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ADDITIONAL PROBLEMS Attempt as many as you can There may be problems that were not covered in your class this semester – ignore those INVENTORY MANAGEMENT: 1. The following table contains figures on the annual volume and unit costs for a random sample of 16 items for a list of 2000 inventory items at a health care facility. Item Unit Cost Usage 1 75 3 2 30 274 3 20 397 4 13 139 5 24 41 6 20 123 7 24 379 8 19 372 9 22 114 10 55 3270 11 105 594 12 55 482 13 35 198 14 40 37 15 12 215 16 20 116 Develop an ABC classification for these items. 2. A large bakery buys flour in 25-pound bags. The bakery uses an average of 4860 bags a year. Preparing an order and receiving a shipment of flour involves a cost of $4/order. Annual carrying costs are $30 per bag. a. Determine the economic order quantity. b. What is the average number of bags on hand? c. How many orders per year will there be? d. Compute the total cost of ordering and carrying flour. e. If ordering costs were to increase by $1 per order, how much would that affect the total annual cost? Solution: 36 bags, 18 bags, 135, $1080, increase by $127.48 3. A hardware store sells approximately 27,000 cans of a white paint a month. Because of storage limitations, a lot size of 4000 cans has been used. Monthly holding cost is 18 cents per can, and reordering cost is $60 per order. The company operates an average of 20 days a month. a. What penalty is the company incurring by its present order size on annual costs? b. The manager would prefer ordering 10 times each month but would have to justify any change in order size. One possibility is to simplify order processing to reduce the ordering cost (say using web orders). What ordering cost would enable the manager to justify ordering every other day? Solution: $16, $52.06 4. (If quantity discounts have been covered in class) A mail order house uses 18000 boxes a year. Carrying costs are 20 cents per box a year, and the ordering costs are $32. The following price schedule applies. Number of Boxes Price per box 1000 to 1999 $1.25 2000 to 4999 $1.20 5000 to 9999 $1.18 10000 or more $1.15 Determine the optimal order quantity. Solution: 10,000 boxes 5. Suppose the expected demand during lead time for a particular item is 300 units, with a standard deviation of daily demand of 30 . Suppose the lead time is 4 days. a. What is the reorder point that provides a 1% risk of stock out during lead time? b. The safety stock needed to provide a 1% risk of stock out during lead time? c. The safety stock needed to provide a 7% risk of stock out during lead time? Solution: 439.8, 139.8, 88.55 6. In the above problem, if a reorder point of 275 is used, what is the probability of stock out? Solution: 66.15% SIMULATION: 1. A car rental agency has collected data on the demand for luxury-class automobiles over the past 25 days. The data are shown below. Daily Demand Number of days 7 2 8 5 9 8 10 7 11 3 Total 25 Because customers drop cars at another location, the agency only has 9 cars available currently. Assume single day rentals. a. Use the following five random numbers to generate 5 days of demand for the rental agency: 15 48 71 56 90. b. What is the average number of cars rented for the 5 days? c. How many rentals will be lost over the 5 days? d. What is the average daily demand for the 5 days? Solution: b. 8.8 cars, c. 1 lost on day 3, 2 lost on day 5, d. 9.4 cars 2. A service technician for a major photocopier company is trained to service two models of copier: the X100 and the X200. Approximately 60% of the technician's service calls are for the X100, and 40% are for the X200. The service time distributions for the two models are as follows: X100 Copier X200 Copier Time (mins) Relative Freq Time (mins) Relative Freq 25 0.50 20 0.40 30 0.25 25 0.40 35 0.15 30 0.10 40 0.10 35 0.10 a. Show the random number intervals that can be used to simulate the type of machine to be serviced and the length of the service time for each model. b. Simulate 20 service calls. What is the total service time spent on the 20 calls? Solution: about 535 minutes See Excel solution on website 3. Three discount pharmacies (Super Z, Devco, and Floorgreen) compete for business in a suburban area. Customers often make a purchase at one of the stores and then make their, next purchase at another store. The matrix below shows the probability that a customer will switch stores from one purchase to the next. Current Next Purchase Total Purchase Super Z DevCo Floorgreen Super Z 0.70 0.10 0.20 1.00 Devco 0.30 0.55 0.15 1.00 Floorgreen 0.10 0.10 0.80 1.00 a. Show the probability distribution and the intervals of random numbers that could be used to generate the next purchase for a customer who last made a purchase at Super Z. b. Repeat part (a) for a customer who last made a purchase at Devco. c. Repeat part (a) for a customer who last made a purchase at Floorgreen. d. Gary Hatcher made his last purchase at Super Z. Use the following four random numbers to simulate the store at which he makes his next four purchases: 42, 81, 16, 57. Solution: See Excel solution on website 4. The New York City corner newsstand orders 250 copies of The New York Times daily. Primarily due to weather conditions, the demand for newspapers varies from day to day. The probability distribution of the demand for newspapers is as follows: # of newspapers 150 175 200 225 250 Probability 0.10 0.30 0.30 0.20 0.10 The newsstand makes a 15-cent profit on every paper sold, but it loses 10 cents on every paper unsold by the end of the day. Use 10 days of simulated results to determine whether the newsstand should order 200, 225, or 250 papers per day. What is the average daily profit the newsstand can anticipate based on your recommendation? Revenue Management in Supply Chains: (if covered in class) 1. Felgas, a manufacturer of felt gaskets, has production capacity of 1000 units per day. Currently, the firm sells production capacity for $5 per unit. At this price, all the production capacity is booked about one week in advance. A group of customers have said that they would be willing to pay $15 per unit if only Felgas accepted their orders on the last day. The demand from this high paying segment is normally distributed with a mean of 250 and a standard deviation of 100. How much capacity should Felgas reserve for the last day? Solution: 293 units of capacity See Excel solution on website Bonus: Suppose the high paying demand distribution was uniformly distributed between 175 and 325, how much capacity should be reserved? Solution: 275 units of capacity (Hint: use the same methodology as used for Normal, except that now you cannot use a Normal table. For uniform distribution you really don’t need a table). 2. A manufacturer sources several components from China and has monthly transportation needs that are normally distributed with a mean of 10 million units and a standard deviation of 4 million. The manufacturer must decide on the portfolio of transportation contracts to carry. A long term bulk contract costs $10,000 per month for a million units. Transportation capacity is also available in the spot market at an average price of $12,500 per million units. How much transportation capacity should the manufacturer sign a long-term bulk contract for? Solution: 6.63 million units See Excel solution on website WAITING LINES (Note: ta=1/arrival rate and ts=1/service rate. If no ca or cs is given, assume them to be 1 (Poisson)) 1. The reference desk of a university library receives requests for assistance. Assume that a Poisson probability distribution with a mean rate of 10 requests per hour can be used to describe the arrival pattern and that service times follow the exponential probability distribution with a mean service rate of 12 requests per hour. a. What is the average number of requests that will be waiting for service? b. What is the average waiting time in minutes before service be-ins? c. What is the average time at the reference desk in minutes (waiting time plus service time)? Solution: 4.17, 0.42hrs, 0.5hrs 2. Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.5 customers arrive each hour. One design consultant is available to answer customer questions and make product re-commendations. The consultant averages 10 minutes with each customer. a. Compute the length of the queue and the waiting times, assuming Poisson arrivals and exponential service times. b. Service goals dictate that an arriving customer should not wait for service more than an average of 5 minutes. Is this goal being met? What action do you recommend? c. If the consultant can reduce the average time spent per customer to 8 minutes, what is the mean service rate? Will the service goal be met? Solution: a. 0.297, 7.14mins b. No. Increase service rate or hire a second consultant c. 7.5 customers/hr, Lq=0.1667, Wq=4 mins. Yes. 3. Fore and Aft Marina is a newly planned marina that will be located on the Ohio River near Madison, Indiana. Assume that Fore and Aft has decided to build a docking facility where one boat at a time can stop for gas and servicing. Assume that arrivals follow a Poisson probability distribution, with a mean of 5 boats per hour, and that service times follow exponential probability distribution, with a mean of 10 boats per hour. Consider the following questions: a. What is the average number of boats that will be waiting for service? b. What is the average time a boat will spend waiting for service? c. What is the average time a boat will spend at the dock? d. If you were the management of Fore and Aft Marina, would you be satisfied with service level your system will be providing? Solution: 0.5 boats, 6 mins, 12 mins 4. The management of the Fore and Aft Marina in the above problem wants to investigate the possibility of enlarging the docking facility so that two boats can stop for gas and servicing simultaneously. Assume that the mean arrival rate is 5 boats per hour and that the mean service rate for each of the channels is 10 boats per hour. a. What is the average number of boats that will be waiting for service? b. What is the average time a boat will spend waiting for service? c. What is the average time a boat will spend at the dock? d. If you were the manager of Fore and Aft Marina, would you be satisfied with the - service level your system will be providing? Solution: 0.03 boats, 0.4 mins, 6.4 mins 5. Consider a two-channel waiting line with Poisson arrivals and exponential service times. The mean arrival rate is 14 units per hour, and the mean service rate is 10 units per hour for each channel. a. What is the average number of units in the system? b. What is the average time a unit waits for service? c. What is the average time a unit is in the system? Solution: 1.35 customers in queue, 2.75 in system; 5.76 mins, 11.76 mins 6. Melvin's Market has an "express" checkout for customers with twelve items or less. The inter-arrival time for customers at the express checkout has an exponential probability distribution with a mean time between arrivals of 90 seconds. The checkout time for a customer at the express checkout has an exponential probability distribution with a mean checkout time that depends on whether a cashier has the help of a bagger. With a bagger's help, the average time a cashier needs to check out a customer is 50 seconds; without a bagger's help, the average time is 72 seconds. Consider the situations in which a cashier has and does not have a bagger's help, and construct a table that compares these situations with respect to the following operating characteristics: a. The average number of customers in queue at the express checkout. b. The average total time a customer spends at the express checkout. Solution: With bagger: =40, =72, Lq=0.69, Wtot=1.88 mins Without bagger: =40, =50, Lq=3.2, Wtot=6.0 mins 7. To promote its reputation for fast service, Earl's While-U-Wait Automotive Tune-up Shop promises to reduce a customer's bill by $0.20 for every minute the customer must wait until his or her car's tune-up is finished. The inter-arrival time for Earl's customers has an exponential probability distribution with a mean arrival rate of 5 customers per hour. The time required by a mechanic to perform a tune-up has an exponential probability distribution with a mean tune-up rate of 2 cars per hour. Earl is considering maintaining a staff of 3, 4, or 5 mechanics. A mechanic's salary is $20 per hour. Define Earl's total hourly cost as the sum of two components: (1) the cost per hour of the mechanics and (2) the profit lost per hour because of reductions of customers' bills. Estimate the total hourly cost if Earl employs 3, 4, or 5 mechanics. Which number of mechanics results in the lowest total? Solution: $132.13, $116.40, $131.56. Choose 4 mechanics See Excel solution on website Material Requirements Planning: (if covered in class) 1. A table is assembled using three components, 2X of Wood sections, 3X of Braces and 4X of Legs. The company that makes the table wants to ship 100 units at the beginning of day 4, 150 units at the beginning of day 5, and 200 units at the beginning of day 7. Receipts of 100 wood sections are scheduled at the beginning of day 2. There are 120 legs on hand. An additional 10% of the order size of legs is added for safety stock. There are 60 braces on hand and no safety stock is required for it. Lead time in days for all items is as follows: If quantity ordered is 1-200, lead time is 1 day, for 201-550 it is 2 days, and for 551-999 it is 3 days. Prepare an MRP plan using lot-for-lot ordering. Solution: See Excel solution on website 2. The BOM for an item is as follows. Product A consists of B (1X) and D(2X). Every B in turn consists of C(2X), and every D consists of B(1X). If the Master Production Schedule of A has a requirement of 500 units in week 6, and the lead time for assembly of A is 1 week, develop the MRP plan for B,C, and D for the next 6 weeks given the following information. Item Data category B C D Lot sizing rule FP(3) MOQ=1500 L4L Safety Stock 50 100 0 Lead time 1 week 1 week 2 weeks Sch Receipts None 2000 (week 1) None Beg. Inventory 50 200 0 Solution: See Excel solution on website LINEAR PROGRAMMING 1. Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the table at the top of the next page. Production Time (hours) Model Cutting and Finishing Pack and Ship Profit/Glove Sewing Regular glove 1 1/2 1/8 $5 Catcher’s glove 2/3 1/3 1/4 $8 Assuming that the company is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution. How many gloves of each model should Kelson manufacture? c. What is the total profit contribution Kelson can earn with the above production quantities? d. How many hours of production time will be scheduled in each department? Solution: b. 500, 150 c. $3700 d. 600, 300, 100 See Excel solution on website 2. Yard Care, Inc., manufactures a variety of lawn care products, including two well- known lawn fertilizers. Each fertilizer product is a blend of two raw materials known as K40 and K50. During the current production period, 900 pounds of K40 and 400 pounds of K50 are available. Each pound of the product known as "Green Lawn" uses 3/5 pound of K40 and 2/5, pound of K50. Each pound of the product known as "Lawn Care" uses 3/4 pound of K40 and 1/4 pound of K50. In addition, a current limit on the availability of packaging materials restricts the production of Lawn Care to a maximum of 500 pounds. Assume that the profit contribution for both products is $3 per pound. a. What is the linear programming model for this problem? b. Find the optimal solution. How many pounds of each product should be manufactured? Solution: b. 687.5, 500. 3562.5 See Excel solution on website 3. Greentree Kennels, Inc., provides overnight lodging for a variety of pets. A particular feature at Greentree is the quality of care the pets receive, including excellent food. The kennel's dog food is made by mixing two brand-name dog food products to obtain what the kennel calls the "well-balanced dog diet." The data for the two dog foods are as follows: Dog food Cost/ounce Protein% Fat% Bark bits $0.06 30 15 Canine Chow $0.05 20 30 If Greentree wants to be sure that the dogs receive at least 5 ounces of protein and at least 3 ounces of fat per day, what is the minimum cost mix of the two dog food products? Solution: 15oz, 2.5 oz $1.025 See Excel solution on website 4. Photo Chemicals produces two types of photographic developing fluids. Both products cost Photo Chemicals $1 per gallon to produce. Based on an analysis of current inventory levels and outstanding orders for the next month, Photo Chemicals' management has specified that at least 30 gallons of product 1 and at least 20 gallons of product 2 must be produced during the next 2 weeks. Management has also stated that an existing inventory of highly perishable raw material required in the production of both fluids must be used within the next 2 weeks. The current inventory of the perishable raw material is 80 pounds. While more of this raw material can be ordered if necessary, any of the current inventory that is not used within the next 2 weeks will spoil - hence, the management requirement that at least 80 pounds be used in the next 2 weeks. Furthermore, it is known that product 1 requires 1 pound of this perishable raw material per gallon and product 2 requires 2 pounds of the raw material per gallon. Since Photo Chemicals' objective is to keep its production costs at the minimum possible level, the firm's management is looking for a minimum-cost production plan that uses all the 80 pounds of perishable raw material and provides at least 30 gallons of product 1 and at least 20 gallons of product 2. What is the minimum-cost solution? Solution: 30, 25 $55 See Excel solution on website 5. A millionaire wants to invest $150 million by purchasing some or all of the following properties: a shopping center that costs $90 million, a professional basketball franchise that costs $50 million, and a 20-story office building that costs $100 million. The annual return on the shopping center is $10 million, from the basketball franchise is $4 million, and from the office building is $ 12 million. The investor has hired a manager who works 50 hours per week. The time required to oversee operations of the shopping center is 30 hours, for the basketball franchise is 10 hrs, and the office building is 20 hours. Because of potential problems due to traffic conditions at the shopping center and fan reaction to the basketball team, the investor wishes to invest in either the shopping center or the basketball franchise, not both. What would be his optimal investment in the alternatives? Solution: $16 million; Basketball franchise and Office building See Excel solution on website 6. A beer company has breweries in two cities and has distributors in six states. The monthly capacities in the breweries, the monthly demand per state, and shipping costs per barrel are shown in the table below. Shipping cost per barrel Tampa St. Louis Demand Tennessee 2.5 1.25 1600 Georgia 1.75 3.25 1800 North Carolina 3 2 1500 South Carolina 2.25 2.75 950 Kentucky 4 1 2000 Virginia 3.75 3.25 1400 Capacity 3500 5000 How should the firm distribute its product at minimum total cost? Solution: Total cost $14,825; Ship 1600 1800 1500 950 2000 650 See Excel solution on website Project Management: 1. Consider the following project Immediate Activity Times Cost to Activity Predecessor Optimistic Most Pessimistic crash/day Likely A - 2 6 8 500 B - 3 3 6 600 C - 5 7 12 1000 D A,B 3 6 6 250 E B 3 6 8 100 F D,C 5 6 9 350 G E,C 4 8 9 700 H F,G 8 12 13 450 Draw the network, specify the critical path, figure out the ES and LS for each activity and their slacks. Round numbers as necessary. Solution: See Excel solution on website 2. What is the probability that the above project will be complete in 30 days or less? In 27 days or less? Solution: See Excel solution on website 3. What activities would you crash to reduce the duration of the above project by 2 days. Solution: See Excel solution on website Quality Management: 1. For the following process, find the control limits. Observations Sample 1 2 3 4 5 6 1 5.02 5.02 4.94 4.99 4.96 4.90 2 5.01 5.03 5.07 4.95 4.96 4.96 3 4.99 5.00 4.93 4.92 4.99 4.99 4 5.03 4.91 5.01 4.98 4.89 4.93 5 4.95 4.92 5.03 5.05 5.01 4.95 6 4.97 5.06 5.06 4.96 5.03 4.87 7 5.05 5.01 5.10 4.96 4.99 5.00 8 5.09 5.10 5.00 4.99 5.08 5.02 9 5.14 5.10 4.99 5.08 5.09 4.94 10 5.01 4.98 5.08 5.07 4.99 4.99 Is the process in control with respect to both mean and range in all periods? Solution: See Excel solution on website 2. In controlling the number of defectives, you take samples of size 100, and get the following number of defectives Sample #Defectives 1 5 2 3 3 6 4 7 5 4 6 6 7 8 8 4 9 5 10 8 11 3 12 4 13 5 14 6 15 6 16 7 17 5 18 3 19 5 20 6 Draw 3 control limits for this process. Is the process in control? Solution: See Excel solution on website 3. In controlling defects in a particular process, the count of defects were as shown below. Sample #Defects 1 12 2 8 3 16 4 14 5 10 6 11 7 9 8 14 9 13 10 15 11 12 12 10 13 14 14 17 15 15 Is the process in control? Draw 3 control limits for this process Solution: See Excel solution on website