Review of Inventory Models. Recitation, Feb. 4 Guillaume Roels J Supply Chain Planning

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1 Review of Inventory Models Recitation, Feb. 4 Guillaume Roels J Supply Chain Planning

2 Why hold inventories? Economies of Scale Uncertainties Demand Lead-Time: time between order and delivery Supply Transportation Smoothing (Seasonality) Speculation

3 Inventory Costs Holding Cost Cost of Capital, Warehouse, Taxes and Insurance, Obsolescence Order Cost Fixed and variable Penalty Cost Lost sale vs. Backorder Consider only costs that are relevant to the ordering decision

4 Outline Newsboy 1-period Random demand (Stochastic) Shortages allowed Variable costs only No Lead Time EOQ Multiple periods Known demand (Deterministic) Constant Demand No Shortages Fixed and variable order costs No Lead Time

5 Newsboy 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. How many copies would you recommend him to order? Example from Nahmias, Production and Operations Analysis

6 Other applications Short product life cycles / Long lead times Computers Apparel Fresh products Fresh food, newspapers Services Airline industry

7 Newsboy Model: Notations Random Demand: D Ordering decision: Q Unit Selling Price: p Unit Purchase Cost: c Objective: Find Q that maximizes Expected Profit, E[π]

8 Review of Optimization Max f(x) First-Order Conditions f (x*)=0 Second-Order Conditions f (x*) 0 0 0

9 Max E[π] = p E[min{D,Q}] c Q First-Order Conditions (E[π]) = p E[(min{D,Q}) ] c = p P(D Q)-c = 0 since min{d,q}= D when D Q (min(q,d)) =0 Q when Q D (min(q,d)) =1 Second Order Conditions One can check that (E[π]) = p (P(D Q)) 0 Order Q* such that P(D Q*) = c/p

10 Distribution Function Suppose that demand has cdf F(x), i.e., F(x)=P(D x) Therefore, P(D Q*)=c/p 1-P(D Q*)=c/p 1-F(Q*)=c/p F(Q*)=(p-c)/p Ratio (p-c)/p is a probability (btw 0 and 1) and is called the critical fractile

11 Generalization c U : Underage Cost (when D Q) In the example, opportunity cost, p-c Loss of goodwill c O : Overage Cost (when D Q) In the example, c Salvage value Min c U E[max{D-Q, 0}] + c O E[max{Q-D, 0}] Solving for Q, F(Q*)=c U /(c U + c O )

12 How to find Q*: F(Q) Graphical Representation 1 c U /(c U +c O ) 0 Q* Q

13 How to find Q*: Analytical Derivation Uniform Demand between [A,B] F(x)=(x-A)/(B-A) A x B Solve (Q*-A)/(B-A)=c U /(c U + c O ), i.e. Q*=A+ (B-A) c U /(c U + c O )

14 How to find Q*: Excel Normal Demand Q*=NORMINV(µ, σ, c U /(c U + c O )) F(Q*)=c U /(c U + c O ) Q*=F -1 (c U /(c U + c O )) Alternatively, use standardized normal Q*=µ + (z*) σ where z*=normsinv(c U /(c U + c O ))

15 How to find Q*: Tables Example: c U =p-c= = $.50 c O =c= $.25 Critical Fractile = c U /(c U + c O ) = 0.67 Standardized Normal Table z*=0.43 z Q*= µ + (z*) σ=10+(0.43) 5 = 12.15

16 Service Levels Shortage Penalty P(D Q*) = 1 - F(Q*) = c O /(c U + c O ) Example: Fill Rate E[min{D,Q*}]/E[D] Example: 89% (from tables or simulation)

17 Extensions Initial Inventory I Order Q* - I if I Q*, 0 otherwise Q* is called the Base Stock and represents the target inventory level Discrete demand Order quantity: Round Up Q* Multiple periods Fixed cost Many applications in Supply Contracts

18 Outline Newsboy 1-period Random demand (Stochastic) Shortages allowed Variable costs only No Lead Time EOQ Multiple periods Known demand (Deterministic) Constant Demand No Shortages Fixed and variable order costs No Lead Time

19 EOQ Example Number 2 pencils at the campus bookstore are sold at a fairly steady rate of 60 per week. The pencils cost the bookstore 2 cents each and sell for 15 cents each. It costs $3 to initiate an order, and holding costs are based on an annual interest rate of 25 percent. Determine the optimal number of pencils for the bookstore to purchase and the time between placement of orders. Example from Nahmias, Production and Operations Analysis

20 Intuition Trade-Off: Spread the fixed ordering cost over many items Avoid high inventory costs Replenishment from An outside vendor Internal production

21 Application Steady Demand / Large Fixed Cost Industries Manufacturing: Automobile, Electrical Appliances, Chemical Products (Lot Sizes) Retail: Slow-moving items (pencils, bathroom tissue )

22 EOQ Notations EOQ = Economic Order Quantity Constant Demand Rate: λ Fixed order cost: K Variable order cost: c Inventory holding cost: h Interest rate: i Order quantity: Q Time between orders: T

23 Evolution of Inventory Inventory position Q time T Order when inventory position reaches zero Order the same amount each time

24 Cost components (1) Inventory holding cost h = i * c (cost of capital) Over a replenishment cycle: Start from Q Ends at 0 Decreases steadily Average inventory = Q/2 Average inventory cost = h Q/2

25 Cost components (2) Per replenishment cycle: Fixed cost: K Variable cost: c Q Length of a cycle: Order size: Q units Demand rate: λ units/year Time between orders T = Q/λ Average order cost = 1/T (K + cq) = K λ/q + c λ

26 Min h Q/2 + K λ/q + c λ First Order Conditions: h/2 - K λ/q 2 = 0 Second Order Conditions: 2 K λ/q 3 0 Hence, order Q*= 2Kλ h

27 Optimization Optimal Cost: Inventory Cost: h Q*/2 = Fixed Order Cost: Kλ/Q*= 2Kλh 2Kλh Total Cost=c λ + 2 2Kλh

28 Graphical View Q inventory fixed cost total cost cost

29 Example λ = 60 units/week = 3,120 units/year K= $3, c =$0.02, h=i c=(.25) (.02) = $0.005/(unit)/(year) 2Kλ (2)(3)(3,120) Q*= = = 1935 units h T=Q/λ=1,935/3,120=0.62 years =32 weeks Work in the same units!

30 Observations Very robust Can round up or down with loosing much Independent of selling price Dependent of purchase cost only through holding cost.

31 Extensions Lead-time L same ordering quantity Order L periods in advance, when stock reaches L/λ. Finite production rates Quantity discounts Supply Chain Application: Determine the lot sizes of all stages in the supply chain (global view).

32 Summary Newsboy 1-period Random demand (Stochastic) Shortages allowed Variable costs only No Lead Time F( Q*) = c U cu + c O EOQ Multiple periods Known demand (Deterministic) Constant Demand No Shortages Fixed and variable order costs No Lead Time Q* = 2Kλ h

33 Newsboy Example (1) The buyer for Needless Markup, a famous high end department store, must decide on the quantity of a highpriced 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 $ Any handbags not sold by the end of the season are purchased by a discount firm for $ 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. Example from Nahmias, Production and Operations Analysis

34 Newsboy Example (2) 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? c U = ( ) = $ (lost margin) c O = (28.50 (1.4) ) = $19.90 (purchase cost + inventory holding cost salvage value) Critical Fractile = c U /(c U + c O ) =.859 Demand is Uniform between 50 and 250 Q*= 50 +(250-50) *(.859) =222 units

35 EOQ Example (1) The Rahway, New Jersey, plant of Metalcase, a manufacturer of office furniture, produce metal desks at a rate of 200 per month. Each desk requires 40 Phillips head metal screws purchased from a supplier in North Carolina. The screws cost 3 cents each. Fixed delivery charges and costs of receiving and storing shipments of the screws amount to about $100 per shipment, independent of the size of the shipment. The firm uses a 25 percent interest rate to determine holding costs. Metalcase would like to establish a standing order with the supplier and is considering several alternatives. What standing order size should they use? Example from Nahmias, Production and Operations Analysis

36 EOQ Example (2) λ = (200)(40)(12)=96,000 units/year K=$100, h=(.25)(0.03)= Kλ (2)(100)(96,000) Q* = = = h ,597 Cycle time T = Q/ λ =.53 year