,200?

Summary Introduction

To determine: The total cost of the 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 19P

The total cost of the plan is $2,608,300.

Explanation of Solution

Given information:

Regular time cost per unit is given as $1,000, overtime cost is $1,300, and subcontract cost is $1,800 per unit. The capacity and demand are given as follows:

Month	Demand	Regular time capacity	Overtime capacity	Subcontract capacity
Initial	0	0	0	0
January	255	235	20	12
February	294	255	24	16
March	321	290	26	15
April	301	300	24	17
May	330	300	30	17
June	320	290	28	19
July	345	300	30	19
August	340	290	30	20

Calculate the total cost:

Table 1

Working note:

Demand, regular time capacity, overtime capacity, and subcontract capacity are given.

Determine regular time, overtime, and subcontract units for the month of January:

It is given that the demand for the month of January is 255 units. 235 units would be produced in the regular time (as the regular time capacity is 235). Remaining 20 units would be produced in the overtime (as the overtime capacity is 20). Hence, regular time unit is 235 units and overtime unit is 20 units.

Determine regular time, overtime, and subcontract units for the month of February:

It is given that the demand for the month of February is 294 units. 255 units would be produced in the regular time (as the regular time capacity is 255). From the remaining 39 units, 24 units would be produced in the overtime (as the overtime capacity is 24) and 15 units would be produced using subcontract (as the subcontract capacity is 16). Hence, regular time unit is 255 units, overtime unit is 24 units, and subcontract unit is 15 units.

Determine regular time, overtime, and subcontract units for the month of March:

It is given that the demand for the month of March is 321 units. 290 units would be produced in the regular time (as the regular time capacity is 290). From the remaining 31 units, 26 units would be produced in the overtime (as the overtime capacity is 26) and 5 units would be produced using subcontract (as the subcontract capacity is 15). Hence, regular time unit is 290 units, overtime unit is 26 units, and subcontract unit is 5 units.

Note: The procedure continues for all months.

Increase or decrease for the month of January:

There would be no increase or decrease for the month of January as there is no period compare.

Increase or decrease for the month of February:

Production in the month of January is 235 units and production in the month of February is 255 units. There is an increase of 20 units. Hence, there would be 20 units increase in February.

Increase or decrease for the month of March:

Production in the month of February is 255 units and production in the month of March is 290 units. There is an increase of 35 units. Hence, there would be 35 units increase in March.

Increase or decrease for the month of April:

Production in the month of March is 290 units and production in the month of April is 300 units. There is an increase of 10 units. Hence, there would be 10 units increase in April.

Increase or decrease for the month of May:

Production in the month of April is 300 units and production in the month of May is 300 units. As there is no change in the production units, there would not be any increase or decrease.

Increase or decrease for the month of June:

Production in the month of May is 300 units and production in the month of June is 290 units. There is a decrease of 10 units. Hence, there would be 10 units decrease in June.

Increase or decrease for the month of July:

Production in the month of June is 290 units and production in the month of July is 300 units. There is an increase of 10 units. Hence, there would be 10 units increase in July.

Increase or decrease for the month of August:

Production in the month of July is 300 units and production in the month of August is 290 units. There is a decrease of 10 units. Hence, there would be 10 units decrease in August.

Summary of the table:

Type	Units	Cost
Regular time	2,260	$2,260,000
Overtime	189	$245,700
Subcontract	57	$102,600
Holding	0	0
Shortage	0	0
Increase	75	0
Decrease	20	0
		$2,608,300

Table 2

Working note:

Total units of regular time, overtime, subcontract, shortage, holding, increase, and decrease were calculated in theTable 1.

Calculate the regular time cost:

Regular time cost is calculated by multiplying regular time unit and the given regular time cost per unit. Hence, the regular time cost is $2,260,000.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time units}\times \text{Regular time cost per unit}\\ =2,260\text{ units}\times \text{\$1,000}\\ \text{=\$2,260,000}\end{array}$

Calculate the overtime cost:

Overtime cost is calculated by multiplying overtime unit and the given overtime cost per unit. Hence, the overtime cost is $245,700.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime units}\times \text{Overtime cost per unit}\\ =189\text{ units}\times \text{\$1,300}\\ \text{=\$245,700}\end{array}$

Calculate the subcontract cost:

Subcontract cost is calculated by multiplying subcontract unit and the given subcontract cost per unit. Hence, the subcontract cost is $102,600.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract units}\times \text{Subcontract cost per unit}\\ =57\text{ units}\times \text{\$1,800}\\ \text{=\$102,600}\end{array}$

Calculate the increase and decrease cost:

As there is no increase or decrease cost per unit, it would remain 0.

Calculate the total cost:

Total cost is calculated by adding regular time cost, overtime cost, subcontract cost, holding cost, shortage cost, increase cost, and decrease cost.

$\begin{array}{c}\text{Total cost}=\left\{\begin{array}{l}\text{Regular time cost}+\text{Overtime cost}+\text{Subcontract cost}+\\ \text{Holding cost}+\text{Shortage cost}+\text{Increase cost}+\text{Decrease cost}\end{array}\right\}\\ =\$2,260,000+\$245,700+\$102,600+\$0+\$0+\$0+\$0\\ \text{=\$2,608,300}\end{array}$

Hence, the total cost of the plan is $2,608,300.

Summary Introduction

To determine: The total cost of the 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 19P

The total cost of the plan is $2,585,100.

Explanation of Solution

Given information:

Regular time cost per unit is given as $1,000, overtime cost is $1,300, and subcontract cost is $1,800 per unit. The capacity and demand are given as follows:

Month	Demand	Regular time capacity	Overtime capacity	Subcontract capacity
Initial	0	0	0	0
January	255	275	20	12
February	294	275	24	16
March	321	275	26	15
April	301	275	24	17
May	330	275	30	17
June	320	275	28	19
July	345	275	30	19
August	340	275	30	20

Calculate the total cost:

Table 3

Working note:

Demand, regular time capacity, overtime capacity, and subcontract capacity are given.

Determine regular time, overtime, and subcontract units for the month of January:

It is given that the demand for the month of January is 255 units. Whole 255 units would be produced in the regular time (as the regular time capacity is 275). Hence, regular time unit is 255 units.

Determine regular time, overtime, and subcontract units for the month of February:

It is given that the demand for the month of February is 294 units. 275 units would be produced in the regular time (as the regular time capacity is 275). Remaining 19 units would be produced in the overtime (as the overtime capacity is 24). Hence, regular time unit is 275 units and overtime unit is 19 units.

Determine regular time, overtime, and subcontract units for the month of March:

It is given that the demand for the month of March is 321 units. 275 units would be produced in the regular time (as the regular time capacity is 290). From the remaining 46 units, 26 units would be produced in the overtime (as the overtime capacity is 26) and 15 units would be produced using subcontract (as the subcontract capacity is 15). Hence, regular time unit is 275 units, overtime unit is 26 units, and subcontract unit is 15 units. Remaining 5 units is a shortage.

Note: The procedure continues for all months.

Increase or decrease for the month of January:

There would be no increase or decrease for the month of January as there is no period compare.

Increase or decrease for the month of February:

Production in the month of January is 255 units and production in the month of February is 275 units. There is an increase of 20 units. Hence, there would be 20 units increase in February.

Increase or decrease from the month of March to August:

As the production for regular time remains the same, there would be no increase or decrease from the month of March to August.

Summary of the table:

Type	Units	Cost
Regular time	2,180	$2,180,000
Overtime	187	$243,100
Subcontract	90	$162,000
Holding	0	0
Shortage	49	0
Increase	20	0
Decrease		0
		$2,585,100

Table 4

Working note:

Total units of regular time, overtime, subcontract, shortage, holding, increase, and decrease were calculated in theTable 3.

Calculate the regular time cost:

Regular time cost is calculated by multiplying regular time unit and the given regular time cost per unit. Hence, the regular time cost is $2,180,000.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time units}\times \text{Regular time cost per unit}\\ =2,180\text{ units}\times \text{\$1,000}\\ \text{=\$2,180,000}\end{array}$

Calculate the overtime cost:

Overtime cost is calculated by multiplying overtime unit and the given overtime cost per unit. Hence, the overtime cost is $243,100.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime units}\times \text{Overtime cost per unit}\\ =187\text{ units}\times \text{\$1,300}\\ \text{=\$243,100}\end{array}$

Calculate the subcontract cost:

Subcontract cost is calculated by multiplying subcontract unit and the given subcontract cost per unit. Hence, the subcontract cost is $162,000.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract units}\times \text{Subcontract cost per unit}\\ =90\text{ units}\times \text{\$1,800}\\ \text{=\$162,000}\end{array}$

Calculate the increase and decrease cost:

As there is no increase or decrease cost per unit, it would remain 0.

Calculate the total cost:

Total cost is calculated by adding regular time cost, overtime cost, subcontract cost, holding cost, shortage cost, increase cost, and decrease cost.

$\begin{array}{c}\text{Total cost}=\left\{\begin{array}{l}\text{Regular time cost}+\text{Overtime cost}+\text{Subcontract cost}+\\ \text{Holding cost}+\text{Shortage cost}+\text{Increase cost}+\text{Decrease cost}\end{array}\right\}\\ =\$2,180,000+\$243,100+\$162,000+\$0+\$0+\$0+\$0\\ \text{=\$2,585,100}\end{array}$

Hence, the total cost of the plan is $2,585,100.

Summary Introduction

To determine: The total cost of the plan, if the overtime increased to $1,400.

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 19P

The total cost of the plan is $2,627,200.

Explanation of Solution

Given information:

Regular time cost per unit is given as $1,000, overtime cost is $1,400, and subcontract cost is $1,800 per unit. The capacity and demand are given as follows:

Month	Demand	Regular time capacity	Overtime capacity	Subcontract capacity
Initial	0	0	0	0
January	255	235	20	12
February	294	255	24	16
March	321	290	26	15
April	301	300	24	17
May	330	300	30	17
June	320	290	28	19
July	345	300	30	19
August	340	290	30	20

Calculate the total cost:

Table 5

Working note:

Demand, regular time capacity, overtime capacity, and subcontract capacity are given.

Determine regular time, overtime, and subcontract units for the month of January:

It is given that the demand for the month of January is 255 units. 235 units would be produced in the regular time (as the regular time capacity is 235). Remaining 20 units would be produced in the overtime (as the overtime capacity is 20). Hence, regular time unit is 235 units and overtime unit is 20 units.

Determine regular time, overtime, and subcontract units for the month of February:

It is given that the demand for the month of February is 294 units. 255 units would be produced in the regular time (as the regular time capacity is 255). From the remaining 39 units, 24 units would be produced in the overtime (as the overtime capacity is 24) and 15 units would be produced using subcontract (as the subcontract capacity is 16). Hence, regular time unit is 255 units, overtime unit is 24 units, and subcontract unit is 15 units.

Determine regular time, overtime, and subcontract units for the month of March:

It is given that the demand for the month of March is 321 units. 290 units would be produced in the regular time (as the regular time capacity is 290). From the remaining 31 units, 26 units would be produced in the overtime (as the overtime capacity is 26) and 5 units would be produced using subcontract (as the subcontract capacity is 15). Hence, regular time unit is 290 units, overtime unit is 26 units, and subcontract unit is 5 units.

Note: The procedure continues for all months.

Increase or decrease for the month of January:

There would be no increase or decrease for the month of January as there is no period compare.

Increase or decrease for the month of February:

Production in the month of January is 235 units and production in the month of February is 255 units. There is an increase of 20 units. Hence, there would be 20 units increase in February.

Increase or decrease for the month of March:

Production in the month of February is 255 units and production in the month of March is 290 units. There is an increase of 35 units. Hence, there would be 35 units increase in March.

Increase or decrease for the month of April:

Production in the month of March is 290 units and production in the month of April is 300 units. There is an increase of 10 units. Hence, there would be 10 units increase in April.

Increase or decrease for the month of May:

Production in the month of April is 300 units and production in the month of May is 300 units. As there is no change in the production units, there would not be any increase or decrease.

Increase or decrease for the month of June:

Production in the month of May is 300 units and production in the month of June is 290 units. There is a decrease of 10 units. Hence, there would be 10 units decrease in June.

Increase or decrease for the month of July:

Production in the month of June is 290 units and production in the month of July is 300 units. There is an increase of 10 units. Hence, there would be 10 units increase in July.

Increase or decrease for the month of August:

Production in the month of July is 300 units and production in the month of August is 290 units. There is a decrease of 10 units. Hence, there would be 10 units decrease in August.

Calculate the total cost if the overtime cost per unit increased to $1,400:

Type	Units	Cost
Regular time	2,260	$2,260,000
Overtime	189	$264,600
Subcontract	57	$102,600
Holding	0	0
Shortage	0	0
Increase	75	0
Decrease	20	0
		$2,627,200

Table 6

Working note:

Total units of regular time, overtime, subcontract, shortage, holding, increase, and decrease were calculated in theTable 5.

Calculate the regular time cost:

Regular time cost is calculated by multiplying regular time unit and the given regular time cost per unit. Hence, the regular time cost is $2,260,000.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time units}\times \text{Regular time cost per unit}\\ =2,260\text{ units}\times \text{\$1,000}\\ \text{=\$2,260,000}\end{array}$

Calculate the overtime cost:

Overtime cost is calculated by multiplying overtime unit and the given overtime cost per unit. Hence, the overtime cost is $264,600.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime units}\times \text{Overtime cost per unit}\\ =189\text{ units}\times \text{\$1,400}\\ \text{=\$264,600}\end{array}$

Calculate the subcontract cost:

Subcontract cost is calculated by multiplying subcontract unit and the given subcontract cost per unit. Hence, the subcontract cost is $102,600.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract units}\times \text{Subcontract cost per unit}\\ =57\text{ units}\times \text{\$1,800}\\ \text{=\$102,600}\end{array}$

Calculate the increase and decrease cost:

As there is no increase or decrease cost per unit, it would remain 0.

Calculate the total cost:

Total cost is calculated by adding regular time cost, overtime cost, subcontract cost, holding cost, shortage cost, increase cost, and decrease cost.

$\begin{array}{c}\text{Total cost}=\left\{\begin{array}{l}\text{Regular time cost}+\text{Overtime cost}+\text{Subcontract cost}+\\ \text{Holding cost}+\text{Shortage cost}+\text{Increase cost}+\text{Decrease cost}\end{array}\right\}\\ =\$2,260,000+\$245,700+\$264,600+\$0+\$0+\$0+\$0\\ \text{=\$2,627,200}\end{array}$

Hence, the total cost of the plan is $2,627,200.

Calculate the total cost if the overtime cost per unit increased to $1,200:

Type	Units	Cost
Regular time	2,260	$2,260,000
Overtime	189	$226,800
Subcontract	57	$102,600
Holding	0	0
Shortage	0	0
Increase	75	0
Decrease	20	0
		$2,589,400

Table 7

Working note:

Total units of regular time, overtime, subcontract, shortage, holding, increase, and decrease were calculated in theTable 6.

Calculate the regular time cost:

Regular time cost is calculated by multiplying regular time unit and the given regular time cost per unit. Hence, the regular time cost is $2,260,000.

$\begin{array}{c}\text{Regular time cost}=\text{Regular time units}\times \text{Regular time cost per unit}\\ =2,260\text{ units}\times \text{\$1,000}\\ \text{=\$2,260,000}\end{array}$

Calculate the overtime cost:

Overtime cost is calculated by multiplying overtime unit and the given overtime cost per unit. Hence, the overtime cost is $264,600.

$\begin{array}{c}\text{Overtime cost}=\text{Overtime units}\times \text{Overtime cost per unit}\\ =189\text{ units}\times \text{\$1,200}\\ \text{=\$226,800}\end{array}$

Calculate the subcontract cost:

Subcontract cost is calculated by multiplying subcontract unit and the given subcontract cost per unit. Hence, the subcontract cost is $102,600.

$\begin{array}{c}\text{Subcontract cost}=\text{Subcontract units}\times \text{Subcontract cost per unit}\\ =57\text{ units}\times \text{\$1,800}\\ \text{=\$102,600}\end{array}$

Calculate the increase and decrease cost:

As there is no increase or decrease cost per unit, it would remain 0.

Calculate the total cost:

Total cost is calculated by adding regular time cost, overtime cost, subcontract cost, holding cost, shortage cost, increase cost, and decrease cost.

$\begin{array}{c}\text{Total cost}=\left\{\begin{array}{l}\text{Regular time cost}+\text{Overtime cost}+\text{Subcontract cost}+\\ \text{Holding cost}+\text{Shortage cost}+\text{Increase cost}+\text{Decrease cost}\end{array}\right\}\\ =\$2,260,000+\$245,700+\$226,800+\$0+\$0+\$0+\$0\\ \text{=\$2,589,400}\end{array}$

Hence, the total cost of the plan is $2,589,400.