Solve and show formula used and final answer (no excel)

Question 3. Loading Dock Planning for a Regional Warehouse The manager of a regional warehouse is tasked with the important decision of determining the optimal number of loading docks to request for the new facility to minimize the overall costs associated with driver-truck combinations and dock operations. With trucks arriving at the rate of four per day, the manager must carefully consider all factors to make the best decision. The manager has been informed that each driver-truck combination costs $300 per day (regardless of whether the truck is waiting to be loaded or is in service) and that each dock, along with its loading crew, costs $1,100 per day. The manager must determine the number of docks to request in order to minimize the overall costs while still ensuring an efficient loading process for drivers. It is assumed that the arrival rate follows a Poisson distribution. Additionally, it is assumed that each dock can handle five trucks per day and that the loading rate follows an exponential distribution. The manager has approached you as a consultant to provide recommendations and insights based on the information above. Your task is to analyze the available data and provide answers to the following questions: (a) Determine the optimal number of docks to request in order to minimize the daily sum of dock costs and driver-truck costs. In other words, assuming the loading rate of five trucks per day per dock, calculate the number of docks needed to minimize the overall costs. [Hint: Calculate I the overall costs for various options of requesting 1 dock, two docks, and three docks or higher. Compare the costs and choose the option with the minimum cost.] (b) An employee has proposed adding a new equipment that would increase the loading rate to 5.71 trucks per day. This would come at an additional cost of $100 per day for each dock. Consider the optimal number of docks you obtained in part (a). Now, evaluate this investment on those docks (that dock) and determine whether it would be justifiable based on the costs and benefits associated with it. Thus, you need to analyze whether by adding this investment, your optimal overall cost obtained in part (a) decreases or not. (c) If the manager decides to invest in the new equipment, the next step would be to calculate the average waiting time in the queue for the drivers. How much is this average in-queue waiting time? This would provide valuable information for the drivers and help the manager make future decisions related to dock planning