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									Operations Management

   Inventory Management

 Inventory Definitions
 Processes for Inventory Management
 Economic Order Quantity Inventory
 Reorder Point/Reorder Quantity Inventory
 Single Period Inventory Models
Inventory Definitions
          Inventory is a stock or store of
Demands determine the type of inventory               Independent Demand
an organization will carry.                           (e.g., customer demand
                                                      for finished products)

                             A                          Dependent Demand
                                                        (e.g., necessary parts
                                                        to assemble a
                                                            finished good)
               B(4)                     C(2)

        D(2)          E(1)       D(3)          F(2)

               Independent demand is uncertain
                 Dependent demand is certain
       Each firm carries types of inventories
       relevant to its production demands

 Raw materials & purchased parts
 Partially completed goods – called
  work in process (WIP)
 Finished-goods inventories
   manufacturing firms
 Merchandise
   retail stores
      Each firm carries types of inventories
      relevant to its production demands

 Replacement parts, tools, & supplies
 Goods-in-transit to warehouses or
       Inventory is carried for many different
       reasons across different industries

 To meet anticipated demand
   Meeting demand in a timely manner enhances
    customer satisfaction

 Smooth production requirements across seasons
   Produce one season, sell in next (or throughout year)

 To decouple successive operations and maintain
  continuity of production
   Protects against machine breakdowns

 To protect against stock-outs
   Vendors do not always deliver on time
       Inventory is carried for many different
       reasons across different industries

 To take advantage of order cycles to minimize
  purchasing and inventory costs
   Minimum order size requirements, full truck loads

 To help hedge against price increases
   Buy now at low price, store goods for future use

 To permit operations to operate
   Operations require a certain amount of WIP inventory

 To take advantage of quantity discounts
   Vendors often give discounts when ordering large
       Operations Strategy

 Having too much inventory is not good
   Tends to hide problems
   Makes it easier to live with (i.e. ignore) problems than
    to eliminate them
   Costly to maintain large stocks of inventories
   Opportunity costs of potentially doing something else
    with the money tied up in inventory

 Wise objectives
   Reduce lot sizes
   Reduce safety stock
   Reduce ordering costs and holding costs
      These are difficult to calculate, and often underestimated,
       leading to higher order sizes
      Objective of Inventory Control

 To achieve satisfactory levels of customer
  service while keeping inventory costs
  (costs of ordering and carrying inventory)
  within reasonable bounds
Processes for Inventory
       An effective inventory management
       approach will have certain information

 A system to keep track of inventory on hand and
  on order
 A reliable forecast of demand
 Knowledge of order lead times and lead time
 Reasonable estimates of
   Holding costs
   Ordering costs
   Shortage costs
 A classification system for inventory items
       Inventory Counting Systems

 Periodic System
   Physical count of items made at periodic intervals
      Walgreens (1987) – manager walked around weekly, ordered
       everything needed across whole store

 Perpetual Inventory System
   System that keeps track of removals from inventory
    continuously, thus monitoring
    current levels of each item
      2005 – grocery store scanners (bar codes)
      2005 – RFID-based systems
        Perpetual inventory counting systems
        range from low-tech to high-tech

 Two-Bin System - Two containers of inventory; reorder
  when the first bin is empty

 Universal Product Code (UPC) –
  Bar code printed on a label that   0
  has information about the item
  to which it is attached                214800 232087768

 Radio Frequency Identification (RFID) – Computer chip
  embedded in a label on side of package, cases, or
Radio Frequency Identification (RFID) used
in tags, chips, implants, and wristbands
RFID tags are activated by RFID
reader devices
                Example of RFID Use:
                Metro Future Store

                                      RFID Loyalty

RFID Portal for Receiving Inventory           RFID Enabled Tools for Counting Inventory

                            Information Kiosks and
                          Terminals to Find/Advertise
Labels for Dynamic Pricing    RFID Tagged Items       Portal for Automatic Checkout
         Example of RFID Use:
         Metro Future Store

                                        RFID-enabled Advertisements

                 RFID-based Inventory
Smart Checkout   Picking

                                          Store Manager’s
                                          Work Bench
       Lead time must be matched
       against expected demand
 Lead time: time interval between ordering
  and receiving the order

 If we expect that demand will occur on a
  certain day in the future, we will need to
  place an order several days earlier, and
  account for:
   Lead time
   Lead time variability
        Managers must estimate several
        types of inventory-related costs
 Holding (carrying) costs: cost to carry an item in
  inventory for a length of time, usually a year
    Annual cost of 20%-40% of value (unit price) of an

 Ordering costs: costs of ordering and receiving
    fixed dollar amount per order, regardless of order size

 Shortage costs: costs when demand exceeds
    difficult to calculate – often assumed
      ABC Classification System

Classifying inventory according to some
measure of importance and allocating
control efforts accordingly.
A - very important
B - mod. important               A
C - least important   $ value
                      of items

                                 Few               Many
                                       Number of Items
       Cycle Counting

 A physical count of items in inventory. Counts
  are conducted periodically.
   A items counted frequently
   B items counted less frequently
   C items counted least frequently

 Cycle counting management trades off inventory
  accuracy against costs of counting
   How much accuracy is needed?
   When should cycle counting be performed?
   Who should do it?
Economic Order Quantity
   Inventory Models
      Economic Order Quantity (EOQ)
 Economic order quantity (EOQ) model
 Economic production quantity (EPQ)
 Quantity discount model
         Assumptions of EOQ Model

   Only one product is involved
   Annual demand requirements are known
   Demand is even throughout the year
   Lead time does not vary
   Each order is received in a single delivery
   There are no quantity discounts
           The Inventory Cycle
                                      Profile of Inventory Level Over Time
   Q             Usage
Quantity           rate
on hand


       Receive       Place Receive      Place Receive
       order         order order        order order
                          Lead time
   Total Cost Under the Economic
   Order Quantity Assumptions

             Annual     Annual
Total cost = carrying + ordering
             cost       cost
             Q       +   DS
      TC =     H
             2           Q
              Cost Minimization Goal

              The Total-Cost Curve is U-Shaped
                        Q   D
                    TC  H  S
Annual Cost

                        2   Q

                                                   Ordering Costs

                                                     Order Quantity
                     QO (optimal order quantity)
          Deriving the EOQ

  Using calculus, we take the derivative of
  the total cost function and set the
  derivative (slope) equal to zero and
  solve for Q.

           2DS     2( Annual Demand )(Order or Setup Cost )
Q OPT =        =
            H                 Annual Holding Cost
          Minimum Total Cost

   The total cost curve reaches its
   minimum where the carrying and
   ordering costs are equal.

          2DS     2( Annual Demand )(Order or Setup Cost )
Q OPT =       =
           H                 Annual Holding Cost
       Economic Production Quantity
 Relevant when production is done in batches or

 Capacity to produce a part exceeds the part’s
  usage or demand rate

 Assumptions of EPQ are similar to EOQ except
  orders are received incrementally during
        Economic Production Quantity
        (EPQ) Assumptions
   Only one item is involved
   Annual demand is known
   Usage rate is constant
   Usage occurs continually
   Production rate is constant
   Lead time does not vary
   No quantity discounts
             Economic Production
             Quantity (EPQ)


                          & Usage
 & Usage

                 Usage                Usage
         Economic Run Size
Solving a similar equation as for EOQ, we get the following
equation for the optimal run size for EPQ:

                       2DS            p
           Q0 
                        H            p u

   p = production or delivery rate
   u = usage rate
      Quantity Discounts

 Volume (Per Unit) Discounts
   1 to 49 = $10/unit
   50 to 100 = $9/unit
   100 and up = $8/unit

 Case Discounts
   Single units = $10/unit
   Case of 10 = $90 = $9/unit
         Quantity Discounting
         Total Costs with Purchasing Cost
Quantity discounts are price reductions offered to customers
to induce them to buy in large quantities.

     Annual     Annual
                        + Purchasing
TC = carrying + ordering cost
     cost       cost

     Q                 +     DS          +     PD
TC =   H
     2                       Q
The buyer’s goal is to select the order quantity that will
minimize total cost.
           Taking the derivative with respect to
           Q doesn’t change the EOQ formula

               Adding Purchasing cost   TC with PD
               doesn’t change EOQ

                                        TC without PD


       0          EOQ                        Quantity
             Total Cost with Constant
             Carrying Costs

Total Cost

                              TCc        Price

                              CC a,b,c


                    EOQ             Quantity
Reorder Point, Reorder Quantity
      Inventory Models
      Reorder point models

 Goal is to place an order when the amount
  of inventory on hand is still sufficient to
  satisfy demand during the time it takes to
  receive that order (i.e., the lead time)
      When to Reorder with EOQ
 Reorder Point - When the quantity on
  hand of an item drops to this amount, the
  item is reordered

 Safety Stock - Stock that is held in excess
  of expected demand due to variable
  demand rate and/or lead time.

 Service Level - Probability that demand
  will not exceed supply during lead time.
                 The reorder point leaves you with
                 lead time inventory plus safety stock

                                     Maximum probable demand
                                     during lead time

                                          Expected demand
                                          during lead time

 Safety stock reduces risk of
 stockout during lead time                Safety stock
                                LT                       Time
        How do we determine the reorder
        point quantity?
 Calculation accounts for …
     The rate of demand
     The lead time
     Demand and/or lead time variability
     Stock-out risk (safety stock)
         Assuming demand and lead time
         are constant (as in EOQ)
 Reorder Point Quantity
   ROP = d X LT
        d = demand rate (units per time)
        LT = lead time (in same units of time)

 Example
     Usage = 12 order forms/day
     Lead time = 7 days
     ROP = (12 forms/day)(7 days) = 84 order forms
     Reorder when 84 order forms are left
  When lead times are random, reorder
  point (ROP) must be adjusted

                         The ROP based on a normal
                         Distribution of lead time demand

               Service level
                                         Risk of
                                       a stockout
              Probability of
               no stockout

                               ROP      Quantity
            demand        Safety
                    0              z     z-scale

ROP = expected demand during lead time + safety stock
         When we have an estimate of standard
         deviation of demand during lead time

  ROP  expected demand during lead time  z dLT

 Example
     Demand during lead time = 84 forms
     σdLT = 2
     ROP = 84 forms + zσdLT = 84 + 1.96(2) = 88 forms
     Reorder when 88 order forms are left
       When only demand is variable

         ROP  d  LT  z LT d
 Example
   Usage = 12 order forms/day; σd = 3
   Lead time = 7 days
   ROP = (12 forms/day)(7 days) + 1.96(7)0.5(3) = 84 +
    15.5 = 90
   Reorder when 90 order forms are left
         When only lead time is variable

          ROP  d  L T  zd LT

 Example
     Usage = 12 order forms/day
     Average Lead time = 7 days; σLT = 1
     ROP = (12)(7) + 1.96(12)(1) = 84 + 24 = 108
     Reorder when 108 order forms are left
       When both demand and lead
       time are variable

      ROP  d  L T  z L T  d  d 2 LT
                              2        2

 Example
   Average Usage = 12 order forms/day; σd = 3
   Average Lead time = 7 days; σLT = 1
   ROP = (12)(7) + 1.96[(7)(9) + (144)(1)] = 84 +
    1.96(14.4) = 84 + 27.7 = 112
   Reorder when 112 order forms are left
Single Period Inventory Model
  (“The Newsboy Problem”)
       Single Period Inventory Model

 Single period model: model for ordering of
  perishables and other items with limited useful
   how many newspapers should a newsboy on a street
    corner stock for a specific day?
   magazines
   fresh fruits
   fresh vegetables
   seafood
   cut flowers
   commemorative t-shirts and souvenirs
   spare parts
       Single period model balances costs
       of a shortage against excess costs

 Shortage cost: generally the unrealized
  profits per unit (plus loss of customer
   Cshortage = Cs = revenue per unit – cost per unit

 Excess cost: difference between purchase
  cost and salvage value of items left over
  at the end of a period
   Cexcess = Ce = original cost/unit – salvage value/unit
       Single Period Model

 Continuous stocking levels
   Demand can be approximated using a continuous
   Identifies optimal stocking levels
   Optimal stocking level balances unit shortage and
    excess cost

 Discrete stocking levels
   Demand can be approximated using a discrete
   Service levels are discrete rather than continuous
   Desired service level is equaled or exceeded
      Continuous stocking level assuming a
      uniform demand distribution

        Service Level 
                        C s  Ce

 Service level represents the probability
  that demand will not exceed the stocking
            Continuous stocking level assuming a
            uniform demand distribution
 Example: The movie “Gigli 2” will be released soon. A (somewhat
  crazy) retailer wants to determine the number of commemorative t-
  shirts to stock. Based on the rousingly successful “Gigli”, we have:
       Demand = uniform(1, 10)
       Cost/unit = $5 per t-shirt
       Revenue = $10 per t-shirt
       Salvage value = $1 per t-shirt

 Cs = revenue/unit – cost/unit = $10 - $5 = $5
 Ce = cost/unit – salvage value/unit = $5 - $1 = $4
 Service Level = Cs/(Cs+Ce) = ($5)/($5 + $4) = 0.555

 The optimal stocking level must satisfy demand 55% of
  the time
     Soptimal = 1 + 0.55(10-1) = 1 + 5 = 6 t-shirts
               Discrete stocking levels involve inverse
               transform from service level to order units
Here, we have a discrete uniform distribution, probability of each demand equals 0.10

    Probability       1.0
   Service Level

Cs/(Cs+Ce) = 0.55

                                                                                       “Gigli 2”
                       0                                                               t-shirts
                             1     2      3    4    5     6     7     8       9   10

              Again, the optimal decision is to stock 6 “Gigli 2” t-shirts.

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