Problem 1: Process Analysis
You manage an assembly line consisting of 6 work stations (referred to as Work Station 1, Work Station 2, ... , Work Station 6) in series, each manned by a single worker (Alex, Barbara, Chuck, Donna, Ed, and Fran). Each unit of product is processed first at Work Station 1, then at Work Station 2, and onward through the six work stations in numerical order. For reasons not completely understood, the amount of time required to complete the work at each work station depends on which worker is assigned to that station. The following table gives the number of minutes each worker requires to complete the tasks associated with each work station.
|
Workers
|
Work Stations
|
|
1
|
2
|
3
|
4
|
5
|
6
|
|
Alex
|
10
|
35
|
41
|
45
|
15
|
23
|
|
Barbara
|
33
|
40
|
24
|
19
|
49
|
13
|
|
Chuck
|
16
|
20
|
39
|
32
|
18
|
48
|
|
Donna
|
47
|
12
|
14
|
26
|
44
|
33
|
|
Ed
|
25
|
43
|
37
|
31
|
27
|
21
|
|
Fran
|
50
|
28
|
22
|
11
|
38
|
42
|
Therefore, if Alex is assigned to Work Station 1, he will require a total of 10 minutes to complete the tasks at that work station for every unit of product. In contrast, he will require 45 minutes if assigned to Work Station 4. Similarly, Donna will require 47 minutes if assigned to Work Station 1, but only 12 minutes if assigned to Work Station 2. Clearly, one worker must be assigned to each work station, and these assignments are considered permanent for the foreseeable future.
Assume that the system is always busy (i.e., there is always demand for this product), and there is no transient state (so, e.g., workers can stop work on a unit of product at the end of a shift and resume work at precisely the same point the next day). Also assume that no inventory is allowed between work stations; instead, workers who complete their tasks faster than the bottleneck work station accumulate idle time.
a. To which work station should each worker be assigned in order to minimize the flow time of the assembly line? What is the associated flow time?
Alex _____ Barbara _____ Chuck _____ Donna _____ Ed _____ Fran _____
Flow Time (minutes) __________
b. To which work station should each worker be assigned in order to maximize the flow rate of the assembly line? What is the associated flow rate?
Alex _____ Barbara _____ Chuck _____ Donna _____ Ed _____ Fran _____
Flow Rate (units/hour) _____________
a. To which work station should each worker be assigned in order to minimize the idle time of the assembly line? What is the associated idle time?
Alex _____ Barbara _____ Chuck _____ Donna _____ Ed _____ Fran _____
Idle Time (minutes/unit) _____________
Batching Problem
You order two products from the same supplier. The annual demand for Product 1 is 10,000 units and the annual demand for Product 2 is 20,000 units. Note that demand for both products is constant throughout the year. The holding cost is the same for both products, $1 per unit per year. However, you incur a fixed cost of $200 each time you order Product 1, and a fixed cost of $100 each time you order Product 2. These fixed order costs are independent of the size of the order.
a. How many units of Product 1 and Product 2 should be ordered at a time in order to minimize total holding + order cost?
Number of Units of Product 1 to Order _____________
Number of Units of Product 2 to Order _____________
b. Suppose that the supplier insists that orders for Product 1 and Product 2 be coordinated so that they can be shipped at the same time (still incurring the fixed cost of $200 and $100, respectively). Given this requirement, how many units of Product 1 and Product 2 (to the nearest integer) should be ordered at a time in order to minimize total holding + order cost? (Hint: the products must be ordered the same number of times per year)
Number of Units of Product 1 to Order _____________
Number of Units of Product 2 to Order _____________
Problem 3: Queuing
A bank has two tellers to service customers who want to conduct their banking business in person. The first teller is dedicated exclusively to personal banking customers who want access to their (non-commercial) bank accounts. The second teller is dedicated exclusively to commercial banking customers who want access to their business bank accounts. The bank maintains separate waiting lines for the different types of customers, and strictly adheres to a policy that personal banking customers will only be served by the first teller, and commercial banking customers will only be served by the second teller.
Customers of each type arrive randomly throughout the business day. On average, 30 personal banking customers and 10 commercial banking customers arrive at the bank per hour, and the time between arrivals for both types of customers is exponentially distributed. Teller 1 is capable of processing 40 personal banking customers per hour, while the more complex transactions required by business customers limit Teller 2 to servicing only15 customers per hour on average. Service times for both tellers have been observed to follow an exponential distribution.
a. How long on average is a personal banking customer in the bank? On average how many personal banking customers are waiting for service at any given time?
Average Time in the Bank (minutes) _________
Average Number Waiting (customers) _________
b. How long on average is a commercial banking customer in the bank? On average how many commercial banking customers are waiting for service at any given time?
Average Time in the Bank (minutes) _________
Average Number Waiting (customers) __________
c. The bank decides to cross-train the two tellers so that they are each capable of processing both personal and commercial banking customers. Customers now enter a single queue and are served by the first available teller. All of the information given about customer arrivals and service times remains unchanged.
Under the new configuration, how long on average is a banking customer in the bank? On average how many banking customers are waiting for service at any given time?
Average Time in the Bank (minutes) _________
Average Number Waiting (customers) _________
Problem 4: Newsboy
Goop Inc. needs to order a raw material to make a special polymer. The demand for the polymer is forecasted to be normally distributed with a mean of 250 gallons and a standard deviation of 125 gallons. Goop sells the polymer for $25 per gallon. Goop purchases raw material for $10 per gallon and Goop must spend $5 per gallon to dispose of all unused raw material due to government regulations. One gallon of raw material yields one gallon of polymer. If demand is higher than Goop can make, then Goop sells only what they can make and the remainder of demand is lost.
a. Suppose Goop purchases 150 gallons of raw material. What is the probability that they will run out of raw material?
Probability of Running Out (%) ______________
b. Suppose Goops orders 300 gallons of raw material. How many gallons of demand on average would remain unfulfilled?
Average Demand Unfulfilled (gallons) _____________
c. Suppose Goop orders 400 gallons of raw material. How much can they expect to spend on disposal costs?
Disposal Costs ($) _____________
d. Suppose Goop wants to ensure that there is a 92% probability that they will be able to meet all customer demand. How many gallons of raw material should they purchase?
Raw Material to Purchase (gallons) _____________
e. How many gallons of raw material should Goop purchase if the objective is to maximize expected profit? What is the expected profit associated with this purchase?
Raw Material to Purchase (gallons) _____________
Expected Profit ($) _____________
Problem 5: Risk Pooling
The UGA Bookstore stocks two types of cashmere sweaters. The two sweaters are identical in every way except on the first sweater is stitched UGA FOOTBALL while on the second is stitched DAWGS FOOTBALL (we'll refer to these two types as UGA sweaters and DAWGS sweaters). Both sweaters retail for $100 apiece and cost the Bookstore $40 to procure. Because the procurement lead time is long relative to the length of the football season, the Bookstore places a single order to cover anticipated sales for the entire season. Any sweaters left over at the end of the season are shipped to a reseller for $20 apiece. The demand for UGA sweaters is normally distributed with a mean of 1000 and a standard deviation of 400. The demand for DAWGS sweaters is normally distributed with a mean of 800 and a standard deviation of 300. It's been noted that in previous years when the demand for one type of sweater is high, the demand for the other type of sweater is low, leading the Bookstore to estimate the correlation between the two sweaters at -0.40.
a. How many UGA sweaters should the Bookstore order for the season to maximize expected profit? What is the expected profit?
Number of UGA Sweaters to Order (units) __________________
Expected Profit ($) __________________
b. How many DAWGS sweaters should the Bookstore order for the season to maximize expected profit? What is the expected profit?
Number of DAWGS Sweaters to Order (units)__________________
Expected Profit ($) __________________
c. The Bookstore's manager is offered the opportunity to replace the UGA and DAWGS sweaters with a new sweater on which UGA DAWGS FOOTBALL is stitched (which we'll refer to as a UGA DAWGS sweater). The manager believes that all customers who would otherwise have demanded a UGA or a DAWGS sweater will instead buy the UGA DAWGS sweater, i.e., no sales will be lost by stocking UGA DAWGS sweaters and discontinuing the UGA and DAWGS lines. In addition, all of the cost and demand information previously given remains the same. How many UGA DAWGS sweaters should the Bookstore order for the season to maximize expected profit? What is the expected profit?
Number of UGA DAWGS sweaters to Order (units) __________________
Expected Profit ($) __________________
Problem 6: Supply Chain Coordination
A supplier manufactures a product at cost m = $40 and sells it to a retailer at wholesale price c = $100. The retailer sells this product to consumers at retail price p = $200. The replenishment lead time for this product is long relative to the selling season, so the retailer only has one opportunity to stock the product. Demand for the product is normally distributed with a mean of 1000 units and a standard deviation of 400 units. Units unsold at the end of the season are salvaged at price v = $15.
a. How many units of the product should the retailer order to maximize its expected profit? What is the associated retailer's expected profit and the supplier's expected profit?
Number of Units to Order (units) ____________
Retailer's Expected Profit ($) ____________
Supplier's Expected Profit ($) ____________
b. How many units of the product should the retailer order to maximize total supply chain expected profit? What is the associated retailer's expected profit and the supplier's expected profit?
Number of Units to Order (units) ____________
Retailer's Expected Profit ($) ____________
Supplier's Expected Profit ($) ____________
c. To motivate the retailer to buy more product, the supplier offers the following deal: the wholesale price is reduced to c = $80, but the retailer must pay the supplier an additional $40 for every unit that it sells. The supplier is also willing to buy back units that the retailer is unable to sell at buyback price b. These units are then salvaged by the supplier. What value of b should the supplier offer to maximize total supply chain expected profit? What is the associated retailer's expected profit and the supplier's expected profit?
Optimal Buyback Price ($) ____________
Retailer's Expected Profit ($) ____________
Supplier's Expected Profit ($) ____________