Chapter 20, Problem 30P

Bountiful Manufacturing produces two types of bike frames (Frame X and Frame Y). Frame X passes through four processes: cutting, welding, polishing, and painting. Frame Y uses three of the same processes: cutting, welding, and painting. Each of the four processes employs 10 workers who work eight hours each day. Frame X sells for $40 per unit, and Frame Y sells for $55 per unit. Materials is the only unit-level variable expense. The materials cost for Frame X is $20 per unit, and the materials cost for Frame Y is $25 per unit. Bountiful’s accounting system has provided the following additional information about its operations and products:

Bountiful’s management has determined that any production interruptions can be corrected within two days.

Required:

1. 1. Assuming that Bountiful can meet daily market demand, compute the potential daily profit. Now, compute the minutes needed for each process to meet the daily market demand. Can Bountiful meet daily market demand? If not, where is the bottleneck? Can you derive an optimal mix without using a graphical solution? If so, explain how.
2. 2. Identify the objective function and the constraints. Then, graph the constraints facing Bountiful. Determine the optimal mix and the maximum daily contribution margin (throughput).
3. 3. Explain how a drum-buffer-rope system would work for Bountiful.
4. 4. Suppose that the Engineering Department has proposed a process design change that will increase the polishing time for Frame X from 15 to 23 minutes per unit and decrease the welding time from 15 minutes to 10 minutes per unit (for Frame X). The cost of process redesign would be $10,000. Evaluate this proposed change. What step in the TOC process does this proposal represent?

To determine

Ascertain the potential daily profit and calculate the minutes of each process needed to meet the daily market demand. Explain whether Company B could meet the daily market demand and if not state the bottleneck that is placed and derive an optimal mix without using a graphical solution and explain the manner in which it is derived.

Explanation of Solution

Theory of constraint: Making money in the future by managing constraints is the main goal of the theory of constraints. The theory of constraints (TOC) recognizes the performance of origination with constraints volume and it develops a specific approach to focus the system level effects of the continuous improvement.

Ascertain the potential daily profit:

Particulars	Frame X	Frame Y
Sales	$40	$55
Materials	20	25
Contribution margin per unit (a)	$20	$30
Market demand (b)	200	100
Daily profit $\left(\text{a}\times \text{b}\right)$	$4,000	$3,000
Total potential daily profit	$7,000$\left(\text{\$4,000}+\text{\$3,000}\right)$

Table (1)

Calculate the minutes of each process needed to meet the daily market demand. Explain whether Company B could meet the daily market demand and if not state the bottleneck that is placed and derive an optimal mix without using a graphical solution and explain the manner in which it is derived:

Process	Resource Demands	Resource Supply
Cutting:
For Frame X	$\left(15\times 200=3,000\right)$
For Frame Y	$\left(10\times 100=1,000\right)$
	4,000	4,800
Welding:
For Frame X	$\left(15\times 200=3,000\right)$
For Frame Y	$\left(30\times 100=3,000\right)$
	6,000	4,800
Polishing:
For Frame X	$\left(15\times 200=3,000\right)$	4,800
Painting:	$\left(\text{10}\times \text{200}=\text{2,000}\right)$
	$\left(15\times 100=1,500\right)$
	3,500	4,800

Table (2)

Company B cannot meet daily demand. Even though all other processes have excess capacity, the welding process has a low capacity and it requires 6,000 minutes however only 4,800 is available. Thus, welding is the bottleneck. The contribution margin per unit of welding resource (minutes) for each product is computed as follows:

For Frame X:

$\begin{array}{c}\text{Contibution margin per minutes}=\frac{\text{Contibution margin per unit}}{\text{usage per unit}}\\ =\frac{\$20}{15}\\ =\$1.33\text{per minute}\end{array}$

For Frame Y:

$\begin{array}{c}\text{Contibution margin per minutes}=\frac{\text{Contibution margin per unit}}{\text{usage per unit}}\\ =\frac{\$30}{30}\\ =\$1.00\text{per minute}\end{array}$

From this calculation it is understood that the company should produce Frame X first. Therefore, out of 4,800 minutes, 3,000 minutes $\left(15\times 200\right)$ of welding will be allocated to Frame X and the remaining 1,800 minutes are used to produce Frame Y, in which it could produce 60 units of Frame Y $\left(\frac{1,800}{30}\right)$. Therefore, the optimal mix for Frame X is 200 units and for Frame Y is 60 units, and this as a result would produce a daily contribution of $5,800$\left(\$20\times 200\right)+\left(\$30\times 60\right)$.

To determine

Identify the objective function and the constraints and then draw a graph for the constraints. Ascertain the optimal mix and the maximum daily contribution margin (throughput).

Explanation of Solution

Contribution margin: Contribution margin is a measurement of performance where only revenue and variable costs are taken into consideration. Hence, this measurement is useful in the evaluation of the probable outcomes of decisions including pricing decisions and other marketing strategies that affect primarily revenue and variable costs.

Identify the objective function and the constraints and then draw a graph for the constraints. Ascertain the optimal mix and the maximum daily contribution margin (throughput):

Figure (1)

Corner point	X	Y	$Z=\$20X+\$30Y$
A	0	0	$0
B	0	100	$3,000
C	120	100	$5,400
D	200	60	$5,800
E	200	0	$4,000

Table (3)

Objective function:

$\text{Max Z}=\text{\$20X}+\$30\text{Y}$

Subject to:

$\begin{array}{l}\text{15X}+\text{10Y}<4\text{,800 }\\ \text{15X}+3\text{0Y}<4\text{,800 }\\ \text{ 15X}<4\text{,800 }\\ \text{10X}+\text{15Y}<4\text{,800 }\end{array}$

$\begin{array}{l}\text{ X}<2\text{00 }\\ \text{ Y}<1\text{00}\\ \text{ X}>0\\ \text{ Y}>0\end{array}$

To determine

Explain the manner in which a drum-buffer-rope system would work for company B.

Explanation of Solution

The welding process is the drummer in this case and, for the entire plant the drummer sets the production rate. As per the optimal mix, the plant should produce 200 units of Frame X and 60 units of Frame Y each day. In order to ensure that these rates are not exceeded by the cutting process, the release of materials is tied to the maximum production rate of the welding process. Finally, in front of the welding process, a time buffer is set up to protect throughput. Thus, this buffer would contain of 400 cut units of Frame X and 200 cut units of Frame Y for two days.

To determine

Evaluate the given proposed change and define the steps in the theory of constrains (TOC) process that is represented by the given proposal.

Explanation of Solution

Process	Resource Demands	Resource Supply
Cutting:
For Frame X	$\left(15\times 200=3,000\right)$
For Frame Y	$\left(10\times 100=1,000\right)$
	4,000	4,800
Welding:
For Frame X	$\left(10\times 200=2,000\right)$
For Frame Y	$\left(30\times 100=3,000\right)$
	5,000	4,800
Polishing:
For Frame X	$\left(23\times 200=4,600\right)$	4,800
Painting:	$\left(\text{10}\times \text{200}=\text{2,000}\right)$
	$\left(15\times 100=1,500\right)$
	3,500	4,800

Table (4)

In this case, the redesigning process would reduce the welding time of Frame X from 3,000 minutes to 2,000 minutes and also it increases the polishing time of Frame X from 3,000 minutes to 4,600 minutes. Thus, this decreases the excess capacity of polishing process as well as releases 1,000 minutes of scarce resource in welding process. Thus these 1,000 minutes of scarce resource could be used to produce an additional 33 units of Frame Y $\left(\frac{1,000}{30}\right)$, and this as a result would produce a daily contribution margin of $900. To recover the $10,000 needed for redesign, the company would take 10.1 workdays $\left(\frac{\$10,000}{\$990}\right)$.