The Wharton School The University of Pennsylvania
Quarter II Fall 1999
PRACTICE PROBLEM SET
Topic 1: Basic Process Analysis Problem 1: Consider the following three-step production process: Raw Material
Finished Units
A
B
C
3 min. / unit
5 min. / unit
2 min. / unit
The numbers below each operation refer to the time necessary to process one unit of the product at that step (i.e., the activity time). The process is staffed by three operators, with one operator assigned to each of the three steps. steps. Assume there is no variability variability in the process, the operators are capable of working at 100% efficiency, and no in-process inventory is permitted between activities. a)
What is the capacity of the process?
b) What is the utilization of the operator at Step A?
c) What is the direct labor content of a unit produced by this process?
d) Suppose each operator receives regular wages of $12/hour for an 8 hour day, and $18/hour for any time over 8 hours. Twelve hours per day is the maximum work time permitted for each operator. A finished unit sells for $10 (with unlimited demand a t that price) and total cost per unit is $6 plus the cost of direct labor. i)
Should you use overtime?
ii)
Would you recommend hiring a skilled operator whose wage rate was $20/hour (regular time) if he could perform Step B at an accelerated activity time of 4 minutes / unit?
Problem 2: Consider the following modification of the process described in Problem 1.
The process is identical except that now a rework step has been added for units that do not pass inspection at step C. Rework requires 10 minutes/unit and is staffed by an additional operator. Raw Material
Finished Units Pass Inspection
A
B
C
3 min. / unit
5 min. / unit
2 min. / unit
Rework Fail Inspection
10 min. / unit
What is the maximum fraction of products that can be defective before rework becomes the bottleneck of this process?
Problem 3: Consider another modification of the process described in Problem 1. The
process is identical except that now 2 units of in-process inventory are placed between steps A and B (as shown below). Other than the maximum of two units in this buffer and up to one unit at each of steps A, B, and C, there is no room to store work in process inventory. Raw Material
Finished Units Buffer (2) A
B
C
3 min. / unit
5 min. / unit
2 min. / unit
What is the process flow time for this process if work is pushed into the system (not counting time in either raw materials inventory or finished goods inventory)?
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An Example: Analysis of a Bakery Process1
In a bakery, croissants are made using the process depicted in Figure 12. The dough and filling for the croissants are prepared separately. The parallel processes of making dough and mixing filling are dependent, both must be completed before the croissants can be filled, folded and baked. All operations are done in batches of 50 croissants. These parallel processes are of a fundamentally different nature than those we have seen so far, for example in the Ome plant at Toshiba. Mixing the filling for a batch of 50 croissants is relatively quick, taking only 10 minutes. However, preparing the dough involves three steps: mixing, proofing, and rolling & cutting. Proofing with an activity time of 15 minutes per 50 croissants, is the slowest step in the process of preparing the dough. Work in Process Dough
Raw Material Dough
Mix 5 min / batch Raw Material: Filling
Roll & Cut
Proof 15 min / batch
Finished Units
5 min / batch
Work in Process Filling
Fill & Fold
Bake
5 min / batch
20 min / batch
Pack 10 min / batch
Mix Filling 10 min / batch
15 min / batch
Figure 1: Process Flow Diagram and Activity Time for Croissant-Making
The throughput of the two sub-process, preparing the dough and mixing the filling, is determined by the lower capacity of the sub-processes, which is preparing the dough. Dough preparation can come up with a batch of 50 croissants every 15 minutes. Now that we have determined the capacity of these parallel subprocesses, we can determine the capacity of the entire process. The bottleneck of the entire process including assembly is the baking step. No matter how fast all the other steps can be completed, the oven can only bake one batch of 50 croissants every 20 minutes. As packaging 50 croissants only takes a total of 10 minutes, the packaging operation incurs an idle time of 10 minutes while it waits for the next batch to come out of the oven. A new batch of 50 croissants completes the process every 20 minutes. The bakery's overall daily capacity of croissants, assuming it operates the oven eight hours a day, is 24 batches, or 1200 croissants. 1
This is a description of a Process Analysis rather than a practice problem. The purpose of this example is to help reinforce some of the concepts introduced in the first set of class sessions. 2 This note is based on a Harvard Teaching Note on process analysis. It has been adapted to be consistent with the terminology used in OPIM631.
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We noted in the previous paragraph that "a new batch of 50 croissants will complete the process every 20 minutes", but how long does it take for a single batch to complete the entire process, beginning in mixing and ending in packaging? The answer will be the process flow time. The bakery begins making a new batch of dough every 20 minutes. Moving faster than this would yield a pile-up of inventory in front of the oven. Under this schedule, as shown in the time-phased Gantt chart in Figure 2, Batch #2 finishes 20 minutes after Batch #1, but each batch takes 60 minutes to flow from start to finish. This is the process flow time for making croissants. Note what would happen if we started mixing Batch #2 immediately after Batch #1 was mixed. The process flow time for Batch #2 would increase to 75 minutes. The management of the bakery would not choose to generally operate in this way (starting a new batch in the mixer as soon as it becomes free) in order to avoid building up piles of dough in front of the proofer. Mix
Batch #1
Proof
Batch #2
Roll & Cut Mix Filling Fill & Fold Bake Pack 5 10
20
30
40
50
60
70
80
Minute
Figure 2: Gantt Chart for Croissant Making
If the bakery begins mixing the filling exactly 15 minutes after it begins mixing the dough (as shown in Figure 2) then the filling and the dough will be ready for the fill & fold step at the same time. However, if the bakery begins mixing the filling at the same time it begins mixing the dough, the filling will spend 15 minutes as work-in-process inventory while it waits for the dough to be prepared. In either case the total manufacturing lead time for croissant making will remain 60 minutes. How much work in process inventory is in the bakery? This is not an easy question, since some of the work centers (e.g. mixing or packing) are sometimes empty (inventory=0) and sometimes not. Fortunately, we can turn to Little's law, which gives us the basic relationship: Inventory=Throughput * Process Flow Time. Both, flow time (60min) and throughput (150 croissants/hour=2.5 croissants/minute), have been computed above. So we can write: Inventory = 2.5 croissants/min. * 60 min. = 150 croissants.
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The Wharton School The University of Pennsylvania
Quarter II Fall 1999
PRACTICE PROBLEM SET
Topic 2: Process Analysis with Batching
Problem 1: Professors Pearson, Fisher, Terwiesch, and Ulrich are thinking of buying a
pizza carry-out shop in West Philly and have asked you to help them evaluate the current operation. The shop, OPIM's Pizza, currently operates in a build-to-order fashion with 4 employees: an order taker who receives/processes orders and does bookkeeping, a pizza tosser who prepares the pizza shell, a pizza assembler who customizes the pizza, and one chef who supervises the placement of the pizza in the oven. The production process begins when an order is transmitted to a computer in the kitchen. Order processing takes a negligible amount of time. It takes one half of a minute for the pizza assembler to prepare the pizza shell to give to the pizza tosser. The pizza tosser then tosses the shell for two minutes to obtain proper shape and consistency. The tossing operation is highly skilled, and can only be performed by the tosser. It then takes one minute for the pizza assembler to customize the pizza by adding the correct sauce and ingredients (e.g., cheese, pepperoni, mushrooms). The secret to OPIM's famous taste is a three minute wait prior to placing the pizza in the oven. This wait allows the flavors of the ingredients to blend. After this three minute delay, the chef places the pizza in one of six ovens (with a capacity of one pizza each) for a baking time of 10 minutes. (a) What is the capacity of the current cooking process? Specifically, what is the maximum number of pizza's that OPIM's can produce each hour (ignoring the rampup phase in the first hour)? (b) Professor Fisher has suggested producing pizza shells for inventory before each day's hours of operations. That is, the pizza tosser would arrive prior to the business hours and perform all the work associated with pizza shell p roduction. These shells would then be available to the assembler as needed during store hours. How would producing pizza shells to inventory change your answer to question (a)? (c) Professor Pearson would like to increase potential demand by offering pizza delivery direct to the customer. Demand in the past has been 20 pizzas per hour but Professor Ulrich is concerned that the current production process will not be able to support this demand increase. What (if any) changes will be needed in the cooking process to support a demand of 40 pizzas/hr (after the first hour startup period)? Assume that pizza shells are made to inventory as indicated in question (b).
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Problem 2: Metal window boxes are manufactured in 5 basic colors in a small plant in
northern Pennsylvania. The manufacturing process consists of 3 basic operations: stamping, painting, and assembly, as shown below. S3 S2
S1 Stamping
Painting
Assembly
Retailer
Each window box is made up of three pieces: a base (one part A) and two sides (two part Bs). The parts are fabricated by a single stamping machine which requires a setup time of 120 minutes whenever switching between the two part types. Once the machine is setup, the activity time for each part A is 1 minute while the activity time for each part B is only 30 seconds. Currently, the stamping machine rotates its production between one batch of 360 for part A and one batch of 720 for part B. Completed parts move from the stamping machine onto the painting station only after the entire batch is complete. At the painting station, parts are painted by a robot in 1 of 5 colors. The robot takes 30 seconds to paint one part A and 10 seconds to paint one part B. The robot can easily switch between painting the two parts, but a switch in c olor does require 20 minutes for setup. Once a piece is painted, it must wait 120 minutes to dry before moving to assembly. The painting robot is currently programmed to change color every time it finishes 360 component sets (i.e., 360 of part A and 720 of part B) and all parts move to assembly as a batch. At assembly, parts of the same color are assembled manually to form the finished product. One base (part A) and two sides (two part Bs), as well as a number of small purchased components, are required for each unit of final product. Each product requires 27 minutes of labor time to assemble. The factory runs one 8 hour shift per day, five days per week. There are currently 15 workers: twelve assembly workers, two operators for the stamping machine, and one operator for the robot. There is sufficient demand to sell every box the system can make. (a) Identify the bottleneck for the current process and calculate the maximum daily flow rate in terms of the # of units of final product produced per day. (b) Suppose we begin the day with no work in process. What is the flow time for the first batch completed? What other factors might affect flow time for other batches as the day progresses? (c) What impact would doubling the batch sizes at the stamping operation have on the total daily flow rate? Explain qualitatively what other, possibly negative, effects this would have.
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(d) Management is considering investing in a setup reduction program for the stamping machine. Suppose the setup time could be reduced to 20 minutes, with some additional investment cost. What would happen to the flow rate if the batch sizes remained the same (i.e., 360 for part A and 720 for part B)? What would be the potential benefits of such a reduction in setup time if the batch sizes were allowed to change?
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The Wharton School The University of Pennsylvania
Quarter II Fall 1999
PRACTICE PROBLEM SET
Topic 3: Process Analysis with Variability Problem 1: While most bank branches and post offices use the "snake" queue (i.e. one
common queue for all "servers"), you may have encountered a service operation in which a separate queue forms outside of each server's window. Furthermore, you may have noticed that, in order to impose a certain amount of order in the waiting area, the operation had set up velvet ropes that kept waiting customers from constantly jockeying among the windows' queues. Now, consider the following situation. Customers arrive to a branch of the Wynnewood Bank every 6 minutes, on average, and the standard deviation of the inter-arrival time is also 6 minutes. Wynnewood bank has three branches with identical demand and services. Each of the branches has one clerk, who can serve a customer in 4 minutes, on average. The standard deviation of the service times is 2 minutes. (a) Suppose the branches use a first come first serve policy. Use the formula in Chapter 8 (MBPF) to estimate the average customer wait before being served. (b) Wynnewood Bank is considering consolidating all three branch locations in a "superbranch" at the newly constructed Wynnewood Square. Assume the demand for this new branch would be the sum of all three locations (i.e. inter-arrival time is 2 min, standard deviation is 2 minutes). What would be the average customer wait? (c) Now consider Merion Bank which has two clerks. At the moment the bank uses specialized queues, one for credit counseling, (average service time 10 min, standard deviation 0 minutes) and one for cash withdrawals (average service time 2 min, standard deviation 0 minutes). There are five times more customers coming for withdrawals than for credit counseling (15 minutes inter-arrival time for credit counseling, 3 minutes inter-arrival time for withdrawals; in both cases the coefficient of variation is 1). The head of Merion Bank heard about the benefits Wynnewood Bank obtained from pooling demand and is now considering pooling the demand for his two clerks. Do you think this is a good idea? Advanced question: Can you use queuing theory to support his idea (or to reject it)?
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Problem 2: The Greek Lady is concerned about the amount of time customers must wait
to be served outside her sandwich truck. To determine the magnitude of the problem, customer inter-arrival data was collected between 11:30 am and 1:30 p.m. over a series of weeks. After sorting through the data, The Greek Lady finds that 35 customers on average arrive each hour with a standard deviation in customer inter-arrival times of 2.5 minutes. The Greek Lady estimates that it takes an average of 1.5 minutes to process an order with a standard deviation of 1 minute. a) What fraction of the time is the Greek Lady serving customer? b) How long do customers spend waiting on average in queue? c) What is the average number of customers in the system (i.e., being processed and waiting)? d) How would your answers change if all orders took exactly 1.5 minutes to fill?
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The Wharton School The University of Pennsylvania
Quarter II Fall 1999
PRACTICE PROBLEM SET
Getting Ready for the Exam CASE:
EXECUTIVE S HIRT
Executive Shirt is a batch flow operation. The case was used as the final exam for the course in 1996 and so is a good way for you to prepare for the final. Before doing any analysis, review each of the exhibits in the case to be sure that you understand what information is given. Assume 8 hours per day and 20 days per month of operation. You may also assume that the process steps are completed in the order given in Exhibit 3. Assume that under Mike’s plan, cutting for regular shirts continues to be performed on a stack of up to 60 layers of fabric. These parts are then d ivided into batches of 5 shirts immediately after cutting. Questions:
(1) What is the Process Capacity for the current and the two proposed plans? Calculate the average Flow Time of a shirt through the production process. Repeat this calculation for regular shirts and for custom shirts. (2) What improvements could you make to Mike’s plan to further reduce the Flow Time (and therefore delivery time) for custom shirts? (3) Suggest possible modifications to Ike’s plan to improve its productivity (i.e., to reduce costs). (4) Which plan would you pursue (Mike’s plan, Ike’s plan, an improved version of either plan, or some other plan of your own design)? Consider how these plans would be positioned graphically in a two-dimensional plot of unit cost versus order response time. CASE:
VAL DE CANON BICYCLE MANUFACTURING
This case provided the foundation for the 1998 exam. It will be handed out in class, including the other questions of the 1998 exam. This is the final prep!
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