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## Inventory Control

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**Inventory Control**Henry C. Co Technology and Operations Management, California Polytechnic and State University**Inventory**• Inventory: stockpiles of raw materials, components, semi-finished or finished goods waiting to be processed, transported or used at a point of the supply chain. • Reasons to have inventories • Improving service level, • Reducing overall logistics costs, • Coping with randomness in demand and lead times • Making seasonal items available all year • Speculating on price patterns, etc. • Annual inventory holding cost can be 30% of the value of the materials kept in stock, or even more. Inventory Management Problems**Relevant Cost**• Procurement costs • Holding costs • Shortage costs • Obsolescence costs Inventory Management Problems**Inventory Management Models**• Deterministic vs. stochastic models • Fast- vs. slow-moving items • No. of stocking points • No. of commodities • Instantaneous resupply v. non-instantaneous resupply • Discrete vs. continuous; finite vs. infinite horizon; shortage allowed/disallowed etc. The Economic Order Quantity model you learned in TOM 531 is an example of a “single stocking point, single-commodity, instantaneous resupply, shortage not allowed, continuous model with infinite horizon.” Inventory Management Problems**The EOQ Model in TOM 531**• The total cost curve reaches its minimum where the carrying and ordering costs are equal. • EOQ represents trade-off between fixed cost associated with production or procurement against inventory holding costs. D = Rate of demand, units/year S = Fixed cost of procurement, $/order v = Variable cost of procurement. H = $/unit/year holding cost Q = Quantity ordered, units Inventory Management Problems**Instantaneous Resupply**Inventory Management Problems**Annual Cost**QO The Classical EOQ Model • The total cost curve reaches its minimum where the carrying and ordering costs are equal. Ordering Costs Order Quantity (Q) (optimal order quantity) Inventory Management Problems**Using calculus, we take the derivative of the total cost**function (TC) and set the derivative (slope) equal to zero and solve for Q. Inventory Management Problems**Try This!**Al-Bufeira Motors manufactures spare parts for aircraft engines in Saudi Arabia. Its component Y02PN, produced in a plant located in Jiddah, has a demand of 220 units per year and a unit production cost of $1200. Manufacturing this product requires a time-consuming set-up that costs $800. The current annual interest rate p is 18%, including warehousing costs. Shortages are not allowed. Inventory Management Problems**Suppose production rate = p= 200 units/day, and demand rate**= d = 80 units/day. • Since p>d, inventory will increase at __________ (p-d) units/day. • Suppose current inventory is 0. • In 10 days, the inventory level would be 10 days * 120 unit/day = _____ units. • In 20 days, the inventory level would be 20 days * 60 unit/day = _____ units. • etc. Inventory Management Problems**The machine produces a batch, then stops, then resumes**production at some later time when the inventory of this item is low. This is call batch production. • Batch production is very common in industry. • When a machine is used to produce two or more products, one product at a time. • One decision the production manager has to make is when to start producing each product, and when to stop. • The run time is the amount of time the machine is producing a batch. • Producing at 200 units/day, if we want to produce 2,000 units per batch, the run time is _____ days. Inventory Management Problems**Maximum Inventory Level**• If current inventory level is 0, what is the inventory at the end of the run time? • Since inventory will be rising at (200- 80 =120) units/day, the inventory level will be _____ units in 10 days. • The inventory at the end of the run time is the maximum inventory. It is equal to (p-d)*t = _____ units. • The machine produced 2,000 units in 10 days, and the maximum inventory level is only 1,200. Why? • After completing a batch, how long will it take to deplete the inventory? • Answer: It will take (p-d)*t/d = _____ days to deplete the inventory. This is the off-time. Inventory Management Problems**Number of Runs Per Year**• If annual = D = 24,000 units, how many runs do we produce each year? • Answer: Since we are producing Q = 2,000 units per batch, there will be D/Q = _____ batches per year. • In other words, there are D/Q = 12 cycles per year. In each cycle, there is a period of time the machine is producing the product (the run time), and a period to allow the inventory to deplete (the off time). Inventory Management Problems**Average Inventory**• What is the average inventory level? • During the run time, the inventory level rises from 0 to the maximum level of (p-d)*t = _____ units. • During the off time, the inventory level drops from a maximum of (p-d)*t units to 0. • The average inventory level therefore = _____ units. • Since p*t = Q, then t = Q/p. We can rewrite the expression for the average inventory as (p-d)*t/2 = (p-d)*(Q/p)/2 = (1-d/p)Q/2. Inventory Management Problems**Tradeoff**• What is the average inventory if the batch size equals the annual demand D = 24,000 units? How many batches do we have to run per year? • Answer: The average inventory = (1-d/p)Q/2 = _____ units. We need to run one batch per year. • What is the average inventory if the batch size equals the weekly demand of 480 units (assuming 50 weeks/year)? How many batches do we have to run per year? • Answer: _____ units; Run _____ batches per year. Inventory Management Problems**Optimal Tradeoff**• Suppose the cost to carry one unit of inventory for one year is H. Since the average inventory level is (1-d/p)Q/2, the annual inventory-carrying cost is H*(1-d/p) Q/2. • Suppose the cost to set-up the machine to produce a batch is S. Since we need to run D/Q batches per year, the annual set-up cost is S*D/Q. • Adding the two costs, we have H*(1-d/p) Q/2 + S*D/Q. Using calculus, the optimal batch size is Inventory Management Problems**Try This!**• Golden Food distributes tinned foodstuff in Great Britain. In a warehouse located in Birmingham, the demand rate d for tomato purée is 400 pallets a month. The value of a pallet is c = ￡2500 and the annual interest rate p is 14.5% (including warehousing costs). Issuing an order costs ￡30. The replenishment rate r is 40 pallets per day. Shortages are not allowed. Inventory Management Problems**Quantity Discounts-On-All-Units**Inventory Management Problems**Try this!**• Maliban runs more than 200 stationery outlets in Spain. The firm buys its products from a restricted number of suppliers and stores them in a warehouse located near Sevilla. Maliban expects to sell 3000 boxes of the Prince Arthur pen during the next year. The current annual interest rate p is 30%. Placing an order costs €50. The supplier offers a box at €3, if the amount bought is less than 500 boxes. The price is reduced by 1% if 500–2000 boxes are ordered. Finally, if more than 2000 boxes are ordered, an additional 0.5% discount is applied. Inventory Management Problems**Single Period Stochastic Models**Short product life cycles / Long lead times Computers Apparel Fresh products Fresh food, newspapers Services Airline industry**These models have the objective of properly balancing the**cost of Underage – having not ordered enough products vs. Overage – having ordered more than we can sell • These models apply to problems like: • Planning initial shipments of ‘High-Fashion’ items • Amount of perishable food products • Item with short shelf life (like the daily newspaper) • Because of this last problem type, this class of problems is typically called the “Newsboy” problem Inventory Management Problems**Stochastic Model 1: The Newsboy Model**• At the start of each day, a newsboy must decide on the number of papers to purchase. Daily sales cannot be predicted exactly, and are represented by the random variable, D. • The newsboy must carefully consider these costs: • cU: underage cost (when D≥S). This is the unit opportunity cost; for example, unit revenue r - unit cost c, i.e., (r-c) • cO: overage cost (when D≤S). This is the unit cost of overstocking; for example, unit cost c - unit salvage value u, i.e., (c-u). Inventory Management Problems**The objective is to Minimize the expected cost:**cu E[max{D-S, 0}] + co E[max{S-D, 0}] Solving for Q, the optimal order quantity S satisfies the following condition: • Equation (4-36) can be rewritten as: Inventory Management Problems**Graphical Representation**Inventory Management Problems**Uniform Demand Between [A,B]**Inventory Management Problems**Try this on Excel!**• Emilio Tadini & Sons is a hand-made shirt retailer, located in Rome (Italy), close to Piazza di Spagna. This year Mr. Tadini faces the problem of ordering a new bright color shirt made by a Florentine firm. • He assumes that the demand is uniformly distributed between 200 and 350 units. • The purchasing cost is c = €18 while the selling price is r = €52 and the salvage value is u = €7. Thus co = 34 and cu = 11. • Hence, Mr. Tadini should order S = 313 units. Inventory Management Problems**Another Example …**• The buyer for Needless Markup, a famous “high end” department store, must decide on the quantity of a high-priced women’s handbag to procure in Italy for the following Christmas season. • The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150.00. • Any handbags not sold by the end of the season are purchased by a discount firm for $20.00. • In addition, the store accountants estimate that there is a cost of $.40 for each dollar tied up in inventory, as this dollar invested elsewhere could have yielded a gross profit. • Assume that this cost is attached to unsold bags only. Suppose that the sales of the bags are equally likely to be anywhere from 50 to 250 handbags during this season. • Based on this, how many bags should the buyer purchase? Example from Nahmias, Production and Operations Analysis Inventory Management Problems**Distribution of Demand Is Normal**Inventory Management Problems**Example**• Every week, the owner of a newsstand purchases a number of copies of The Computer Journal. • Weekly demand for the Journal is normally distributed with mean 10 and standard deviation 5. • He pays 25 cents for each copy and sells each for 75 cents. Question: How many copies should he order? Example from Nahmias, Production and Operations Analysis Inventory Management Problems**Using NORMSINV**Inventory Management Problems**Using NORMINV**Inventory Management Problems**Using Table**Inventory Management Problems**Stochastic Model 2**• (s, S) Policy for Single Period : • If there is an initial inventory q0 and a fixed reorder cost k, the optimal replenishment policy can be obtained as follows. If q0 ≥S, no reorder is needed. • Otherwise, the best policy is to order S −q0, provided that the expected revenue associated with this choice is greater than the expected revenue associated with not producing anything. • Hence, two cases can occur: (i) if the expected revenue ρ(S) − k − cq0 associated with reordering is greater than the expected revenue ρ(q0) − cq0 associated with not reordering, then S − q0 units have to be reordered; (ii) otherwise, no order has to be placed. Inventory Management Problems**Stochastic Model 3**• In reorder point policy (fixed order quantity), inventory level monitored continuously. As soon as its net value I(t) (amount in stock - unsatisfied demand + orders placed but not yet received) reaches a reorder point l, a constant quantity q is ordered. Inventory Management Problems**Stochastic Model 4**• In the reorder cycle policy (periodic review policy) stock level is reviewed periodically at time instants ti (ti+1 = ti + T , T≥0). At time ti, qi = S − I (ti) units are ordered. The order-up-to-level S represents the maximum inventory level in case lead time tl is negligible. Inventory Management Problems**Stochastic Model 5**• The (s, S) inventory policy is a natural extension of the (s, S) policy illustrated for the one-shot case. At time ti , S −I(ti) items are ordered if I(ti) < s. If s is large enough (s → S), the (s, S) policy is similar to the reorder cycle inventory method. On the other hand, if s is small (s → 0), the (s, S) policy is similar to a reorder level policy with a reorder point equal to s and a reorder quantity q S. Inventory Management Problems**Stochastic Model 6**• The two-bin policy is a variant of the reorder point inventory method where no demand forecast is needed, and the inventory level does not have to be monitored continuously. The items in stock are assumed to be stored in two identical bins. As soon as one of the two becomes empty, an order is issued for an amount equal to the bin capacity. Browns supermarkets make use of the two-bin policy for tomato juice bottles. The capacity of each bin is 400 boxes, containing 12 bottles each. In a supermarket close to Los Alamos (New Mexico, USA) the inventory level on 1 December last was 780 boxes of 12 bottles each. Last 6 December, the inventory level was less than 400 boxes and an order of 400 boxes was issued (see Table 4.2). The order was fulfilled the subsequent day. Inventory Management Problems