INVENTORY MODELING Items in inventory in a store Items waiting to be shipped Employees in a firm Computer information in computer files Etc.

COMPONENTS OF AN INVENTORY POLICY Q = the amount to order (the order quantity) R = when to reorder (the reorder point)

BASIC CONCEPT Balance the cost of having goods in inventory to other costs such as: –Order Cost –Purchase Costs –Shortage Costs

HOLDING COSTS Costs of keeping goods in inventory –Cost of capital –Rent –Utilities –Insurance –Labor –Taxes –Shrinkage, Spoilage, Obsolescence

Holding Cost Rate Annual Holding Cost Per Unit These factors, individually are hard to determine Management (typically the CFO) assigns a holding cost rate, H, which is a percentage of the value of the item, C Annual Holding Cost Per Unit, C h C h = HC (in $/item in inv./year)

PROCUREMENT COSTS When purchasing items, this cost is known as the order cost, C O (in $/order) These are costs associated with the ordering process that are independent of the size of the order-- invoice writing or checking, phone calls, etc. – Labor –Communication –Some transportation

PROCUREMENT COSTS When these costs are associated with producing items for sale they are called set- up costs (still labeled C O -- in $/setup) Costs associated with getting the process ready for production (regardless of the production quantity) –Readying machines –Calling in, shifting workers –Paperwork, communications involved

PURCHASE/PRODUCTION COSTS These are the per unit purchase costs, C, if we are ordering the items from a supplier These are the per unit production costs, C, if we are producing the items for sale

CUSTOMER SATISFACTION COSTS Shortage/Goodwill Costs associated with being out of stock –goodwill –loss of future sales –labor/communication Fixed administrative costs =  ($/occurrence) Annualized Customer Waiting Costs = C s ($/item short/year)

BASIC INVENTORY EQUATION (Total Annual Inventory Costs) = (Total Annual Order/Setup-Up Costs) + (Total Annual Holding Costs) + (Total Annual Purchase/Production Costs) + (Total Annual Shortage/Goodwill Costs) This is a quantity we wish to minimize!!

REVIEW SYSTEMS Continuous Review -- –Items are monitored continuously –When inventory reaches some critical level, R, an order is placed for additional items Periodic Review -- –Ordering is done periodically (every day, week, 2 weeks, etc.) –Inventory is checked just prior to ordering to determine an order quantity

TIME HORIZONS Infinite Time Horizon –Assumes the process has and will continue “forever” Single Period Models –Ordering for a one-time occurence

EOQ-TYPE MODELS EOQ (Economic Order Quantity-type models assume: Infinite Time Horizon Continuous Review Demand is relatively constant

THE BASIC EOQ MODEL Order the same amount, Q, each time Reordering is instantaneous Demand is relatively constant at D items/yr. Infinite Time Horizon/Continuous Review No shortages –Since reordering is instantaneous

Economic Order Quantity On-hand inventory (units) Time Average cycle inventory QQ—2QQ—2 1 cycle Receive order Inventory depletion (demand rate) Figure 13.2

THE EOQ COST COMPONENTS Total Annual Order Costs: –(Cost/order)(average # orders per year) = C O (D/Q) Total Annual Holding Costs: –(Cost Per Item in inv./yr.)(Average inv.) = C h (Q/2) Total Annual Purchase Costs: –(Cost Per Item)(Average # items ordered/yr.) = CD

Economic Order Quantity Annual cost (dollars) Lot Size (Q) Ordering cost Holding cost Total cost = Holding Cost + Set-up Cost Figure 13.3

THE EOQ TOTAL COST EQUATION TC(Q) = C O (D/Q) + C h (Q/2) + CD This a function in one unknown (Q) that we wish to minimize

SOLVING FOR Q* TC(Q) = C O (D/Q) + C h (Q/2) + CD

THE REORDER POINT, r* Since reordering is instantaneous, r* = 0 MODIFICATION -- fixed lead time = L yrs. r* = LD But demand was only approximately constant so we may wish to carry some safety stock (SS) to lessen the likelihood of running out of stock Then,r* = LD + SS

TOTAL ANNUAL COST The optimal policy is to order Q* when supply reaches r* TC(Q*) = C O D/Q* + (C h /2)(Q*) + CD + C h SS fixed safety cost stock cost The optimal policy minimizes the total variable cost, hence the total annual cost

TOTAL VARIABLE COST CURVE Ignoring fixed costs and safety stock costs:

EXAMPLE -- ALLEN APPLIANCE COMPANY Juicer Sales For Past 10 weeks Using 10-period moving average method, D = ( …+ 130)/10 = 120/ wk = 6240/yr

ALLEN APPLIANCE COSTS Juicers cost $10 each and sell for $11.85 Cost of money = 10% Other misc. costs associated with inventory = 4% Labor, postage, telephone charges/order = $8 Workers paid $12/hr min. to unload an order H = =.14; C h =.14(10) = $1.40 C O = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12

OPTIMAL ORDER QUANTITY FOR ALLEN

OPTIMAL QUANTITIES Total Order Cost = C O D/Q* = (12)(6240)/327 = $ Total Holding Cost = (C h /2)Q* = (1.40/2)(327) =$ –(Total Order Cost = Total Holding Cost -- except for roundoff) # Orders Per Year = D/Q* = 6240/327 = Time between orders (Cycle Time) = Q*/D = 327/6240 =.0524 years = 2.72 weeks

TOTAL ANNUAL COST Total Variable Cost = Total Order Cost + Total Holding Cost = $ $ = $ Total Fixed Cost = CD = 10(6240) = $62,400 Total Annual Cost = $ $62,400 = $62,857.89

WHY IS EOQ MODEL IMPORTANT? No real-life model really is an EOQ model Many models are variants of EOQ-type models Many situations can be approximated by EOQ models The EOQ model is relatively insensitive to some pretty major errors in input parameters

INSENSIVITY IN EOQ MODELS We cannot affect fixed costs, only variable costs TV(Q) = C O D/Q + (C h /2)(Q) Now, suppose D really = 7500 (>20% error) We did not know this and got Q* = 327 TV(327) = ((12)(7500))/327 + (1.40/2)(327) =$ Q* should have been: SQRT(2(12)(7500)/1.40) = 359 TV(359) = ((12)(7500))/359 + (1.40/2)(359) =$ This is only a 0.4% increase in the TVCost

DETERMINING A REORDER POINT, r* (Without Safety Stock) Suppose lead time is 8 working days The company operates 260 days per year r* = LD where L and D are in the same time units L = 8/260 .0308 yrs D = 6240 /year r* =.0308(6240)  192 OR, L = 8 days; D/day = 6240/260 = 24 r* = 8(24) = 192

ACTUAL DEMAND DISTRIBUTION Suppose we can assume that demand follows a normal distribution –This can be checked by a “goodness of fit” test From our data, over the course of a week, W, we can approximate  W by (105 + … + 130) = 120  W 2  s W 2 = (( … ) - 10(120) 2 )/9  83.33

DEMAND DISTRIBUTION DURING 8 -DAY LEAD TIME Normal 8 days = 8/5 = 1.6 weeks, so  L = (1.6)(120) = 192  L 2  (1.6)(83.33) =  L 

SAFETY STOCK Suppose we wish a cycle service level of 99% –WE wish NOT to run out of stock in 99% of our inventory cycles Reorder point, r* =  L + z.01  L = (11.55)  = 27 units = safety stock = 2.33(11.55) Safety stock cost = C h SS = 1.40(27) = $37.80 –This should be added to the TOTAL ANNUAL COST

OTHER EOQ-TYPE MODELS Quantity Discount Models Production Lot Size Models Planned Shortage Model ALL SEEK TO MINIMIZE THE TOTAL ANNUAL COST EQUATION

QUANTITY DISCOUNTS All-units vs. incremental discounts ALL UNITS DISCOUNTS FOR ALLEN QuantityUnit Cost < 300 $ $ $ $ 9.40  5000 $ 9.00

PIECEWISE APPROACH For each piece of the total cost equation, the minimum cost for the piece is at an end point or at its Q* If Q* for a piece lies: –above the upper interval limit -- ignore this piece –within this piece -- it is optimal for this piece –below the lower interval limit -- the lower interval limit is optimal for this piece Calculate the total annual cost using the best value for Q for each piece, and choose the lowest

QUANTITY DISCOUNT APPROACH FOR ALLEN When C changes, only C h changes in the formula for Q* since C h =.14C QuantityUnit Cost C h Q* Best Q TC < 300 $10.00 $ $ 9.75 $ $61, $ 9.50 $ $59, $ 9.40 $ $59,389  5000 $ 9.00 $ $59,325 ORDER 5000

OTHER CONSIDERATIONS 5000 is 5000/6240 =.8 years = 9.6 months supply –May not wish to order that amount Company policy may be: DO NOT ORDER MORE THAN A 3-MONTHS SUPPLY = 6240/4 = 1560 If that is the case, since 1560 is in the interval from and the best Q in that interval is 1000, 1000 should be ordered

PRODUCTION LOT SIZE PROBLEMS We are producing at a rate P/yr. That is greater than the demand rate of D/yr. Inventory does not “jump” to Q but builds up to a value I MAX that is reached when production is ceased Length of a production time = Q/P I MAX = P(Q/P) - D(Q/P) = (1-D/P)Q Average inventory = I MAX /2 = ((1-D/P)/2)Q

Economic Production Quantity On-hand Inventory Time Figure G.1

Economic Production Quantity Production quantity On-hand Inventory Q Time Figure G.1

Economic Production Quantity Production quantity On-hand Inventory Q Time Figure G.1 Demand during production interval p - d

Economic Production Quantity Production quantity Demand during production interval On-hand Inventory Q Time p - d Figure G.1

Economic Production Quantity Production quantity Demand during production interval On-hand Inventory Q Time p - d Figure G.1 Production and demand Demand only TBO

Economic Production Quantity Production quantity Demand during production interval Production and demand Demand only TBO On-hand Inventory Q Time p - d Figure G.1

Economic Production Quantity Production quantity Demand during production interval Maximum inventory Production and demand Demand only TBO On-hand Inventory Q Time I max p - d Figure G.1

PRODUCTION LOT SIZE -- TOTAL ANNUAL COST C O = Set-up cost rather than order cost Set-up time for production  lead time Q = The production lot size TC(Q) = C O (D/Q) + C h ((1-D/P)/2)Q + CD

OPTIMAL PRODUCTION LOT SIZE, Q* TC(Q) = C O (D/Q) + C h ((1-D/P)/2)Q + CD

EXAMPLE-- Farah Cosmetics Production Capacity 1000 tubes/hr. Daily Demand 1680 tubes Production cost $0.50/tube (C = 0.50) Set-up cost $150 per set-up (C O = 150) Holding Cost rate: 40% (C h =.4(.50) =.20) D = 1680(365) = 613,200 P = 1000(24)(365) = 8,760,000

OPTIMAL PRODUCTION LOT SIZE

TOTAL ANNUAL COST TOTAL ANNUAL COST = TV(Q) = C O (D/Q) + C h ((1-D/P)/2)Q = (150)(613,200/31,449) +.2(1-613,200/8,760,00)(31,449) = $5,850 TC(Q) = TV(Q) + CD = 5, (613,200) = $312,450

OTHER QUANTITES Length of a Production run = Q*/P = 31,449/8,760,000 =.00359yrs. =.00359(365) = 1.31 days Length of a Production cycle = Q*/D = 31,449/613,200 = yrs. = (365) = days Number of Production runs/yr. = D/Q* = 19.5 I MAX = (1-613,200/8,760,00)(31,449) = 29,248

PLANNED SHORTAGE MODEL Assumes no customers will be lost because of stockouts Stockout costs: –  -- fixed administrative cost/stockout –C s -- annualized cost per unit short Acts like a holding cost in reverse We plan on being short by S items when an order of size Q comes in

PROPORTION OF TIME OUT OF STOCK T 1 = time of a cycle with inventory T 2 = time of a cycle out of stock T = T 1 + T 2 = time of a cycle I MAX = Q-S Proportion of time in stock = T 1 /T = (Q-S)/Q Proportion of time out of stock = T 2 /T = S/Q Avg. inventory = ((Q-S)/Q)((Q-S)/2) = (Q-S) 2 /2Q Average Stockouts = (S/Q)(S/2) = S 2 /2Q

TOTAL ANNUAL COST EQUATION TC(Q,S) = C O (D/Q) + C h ((Q-S) 2 /2Q) +  S(D/Q) + C s (S 2 /2Q) + CD Take partial derivatives with respect to Q and S and set = 0. We get two equations in the two unknowns Q and S.

OPTIMAL ORDER QUANTITY, Q* OPTIMAL # BACKORDERS, S*

EXAMPLE SCANLON PLUMBING Saunas cost $2400 each (C = 2400) Order cost = $1250(C O = 1250) Holding Cost = $525/unit /yr.(C h = 525) Backorder Good will Cost $20/wk (C S = 1040) Backorder Admin. Cost = 10/order(  = 10) Demand = 15/wk(D=780)

RESULTS

What IF Lead Time Were 4 Weeks? Demand over 4 weeks = 4(15) = 60 Want order to arrive when there are 20 backorders. Thus order should be placed when there are = 40 saunas left in inventory

Part III: Single-Period Model: Newsvendor Used to order perishables or other items with limited useful lives. –Fruits and vegetables, Seafood, Cut flowers. –Blood (certain blood products in a blood bank) –Newspapers, magazines, … Unsold or unused goods are not typically carried over from one period to the next; rather they are salvaged or disposed of. Model can be used to allocate time-perishable service capacity. Two costs: shortage (short) and excess (long).

Single-Period Model Shortage or stockout cost may be a charge for loss of customer goodwill, or the opportunity cost of lost sales (or customer!): C s = Revenue per unit - Cost per unit. Excess (Long) cost applies to the items left over at end of the period, which need salvaging C e = Original cost per unit - Salvage value per unit. (insert smoke, mirrors, and the magic of Leibnitz’s Rule here…)

The Single-Period Model: Newsvendor How do I know what service level is the best one, based upon my costs? Answer: Assuming my goal is to maximize profit (at least for the purposes of this analysis!) I should satisfy SL fraction of demand during the next period (DDLT) If C s is shortage cost/unit, and C e is excess cost/unit, then

Single-Period Model for Normally Distributed Demand Computing the optimal stocking level differs slightly depending on whether demand is continuous (e.g. normal) or discrete. We begin with continuous case. Suppose demand for apple cider at a downtown street stand varies continuously according to a normal distribution with a mean of 200 liters per week and a standard deviation of 100 liters per week: –Revenue per unit = $ 1 per liter –Cost per unit = $ 0.40 per liter –Salvage value = $ 0.20 per liter.

Single-Period Model for Normally Distributed Demand C s = 60 cents per liter C e = 20 cents per liter. SL = C s /(C s + C e ) = 60/( ) = 0.75 To maximize profit, we should stock enough product to satisfy 75% of the demand (on average!), while we intentionally plan NOT to serve 25% of the demand. The folks in marketing could get worried! If this is a business where stockouts lose long-term customers, then we must increase C s to reflect the actual cost of lost customer due to stockout.

Single-Period Model for Continuous Demand demand is Normal(200 liters per week, variance = 10,000 liters 2 /wk) … so  = 100 liters per week Continuous example continued: –75% of the area under the normal curve must be to the left of the stocking level. –Appendix shows a z of 0.67 corresponds to a “left area” of –Optimal stocking level = mean + z (  ) = (0.67)(100) = 267. liters.

Single-Period & Discrete Demand: Lively Lobsters Lively Lobsters (L.L.) receives a supply of fresh, live lobsters from Maine every day. Lively earns a profit of $7.50 for every lobster sold, but a day-old lobster is worth only $8.50. Each lobster costs L.L. $ (a) what is the unit cost of a L.L. stockout? C s = 7.50 = lost profit (b) unit cost of having a left-over lobster? Ce = = cost – salvage value = 6. (c) What should the L.L. service level be? SL = C s /(C s + C e ) = 7.5 / ( ) =.56 (larger C s leads to SL >.50) Demand follows a discrete (relative frequency) distribution as given on next page.

Lively Lobsters: SL = C s /(C s + C e ) =.56 Demand follows a discrete (relative frequency) distribution: Result: order 25 Lobsters, because that is the smallest amount that will serve at least 56% of the demand on a given night.