$$11$

$$9$

$$5$

$$4$

Step $1$

Step $1$

Station W

Station W

Step $2$

Step $2$

Station Y

Station Z

Product A

Product B

Raw materials

Raw materials

$(10$ min$)$

$(7$ min$)$

$(12$ min$)$

$(12$ min$)$

Purchased part

Step $1$

Product C

Step $2$

Station X

Purchased part

Purchased part

Raw materials

Station W

$$16$

$$5$

$(15$ min$)$

$(11$ min$)$

In the diagram, a flowchart represents the manufacturing process of products A$,$ B$,$ and C in different rows. From top to bottom, rows are arranged for Product A$,$ Product B$,$ and Product C$,$ respectively. Each row begins with an inverted triangle containing the dollar amount for the raw materials that go into the process. An arrow that emerges from the triangle is followed to the right by a series of rectangles for each step in the process. Each rectangle contains the step number, station name where that step takes place, and duration in minutes for that step. Between steps $2$ and $3,$ each process has a purchased part added to the flow as indicated by an inverted triangle with the dollar amount for the purchased part inside. The final step in each process is followed by an arrow to a summary process that shows the product name, price per unit, and demand in units per week for that product. For product A$,$ a value of $$11$ is displayed inside the initial triangle. For Step $1,$ Station W is used and the step takes $10$ minutes. For Step $2,$ Station Z is used and the step takes $12$ minutes. After step $2,$ a purchased part costing $$4$ is added to the flow. For Step $3,$ Station X is used and the step takes $8$ minutes. The final output arrow points to summary process that reads Product: A$,$ Price: $$105$ per unit, and Demand: $53$ units per week. For product B$,$ a value of $$9$ is displayed inside the initial triangle. For Step $1,$ Station W is used and the step takes $7$ minutes. For Step $2,$ Station Y is used and the step takes $12$ minutes. After step $2,$ a purchased part costing $$5$ is added to the flow. For Step $3,$ Station Z is used and the step takes $9$ minutes. The final output arrow points to summary process that reads Product: B$,$ Price: $$95$ per unit, and Demand: $159$ units per week. For product C$,$ a value of $$16$ is displayed inside the initial triangle. For Step $1,$ Station X is used and the step takes $11$ minutes. For Step $2,$ Station W is used and the step takes $15$ minutes. After step $2,$ a purchased part costing $$5$ is added to the flow. For Step $3,$ Station Y is used and the step takes $6$ minutes. The final output arrow points to summary process that reads Product: C$,$ Price: $$120$ per unit, and Demand: $53$ units per week.

Step $3$

Step $3$

Station X

Station Z

Product: A

Product: B

$(8$ min$)$

$(9$ min$)$

Price: $$105/$unit

Demand: $53$ units$/$wk

Demand: $159$ units$/$wk

Price: $$95/$unit

Step $3$

Station Y

$(6$ min$)$

Product: C

Price: $$120/$unit

Demand: $53$ units$/$wk

x y graph

Part $2$

The senior management team wants to improve the profitability of the firm by accepting the right set of orders.$$ Currently, decisions are made using the traditional$$ method, which is to accept as much of the product with the highest contribution margin as possible$($up to the limit of its$$ demand$),$ followed by the next highest contribution$$ product, and so on until all available capacity is utilized. Because the firm cannot satisfy all the$$ demand, the product mix must be chosen carefully. Jay$$ Perry, the newly promoted production$$ supervisor, is knowledgeable about the theory of constraints and the$$ bottleneck$-$based method for scheduling. He believes that profitability can indeed be improved if bottleneck resources are exploited to determine the product mix.

What is the change in profits$$ if$,$ instead of the traditional method that YPI has used thus$$ far, the bottleneck method advocated by Jay is used for selecting the product$$ mix?

Part $3$

What is the profit if the traditional method is used for determining$$ YPI's product$$ mix? $($Note: When determining the product$$ mix, if the number of units to produce is not a whole$$ number, be sure to round down to the next whole number before proceeding with any profit$$ calculations.$)$