Problem 7.4Problem 8.4 (M. M. Sprout)d. Upon adding another server (assuming that the fifth line has been added) we haveProblem 8.5 (Heavenly Mercy Hospital)b. Now consider the case where the service time is reduced to 20 minutes but the cost of the equipment and radiologist is $150 per hour. In this case:Problem 7.4 First, is this an EOQ problem? Well, notice that the question dictates that we do a run every two years. That would mean, in a deterministic EOQ setting, that Q must equal two years of mean demand, i.e., 32000. Hence, this question does not give us the freedom to change how often we do a run (which is what EOQ is all about). Thus, the question is whether 32000 is the best quantity we can print every tow years? This thus asks about what the appropriate safety stock (or service level) should be. We know that this is answered via newsvendor logic. Answer these two questions: 1. What is my underage cost (cost of not having enough)? I.e., if I were to stock one more unit, how much could I make? Every catalog fetches sales of $35.00 and costs $5.00 to produce. Thus, the net marginal benefit of each additional unit (MB), or the underage cost, is Cu=p – c = $35 - $5= $30. 2. What is my overage cost? I.e., if I had stocked one less unit, how much could I have saved? The net marginal cost of stocking an additional unit (MC) is Co = c – v = $5 – 0 = $5. Now, we can figure out the optimal service level (or critical fractile): SL = 30/(30+5) = 0.857. The last step is to convert the SL into a printing quantity. Recall that total average demand for 2 years (R) = 32,000 with a standard deviation of 5656.86. The optimal printing quantity, Q* is determined such that Prob(R ≤ Q*) = Cu/( Cu +Co)=0.857. The optimal order quantity Q* = R + z σ where z is read off from the standard Normal tables such that area to the left of z is 0.857. That is, z = 1.07. This gives Q* = 38,053 catalogs. It can be verified that the optimal expected profit (when using Q* = 38053) is larger than $25,000, the fixed cost of producing the catalog.Problem 8.4 (M. M. Sprout) a. We are given: Average arrival rate Ri = 1/4 per minute, Average unit capacity 1/Tp = 1/3 per minute, Number of servers c = 1. Using the Queue.xls spreadsheet, we get Server utilization ρ = Ri / Rp = 0.75, Average waiting time Ti = 9 mins, Average number of customers on hold Ii = 2.25, Average number of customers in the system I = 3. The costs include the CSR wages and the cost of waiting (line charge + waiting cost for customers). We have Hourly wages of CSR = $20 / hour, Line charge = $5 / hour (for all lines used), Customer waiting cost = Average number on hold×60×$2 = 2.25×120 = $270. Hence, the total hourly cost = $20 + $5 + $270 = $295/hour. b. With only four lines and one CSR, we have Average arrival rate Ri = 1/4 per minute, Average unit capacity 1/Tp = 1/3 per minute, Number of servers c = 1, Maximum buffer size K = 3. Using the spreadsheet Queue.xls we get Average waiting time Ti = 3.45 mins, Average # of customers on hold Ii = 0.77, Average number of customers in system I = 1.44, Calls blocked per hour = 1.56. In this case the costs incurred are the CSR wages, the cost of waiting (line charge + waiting cost for customers) and the lost business because of blocked calls. We have Hourly wages of CSR = $20 / hour, Line charge = $5 / hour, Customer waiting cost = Average number on hold×60×$2 = .77×120 = $92.4, Cost of blocking = Calls blocked per hour×$100 =1.56×$100 = $156. This implies that Total hourly cost = $20 + $5 + $92.4 + $156 = $273.4. c. Upon adding another telephone line, we have Average arrival rate Ri = 1/4 per minute, Average unit capacity 1/Tp = 1/3 per minute, Number of servers c = 1, Maximum buffer size, K = 4. Using the Queue.xls spreadsheet we get Average waiting time Ti = 4.33 mins, Average # of customers on hold Ii = 1.005, Average number of customers in system I = 1.70, Calls blocked per hour = 1.08In this case the costs incurred are the wages of the CSR, the cost of waiting (line charge + waiting cost for customers) and the lost business because of blocked calls. We have Hourly wages of CSR = $20 / hour, Line charge (of existing lines) = $5 / hour, Customer waiting cost = Average number on hold×60×$2 = 1.005×120 = $120.6, Cost of blocking = Calls blocked per hour×$100 =1.08×$100 = $108. Excluding the cost of the new line we have Total cost per hour = $20 + $5 + $120.6 + $108 = $253.6. As long as the cost of the new line is less than $273.4 (cost with 4 lines) - $253.6 (cost with 3 lines) = $19.8 / hour, it pays to install the new line. d. Upon adding another server (assuming that the fifth line has been added) we have Average arrival rate Ri = 1/4 per minute, Average unit capacity 1/Tp = 1/3 per minute, Number of servers c = 2, Maximum buffer size K = 3. Using the Queue.xls spreadsheet we get Average waiting time Ti = 0.42 mins, Average # of customers on hold Ii = 0.105 Average number of customers in system I = 0.85, Calls blocked per hour = 0.105. In this case the costs incurred are the wages of the CSR, the cost of waiting (line charge + waiting cost for customers) and the lost business because of blocked calls. We have Hourly wages of CSRs = $40 / hour, Line charge = $5 / hour, Customer waiting cost = Average number on hold×60×$2 = .105×120 = $12.6, Cost of blocking = Calls blocked per hour×$100=0.105×$100 = $10.5. In this case we have Hourly cost of system = $40 + $5 + $12.6 + $10.5 = $68.1/hour. This is a significant reduction in cost. The new CSR should thus be hired.Problem 8.5 (Heavenly Mercy Hospital) a. We have: Average arrival rate Ri = 18 per hour = 0.3 per minute, Average unit capacity 1/Tp = 2 per hour = 1/30 per minute, Number of servers c: To be determined, Cost per server = $100 per hour, Desired average time in system = 40 minutes. To plan staffing, we know that we should have a utilization of less than 100%, thus: Utilization = inflow/capacity = 18/hr/(c*2/hr) < 1 so that c > 9 Increasing the number of servers from 10 upward, we have the following results: (using the Performance.xls spreadsheet with K=100): Number of Servers (c) Avg. Number in System (I) Avg. Time in System (T) 10 15 50 minutes 11 11 36 minutes 12 10 32.7 minutes Thus hiring 11 servers achieves a turnaround time of …