Assignment #$3$ Simulation $(10$ marks$)$ Topic: Balancing Process Capacity Objective: A car wash company wants to maximize its capacity utilization and total profits for the eight$-$hour workday $(9$:$00$ am to $5$:$00$ pm$).$ The operations manager of this company should work on two scenarios $($challenges$)$: $($a$)$ the no variation and $($b$)$ the variation. In the former scenario, there is no variability in demand and process $($supply$),$ while in the latter scenario, variability is present. Key concepts: Capacity utilization; Cycle time; Throughput: flow rate $($or throughput rate$)$ and flow time $($or throughput time$)$; Inventory $($or work$-$in$-$process$)$; Little$$s law; Variability in demand and process Specifications: The customer arrival rate varies according to the time of day, with a peak in demand from $12$:$00$ pm to $1$:$00$ pm$.$ Serving as the operations manager of the car wash company, you are responsible for allocating the company$$s entire budget of $$3$ million to the three stages of the process $($i$.$e$.,$ the vacuum, the machine wash, and$/$or the hand dry stations$)$ to maximize efficiency and thereby overall profitability. Questions $1.$ Please work on two scenarios$/$challenges $($i$.$e$.,$ no variation and variation$)$ and run the simulations as many times as you want to identify the: $($a$)$ maximum profit, $($b$)$ optimal number of cars should be washed to maximize profits, $($c$)$ optimal budget $($available budget: $$3$ million$)$ allocation to the three stages. Tip: the optimal budget allocation for Challenge $1($i$.$e$.,$ no variation$)$ is as follows: Vacuum: $$1,000,000$; Wash: $$500,000$; Hand Dry: $$1,500,000$; Profit: $$825.$ Challenge $2$:

Process information before investment:

Vacuum and Trash Removal: Min $22\mathrm{.}5,$ Mode $36\mathrm{.}0,$ Max $40\mathrm{.}5,$ Upgrade: $$1$M buys $+1\mathrm{.}5$ cars$/$ hr

Machine Wash: Min $18\mathrm{.}8,$ Mode $30\mathrm{.}0,$ Max $33\mathrm{.}8,$ Upgrade: $$1$M buys $+15$ cars$/$hr

Hand Dry and Windows: Min $21\mathrm{.}0,$ Mode $24\mathrm{.}0,$ Max $33\mathrm{.}0,$ Upgrade: $$1$M buys $+9$ cars$/$ hr$.$ how can I calculate the best way to distribute the $$3$M investment to maximize profits with the variability in Challenge $2?$