Chapter 13, Problem 17P

a)

Summary Introduction

To determine: The optimal plan using the transportation method.

Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.

Answer to Problem 17P

The optimal plan using the transportation method has been developed.

Explanation of Solution

Given information:

The following information has been given:

Quarter	Forecast (units)	Regular time	Overtime	Subcontract
1	500	400	80	100
2	750	400	80	100
3	900	800	160	100
4	450	400	80	100

Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.

Develop optimal plan using transportation model:

Develop cost matrix:

Cost matrix	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Supply
Beginning inventory	0.2	0.4	0.6	0.8	1	250
Regular time 1	1	1.2	1.4	1.6	1.8	400
Over time 1	1.5	1.7	1.9	2.1	2.3	80
Subcontract 1	2	2.2	2.4	2.6	2.8	100
Regular time 1	1.5	1	1.2	1.4	1.6	400
Over time 1	2	1.5	1.7	1.9	2.1	80
Subcontract 1	2.5	2	2.2	2.4	2.6	100
Regular time 1	2	1.5	1	1.2	1.4	400
Over time 1	2.5	2	1.5	1.7	1.9	80
Subcontract 1	3	2.5	2	2.2	2.4	100
Regular time 1	2.5	2	1.5	1	1.2	400
Over time 1	3	2.5	2	1.5	1.7	80
Subcontract 1	3.5	3	2.5	2	2.2	100
Demand	500	750	900	450		2570
						2600

Excel worksheet to generate the above table:

Develop optimal plan:

Optimal plan	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Dummy
Beginning inventory	100	150
Regular time 1	400
Over time 1		80
Subcontract 1						100
Regular time 1		400
Over time 1		80
Subcontract 1						100
Regular time 1			800
Over time 1		40	100			20
Subcontract 1						100
Regular time 1				400
Over time 1				50		30
Subcontract 1						100
Demand	500	750	900	450

The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.

b)

Summary Introduction

To determine: The total cost of the optimal plan.

Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.

Answer to Problem 17P

The optimal cost of the plan is $2,641.

Explanation of Solution

Given information:

The following information has been given:

Quarter	Forecast (units)	Regular time	Overtime	Subcontract
1	500	400	80	100
2	750	400	80	100
3	900	800	160	100
4	450	400	80	100

Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.

Develop optimal plan using transportation model:

Develop cost matrix:

Cost matrix	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Supply
Beginning inventory	0.2	0.4	0.6	0.8	1	250
Regular time 1	1	1.2	1.4	1.6	1.8	400
Over time 1	1.5	1.7	1.9	2.1	2.3	80
Subcontract 1	2	2.2	2.4	2.6	2.8	100
Regular time 1	1.5	1	1.2	1.4	1.6	400
Over time 1	2	1.5	1.7	1.9	2.1	80
Subcontract 1	2.5	2	2.2	2.4	2.6	100
Regular time 1	2	1.5	1	1.2	1.4	400
Over time 1	2.5	2	1.5	1.7	1.9	80
Subcontract 1	3	2.5	2	2.2	2.4	100
Regular time 1	2.5	2	1.5	1	1.2	400
Over time 1	3	2.5	2	1.5	1.7	80
Subcontract 1	3.5	3	2.5	2	2.2	100
Demand	500	750	900	450		2570
						2600

Excel worksheet to generate the above table:

Develop optimal plan:

Optimal plan	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Dummy
Beginning inventory	100	150
Regular time 1	400
Over time 1		80
Subcontract 1						100
Regular time 1		400
Over time 1		80
Subcontract 1						100
Regular time 1			800
Over time 1		40	100			20
Subcontract 1						100
Regular time 1				400
Over time 1				50		30
Subcontract 1						100
Demand	500	750	900	450

The given demand and supply should be splitted and the remaining supply and demand should be used as a dummy value.

Calculate the total optimal cost:

It is calculated by adding the multiple of values in the optimal plan table and the value in the cost matrix to the respective value.

$\begin{array}{c}\text{Total optimal cost}=\left[\begin{array}{l}\left(100\times 0.2\right)+\left(400\times 1\right)+\left(150\times 0.4\right)+\left(80\times 1.7\right)+\left(400\times 1\right)+\\ \left(80\times 1.5\right)+\left(40\times 2\right)+\left(800\times 1\right)+\left(100\times 1.5\right)+\left(400\times 1\right)+\left(50\times 1.5\right)\end{array}\right]\\ =20+400+60+136+400+120+80+800+150+400+75\\ =\$2,641\end{array}$

Hence, the total optimal cost is $2,641.

c)

Summary Introduction

To determine: The number of units remained unused in regular time capacity.

Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.

Answer to Problem 17P

No, the regular time capacity remains unused.

Explanation of Solution

Given information:

The following information has been given:

Quarter	Forecast (units)	Regular time	Overtime	Subcontract
1	500	400	80	100
2	750	400	80	100
3	900	800	160	100
4	450	400	80	100

Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.

Develop optimal plan:

Optimal plan	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Dummy
Beginning inventory	100	150
Regular time 1	400
Over time 1		80
Subcontract 1						100
Regular time 1		400
Over time 1		80
Subcontract 1						100
Regular time 1			800
Over time 1		40	100			20
Subcontract 1						100
Regular time 1				400
Over time 1				50		30
Subcontract 1						100
Demand	500	750	900	450

The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.

Determine whether the regular time capacity remain unused:

From the above table, it is clear that no regular time capacity remains unused. All the regular time capacity has been used in respective quarters.

d)

Summary Introduction

To determine: The extent of backordering in units and dollars.

Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.

Answer to Problem 17P

The total unit of the backordered is 40 units and the total cost of producing the backorders are $20.

Explanation of Solution

Given information:

The following information has been given:

Quarter	Forecast (units)	Regular time	Overtime	Subcontract
1	500	400	80	100
2	750	400	80	100
3	900	800	160	100
4	450	400	80	100

Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.

Develop optimal plan:

Optimal plan	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Ending inventory	Dummy
Beginning inventory	100	150
Regular time 1	400
Over time 1		80
Subcontract 1						100
Regular time 1		400
Over time 1		80
Subcontract 1						100
Regular time 1			800
Over time 1		40	100			20
Subcontract 1						100
Regular time 1				400
Over time 1				50		30
Subcontract 1						100
Demand	500	750	900	450

The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.

Determine the extent backordering in units and dollars:

The colored cell is the only cell in the optimal plan, which is used for backordering. Hence, the backordering in units is 40 units. It is given that backorder cost is $0.50 per unit per quarter.

$\begin{array}{c}\text{Extent of backorder in dollars}=\text{Number of backorder units}\times \text{Backorder cost per unit}\\ =40\text{ units}\times \$0.50\\ =\$20\end{array}$

Hence, the total unit backordered is 40 units and the total costs of producing the backorders are $20.