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					Inventory Management
     Operations Management
      Dr. Mark P. Van Oyen

“You’ve got to WIP it, WIP it good!” Devo
         file: inven-lec.ppt                1
Types of Inventory
   1. Raw materials and purchased parts.
   3. Finished goods.
   – Supplier
   – Distributor
   – Retailer
 4. Replacement parts, tools, and supplies.
We are NOT talking about: Work in process (WIP).
    – Partially completed goods.
    – Goldratt warns about using “Economic Batch Quantity”
      for shop floor lot-sizing… which is our EOQ as you’ll
Types of Inventory Demand
 Independent    demand items have demands which
    are not linked to the demand for other products.
    (Inventory focus)
    – Finished goods or end items (Saturns, laser pointers,
    – These demands are typically forecasted.
We are NOT talking about
   Dependent demand items = Subassemblies or
    component parts.
    – Derived from the demand of the independent demand items
      (finished goods)
    – Often dealt with using PUSH (MRP II, ERP) or PULL
      (Kanban, CONWIP, Bucket Brigades, etc.)
                        Independent demand (Inventory).
                           Dependent demand (MRP)

                                     Independent Demand is usually
                                     for end products (forecasted, uncertain)

                    X                    Dependent Demand (MRP can
                                         back-figure release times given
                                         due dates for independent demands)
         B(2)              C(1)
  D(3)        E(1) E(2)           F(2)
                                                         AA              Dependent

Figure 14-6 of Stevenson
                                                d              e

                                           f1       f2    g1       g2

    Functions of Inventory
   To meet anticipated demand.
   To protect against stockouts. (To meet unanticipated demand)
   To hedge against price increases, or to take advantage of quantity
   To smooth production requirements.
     – Level aggregate production plan is an example.
   To take advantage of order cycles.
     – Efficiency in fixed costs of ordering or producing (e.g. share
        delivery truck)
   To de-couple operations.
     – WIP inventory between successive manufacturing steps or
        supply chain echelons (buffer stock) permits operations to
        continue during periods of breakdowns/strikes/storms.
   Despite “Zero Inventory” zealots, an appropriate amount of
    inventory is a good thing in almost all systems!
Tradeoffs in the Size of
    Inventories that are too high are expensive to carry, and
     they tie up capital.
    Warehousing, transportation
    Greater risk of defects, even when carefully inventoried
    Market shifts leave seller with many unwanted parts

    Inventories that are too low can result in
    reduced operational efficiency (machine starvation)
    poor service (delayed service, product substitution)
    lost sales (customer refuses raincheck, finds a more
     reliable supplier)
Objectives/Controls of Inventory
     Achieve satisfactory levels of customer service while
      keeping inventory costs under control.
       – Fill rate (probability of meeting a demand from
       – Inventory turnover ratio = Annual Sales / Average
         Inventory Level = D / WIP
       – Others not focused one here: Average Backlog,
         Mean time to fill order (Cycle Time)
     Profit maximization by achieving a balance in stocking,
      avoiding both over-stocking and under-stocking.
   Controls:
      – Decide how much to order/produce (Q)
      – Decide when to order/produce (ROP)
The Inventory Cycle

               Average                Profile of Inventory Level Over Time
               Usage                Actual
Quan             rate               Usage
on hand


     Receive      Place   Receive        Place   Receive
     order        order   order          order   order

                    Lead time
   Holding costs or carrying costs (applied per unit of inventory)
     – Associated with keeping inventory for a period of time.
     – Capital tied up
     – Warehousing and transportation
     – Perishable products -OR- expected risk of damage to product
     – Note that easiest calculation is (Holding cost rate/unit/yr) * (Average
       annual inventory level)
   Ordering OR production costs (Fixed order set up + variable unit costs)
     – For ordering and receiving inventory if ordering from a supplier,
       (Delivery charges, postage & handling, cost of purchasing dept.
     – OR production costs if we are the manufacturer .
   Shortage costs. (used in Newsvendor Model with demand uncertainty)
     – Demand exceeds supply of inventory and a stockout occurs.
     – Opportunity costs (lost sales and LOST CUSTOMERS!)
     – Loss of good will, or, may need to substitute higher-cost item!
The Inventory Management
   Determine Inventory Policy:
    How much to order or make? (Q)
    When to order or make? (Reorder point)
    How much to store in safety stock?
   To:
    Minimize the cost of the inventory system

How Much to Order: Economic Order
Quantity (EOQ) Models
   Objective:
    Identify the optimal order quantity that minimizes the
    sum of certain annual costs that vary with order size.
   Assumption: Fixed order quantity systems (don’t
    change order size dynamically over time)
   We will consider 3 models:
     – I. EOQ: all items delivered as a batch
     – II. EOQ with gradual deliveries
     – III. EOQ with quantity discounts (uses Model I

Model I: (Basic) EOQ
     “How Many Parts to Make at Once” by Ford Harris
       – Original 1913 version of this model.
     Very simple - makes a lot of restrictive assumptions.
      It’s not realistic at first glance, but it is! Its usefulness
      keeps it alive!
     Nicely illustrates basic tradeoffs that exist in any
      inventory management problem.

     Basic Model Scenario:
     Own a warehouse from which parts are demanded by
     Periodically run out of parts and have to replenish
      inventory by ordering from suppliers.               12
EOQ Model I - Assumptions
   Demand, D, is known and constant.
   Purchase Price, P, is known and constant.
   Fixed setup cost per order, S, independent of Q
   Holding cost C=H per unit per year.
   Lots of size Q are delivered in full (Production
    Perspective: produce items and hold them in FGI
    until production is completed, at which time we
    ship them out).
   There will be no stockouts, no backorders, no
   Lead Time is known and constant.
Basic EOQ Model Assumptions

  D = Demand rate is known and constant. (units/yr)
       [e.g. Sell brake pads with D = 60 pads/wk * 52]
  P = Unit production/purchase cost - not counting
   setup or inventory costs ($/unit) [e.g. 2 $/pad]
  S = Order set-up cost, constant per order ($/order)
               [e.g. 28.85 $/order]
  C = H = Average annual carrying cost per unit
   ($/unit/yr.) [25% of purchase price/year = 0.50 $/yr]
  Q = order quantity = decision variable
       [e.g. ?? Brake pads]
Inventory Cycle for Basic EOQ

 Q = 600              D = 60*52 pads/yr


                                      Q/D = Length
             10 wk.
                                      of Order Cycle
Total Annual Cost Calculation
  Order Frequency = F = D/Q = reciprocal of period
         e.g. (60 pads/wk)* 52 /600 pads = 52/10.
  Annual ordering cost is (# orders/year) * (order
   setup cost) + (unit purch. cost) * (annual demand)
   = (D/Q)S + PD
  Average inventory per order cycle: (Q+0)/2 = Q/2.
   why? use geometry to get average inv. Level (vs. calc.)
  Annual carrying cost is (Q/2) C.
  Total annual Stocking Cost (TSC)
     = [annual carrying cost + annual ordering cost]:
     TSC = (D/Q)S + PD + (Q/2) C
 Cost Minimization Goal

                   The Total-Cost Curve is U-Shaped
     Annual Cost

                                                           Ordering Costs

                          QO (optimal order quantity) Order Quantity
Economic Order Quantity (EOQ)
  There  is a tradeoff between carrying costs
   and ordering costs!
  EOQ is the value of Q that minimizes TSC.
  In this sense, the Q determined by the EOQ
   formula is OPTIMAL given a simplistic
  FYI only: EOQ is found by either:
   – Using calculus and solving (d/dQ)TSC = 0, the points of graph with
     zero slope are either local maxima & minima, or,
   – Observing that the EOQ occurs where carrying and ordering costs are
     equal, i.e., by solving (Q/2) C = (D/Q)S. [this is not a general method!]
EOQ Model I- Development
It turns out that
  Costs are minimized where ordering and
  carrying costs are equal.

 D  Q     2 DS
S C Q 
 Q  2       C
EOQ Model I- Example
Demand = 60 pads/wk
                          2 DS
Ordering Cost, S =
Unit Carrying Cost =
  25% of purchase         2(28.85)(60)(52)
  price/yr.             
Unit purchase price,
  P = $2.00/pad          600 pads
Model II: EOQ for Gradual
 Inbasic EOQ model I, orders are assumed to be
  delivered instantaneously & all at once.
 Sometimes inventories build up gradually:
  Gradual Deliveries.
   – when the firm is both a producer and user.
   – Pipelines (think of water slowly filling a bathtub when
     the faucet is open all the way but the drain is open)
   – Many small shipments made as the products become
     available (multiple Amazon shipments for one order
     consisting of 12 books; The Goal illustration – getting
     the customer to take multiple smaller lots)
EOQ Model II - Assumptions
  Orders  are delivered (or produced) gradually at a
   rate of p units per day, where we choose “day”
   = convenient time unit as an example
  Demand (annual) is known and has a constant
   rate. d = usage rate (units/day). Units on d and
   p must match!
  Simplify Life: Convert time unit on p to
   “years”, then d = D = units/year demand rate.
    Lead Time is known and constant.
    Purchase Price is known and constant.

 Model II: EOQ for Gradual
Let “TIME” be a convenient time period (day,
  week, month, year)
 p = production delivery rate (units/TIME).
  E.g. = 150 pads/wk.
 d = usage rate (units/ TIME).
  – d is just a conversion of D in units/year to units/
Illustration of tip: TIME = year, so p = 150*52
   pads/Year.,       d= D.
 Model II: EOQ for Gradual
 Delivery   period = Q/p = (TIME to deliver all Q
 products ordered) = 600/(150*52) = 4/52 years
 Maximum    Inventory level reached =
 Imax = (flow rate in - flow rate out) * (time Q)
      = (p-d)(Q/p) in units of product
      = (150 - 60) (52 wks/yr) ( 4 /52 years)
                p         d

                    Q/p                          24
  Inventory Cycle for EOQ for Gradual
                    p                  Cycle Time = Q/d
                  rate)         Imax
Inventory (buildup
                                          d (usage rate)
  Level   rate)

         Buildup Period = Q/p
EOQ for Gradual Deliveries

TSC   = (Imax/2)C + (D/Q)S + PD =
(Q/2) [(p-d)/p] C + (D/Q)S + PD.

                 2DS             p
 EOQ =                              .
                  C             p-d
EOQ Model II- Example
Demand = 60 pads/wk
Ordering Cost =
                         2 DS     p
  $28.85/order     Q
Unit Carrying Cost =       C     pd
  25% of purchase
                         2(60 * 52)(28.85)     150
  price/yr.           
Unit purchase price =         .25($2)        150  60
  $2.00/pad            775 pads
production rate =
  150 pads/wk
 EOQ Model II- Example Picture

                       Imax = 465
  Level     90/wk                   demand of 60/wk

      5.17 wks = Q/p                   7.75 wks 28
The Concept of Reorder Point (ROP)
  What  inventory level should trigger/ control the
   placement of an order?
  ROP = ReOrder Point = DDLT + SS
               Demand-During-Lead-Time + Safety-Stock
  If   Lead Time is 3 wks., ROP = (60)(3)
                             = 180 and inven. Falling

  If   Lead Time is 10 wks., ROP = (90)(2.92)
                      = 263 and inven. increasing! (this
   is a tricky case because of the very long lead time which
   requires a reorder while the inventory buildup is still in
   progress – it should be a rare case)                       29
III. EOQ with All Units at Quantity Discount

 Purchase   Price is known but varies with the
    amount purchased. The price is the same for all
    units. Also, holding cost may be fraction of price!
   Demand and Lead time both known and constant.
   Lots are delivered in FULL as in Model I.
The problem is solved as a series of problems - one
  for each price break.
(1) solve for EOQ for each possible price and modify
  the Q value to be feasible at that price, then
(2) compute resulting TSC and finally
(3) search for the lowest cost.                     30
Model III: Basic EOQ with Quantity
Discounts (All Units)
  Quantity  discounts offered by the supplier to
   the buyer occur when the unit purchase price
   of the product (actual cost = ac) decreases
   as the quantity purchased (Q) increases.
  Assume orders are received all at once
   (Model I).
  Total Cost = (Q/2)C + (D/Q)S + (D)(ac).
   – Often: C = some percent of (ac) .
  Find   Q = EOQ that minimizes total cost.
Example from G&F 10.3 (p. 387-389)
D  = 10,000, S = 5.50, C = 0.2 (ac).
 Price Schedule:
     Range of Q                    (ac)                C
         1-399                    2.20               0.44
        400-699                   2.00               0.40
         700+                     1.80               0.36

 Assume  orders are received all at once (Model I).
 Note: “all-units” quantity discount.
   – E.g. at Q = 524, ac=2.00 applies to all Q units (not just 400-
     524!)                                                        32
Computing EOQ for Ex.10.3 G&F
EOQ      = 2DS = 2(5.5)10,000 = 500.0 (not feasible - higher!).
    2.20    C        0.44
EOQ      = 2DS = 2(5.5)10,000 = 524.4 (feasible).
    2.00    C        0.40
EOQ      = 2DS = 2(5.5)10,000 = 552.8 (not feasible - too low!).
   1.80     C        0.36
    ac = 2.20 yields optimal EOQ at a level that deserves a better
     price/volume, so this is the one situation in which we can
     disqualify the possibility of this range containing our answer!
    ac = 2.00 yields EOQ = 524.4 which is feasible.
    ac = 1.80 yields EOQ = 552.8 which is NOT feasible. We then
     take the closest quantity that will give us that price range:
      – Q = 700 for ac = 1.80.
 Next step: cost these out - answer has to be one of them!
Computing EOQ for Example 10.3
   TC(Q) = (Q/2)C + (D/Q)S + (D)ac.
  TC(Q  = 524.4) = (524.4/2)(0.40) +
   (10,000/524.4)(5.5) + 10,000(2.00) =
  TC(Q = 700) = (700/2)(0.36) +
   (10,000/700)(5.5) + 10,000(1.8) =
   18,204.57 (min).
  Conclusion: EOQ = 700. (even though the
   EOQ solution was not feasible at that price!)
Inventory Management Under
 Demand      or Lead Time or both uncertain
 Even “good” managers are likely to run out once
  in a while (a firm must start by choosing a service
  level/fill rate)
 When can you run out?
   – Only during the Lead Time if you monitor the
 Solution: build a standard ROP system based on
  the probability distribution on demand during the
  lead time (DDLT), which is a r.v. (collecting
  statistics on lead times is a good starting point!)
The Typical ROP System

                                        Average Demand

ROP   set as demand that accumulates during
       lead time

ROP = ReOrder Point                           Lead Time
The Self-Correcting Effect- A
Benign Demand Rate after ROP
         Hypothetical Demand

                          Average Demand


           Lead Time           Lead Time
What if Demand is “brisk” after
hitting the ROP?
            Hypothetical Demand

                           Average Demand

Stock                             Lead Time
When to Order
  The  basic EOQ models address how much
   to order: Q
  Now, we address when to order.
  Re-Order point (ROP) occurs when the
   inventory level drops to a predetermined
   amount, which includes expected demand
   during lead time (EDDLT) and a safety
   stock (SS):
      ROP = EDDLT + SS.
When to Order

 SS is additional inventory carried to reduce
 the risk of a stockout during the lead time
 interval (think of it as slush fund that we dip into when
 demand after ROP (DDLT) is more brisk than average)
 ROP    depends on:
  –   Demand rate (forecast based).          DDLT,
  –   Length of the lead time.               EDDLT &
  –   Demand and lead time variability. Std. Dev.
  –   Degree of stockout risk acceptable to
      management (fill rate, order cycle Service Level) 40
The Order Cycle Service Level,(SL)

 The  percent of the demand during the lead time
  (% of DDLT) the firm wishes to satisfy. This
  is a probability.
 This is not the same as the annual service
  level, since that averages over all time periods
  and will be a larger number than SL.
 SL should not be 100% for most firms.
  (90%? 95%? 98%?)
 SL rises with the Safety Stock to a point.
                 Safety Stock

                 Maximum probable demand during
                 lead time (in excess of EDDLT)
                 defines SS
                                  Expected demand
                                  during lead time

                               Safety stock (SS)
                    LT                       Time
Variability in DDLT and SS

 Variabilityin demand during lead time (DDLT)
  means that stockouts can occur.
   – Variations in demand rates can result in a temporary
     surge in demand, which can drain inventory more
     quickly than expected.
   – Variations in delivery times can lengthen the time a
     given supply must cover.
 We  will emphasize Normal (continuous)
  distributions to model variable DDLT, but
  discrete distributions are common as well.
 SS buffers against stockout during lead time.
Service Level and Stockout Risk

   Targetservice level (SL) determines how
    much SS should be held.
    – Remember, holding stock costs money.
   SL  = probability that demand will not
    exceed supply during lead time (i.e. there
    is no stockout then).
   Service level + stockout risk = 100%.

Computing SS from SL for Normal
 Example   10.5 on p. 374 of Gaither &
 DDLT is normally distributed a mean of
  693. and a standard deviation of 139.:
  – EDDLT = 693.
  – s.d. (std dev) of DDLT =  = 139..
  – As computational aid, we need to relate this to
    Z = standard Normal with mean=0, s.d. = 1
     » Z = (DDLT - EDDLT) / 
Reorder Point (ROP)

     Service level
                               Risk of
                             a stockout
    Probability of
     no stockout

                     ROP      Quantity
  demand        Safety
          0              z     z-scale

 Area under standard Normal pdf from -  to +z

Z = standard Normal with mean=0, s.d. = 1
Z = (X -  ) /                               z     P(Z  z)
See G&F Appendix A                           0         .5
See Stevenson, second from last page
                                            . 67      .75
                                             .84      .80
           P(Z <z)                          1.28      .90
           Standard                         1.645     .95
                                             2.0      .98
                                            2.33      .99
                                             3.5     .9998
                0          z    z-scale
Computing SS from SL for Normal DDLT
to provide SL = 95%.
   ROP       = EDDLT + SS
              = EDDLT + z ().
       z is the number of standard deviations SS is set above
      EDDLT, which is the mean of DDLT.
     z is read from Appendix B Table B2. Of Stevenson -
      OR- Appendix A (p. 768) of Gaither & Frazier:
      – Locate .95 (area to the left of ROP) inside the table (or as close
        as you can get), and read off the z value from the margins: z =
  Example: ROP = 693 + 1.64(139) = 921
  SS = ROP - EDDLT = 921 - 693. = 1.64(139) = 228
   If we double the s.d. to about 278, SS would double!
   Lead time variability reduction can same a lot of
    inventory and $ (perhaps more than lead time itself!)
           Summary View
             Holding Cost = C[ Q/2 + SS]
             (1) Order trigger by crossing ROP
             (2) Order quantity up to (SS + Q)
Q+SS =
                              Not full due to brisk
Target                        Demand after trigger

Stock                                  Lead Time
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!):
   Cs = Revenue per unit - Cost per unit.
 Excess (Long) cost applies to the items left over
  at end of the period, which need salvaging
   Ce = Original cost per unit - Salvage value per
   (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 Cs is shortage cost/unit, and Ce is excess cost/unit,
               SL 
                    Cs  Ce                            52
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
 Cs = 60 cents per liter
 Ce = 20 cents per liter.
 SL = Cs/(Cs + Ce) = 60/(60 + 20) = 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 Cs to reflect the actual cost of lost
    customer due to stockout.
Single-Period Model for Continuous
    demand is Normal(200 liters per week, variance =
     10,000 liters2/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 0.749
     – Optimal stocking level = mean + z () = 200
       + (0.67)(100) = 267. liters.

Single-Period & Discrete Demand: Lively
   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?
         Cs = 7.50 = lost profit
   (b) unit cost of having a left-over lobster?
         Ce = 14.50 - 8.50 = cost – salvage value = 6.
   (c) What should the L.L. service level be?
         SL = Cs/(Cs + Ce) = 7.5 / (7.5 + 6) = .56 (larger Cs leads to SL >
   Demand follows a discrete (relative frequency) distribution as given on next
  Lively Lobsters: SL = Cs/(Cs + Ce) =.56
                                                          Probability that demand
                                           Cumulative     will be less than or equal to x
     follows a                 Relative     Relative
     discrete                Frequency Frequency
     (relative       Demand     (pmf)         (cdf)
     distribution:     19        0.05              0.05          P(D < 19 )

                       20        0.05              0.10          P(D < 20 )

                       21        0.08              0.18          P(D < 21 )
Result: order 25
     Lobsters,         22        0.08              0.26          P(D < 22 )

     because that is   23        0.13              0.39          P(D < 23 )
     the smallest      24        0.14              0.53          P(D < 24 )
     amount that       25        0.10              0.63          P(D < 25 )
     will serve at
     least 56% of      26        0.12              0.75          P(D < 26 )

     the demand on     27        0.10     you do                 P(D < 27 )
     a given night.    28        0.10     you do                 P(D < 28 )

                       29        0.05              1.00          P(D < 29 )
                            * pmf = prob. mass function

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