1 Inventory Theory The Management of Idle Resources A quantity of commodity held for some time to satisfy some future demand.

2 Why do we need inventory? Economies of batch production Unpredictable or unreliable vendors buffer for imbalanced production lines buffer for machine downtimes safety stock against random demands or uncertain lead- times hedge against poor quality bi-product from production smoothing avoid loss of sales or high cost of backorders fill logistics pipeline - resupply time display goods to potential customers

3 More of Why Inventory? A buffer between supply and demand –Accounts for differences in rates and timing between supply and demand Internal (matter of policy) –Economies of scale –Production smoothing –Customer service External (uncontrollable) –Uncertainty

4 Types of Inventory Raw material and purchased parts inventory In-process inventory Finished goods inventory vendorvendorvendor warehousewarehousewarehouse production customers

5 Demands Independent – demand not related to any other item and primarily influenced by market conditions Dependent – demand for an item is influenced by the demand of another item A demanding manager

6 More Types of Inventory Raw material –Material needing further processing –Components that go into the product as is –Supplies such as glue, screws, ink, thread –Dependent demand Work in process (WIP) –Inventory in the production system waiting to be processed or assembled and may include semi-finished products –Dependent demand Finished goods –Output of the production process or end items –Demand is usually independent –Finished goods from one manufacturing plant may be raw material for another

7 A Taxonomy Inventory Models DeterministicStochastic Single Multi- Static (EOQ)Dynamic periodperiod basicsmoothing backorderdynamic lot sizecontinuous periodic finite productionreviewreview finite production with backorders quantity discount multi-item Repairable

8 Fundamental Questions What to order? When to order? How much to order? When to review? I knew that.

9 Some Inventory Policies Continuous review - order Q when inventory reaches r Continuous review - one for one reordering Periodic review - review inventory every T time periods. If inventory is less than r, order up to R Periodic review - review inventory every T time periods. Order up to R

10 Measures of Effectiveness Monetary –Profit maximization –Cost minimization Supply effectiveness –minimize expected backorders –maximize probability of meeting demands Operational effectiveness –minimize expected downtime –maximize system availability I would like to see less inventory.

11 Conflicting objectives Marketing: I can’t sell from an empty wagon. I can’t keep our customers if we continue to stockout and there is not sufficient product variety. Production: If I can produce in larger lot sizes, I can reduce per unit cost and function efficiently. Purchasing: I can reduce our per unit cost if I buy large quantities in bulk. Finance: Where I am going to get the funds to pay for the inventory? The levels should be lower. Warehousing: I am out of space. I can’t fit anything else in this building.

12 More conflicting objectives

13 Characteristic of Inventory Demand –Constant –Variable –Random variable Lead time –Constant (known) –Random variable Review –Continuous –Periodic Stockouts –Backordered –Lost sales We just don’t have that model in stock. It is backordered but should arrive any day now.

14 Inventory costs Purchasing costs –material cost, unit cost Ordering costs –fixed cost of preparing and monitoring order –receiving and handling Production cost –material and variable manufacturing cost Set-up cost –fixed cost to prepare for manufacture Holding costs –opportunity cost –storage and handling costs –taxes and insurance –pilferage, damage, spoilage, obsolescence, etc. Backorder and lost sales costs Purchase option Production option

15 Representative Holding Costs Costs proportional to the quantity of inventory held. Includes: a) Physical Cost of Space (3%) b) Taxes and Insurance (2 %) c) Breakage Spoilage and Deterioration (1%) d) Opportunity Cost of alternative investment. (10%) (Total: 16%) holding costs = 16 % x unit cost

16 Shortage Costs Loss of revenue for lost demand Costs of bookkeeping for backordered demands Loss of goodwill for being unable to satisfy demands when they occur. Generally assume cost is proportional to number of units of excess demand.

17 Assumptions – general model Demand is known and constant order quantity is not restricted to integers unit cost does not depend upon the order quantity no change in unit cost over time (inflation) each item can be treated independently lead-time is known and constant backorders are permitted infinite planning horizon

18 Notation Decision variables: Q = order quantity or lot size (units) r = reorder point (units) T = time between orders or production runs (cycle time) (yr) b = maximum backorders per cycle (units) Parameters D = demand rate (units per yr)(note that book uses ) P = production rate (units per yr) (P > D) c = unit purchase or production cost ($/unit) K = order or set-up cost ($/order) h = holding cost = ic ($/unit per yr) g = backorder cost ($/unit per yr) g’ = cost per backorder ($/unit) L = lead-time (yr)

19 Basic EOQ Model Additional assumptions: 1.No backorders 2. Instantaneous arrivals time inventory slope = -D T Q

20 Total inventory cost = ordering cost + purchase cost + holding cost order cost per cycle = K; purchase cost per cycle = cQ average inventory per cycle = Q/2 holding cost per cycle = (hT) (Q/2) length of cycle = T = Q/D Total inventory cost per cycle = K + cQ + h(Q/D) (Q/2) Cycles per year = D/Q Total inventory cost per year = (D/Q) x [ K + cQ + h(Q/D) (Q/2) ] = DK/Q + cD + h (Q/2)

21 Cost Minimization

22 Total inventory cost per year = G(Q) = DK/Q + cD + h (Q/2) Find Q that Minimizes G(Q): dG(Q)/dQ = - DK/Q 2 + h/2 = 0 solving:h/2 = DK/Q 2 Q 2 = 2DK/h T* = Q*/D

23 Reorder Point time inventory slope = -D L Q R L R = lead-time demand = LD – mQ* where m = integer [L / T*]

24 G(Q) = DK/Q + cD + h (Q/2)

25 Example – Basic EOQ Monthly demand for plastic bolts used to fasten the wing of the C-5A to the fuselage is 8,000. Each bolt cost $.075. The cost of ordering including delivery charges is $100 per shipment. An annual holding cost of ten percent of the purchase cost is to be used. That is h =.10 (.075) = If L = 2 months = 2/12 =.1667 years, then R =.1667 (96,000) = 16,003

26 The Finite Production Model with Backorders time On-hand inventory T1T1 T2T2 T3T3 I max T4T4 P-D -D -b Q T t p = T 1 + T 2 t d = T 3 + T 4

27 Total Annual Cost Set-uppurchaseholdingbackorder costcostcostcost To solve:

28 Solution

29 number complete cycles cycle time t p = T 1 + T 2 t d = T 3 + T 4 Now I will tell you the story of how the re-order point is calculated. case II case I

30 Re-order Point time On-hand inventory -b L R L = production lead-time Case I.

31 Re-order Point time On-hand inventory -b L R L = production lead-time Case II.

32 Note if g’ = 0 then Q* and b* will be positive. If g’ > 0 then the radicand may be negative. In this case no backorders should be permitted and b* = 0. Note if g = 0 and g’ > 0 then 1) either no backorders should be permitted: D g’ > annual cost with no backorders 2) or no inventory carried (i.e. produce-to-order): D g’ < annual cost with no backorders

33 No backorders permitted

34 The Production Model with no Backorders time on-hand inventory tptp tdtd B t = t p + t d t p = Q/P t d = B/D where B = (Q/P) (P – D) G(Q) = K D / Q + (h /2) (Q /P)[P – D]

35 No backorders permitted (continued) Re-order point:

36 Purchase Option instantaneous arrivals

37 The Backorder Model time slope = -D S -b On-hand inventory t1t1 t2t2 t 1 = b/D; t 2 = (Q-b)/D Q t = t 1 + t 2

38 Purchase Option instantaneous arrivals (continued) Re-order point: On-hand + on-order on-hand number complete cycles cycle time

39 Purchase option No backorders permitted Say. This is just the simple EOQ model isn’t it?

40 EOQ Homework I always work extra problems – not only the ones that are assigned! Text Chapter 4: 4,8,10,12,13,14,17,20,21,22 24,26,27,28,29,30,33,35,40 + Handout