ME 375K Production Engineering Management First Test, Spring 1998 Each problem is 20 points

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1 Name ME 375K Production Engineering Management First Test, Spring 1998 Each problem is 2 points 1. A raw material inventory holds an expensive product that costs $1 for each item. The annual demand for the item is 2, units per year. In order to place an order and receive a replenishment of the item there is an reorder cost of $16. The company has an interest rate used for holding cost computations of 1% a year. The company maintains a safety stock of one month s supply. As an approximation, we assume that the withdrawal rate is constant (not random), replenishment orders arrive instantaneously, and the lead time is. a. Find the optimum lot size. Show on the plot below how the on-hand inventory varies with time when the optimum lot size is used. Show the important quantities on the plot such as the lot size and cycle time of the inventory. Show your scale on the inventory level axis. EOQ = 8, Cycle Time =.48 months, Monthly demand = Safety Stock = 1666, 1 2 In the figure the numbers are rounded to the nearest 8 and the times to the nearest half hear

2 b. The company decides to reorder once a month. The safety stock remains at a one month supply. What is the average residence time for items in this inventory? The reorder quantity is one month s demand or The average inventory = 1667/ = 25 Average residence time = 25/1667 = 1.5 months c. If you could change the reorder cost, in which direction would you change it? How would that affect the optimum lot size? Is this good or bad? Give a qualitative answer. If you could change the reorder cost, you should certainly reduce it. This would make the optimum lot size smaller and the inventory cost smaller. It is a very good idea. 2

3 2. A finished good inventory holds an expensive product that costs $1 for each item. The monthly demand for the item is 2 units. The product is produced in the company s manufacturing facility. The lot size is 1 units, and when the plant is producing the item it is added to inventory at a rate of 5 units per month. A production lot requires one month to pass through the plant. For simplicity assume a deterministic model with no safety stock. a. Show on the plot below, how the finished good inventory varies with time when the lot size is 1 units. Show the important quantities on the plot such as the maximum inventory level and cycle time of the inventory. Show your scale on the inventory level axis. With the lot size equal to 1 units, it requires 1/5 of a month to add this amount to inventory. The cycle time is 1/2 month. The maximum inventory is (5-2)*(1/5) = 3/5 =

4 b. What is the average WIP in the plant? The WIP in the plant is ( flow rate )*( residence time) = 2*1 = 2 units c. What is the average residence time in the finished good inventory? s + (1 D / P)Q/ 2 D = {(1 - (2/5))*(1)/2}/2 =.15 months d. If you could change the rate at which product is added to the inventory from 5 per month, in which direction would you change it? A lot still requires one month to pass through the plant. How would this affect the WIP? How would this affect residence time in the finished goods inventory? If you could change it, you should change in down toward the value of D. This would not affect the WIP in the plant, but it would reduce the residence time in inventory. When P = D, the residence time is zero (with no safety stock). 4

5 3. A distributor maintains an inventory of computers. The cost to the distributor is $1 for each computer. The annual demand for computers is 2 units per year. In order to place an order and receive a replenishment of the item there is an reorder cost of $16. The company has an interest rate used for holding cost computations of 1% a year. The distributor uses a continuous review system, and places an order for the replenishment of his inventory whenever the inventory level reaches a specified reorder point. The lead time for replenishment is.1 year. The demand during the lead time (.1 years) has a Normal distribution whose expected value is 2 units and the standard deviation is 1 units. a. What reorder point will give the distributor a service level equal to 95%? We are looking for r such that P(x r) =.95%. Use the standard Normal table to find F(z) =.95. The closest number in the table is z = Since z = (r - µ)/σ. r = z σ+ µ = 1.64*(1) + 2 = b. Say the distributor decides to place an order four times a year? What is the optimum reorder point? If a customer arrives and the inventory is empty, the sale is backordered at a cost of $1 per unit. With four orders a year, the lot size = 5 units. F(r*) = 1 hq π 2 D From the table F(z) =.75 for z =.68 r* = (.68)*1 + 2 = 268. = 1 - (1)*(.1)(5)/(1)(2) =.75 c. In a time of scarce computers the lead time grows to.2 years. What reorder point will give the distributor a service level equal to 95%? If the lead time grows to.2 years, the mean demand becomes 4 units and the standard deviation becomes sqrt(2)*1 = Use the standard Normal table to find F(z) =.95. The closest number in the table is z = Since z = (r - µ)/σ. r = z σ+ µ = 1.64*(141.2) + 4 = d. A different supplier offers to ship replenishments with a lead time. What reorder point would this option allow. What is the optimum order quantity in this case? With zero lead time, the reorder point can be zero. The expected shortage is also zero. This allows the optimum order quantity of the deterministic case. Q * = 2D( A + C s ) with C s =. Q* = sqrt(2*2*16/1*.1) = 8 units h 5

6 4. Answer the following questions. a. If the releases of a manufacturing plant are scheduled with an MRP system, is the plant using a pull system or a push system? The plant is using a push system. b. If a manufacturing plant uses a Kanban system, is the plant using a pull or a push system? The plant is using a Pull system. c. What distinguishes a pull system from a push system? A push system puts material into the front of the line. The single for a station to begin production is the appearance of raw materials at the station. A pull system, pulls goods from the finished goods end of the line. The signal to produce for a station is an authorization from the following station. d. A dealer in expensive watches has an average demand of one watch a day. Daily demands are independent. The dealer s lead time for replenishment of her inventory is 3 days. What should her reorder point be if she wants at least a 6% chance of not running short before the replenishment arrives? Use a Poisson distribution with 3. Find the number of sales that has a cumulative distribution at least equal to 6%. k p(k) F(k) The reorder point is equal to 3. e. A machine produces 5 different products at a rate of 1 units per month for each product. Each unit of product requires 4 minutes of production time on the machine. The machine operates for 25 minutes a month. The time spent for switching from one product to another is 25 minutes. Assuming all products are produced in equal sized lots, what is the smallest lot size that this situation allows? The total production time for each product is 4 minutes. The total for all five products is 2 minutes. This leaves 5 minutes for setups, or time for 5/25 = 2 setups per month. We use 4 setups for each product. This implies a lot size of 25. 6