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CHAPTE R 11 Inventory Management Introduction to Operations Management 1 What is inventory? An inventory is an idle stock of material used to facilitate production or to satisfy customer needs Introduction to Operations Management 2 Why do we need inventory? Economies of scales in ordering or production - cycle stock – trade-off setup cost vs carrying cost Smooth production when requirements have predictable variability - seasonal stock – trade-off production adjustment cost vs carrying cost Introduction to Operations Management 3 Why do we need inventory? (cont’d) Provide immediate service when requirements are uncertain (unpredicatable variability) -safety stock – trade-off shortage cost versus carrying cost Decoupling of stages in production/distribution systems -decoupling stock – trade-off interdependence between stages vs carrying cost Introduction to Operations Management 4 Why do we need inventory? (cont’d) Production is not instantaneous, there is always material either being processed or in transit - pipeline stock – trade-off cost of speed vs carrying cost Introduction to Operations Management 5 Why carrying inventory Asyou can see from the above list, there are economic reasons, technological reasons as well as management/operations reasons. Introduction to Operations Management 6 Why carrying inventory is costly? Traditionally, carrying costs are attributed to: – Opportunity cost (financial cost) – Physical cost (storage/handling/insurance/theft/obs olescence/…) Introduction to Operations Management 7 Inventory Management Independent Demand A Dependent Demand B(4) C(2) D(2) E(1) D(3) F(2) Independent demand is uncertain. Dependent demand is certain. Introduction to Operations Management 8 Types of Inventories Raw materials & purchased parts Partially completed goods called work in progress Finished-goods inventories – (manufacturing firms) or merchandise (retail stores) Introduction to Operations Management 9 Types of Inventories (Cont’d) Replacement parts, tools, & supplies Goods-in-transit to warehouses or customers Introduction to Operations Management 10 Inventory Counting Systems Periodic System Physical count of items made at periodic intervals Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item Introduction to Operations Management 11 Inventory Counting Systems (Cont’d) Two-Bin System - Two containers of inventory; reorder when the first is empty Universal Bar Code - Bar code printed on a label that has information about the item to which it is attached 0 214800 232087768 Introduction to Operations Management 12 ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A - very important High Annual A B - mod. important $ volume B of items C C - least important Low Few Many Number of Items Introduction to Operations Management 13 What are we tackling in inventory management? Allmodels are trying to answer the following questions for given informational, economic and technological characteristics of the operating environment: – How much to order (produce)? – When to order (produce)? – How often to review inventory? – Where to place/position inventory? Introduction to Operations Management 14 The inventory models Thequantitative inventory management models range from simple to very complicated ones. However, there are two simple models that capture the essential tradeoffs in inventory theory – The newsboy (newsvendor) model – The EOQ (Economic Ordering Quantity) model Introduction to Operations Management 15 The Newsboy Model Thenewsboy model is a single-period stocking problem with uncertain demand: Choose stock level and then observe actual demand. – Tradeoff: overstock cost versus opportunity cost (lost of profit because of under stocking) Introduction to Operations Management 16 The Newsboy Model Shortage cost (Cs) - the opportunity cost for lost of sales as well as the cost of losing customer goodwill. Cs = revenue per unit - cost per unit Excess cost (Ce) - the over-stocking cost (the cost per item not being able to sell) Ce = Cost per unit - salvage value per unit Introduction to Operations Management 17 The Newsboy Model In this model, we have to consider two factors (decision variables): the supply (X), also know as the stock level, and the demand (Y). The supply is a controllable variable in this case and the demand is not in our control. We need to determine the quantity to order so that long-run expected cost (excess and shortage) is minimized. Introduction to Operations Management 18 The Newsboy Model Let X = n and P(n) = P(Y>n). The question is: Based on what should we stock this n-th unit? The opportunity cost (expected loss of profit)for this n-th unit is P(n) Cs The expected cost for not being able to sell this n-th unit (expected loss) (1-P(n))Ce Introduction to Operations Management 19 The Newsboy model Stock the n-th unit if P(n)Cs > (1-P(n))Ce Do not stock if P(n)Cs < (1-P(n))Ce Introduction to Operations Management 20 The Newsboy model The equilibrium point occurs at P(n)Cs = (1-P(n)) Ce Solving the equation P(n) = Ce/(Cs + Ce) Introduction to Operations Management 21 Newsboy model Service level is the probability that demand will not exceed the stocking level and is the key to determine the optimal stocking level. Inour notation, it is the probability that YX(=n), which is given by P(Yn) = 1 - P(n) = Cs/(Cs + Ce) Introduction to Operations Management 22 Example (p.564) Demand for long-stemmed red roses at a small flower shop can be approximated using a Poisson distribution that has a mean of four dozen per day. Profit on the roses is $3 per dozen. Leftover are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked down flowers are sold. What is the optimal stocking level? Introduction to Operations Management 23 Example (solutions) Cumulative frequencies for Cs = $3, Ce = $2 Poisson distribution, mean = 4 Opportunity cost is P(n)Cs Demand (dzn Cumulative Expected loss for over- per day) frequency stocking (1 - P(n)) Ce 0 0.018 1 0.092 P(Y n) = Cs/(Cs + Ce) = 2 0.238 3/((3+2) = 0.6 3 0.434 4 0.629 From the table, it is 5 0.785 between 3 and 4, round up 5 give you optimal stock of 4 dozen. Introduction to Operations Management 24 The Inventory Cycle Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorde r point Time Receive Place Receive Place Receive order order order order order Lead time to Operations Management Introduction 25 How to estimate the inventory costs? Introduction to Operations Management 26 Estimating the inventory costs Q = quantity to order in each cycle S = fixed cost or set up cost for each order D = demand rate or demand per unit time H = holding cost per unit inventory per unit time c = unit cost of the commodity TC = total holding cost per unit time Introduction to Operations Management 27 Estimating the inventory cost The cost per order = S + cQ The length of order cycle = Q/D unit time The average inventory level = Q/2 Thus the inventory carrying cost = (Q/2) H (Q/D) = HQ2/(2D) Introduction to Operations Management 28 Estimating the inventory cost The total cost per order cycle = S + cQ + HQ2/(2D) Thus the total cost per unit time is dividing the above expression by Q/D, I.e., TC = {S + cQ + HQ2/(2D)} {D/Q} = DS/Q + cD + HQ/2 Introduction to Operations Management 29 Total Cost Annual Annual Total cost = carrying + ordering cost cost Q + DS TC = H 2 Q Introduction to Operations Management 30 Deriving the EOQ Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q. 2DS 2(Annual Demand )(Order or Setup Cost) QOPT = = H Annual Holding Cost Introduction to Operations Management 31 Cost Minimization Goal The Total-Cost Curve is U-Shaped Q D TC H S Annual Cost 2 Q Ordering Costs QO (optimal order quantity) Order Quantity (Q) Introduction to Operations Management 32 Minimum Total Cost The total cost curve reaches its minimum where the carrying and ordering costs are equal. 2DS 2(Annual Demand )(Order or Setup Cost) QOPT = = H Annual Holding Cost Introduction to Operations Management 33 Cost Total Costs Adding Purchasing cost TC with PD doesn’t change EOQ TC without PD PD 0 EOQ Quantity Introduction to Operations Management 34 Total Cost with Constant Carrying Costs TCa TCb Total Cost Decreasing TCc Price CC a,b,c OC EOQ Quantity Introduction to Operations Management 35 Example (Example 2, p.541) A local distributor for a national tire company expects to sell approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carrying costs are $16 per tire, and ordering costs are $75. This distributor operate 288 days a year. a. What is the EOQ? b. How many times per year does the store reorder? c. What is the length or an order cycle? d. What is the total ordering and inventory cost? Introduction to Operations Management 36 Solution (Example 2, p.541) D = 9600 tires per year, H = $16 per unit per year, S = $75 a. Q0 = {2DS/H} = {2(9600)75/16} = 300 b. Number of orders per year = D/ Q0 = 9600 / 300 = 32 c. The length of one order cycle = 1 / 32 years = 288/32 days = 9 days d. Total ordering and inventory cost = QH/2 + DS/Q = 2400 + 2400 = 4800 Introduction to Operations Management 37 EOQ with quantity discount This is a variant of the EOQ model. Quantity discount is another form of economies of scale: pay less for each unit if you order more. The essential trade-off is between economies of scale and carrying cost. Introduction to Operations Management 38 EOQ with quantity discount To tackle the problem, there will be a separate (TC) curve for each discount quantity price. The objective is to identify an order quantity that will represent the lowest total cost for the entire set of curves in which the solution is feasible. There are two general cases: – The holding cost is constant – The holding cost is a percentage of the purchasing price. Introduction to Operations Management 39 Quantity discount (constant holding cost) Total cost per cycle = setup cost + purchase cost + carrying cost Setup cost = S Purchase cost = ciQ, if the order quantity Q is in the range where unit cost is ci. Carrying cost = (Q/2)h(Q/D) TC = {S+ciQ+hQ2/(2D)}{D/Q} (annual cost) Introduction to Operations Management 40 Quantity discount (constant holding cost) Differentiating, we obtain an optimal order quantity which is independent of the price of the good 2DS Q0 h The questions is: Is this Q0 a feasible solution to our problem? Introduction to Operations Management 41 Quantity discount (constant holding cost) Solution steps: 1. Compute the EOQ. 2. If the feasible EOQ is on the lowest price curve, then it is the optimal order quantity. 3. If the feasible EOQ is on other curve, find the total cost for this EOQ and the total costs for the break points of all the lower cost curves. Compare these total costs. The point (EOQ point or break point) that yields the lowest total cost is the optimal order quantity. Introduction to Operations Management 42 Example The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying cost are $4 per case a year, and the new price schedule indicates that orders of less than 50 cases will cost $20 per case, 50 to 79 cases will cost $18 per case, 80 to 99 cases will cost $17 per case, and larger orders will cost $16 per case. Determine the optimal order quantity and the total cost. Introduction to Operations Management 43 Example (solutions) 1. The common EOQ: = {2(816)(12)/4} = 70 2. 70 falls in the range of 50 to 79, at $18 per case. TC = DS/Q+ciD+hQ/2 = 816(12)/70 + 18(816) + 4(70)/2 = 14,968 3. Total cost at 80 cases per order TC = 816(12)/80 + 17(816) + 4(80)/2 = 14,154 Total cost at 100 cases per order TC = 816(12)/100 + 16(816) + 4(100)/2 =13,354 The minimum occurs at the break point 100. Thus order 100 cases each time Introduction to Operations Management 44 Quantity discount (h = rc) Usingthe same argument as in the constant holding cost case, Let TCi be the total cost when the unit quantity is ci. TCi = rciQ/2 + DS/Q + ciD Therefore EOQ = {2DS/(rci)} In this case, the carrying cost will decrease with the unit price. Thus the EOQ will shift to the right as the unit price decreases. Introduction to Operations Management 45 Quantity discount (h = rc) Steps to identify optimal order quantity 1. Beginning with the lowest price, find the EOQs for each price range until a feasible EOQ is found. 2. If the EOQ for the lowest price is feasible, then it is the Optimal order quantity. 3. Same as step (3) in the constant carrying case. Introduction to Operations Management 46 Example (p.549) Surge Electric uses 4000 toggle switches a year. Switches are priced as follows: 1 to 499 at $0.9 each; 500 to 999 at $.85 each; and 1000 or more will be at $0.82 each. It costs approximately $18 to prepare an order and receive it. Carrying cost is 18% of purchased price per unit on an annual basis. Determine the optimal order quantity and the total annual cost. Introduction to Operations Management 47 Solution D = 4000 per year; S = 18 ; h = 0.18 {price} Step 1. Find the EOQ for each price, starting with the lowest price EOQ(0.82) = {2(4000)(18)/[(0.18)(0.82)]} = 988 (Not feasible for the price range). EOQ(0.85)= 970 (feasible for the range 500 to 999) Step 2. Feasible solution is not on the lowest cost curve Step 3. TC(970) = 970(.18)(.85)/2 + 4000(18)/970 + .85(4000) = 3548 TC(1000) = 1000(.18)(.82)/2 + 4000(18)/1000 + .82(4000) = 3426 Thus the minimum total cost is 3426 and the minimum cost order size is 1000 units per order. Introduction to Operations Management 48 We have settled the problem of how much to order. The next decision problem is when to order. Introduction to Operations Management 49 Reorder point In the EOQ model, we assume that the delivery of goods is instantaneous. However, in real life situation, there is a time lag between the time an order is placed and the receiving of the ordered goods. We call this period of time the lead time. Demand is still occurring during the lead time and thus inventory is required to meet customer demand. This is why we need to consider when to order! Introduction to Operations Management 50 When to reorder We can reorder based on: – time, say weekly, monthly, etc – quantity of goods on hand Introduction to Operations Management 51 When to Reorder with EOQ Ordering Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level - Probability that demand will not exceed supply during lead time. Introduction to Operations Management 52 Quantity Safety Stock Maximum probable demand during lead time Expected demand during lead time ROP Safety stock LT Time Introduction to Operations Management 53 When to reorder In general, we need to consider the following four factors when deciding when to order: – The rate of demand (usually based on a forecast) – The length of lead time. – The extent of demand and lead time variability. – The degree of stock-out risk acceptable to management. Introduction to Operations Management 54 When to reorder We will just cover two simple cases: – The demand rate and lead time are constant – Variable demand rate and constant lead time. Introduction to Operations Management 55 Reorder Point Service level Risk of a stockout Probability of no stockout ROP Quantity Expected demand Safety stock 0 z z-scale Introduction to Operations Management 56 ROP - constant demand and lead time Let d be the demand rate per unit time and LT be the lead time in unit time. The reorder point quantity is given by ROP = expected demand during lead time + safety stock during lead time In this case, safety stock required is 0 Thus ROP = d (LT) Example: See Example 7 on page 551 Introduction to Operations Management 57 ROP - variable demand rate The lead time is constant. In this case, we need extra buffer of safety stock to insure against stock-out risk. Since it cost money to carry safety stocks, a manager must weigh carefully the cost of carrying safety stock against the reduction in stock-out risk (provided by the safety stock) Introduction to Operations Management 58 ROP - variable demand rate Inother words, he needs to trade-off the cost of safety stock and the service level. Service level = 1 - stock-out risk ROP = Expected lead time demand + safety stock = d (LT) + z Where is the standard deviation of the lead time demand = {LT} d Therefore ROP = d(LT) + z {LT} d Introduction to Operations Management 59 Example (Problem 4, p.569) The housekeeping department of a hotel uses approximately 400 washcloths per day. The actual amount tends to vary with the number of guests on any given night. Usage can be approximated by a normal distribution that has a mean of 400 and a standard deviation of 9 washcloths per day. A linen supply company delivers towels and washcloths with a lead time of three days. If the hotel policy is to maintain a stock-out risk of 2 percent, what is the minimum number of washcloths that must be on hand at reorder time, and how much of that amount can be considered safety stock? Solution: d = 400, LT = 3 days, d = 9 , service level = 1 - risk = 0.98 Introduction to Operations Management 60 Example (solution) Z = 2.055 (from normal distribution table) Thus ROP = 400 (3) + 2.055 (3) d = 1200 + 32.03 = 1232 The safety stock is approximately 32 washcloths, providing a service level of 98%. Introduction to Operations Management 61 Fixed-order-interval model In the EOQ/ROP models, fixed quantities of items are ordered at varying time interval. However, many companies ordered at fixed intervals: weekly, biweekly, monthly, etc. They order varying quantity at fixed intervals. We called this class of decision models fixed-order- interval (FOI) model. Introduction to Operations Management 62 FOI model Why use FOI model? Is it optimal? Whatare decision variables in the FOI model? Introduction to Operations Management 63 FOI model Assuming lead time is constant, we need to consider the following factors – the expected demand during the ordering interval and the lead time – the safety stock for the ordering interval and lead time – the amount of inventory on hand at the time of ordering Introduction to Operations Management 64 FOI model Order quantity Safety stock Time Amount on hand (A) OI Place order LT Receive order Introduction to Operations Management 65 FOI model We use the following notations OI = Length of order interval A = amount of inventory on hand LT = lead time d = Standard deviation of demand Introduction to Operations Management 66 FOI model Amount to order = expected demand during protection interval + Safety stock for protection interval - amount on hand at time of placing order = d(OI LT) z d OI LT A Introduction to Operations Management 67 Example (p.560) The following information is given for a FOI system. Determine the amount to order. d = 30 units per day, d = 3 units per day, LT = 2 days; OI = 7 days Amount on hand = 71 units, Desired serve level = 99%. Solution: z = 2.33 for 99% service level. Amount to order = 30 (7 + 2) + 2.33 (3) (7 + 2) - 71 = 270 + 20.97 - 71 = 220 Introduction to Operations Management 68