In this assignment you will evaluate the effects of using different batch sizes in a production environment that needs to use one machine to make a number of different products. In aggregate, the demand rate for all of the products combined averages about 50 units per hour. If the machine is going to keep up with demand, it then needs to be producing on average 50 units per hour. To keep the analysis simple, we will assume that all products require one minute of production time on the machine per unit. As such the machine is going to spend on average 50 out of every 60 minutes (83.33% of the time) producing these items. Although we have demand occurring simultaneously for many different units, the machine produces the different products in batches. Production orders are released when the inventory level for the item in question hits the item’s order point, or when a kanban card for the item in question is emptied. So the machine will receive an order for a certain quantity (Q) of one item, and produce that quantity. It will then receive an order for a certain quantity of some other item, and product those, etc. Since we are experiencing demand for many different items, it is possible that two or more production orders (each corresponding to production of a different item) are released at about the same time. In this case some jobs will wait in line (queue) for their turn to be processed on the machine. The question we need to address is the order quantity (Q) that will be used when production orders are released. For simplicity we will assume all of the different items produced by this machine use the same order quantity or lot size. As we learned in the EOQ unit, a larger order quantity will result in a larger time between orders. In this unit we will study some additional implications of batches sizes in a constrained capacity production environment. For example, if we use a lot size Q = 500 units per batch, the machine will receive a new production order on average once every 10 hours. (Note that production orders for batches of 500 units, on average 10 hours apart, averages out to producing 50 units per hour.) In the queuing notation we learned earlier, the arrival rate of production orders will be λ = 1/10 = 0.10 batches per hour. At one minute per unit processing time, the batch will require 500 minutes, or 500/60 = 8.333 hours of processing time. In queuing notation, the service rate (while the machine is running, not counting idle time between batches) is μ = 1/8.333 = 0.12 batches per hour. Notice we have a system with arrival rate λ = 0.10 batches per hour, and service rate μ = 0.12 batches per hour. Since we have one machine, the system utilization is then λ/μ = 0.10 / 0.12 = 0.83333, or 83.333% which agrees with what we observed above. Alternatively, we could use a lot size of Q = 200 units per batch. Keeping up with a demand rate of 50 units per hour implies we will be releasing new production orders on average 4 hours apart. (200 / 4 = 50 units per hour). Notice we are now releasing production orders for smaller quantities, but more frequently. The arrival rate is now λ = 1/4 = 0.25 batches per hour. As the production time is still one minute per unit, we now require 200 minutes, or 200/60 = 3.333 hours of machine time to run each batch. The service rate, within the context of a queuing system, is now μ =1/3.333 = 0.30 batches per hour. Notice the system utilization is still λ/μ = 0.25/0.30 = 0.83333. We are still meeting the same product demand and machine workload (50 units per hour on average), we are simply running smaller batches more frequently. Assignment Part 1: Complete Table 1 presented in the attached Lesson 13 Assignment Worksheet. The two examples presented above are entered to help get you started. You may use your calculator and simply enter the results of your calculations into the spreadsheet, or you can enter formulas and let Excel do the work. (Do not look for formulas specific to this analysis in Chapter 13 or elsewhere in the book, they are not in there. Here you are applying basic relationships between batch size and time between orders, etc. that you have studied earlier. You should be able to think through the calculations, as opposed to simply following a formula.) If you are doing the calculations correctly, you should find system utilization λ /μ = 0.83333 in every case in Table 1. As noted above, we are meeting the same demand, but using successively smaller batch sizes and correspondingly more frequent job release. For the average time in queue and average time in system columns, you will use the M/D/1 queuing template provided below. This is similar to the M/M/1 model we used earlier, except the formulas correspond to the situation where the service time (in this case, the machine processing time) is constant. We still have queuing due to the fact that the time between arrivals is random. Enter the correct values of λ and μ for each scenario into the M/D/1 template, and copy your results for time in queue and time in system into Table 1. M/D/1 Excel Template After you have completed Table 1, write a short paper that answers the following: What happens to the average time in queue (waiting in line) and average time in system (waiting in line or being processed) as we move from releasing production orders for large quantities with a long time between orders, to using smaller quantities with a shorter time between orders? When we studied the EOQ model we saw the effect on product inventory levels (average cycle stock = Q/2., “large triangles” vs. “small triangles”,etc.), but now we are investigating the effect on the production process that makes the items in question. Assignment Part 2: Repeat the analysis from Part 1, but now assume there is a 15 minute (0.25 hours) setup time associated with each new batch. Notice, this is an extra 0.25 hours of machine time that is consumed with each batch, regardless of which batch size (Q) is being used. Table 2 shows the effect of the setup time on our Q = 500 policy. We still have an arrival rate of λ = 0.10 batches per hour, since Q = 500 results in on average one production run every 10 hours. The total machine time per batch, however, is now 515 minutes, or 515/60 = 8.5833 hours. As such our service rate is μ = 1/8.5833 = 0.1165 batches per hour. As shown in Table 2, under the Q = 200 policy our arrival rate is still λ = 0.25 batches per hour, which corresponds to one batch every 4 hours on average. The setup time is the same 15 minutes, regardless of whether we produce Q = 500 or Q = 200 or any other quantity. So the total machine time required for the batch of 200 units (setup + run time) is 215 minutes, or 215/60 = 3.5833 hours. We now have a processing rate (while the machine is running) of μ = 1/3.5833 = 0.2791 batches per hour. Notice in Table 1, the machine utilization is 0.83333 in every case, reflecting the fact that the machine spends the same average fraction of time each hour producing the products. In Table 2, however, the machine utilization is λ/μ = 0.10/0.1165 = 0.858 when using Q = 500, and λ/μ = 0.25/0.2791 = 0.896 when using Q = 200. Although the fraction of each hour spent actually running the product is unchanged (still 83.33% of the time), the smaller batch sizes require us to complete the 15 minute setup more often. As a result we see the machine utilization (setup + run time) increase as we decrease the batch size. Complete the remaining rows of Table 2, and use the M/D/1 template to compute the steady state queue times for each scenario in Table 2. Write a short explanation of what you have seen. REMEMBER: The queuing models only work when utilization is less than 100%, or in other words when λ < μ. Do not use the M/D/1 template for cases where λ > μ. Logically, the cases in Table 2 with λ >μ (utilization greater than 1.00) represent cases where the batch size is too small, and the time expended on setups is too large, given the production time requirements on the machine. In other words, the setup time creates a lower bound on the range of batch sizes we can try to use. If we are doing so many setups such that the machine cannot keep up with product demand, we need to use a larger batch size and as a result decrease the number of setups. With Table 2 complete (for those rows where utilization < 100%), write an explanation of your results. In contrast to the results of Table 1, what happens to the queue time for the process as the batch size is decreased? Explain why.