IE 1055/2025 Homework 3 Due Jan. 25, 20171. Problem 2.25 from the text. Assume that part A must pass through machines 1 and 2 to be completed. For example, machine 1 drills a hole and machine 2 puts a taper in the hole made at machine 1. Similarly, part B must pass through both machines 3 and 4 to be completed. Please note that on problem 2.25 that "the desired output 06" means the output of station 6 and that "input 11 to machine 1" is a typo and means theinput to machine 1 (it ought to read "input to machine 1"). (20 pts)a. Determine an equation for the required input for both machines 1 and 3 (not just machine 1 as stated in the problem.)b. Determine the required input for both machines 1 and 3 if the following scrap percentages apply: d1=.10, d2=.05, d3=.05, d4=.02, d5=.04, d6=.01.c. If the desired output of parts at station 6 is 70 parts per hour determine the input rate of parts required at stations 1 and 3 for the problem variation described in part b of this problem. d. How does the answer to part b change if 75% of the defects at machine 1 and 99% of the defects at machine 5 can be reworked? Give both a qualitative and quantitative answer – the quantitative answer will relate I1 to O6 for example but will not be a number such as start with 100 parts because no numeric value is specified for O6.2. 2.23 except change the quantity required to 300 non-defective parts. (10 pts)3. Rework problem 2.23 (still change the quantity required to 300 non-defective parts) if one halfof the defective items can be reworked. Assume that the reworked items also have an 80% probability of being non-defective and that the parts can be reworked indefinitely (thus, the reworked parts have the same probability of being good as the new parts do - these assumptions are similar to what we assumed in the class discussion.) How does your answer compare to the answer for problem 2.23 from problem 2? Discuss any implications. (10 pts)4. You work for SteelPro a fabrication company that specializes in machining specialty alloys. One of your best customers is Bombardier Transportation. Bombardier has asked you to fill an order for 20 of their part 107. Bombardier will pay you $500 for each good part 107 but will only accept exactly 20 good ones – no more, no less. The raw materials required to make a 107 cost $150. Unfortunately you have yield loss issues related to manufacturing the 107. Assume that the 107s are produced independently of each other with the probability of an individual 107 being good being equal to 0.85. Defective 107s can be scrapped for $50. There is also the technical challenge that the way you process the 107s is that you machine the entire batch (whatever batch size you choose) and then you heat treat and further process them and only after the heat treating and further processing do the defects become detectable. Thus, you must determine the batch size to run prior to knowing what your precise yield will be. Assume you only have time to run one batch. (20 pts)i. Given the economics stated in the problem, what is the best batch size to run for an order of 20 107s to maximize the expected profit?ii. Bombardier has indicated that they will order part 107, or a part very similar to it made by similar processes, three times per year. Suppose you could invest$1500 to improve the yield for the 107 and related parts from 0.85 to 0.92, would that be a good idea? Why or why not? 5. (15 pts) Suppose you work for the Pittsburgh Steel Company. Your company specializes in rolling specialty steel alloys and titanium. Material is purchased in slabs that may be up to 6 inches thick and is rolled into plates (about 0.5 inches thick)and sheets (.060 to 0.25 inches thick). Pittsburgh Steel Company's marketing strategy is to fill smaller orders that their competitors cannot produce profitably and to specialize in orders for materials that are difficult to process (like certain steel alloys, etc.) You have been given the assignment of improving the layout of their existing facility. Currently the facility has three hot rolling mills, two large walking beam furnaces (these feed the hot rolling mills), shearing equipment, saws for rough cuttingslab material, a pickling house (used for de-scaling of the material), and two annealing furnaces. a. Describe some of the information that you would need in order to create a new layout for the facility and explain how you would obtain this information. Be as specific as possible. In addition, discuss what some of the special considerations and constraints might be in designing this layout.b. Your boss wants to know if a product, process, or group technology layout would be the best choice in this setting. How would you answer her? How would you arrive at that conclusion?6. Consider a process where a fraction p (if p = .07 then 7 parts out of 100 parts are defective) ofthe items processed turn out defective. Suppose a defective item has two particular types of defects that determine the ability to rework it. Given that an item has been identified as defective, inspector 1 inspects it for the first type of defect. As shown in the Figure on the next page, if he/she finds that the item has the first type of defect, then the item is scrapped (with probability q3), or it is sent back for reprocessing (with probability q1). Otherwise, the defective item is directly passed to the second inspector who inspects it for the second type ofdefect.With probability (1-r), inspector 2 finds that the item has the second type of defect and the item is scrapped. Otherwise, the defective item is considered to be reworkable and it is sent back for reprocessing. It is assumed that an item can be reprocessed indefinitely and that, regardless of the number of times an item has been reprocessed, it turns out defective with probability p. It is also assumed that the two types of defects occur independently. (25 pts)(a) Derive a closed-form expression for O as a function of I and the appropriate probabilities.(b) Resolve part (a) assuming that q2 = 1. (Hint: to avoid carrying over a mistake you might have made in the first question, you are encouraged to view this question as a “new problem” with q2 = 1.)(c) Resolve part (a) assuming that the parts can be reworked at most 2 times. Comment on the difference between your answers to parts (a) and (c).IOp1q1q3q2Scrap by 12r1-rScrap by