Problem

$1$

: Electronics Manufacturing Line

A manufacturing process produces electronic components through four sequential workstations: Soldering, Testing, Packaging, and Labeling. Each workstation has a specific processing time per unit and operates for

$1$

$0$

$hours$

$($

$6$

$0$

$0$

$minutes$

$)$

$per$ day. Due $to$ market demand, the goal $isto$ maximize daily output while respecting capacity constraints. The process has a bottleneck that $limits$ the total number $of$ units produced daily. The profit per unit $is\frac{}{?}$

$($

$\frac{}{?}$

$

$1$

$5$

$\frac{}{?}$

$)$

$,$

$and$ the objective $isto$ calculate the maximum daily profit based $on$ the bottleneck's capacity.

Process Details:

$-$

$Soldering$ Workstation: Processes

$1$

$unit$ every

$3$

$minutes.$

$-$

$Testing$ Workstation: Processes

$1$

$unit$ every

$8$

$minutes.$

$-$

$Packaging$ Workstation: Processes

$1$

$unit$ every

$5$

$minutes.$

$-$

$Labeling$ Workstation: Processes

$1$

$unit$ every

$2$

$minutes.$

$-$

$Operating$ Time:

$6$

$0$

$0$

$minutes$ per day

$($

all workstations operate simultaneously but are constrained $by$ the bottleneck

$)$

$.$

$-$

$Profit$:$\frac{}{?}$

$($

$\frac{}{?}$

$

$1$

$5$

$\frac{}{?}$

$)$

$per$ unit produced and sold.

$-$

$Demand$: Assume demand $is$ sufficient $to$ sell all units produced.

Table: Workstation Processing Times

Formulas:

$-$

$Capacityof$ a Workstation

$($

units per day

$)$

:$\frac{}{?}$

$($

$\frac{}{?}$

mathrm

$\{$

$C$

$\}$

$=$

$\frac{}{?}$

$)$

$Available$ Time$\frac{}{?}$

$($

$\frac{}{?}$

div$\frac{}{?}$

$)$

$Processing$ Time per Unit

$-$

$Available$ Time$\frac{}{?}$

$($

$=$

$6$

$0$

$0$

$\frac{}{?}$

$)$

$minutes$ per day

$-$

$Cis$ rounded down $to$ the nearest integer

$($

since partial units cannot $be$ produced

$)$

$-$

$Bottleneck$ Capacity: The workstation with the lowest capacity

$($

smallest $C$

$)$

$determines$ the process output

$-$

$Daily$ Output: Equal $to$ the bottleneck capacity

$($

units per day

$)$

$-$

$Daily$ Profit: $P\frac{}{?}$

$($

$=$

$\frac{}{?}$

$)$

$Daily$ Output$\frac{}{?}$

$($

$\frac{}{?}$

times$\frac{}{?}$

$)$

$Profit$ per Unit

$-$

$Profit$ per Unit$\frac{}{?}$

$($

$=\frac{?}{?}$

$

$1$

$5$

$\frac{}{?}$

$)$

Question: Determine the bottleneck workstation's capacity and calculate the maximum daily profit

$($

$in$ dollars

$)$

$basedon$ the process's daily output. Provide only the numerical value $of$ the maximum daily profit $as$ your final answer.

Problem $2$: Textile Production Process

A textile manufacturing process produces fabric through three sequential workstations:

Weaving, Dyeing, and Quality Control. Each workstation has a specific processing time per

unit and operates for $12$ hours $(720$ minutes$)$ per day. Due $to$ market demand, the goal $isto$

maximize daily output while respecting capacity constraints. The process has a bottleneck

that $limits$ the total number $of$ units produced daily. The profit per unit $is$ $$8,$ and the

objective $isto$ calculate the maximum daily profit based $on$ the bottleneck's capacity.

Process Details:

Weaving Workstation: Processes $1$ unit every $6$ minutes.

Dyeing Workstation: Processes $1$ unit every $9$ minutes.

Quality Control Workstation: Processes $1$ unit every $4$ minutes.

Operating Time: $720$ minutes per day $(all$ workstations operate simultaneously but are

constrained $by$ the bottleneck$).$

Profit: $$8$ per unit produced and sold.

Demand: Assume demand $is$ sufficient $to$ sell all units prod A food processing facility produces packaged meals throu $2$