Inventory Control Subject to Known Demand

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1 Production and Operation Managements Inventory Control Subject to Known Demand Prof. JIANG Zhibin Department of Industrial Engineering & Management Shanghai Jiao Tong University

2 Contents Introduction Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory System; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate Quantity Discount Models Resource-constrained multiple product system EOQ models for production planning Power-of-two policies

3 Introduction Inventory is the stock of any item or resource used in an organization. Generally, inventory is being acquired or produced to meet the need of customers; An inventory system is the set of policies and controls that monitors levels of inventory or determines what levels should be maintained. Independent demand system Dependent demand system

4 Introduction The fundamental problem of inventory management : When to place order for replenish the stock? How much to order? The complexity of the resulting model depends on the assumptions about the various parameters of the system The major distinction is between models for known demand and random demand.

5 Introduction The current investment in inventories in USA is enormous; It amounted up to $1.57 trillion in the third quarter of 2007; It accounts for 20-25% of the total annual GNP (general net product); There exists enormous potential for improving the efficiency of economy by scientifically controlling inventories;

6 Introduction Inventory model discussed here are most applicable to manufacturing, wholesale, and retail sectors, composing 82 Fig. 4-1 Breakdown of the Total Investment in Inventories in the U.S. Economy (2007) % of the total.

7 Contents Introduction Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory System; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate Quantity Discount Models Resource-constrained multiple product system EOQ models for production planning Power-of-two policies

8 Types of Inventories A natural classification is by value added from manufacturing: Raw materials-resources required in the production or processing activity of the firm. Components-Raw materials or subassemblies that will later be included into a final product. Work-in-process (WIP)-the inventory either waiting in the system for processing or being processed. Finished good-also known as end items-the final products.

9 Contents Introduction Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory System; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate Quantity Discount Models Resource-constrained multiple product system EOQ models for production planning Power-of-two policies

10 Why Hold Inventories For economies of scale-it may be economical to produce a relatively large number of items in each production run and store them for future use. Coping with Uncertainties Uncertainty in demand; Uncertainty in lead time Uncertainty in supply

11 Why Hold Inventories For speculation- Purchase large quantities at current low prices and store them for future use. Cope with considerable fluctuation in price of costly commodities required in large quantities; Cope with labor strike;

12 Why Hold Inventories For Transportation Pipeline inventories is the inventory moving from point to point, e.g., materials moving from suppliers to a plant, from one operation to the next in a plant. It exists for purpose of transportation or materials handling in a plant;

13 Why Hold Inventories Smoothing-Producing and storing inventory in anticipation of peak demand helps to alleviate the disruptions caused by changing production rates and workforce level. Logistics-To cope with constraints in purchasing, production, or distribution of items that may cause a system maintain inventory; Purchase an item in minimum quantities; Logistics of manufacture-zero inventory is impossible in order to keep continuity in manufacturing process; Control costs -More inventory need less and simpler control

14 Why we should control the inventory Inventory Level Machine down Quality Working efficiency Machine down 库 存 水 平 quality Working efficiency delay delay Shut down Production plan WIP lost Shut down Production plan WIP lost (a) Higher inventory level (b) Lower inventory level

15 Contents Introduction Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory System; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate Quantity Discount Models Resource-constrained multiple product system EOQ models for production planning Power-of-two policies

16 Characteristics of Inventory Systems Demand: patterns and characteristics Constant versus variable; Known versus random Lead Time Ordered from the outside; Produced internally

17 Characteristics of Inventory Systems Review patterns Continuous-supermarket; Periodic-regular stock-taking for a grocery store; Excess demand -demand that cannot be filled immediately from stock; backordered or lost. Changing inventory Limited shelf life- perishability; Become obsolete- obsolescence;

18 Contents Introduction Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory System; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate Quantity Discount Models Resource-constrained multiple product system EOQ models for production planning Power-of-two policies

19 Relevant Costs- Holding cost Holding cost (carrying or inventory cost)-the sum of costs that are proportional to the amount of inventory physically on hand at any point in time. Some items of holding costs: Cost of providing the physical space to store the items; Taxes and insurance; Breakage, spoilage, deterioration, and obsolescence; Opportunity cost of alternative investment;

20 Relevant Costs- Holding cost Opportunity cost, also known as cost of capital, turns out to be the most significant one If we have 10,000RMB on hand, and save it in bank for one month at interest rate 0.25/month, then we may earn 25RMB/month from bank; If we use this amount money to buy some goods and store them in warehouse, then we lose 25RMB/month.

21 Relevant Costs- Holding cost Holding cost considers the aggregated interest rate comprised of the four components For example, 28% cost of capital, 2% taxes and insurance, 6% cost of storage, 1% breakage and spoilage, mean 37% total interest charge. We would assess a charge of 37 cents for every dollar that we have invested during a one-year period.

22 Relevant Costs- Holding cost Holding cost h (in terms of dollars per unit per year) h=ic I: the annual interest rate c: the dollar value of one unit of inventory For example, an item valued at $180 would have an annual holding cost of h=0.37*180= Inventory cost fluctuates with time-inventory as a function of time The holding cost incurred at any point in time is proportional to the inventory level at that point in time.

23 Relevant Costs- Holding cost t I Itdt t t 2 ( ( ) )/( 2 1) t1 Fig.4-2 Inventory as a Function of Time

24 Relevant Costs- Order cost It depends on the amount of inventory that is ordered or produced. Two components The fixed cost K: independent of size of order as long as it is not zero; which includes only those costs that are relevant to the current ordering decision, maybe the costs of order generation and receiving, and handling cost; not includes the overhead costs The variable cost c: incurred on per-unit basis; Cx ( ) 0 if x 0; K cx if x 0

25 Relevant Costs- Order cost Fig.4-3 Order Cost Function

26 Relevant Costs- Penalty Cost Also know as shortage cost or stock-out cost-is the cost of not having sufficient stock on hand to satisfy a demand when it occurs. Two interprets: In back-order case: include whatever bookkeeping and/or delay costs may be involved; In lost-sale case: include of loss-of-goodwill cost, a measure of customer satisfaction.

27 Relevant Costs- Penalty Cost Two approaches: Penalty cost, p, is charged per-unit basis. Each time a demand occurs that cannot be satisfied immediately, a cost p is incurred independent of how long it takes to eventually fill the demand. Charge the penalty cost on a per-unit-time basis.

28 The EOQ Model-Basic Model Average holding level is Q/2 Fig 4-4 Inventory Levels for the EOQ Model Considerations: On-hand inventory level at the time zero is zero; An order must be placed at time zero; Q is the size of the order (lot size); Next order is placed just when the inventory level drops to zero; The order cycle T=Q/ The objective is to choose Q to minimize the average cost per unit time (usually, a year)

29 The EOQ Model-Basic Model Express the average annual cost as a function of the lot size Order cost in each order cycle: C(Q)=K+cQ; The average holding cost during one order cycle is hq/2; The average annual cost (suppose there are n cycles in a year) : The annual total ordering cost GQ ( ) ( K cq ) n hq K cq hq K c hq nt 2 Q / 2 Q 2 K h 2K G'( Q) ; G''( Q) 0 for Q 0; 2 3 Q 2 Q Average holding cost for one cycle = that for one year.

30 The EOQ Model-Basic Model Since Q >0,G(Q) is convex function of Q; G(Q) is minimized at Q*--economic order quantity, EOQ G'( Q) 0 Q * 2 K h Notes: Q* is the value of Q where the two curves intersect; The unit order cost, c, does not appear explicitly in the expression of Q*, since c is independent from Q in the C(Q); Because c is constant, it is ignored while computing average cost. Fig.4-5 The Average Annual Cost Function G(Q)

31 The EOQ Model-Basic Model Example 4.1 Pencils are sold at a fairly steady rate of 60 per week; Pencils cost 2 cents each and sell for 15 cents each; Cost $12 to initiate an order, and holding costs are based on annual interest rate of 25%. Determine the optimal number of pencils for the book store to purchase each time and the time between placement of orders Solutions Annual demand rate =60 52=3,120; The holding cost is the product of the variable cost of the pencil and the annual interest-h= =0.05 * 2K ,120 Q h ,870 T Q 3, ,120 yr Back

32 The End!

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