LESSON 11: INVENTORY MODELS (DETERMINISTIC)

1. LESSON 11: INVENTORY MODELS (DETERMINISTIC) Outline • Hierarchy of Production Decisions • Inventory Control Questions • Inventory Control Costs • The Economic Order Quantity (EOQ) model • The EOQ Model Costs • The EOQ • Some Important Characteristics of the EOQ Cost Function

2. Hierarchy of Production Decisions • In Chapter 2, Lessons 4-7, we discuss demand forecasting • Forecasting provides the demand of products in the near future • In Chapter 3, Lessons 8-10, we discuss aggregate planning. • Aggregate production plan is translated into a detail product-wise production plan by Master Production Schedule (MPS). • The next question in the hierarchy of decision making is inventory control.

3. Hierarchy of Production Decisions

4. Inventory Control Questions • Inventory control questions: • Raw materials, components (subassemblies), and finished goods may be purchased or produced in-house. • Timing and amount of purchase and production must be carefully planned. There are two major questions: 1. How much (lot size)? How much to purchase or produce? 2. When? When to purchase or produce?

5. Inventory Control Questions • There can be two extreme strategies: • Small lot sizes and too frequent inventory replenishment • This strategy yields higher ordering costs (if the items are purchased) or setup costs (if the items are produced). • The demand forecast is for a short term and it is more accurate. • Large lot sizes and too infrequent inventory replenishment

6. Inventory Control Questions • Large lot sizes and too infrequent inventory replenishment • This strategy requires a large investment in the inventory and yields higher inventory holding costs. • The demand forecast is for a long term and it is less accurate.

7. Inventory Control Costs • Inventory holding costs, • Opportunity cost of capital • Cost of storage space • Taxes and insurance against fire, theft, and other losses • Breakage, spoilage, deterioration and obsolescence • Example of calculation of inventory holding costs Cost of capital - 15% Taxes and insurance - 2% Storage - 5% Breakage/spoilage - 3% Total - 25%

8. Inventory Control Costs • Inventory holding costs, • Notation: = Annual interest rate = Dollar value of one unit of inventory = holding cost in terms of dollars per unit per year • Then, we have the relationship

9. Inventory Control Costs • Order costs or setup costs, • If the items are purchased, a fixed ordering cost may be incurred each time an order is placed. Similarly, if the items are produced, a fixed setup cost may be incurred each time the production facility is set up to produce the item. • Some examples are bookkeeping expense, order processing fees, transportation costs, receiving costs, handling costs, etc.

10. Inventory Control Costs • Order costs or setup costs, • If there is a variable part of the order cost/setup that depends on the number of units ordered/produced, the variable cost is usually not considered in the order cost. Instead, the variable cost may be included in the cost of the item. • For example, the salary paid to the purchasing clerk does not depend on the number of times orders are placed. Such a cost is an overhead expense, and not a part of the ordering cost. • Notation: = ordering/setup cost per order/setup.

11. Inventory Control Costs • Penalty costs, • Shortages occur when the demand exceeds the amount of inventory on hand. One of two types of costs is charged depending on whether a shortage results in loss of sales or not: • Backorder - if the excess demand is backlogged and fulfilled in a future period, a backorder cost is charged (bookkeeping and/or delay costs). • Lost sales - if the excess demand is lost because the customer goes elsewhere, the lost sales is charged. The lost sales include goodwill and loss of profit margin. So, penalty cost = selling price - unit variable cost + goodwill, if there exists any.

12. Inventory Control Costs • Penalty costs, • In Chapter 4, Lessons 11-15, we assume that the demands are known and fixed and shortages will not take place. So, penalty costs are not considered. • In Chapter 5, Lessons 16-20, we assume that the demands are uncertain and shortages may occur. So, penalty costs are considered in Chapter 5.

13. The EOQ Model • Major assumptions 1. Demand is known and fixed (uniform). The rate of demand is units per year. This assumption is relaxed in Chapter 5, Lessons 16-20, Stochastic Inventory Models. 2. The cost parameters, unit cost, holding cost, and ordering cost are known and fixed. Given that the inventory control decisions are made for a short term, it’s likely that the costs will not change during the planning period. Still, this assumption is a simplification. The costs parameters may change over time. 3. Shortages are not permitted. This assumption is relaxed in Chapter 5, Lessons 16-20, Stochastic Inventory Models.

14. The EOQ Model • Major assumptions 4. The inventory level increases instantaneously at one point of time when an order is received. This assumption is appropriate in the context of purchasing. The EPQ model, discussed in Lesson 12, considers a gradual increase in the inventory level. The EPQ model is appropriate in the context of production. 5. There is no price discount for large order sizes. This assumption is relaxed later. See Lesson 13, the EOQ with price break.

15. The EOQ Model • Constant order size and inventory cycle • Since there is no uncertainty, it’s optimal to plan an order receipt only when the inventory level reaches zero. • Suppose that there is no inventory in the beginning. So, an order will be received in the beginning. • Since the demand and cost parameters do not change over time • If it’s optimal to order units in the beginning, it’s also optimal to order units next time. • The above observation provides two important concepts. One is the constant order size and the other is the inventory cycle.

16. The EOQ Model • Constant order size and inventory cycle • The order size, is chosen to minimize the total inventory control costs. The formula for optimal order quantity is given later. • At the start of each inventory cycle, the inventory level is . The inventory level decreases uniformly. In the end of the inventory cycle the inventory level is zero. So, the length of the cycle is the length of time over which the demand is . Thus, the length of the cycle in years is

17. The EOQ Model Demand rate Order qty, Q Inventory Level Reorder point, R Lead time Lead time 0 Time Order Placed Order Received Order Placed Order Received

18. The EOQ Model • Lead time and reorder point • Sometimes, there may be a lead time associated with the orders. The lead time, is the length of time between order placement and order receipt. • The presence of lead time requires the order be placed some time before it is needed. For example, if there is a lead time of 2 days, the order must be placed when the inventory is sufficient to meet the demand for 2 days. • Reorder point, is the inventory level at the time of order placement. • Since, reorder point must cover the lead time demand,

19. The EOQ Model Costs

20. The EOQ Model Costs Total Cost Slope = 0 Annual cost ($) Minimum total cost Holding Cost = Ordering Cost = Optimal solution, Q*, Economic Order Quantity (EOQ) Order Quantity, Q

21. The EOQ As the annual demand and cost parameters are fixed (see major assumptions), it’s important to analyze the effect of order quantity on the annual ordering, holding and total costs. The previous slide shows such an effect. Annual ordering cost, decreases as the order quantity increases Annual holding cost, increases as the order quantity increases The total cost, . The total cost curve is nearly U-shaped. The total cost is minimum for some order quantity that’s neither too small nor too large.

22. The EOQ

23. The EOQ • Recall that there are two major inventory control questions, how much and when. The EOQ model answers these questions as follows. • How much to order? • Order • When to order? • When the inventory on hand reaches the reorder point

24. Example 1: R & B beverage company has a soft drink product that has a constant annual demand rate of 3600 cases. A case of the soft drink costs R & B $3. Ordering costs are $20 per order and holding costs are 25% of the value of the inventory. R & B has 250 working days per year, and the lead time is 5 days. Identify the following aspects of the inventory policy: a. Economic order quantity

25. b. Reorder point c. Cycle time d. Total annual cost

26. Some notes on Example 1: • The total cost curve is flat near the EOQ value. So, the total cost does not change much because of a little change in the order quantity from the EOQ value. See the discussion under some important characteristics. Since the number of cases of soft drinks is a whole number, the EOQ value has been rounded to the nearest integer. • It’s not a coincidence that the annual holding cost is nearly the same as the annual ordering cost. If the EOQ units are ordered the annual ordering cost is the same as the annual holding cost. See the total cost curve and the discussion under EOQ model costs and some important characteristics. The little difference between two costs is due to the rounding.

27. Some Important Characteristics of the EOQ Cost Function • At EOQ, the annual holding cost is the same as annual ordering cost.

28. Some Important Characteristics of the EOQ Cost Function • If the order quantity is near the EOQ value, the total cost does not change from the optimal value. • See the discussion under the EOQ model costs and the figure showing various costs against order quantity. The total cost curve is flat near EOQ. This supports the point. • The EOQ policy usually provides a good decision even when the cost parameters are little off than the assumed values (and, therefore, the EOQ value is incorrect). So, the EOQ model is insensitive to errors. See text pp. 208-209 for a detail discussion. In this note, the insensitivity is shown with an example.

29. Some Important Characteristics of the EOQ Cost Function

30. Some Important Characteristics of the EOQ Cost Function • If the order quantity is near the EOQ value, the total cost does not change from the optimal value. • Consider the Example 1 data. The assumed values of the annual demand, holding cost and ordering costs are 3600 units, 25% and $20/order respectively. So, the EOQ=438 units.

31. Some Important Characteristics of the EOQ Cost Function • If the order quantity is near the EOQ value, the total cost does not change from the optimal value. • Suppose that the correct value of the annual demand is 3400 units. So, the correct EOQ=426 units and the optimal total annual cost is $319. If the decision maker, being unaware of the correct value of the annual demand, uses an order quantity of 438 units, the total annual cost will be $320 which is less than 0.30% off the optimal value of $319. • Similarly, the order quantity of 438 units, does not produce a large error when holding cost changes to 35% or ordering cost to $30/order.

32. READING AND EXERCISES Lesson 11 Reading: Section 4.1 - 4.5 , pp. 194-208 (4th Ed.), pp. 183-200 (5th Ed.) Exercise: 10 and 12, p. 210 (4th Ed.), pp. 201-202 (5th Ed.)