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Inventory Management




    Introduction to Operations Management
                                            1
          What is inventory?

 An inventory is an idle stock of
 material used to facilitate production
 or to satisfy customer needs




              Introduction to Operations Management
                                                      2
       Why do we need inventory?
 Economies  of scales in ordering or
 production - cycle stock
  – trade-off setup cost vs carrying cost
 Smooth production when requirements
 have predictable variability - seasonal stock
  – trade-off production adjustment cost vs carrying
    cost


                 Introduction to Operations Management
                                                         3
  Why do we need inventory? (cont’d)

 Provide  immediate service when
 requirements are uncertain (unpredicatable
 variability) -safety stock
  – trade-off shortage cost versus carrying cost
 Decoupling of stages in
 production/distribution systems -decoupling
 stock
  – trade-off interdependence between stages vs
    carrying cost
                 Introduction to Operations Management
                                                         4
Why do we need inventory? (cont’d)

 Production  is not instantaneous, there is
 always material either being processed or in
 transit - pipeline stock
  – trade-off cost of speed vs carrying cost




               Introduction to Operations Management
                                                       5
       Why carrying inventory

 Asyou can see from the above list,
 there are economic reasons,
 technological reasons as well as
 management/operations reasons.




            Introduction to Operations Management
                                                    6
  Why carrying inventory is costly?

 Traditionally,   carrying costs are
  attributed to:
  – Opportunity cost (financial cost)
  – Physical cost
    (storage/handling/insurance/theft/obs
    olescence/…)




               Introduction to Operations Management
                                                       7
             Inventory Management
                                                                    Independent Demand



                                A                                      Dependent Demand



              B(4)                             C(2)



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




Independent demand is uncertain. Dependent demand is certain.
                            Introduction to Operations Management
                                                                                          8
         Types of Inventories
 Raw  materials & purchased parts
 Partially completed goods called
  work in progress
 Finished-goods inventories
  – (manufacturing firms)
    or merchandise
    (retail stores)


              Introduction to Operations Management
                                                      9
     Types of Inventories (Cont’d)
 Replacement  parts, tools, & supplies
 Goods-in-transit to warehouses or
  customers




             Introduction to Operations Management
                                                     10
      Inventory Counting Systems

 Periodic   System
  Physical count of items made at periodic intervals
 Perpetual   Inventory System
 System that keeps track
 of removals from inventory
 continuously, thus monitoring
 current levels of each item



                Introduction to Operations Management
                                                        11
Inventory Counting Systems (Cont’d)

 Two-Bin   System - Two containers of
  inventory; reorder when the first is
  empty
 Universal Bar Code - Bar code
  printed on a label that has
  information about the item
  to which it is attached          0


                                                        214800 232087768


                Introduction to Operations Management
                                                                           12
         ABC Classification System

Classifying inventory according to
some measure of importance and
allocating control efforts accordingly.
 A - very important                    High

                                 Annual                  A
 B - mod. important             $ volume                          B
                                of items                                  C
 C - least important
                                       Low
                                                   Few                   Many
                                                             Number of Items
                 Introduction to Operations Management
                                                                                13
        What are we tackling in inventory
                management?
 Allmodels are trying to answer the following
  questions for given informational, economic and
  technological characteristics of the operating
  environment:
   – How much to order (produce)?
   – When to order (produce)?
   – How often to review inventory?
   – Where to place/position inventory?


                 Introduction to Operations Management
                                                         14
          The inventory models
 Thequantitative inventory management
 models range from simple to very
 complicated ones. However, there are two
 simple models that capture the essential
 tradeoffs in inventory theory
  – The newsboy (newsvendor) model
  – The EOQ (Economic Ordering Quantity) model



               Introduction to Operations Management
                                                       15
          The Newsboy Model

 Thenewsboy model is a single-period
 stocking problem with uncertain demand:
 Choose stock level and then observe
 actual demand.
  – Tradeoff: overstock cost versus opportunity
    cost (lost of profit because of under stocking)




               Introduction to Operations Management
                                                       16
           The Newsboy Model

 Shortage  cost (Cs) - the opportunity cost for
 lost of sales as well as the cost of losing
 customer goodwill.
   Cs = revenue per unit - cost per unit
 Excess cost (Ce) - the over-stocking cost
 (the cost per item not being able to sell)
   Ce = Cost per unit - salvage value per unit


              Introduction to Operations Management
                                                      17
           The Newsboy Model
 In this model, we have to consider two
  factors (decision variables): the supply (X),
  also know as the stock level, and the
  demand (Y). The supply is a controllable
  variable in this case and the demand is not
  in our control. We need to determine the
  quantity to order so that long-run expected
  cost (excess and shortage) is minimized.

                Introduction to Operations Management
                                                        18
         The Newsboy Model

 Let  X = n and P(n) = P(Y>n). The
  question is: Based on what should we
  stock this n-th unit?
 The opportunity cost (expected loss of
  profit)for this n-th unit is P(n) Cs
 The expected cost for not being able to
  sell this n-th unit (expected loss)
    (1-P(n))Ce
                 Introduction to Operations Management
                                                         19
           The Newsboy model

 Stock   the n-th unit if P(n)Cs > (1-P(n))Ce

 Do   not stock if P(n)Cs < (1-P(n))Ce




               Introduction to Operations Management
                                                       20
             The Newsboy model
 The   equilibrium point occurs at
   P(n)Cs = (1-P(n)) Ce

 Solving the equation
   P(n) = Ce/(Cs + Ce)




                 Introduction to Operations Management
                                                         21
                Newsboy model

 Service   level is the probability that demand
  will not exceed the stocking level and is the
  key to determine the optimal stocking level.

 Inour notation, it is the probability that
  YX(=n), which is given by

       P(Yn) = 1 - P(n) = Cs/(Cs + Ce)
                  Introduction to Operations Management
                                                          22
                  Example (p.564)
Demand for long-stemmed red roses at a small flower shop can be
approximated using a Poisson distribution that has a mean of four
dozen per day. Profit on the roses is $3 per dozen. Leftover are
marked down and sold the next day at a loss of $2 per dozen. Assume
that all marked down flowers are sold. What is the optimal stocking
level?




                     Introduction to Operations Management
                                                                      23
                 Example (solutions)
                                                  Cumulative frequencies for
   Cs = $3, Ce = $2                              Poisson distribution, mean = 4
   Opportunity cost is P(n)Cs
                                                 Demand (dzn      Cumulative
   Expected loss for over-                      per day)         frequency
    stocking (1 - P(n)) Ce                       0                0.018
                                                 1                0.092
   P(Y  n) = Cs/(Cs + Ce) =                    2                0.238
    3/((3+2) = 0.6                               3                0.434
                                                 4                0.629
   From the table, it is                        5                0.785
    between 3 and 4, round up                5




    give you optimal stock of 4
    dozen.
                      Introduction to Operations Management
                                                                                   24
                   The Inventory Cycle
              Profile of Inventory Level Over Time
Q              Usage
                 rate

Quantity
on hand


Reorde
r
point

                                                                       Time
    Receive         Place      Receive             Place     Receive
    order           order      order               order     order
                        Lead time to Operations Management
                           Introduction
                                                                              25
How to estimate the inventory costs?




              Introduction to Operations Management
                                                      26
Estimating the inventory costs
  Q  = quantity to order in each cycle
   S = fixed cost or set up cost for each order
   D = demand rate or demand per unit time
   H = holding cost per unit inventory per unit
        time
   c = unit cost of the commodity


   TC   = total holding cost per unit time

                   Introduction to Operations Management
                                                           27
Estimating the inventory cost
   The   cost per order = S + cQ

   The   length of order cycle = Q/D unit time

   The average inventory level = Q/2
   Thus the inventory carrying cost
      = (Q/2) H (Q/D) = HQ2/(2D)



                   Introduction to Operations Management
                                                           28
Estimating the inventory cost

   The   total cost per order cycle
      = S + cQ + HQ2/(2D)

    Thus the total cost per unit time is dividing the
    above expression by Q/D, I.e.,


    TC = {S + cQ + HQ2/(2D)} {D/Q}
      = DS/Q + cD + HQ/2

                  Introduction to Operations Management
                                                          29
             Total Cost


             Annual     Annual
Total cost = carrying + ordering
             cost       cost

                    Q                  +          DS
      TC =            H
                    2                             Q


          Introduction to Operations Management
                                                       30
             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)
QOPT =       =
          H               Annual Holding Cost




                 Introduction to Operations Management
                                                          31
               Cost Minimization Goal
              The Total-Cost Curve is U-Shaped

                         Q   D
                     TC  H  S
Annual Cost




                         2   Q




                                                                    Ordering Costs


                     QO (optimal order quantity)             Order Quantity (Q)
                     Introduction to Operations Management
                                                                                     32
            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)
QOPT =       =
          H               Annual Holding Cost




                 Introduction to Operations Management
                                                          33
Cost            Total Costs

           Adding Purchasing cost                     TC with PD
           doesn’t change EOQ



                                                      TC without PD




                                                            PD



       0       EOQ                                          Quantity
              Introduction to Operations Management
                                                                       34
Total Cost with Constant Carrying Costs

                                            TCa

                                             TCb
 Total Cost




                                                            Decreasing
                                               TCc          Price




                                                 CC a,b,c


                                                            OC


                 EOQ                                  Quantity
              Introduction to Operations Management
                                                                         35
Example (Example 2, p.541)

A local distributor for a national tire company expects to sell
approximately 9600 steel-belted radial tires of a certain size and tread
design next year. Annual carrying costs are $16 per tire, and ordering
costs are $75. This distributor operate 288 days a year.
a. What is the EOQ?
b. How many times per year does the store reorder?
c. What is the length or an order cycle?
d. What is the total ordering and inventory cost?




                          Introduction to Operations Management
                                                                           36
Solution (Example 2, p.541)

 D = 9600 tires per year, H = $16 per unit per year, S = $75
 a. Q0 = {2DS/H} = {2(9600)75/16} = 300
 b. Number of orders per year = D/ Q0 = 9600 / 300 = 32
 c. The length of one order cycle = 1 / 32 years = 288/32 days = 9 days
 d. Total ordering and inventory cost             = QH/2 + DS/Q
                                                  = 2400 + 2400 = 4800




                         Introduction to Operations Management
                                                                          37
     EOQ with quantity discount

 This is a variant of the EOQ model.
 Quantity discount is another form of
 economies of scale: pay less for each unit
 if you order more. The essential trade-off
 is between economies of scale and
 carrying cost.



            Introduction to Operations Management
                                                    38
       EOQ with quantity discount
 To tackle the problem, there will be a separate
 (TC) curve for each discount quantity price. The
 objective is to identify an order quantity that will
 represent the lowest total cost for the entire set of
 curves in which the solution is feasible. There are
 two general cases:
  – The holding cost is constant
  – The holding cost is a percentage of the
    purchasing price.

               Introduction to Operations Management
                                                       39
Quantity discount (constant holding cost)

  Total   cost per cycle
     = setup cost + purchase cost + carrying cost
  Setup   cost = S
  Purchase cost = ciQ, if the order quantity Q
   is in the range where unit cost is ci.
  Carrying cost = (Q/2)h(Q/D)


  TC   = {S+ciQ+hQ2/(2D)}{D/Q} (annual cost)
                 Introduction to Operations Management
                                                         40
Quantity discount (constant holding cost)
  Differentiating, we obtain an optimal order quantity
  which is independent of the price of the good

         2DS
    Q0 
          h

  The questions is: Is this Q0 a feasible solution to our
  problem?




                    Introduction to Operations Management
                                                            41
Quantity discount (constant holding cost)

  Solution   steps:
   1. Compute the EOQ.
   2. If the feasible EOQ is on the lowest price
   curve, then it is the optimal order quantity.
   3. If the feasible EOQ is on other curve, find
   the total cost for this EOQ and the total costs
   for the break points of all the lower cost
   curves. Compare these total costs. The point
   (EOQ point or break point) that yields the
   lowest total cost is the optimal order quantity.
                 Introduction to Operations Management
                                                         42
                        Example
   The maintenance department of a large hospital uses
    about 816 cases of liquid cleanser annually. Ordering
    costs are $12, carrying cost are $4 per case a year, and
    the new price schedule indicates that orders of less
    than 50 cases will cost $20 per case, 50 to 79 cases
    will cost $18 per case, 80 to 99 cases will cost $17 per
    case, and larger orders will cost $16 per case.
    Determine the optimal order quantity and the total cost.




                   Introduction to Operations Management
                                                               43
                Example (solutions)
1.    The common EOQ: = {2(816)(12)/4} = 70

2.    70 falls in the range of 50 to 79, at $18 per case.
            TC = DS/Q+ciD+hQ/2
                = 816(12)/70 + 18(816) + 4(70)/2 = 14,968

3.    Total cost at 80 cases per order
        TC = 816(12)/80 + 17(816) + 4(80)/2 = 14,154
     Total cost at 100 cases per order
        TC = 816(12)/100 + 16(816) + 4(100)/2 =13,354

The minimum occurs at the break point 100. Thus order 100 cases
each time
                       Introduction to Operations Management
                                                                  44
       Quantity discount (h = rc)
 Usingthe same argument as in the constant
 holding cost case, Let TCi be the total cost
 when the unit quantity is ci.
    TCi = rciQ/2 + DS/Q + ciD
 Therefore EOQ = {2DS/(rci)}

 In this case, the carrying cost will decrease
 with the unit price. Thus the EOQ will shift to
 the right as the unit price decreases.
                Introduction to Operations Management
                                                        45
          Quantity discount (h = rc)
 Steps   to identify optimal order quantity

 1. Beginning with the lowest price, find the EOQs for
 each price range until a feasible EOQ is found.
 2. If the EOQ for the lowest price is feasible, then it is
 the Optimal order quantity.
 3. Same as step (3) in the constant carrying case.




                   Introduction to Operations Management
                                                              46
                  Example (p.549)

Surge Electric uses 4000 toggle switches a year. Switches are priced
as follows: 1 to 499 at $0.9 each; 500 to 999 at $.85 each; and 1000
or more will be at $0.82 each. It costs approximately $18 to prepare
an order and receive it. Carrying cost is 18% of purchased price per
unit on an annual basis. Determine the optimal order quantity and the
total annual cost.




                      Introduction to Operations Management
                                                                        47
                               Solution
        D = 4000 per year; S = 18 ; h = 0.18 {price}
Step 1. Find the EOQ for each price, starting with the lowest price
EOQ(0.82) = {2(4000)(18)/[(0.18)(0.82)]} = 988                  (Not feasible for the price range).

EOQ(0.85)= 970 (feasible for the range 500 to 999)
Step 2. Feasible solution is not on the lowest cost curve
Step 3. TC(970) = 970(.18)(.85)/2 + 4000(18)/970 + .85(4000) = 3548
        TC(1000) = 1000(.18)(.82)/2 + 4000(18)/1000 + .82(4000) = 3426


Thus the minimum total cost is 3426 and the minimum cost order size is
1000 units per order.

                         Introduction to Operations Management
                                                                                                  48
 We have settled the problem of how much
 to order. The next decision problem is
 when to order.




             Introduction to Operations Management
                                                     49
               Reorder point
 In the EOQ model, we assume that the delivery
  of goods is instantaneous. However, in real life
  situation, there is a time lag between the time
  an order is placed and the receiving of the
  ordered goods. We call this period of time the
  lead time. Demand is still occurring during the
  lead time and thus inventory is required to meet
  customer demand. This is why we need to
  consider when to order!

               Introduction to Operations Management
                                                       50
            When to reorder

 We   can reorder based on:
 – time, say weekly, monthly, etc
 – quantity of goods on hand




              Introduction to Operations Management
                                                      51
When to Reorder with EOQ Ordering

 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.
               Introduction to Operations Management
                                                       52
      Quantity   Safety Stock


                                  Maximum probable demand
                                  during lead time

                                             Expected demand
                                             during lead time



ROP

                                             Safety stock
                        LT                                  Time
                 Introduction to Operations Management
                                                                   53
              When to reorder
 In general, we need to consider the
  following four factors when deciding when
  to order:
  – The rate of demand (usually based on a
    forecast)
  – The length of lead time.
  – The extent of demand and lead time variability.
  – The degree of stock-out risk acceptable to
    management.
                Introduction to Operations Management
                                                        54
When to reorder

   We   will just cover two simple cases:
    – The demand rate and lead time are
      constant
    – Variable demand rate and constant lead
      time.




                 Introduction to Operations Management
                                                         55
Reorder Point



      Service level
                                          Risk of
                                        a stockout
     Probability of
     no stockout

                             ROP        Quantity
Expected
demand                Safety
                      stock
              0                     z     z-scale


Introduction to Operations Management
                                                     56
ROP - constant demand and lead time
 Let  d be the demand rate per unit time and LT
  be the lead time in unit time.
 The reorder point quantity is given by
     ROP = expected demand during lead time +
            safety stock during lead time
 In this case, safety stock required is 0
     Thus ROP = d (LT)

  Example: See Example 7 on page 551
               Introduction to Operations Management
                                                       57
        ROP - variable demand rate
 The  lead time is constant. In this case, we need
  extra buffer of safety stock to insure against
  stock-out risk.
 Since it cost money to carry safety stocks, a
  manager must weigh carefully the cost of carrying
  safety stock against the reduction in stock-out risk
  (provided by the safety stock)




                  Introduction to Operations Management
                                                          58
           ROP - variable demand rate
 Inother words, he needs to trade-off the cost of
  safety stock and the service level.
 Service level = 1 - stock-out risk
 ROP = Expected lead time demand + safety stock
       = d (LT) + z 
       Where  is the standard deviation of the lead time
       demand = {LT} d

  Therefore ROP = d(LT) + z {LT} d

                      Introduction to Operations Management
                                                              59
            Example (Problem 4, p.569)
The housekeeping department of a hotel uses approximately 400
washcloths per day. The actual amount tends to vary with the number of
guests on any given night. Usage can be approximated by a normal
distribution that has a mean of 400 and a standard deviation of 9
washcloths per day. A linen supply company delivers towels and
washcloths with a lead time of three days. If the hotel policy is to maintain
a stock-out risk of 2 percent, what is the minimum number of washcloths
that must be on hand at reorder time, and how much of that amount can
be considered safety stock?


Solution:

        d = 400, LT = 3 days, d = 9 , service level = 1 - risk = 0.98

                          Introduction to Operations Management
                                                                                60
                    Example (solution)

Z = 2.055 (from normal distribution table)


Thus ROP = 400 (3) + 2.055 (3) d = 1200 + 32.03 = 1232


The safety stock is approximately 32 washcloths, providing a service
level of 98%.




                         Introduction to Operations Management
                                                                       61
       Fixed-order-interval model

 In the EOQ/ROP models, fixed quantities
  of items are ordered at varying time
  interval. However, many companies
  ordered at fixed intervals: weekly,
  biweekly, monthly, etc. They order varying
  quantity at fixed intervals. We called this
  class of decision models fixed-order-
  interval (FOI) model.

              Introduction to Operations Management
                                                      62
                     FOI model

 Why    use FOI model?

 Is   it optimal?

 Whatare decision variables in the FOI
  model?


                 Introduction to Operations Management
                                                         63
                  FOI model

 Assuming lead time is constant, we need
 to consider the following factors
  – the expected demand during the ordering
    interval and the lead time
  – the safety stock for the ordering interval and
    lead time
  – the amount of inventory on hand at the time
    of ordering

               Introduction to Operations Management
                                                       64
                         FOI model
                                   Order quantity




                                                              Safety stock

                                                                   Time
Amount on hand (A)
                                        OI
               Place order                        LT
                                        Receive
                                        order
                      Introduction to Operations Management
                                                                             65
               FOI model

 We use the following notations
 OI = Length of order interval
 A = amount of inventory on hand
 LT = lead time
 d = Standard deviation of demand



            Introduction to Operations Management
                                                    66
                  FOI model

 Amount   to order
   = expected demand during protection interval
     + Safety stock for protection interval
     - amount on hand at time of placing order


   = d(OI  LT)  z d OI  LT  A




               Introduction to Operations Management
                                                       67
                       Example (p.560)
The following information is given for a FOI system. Determine the
amount to order.
d = 30 units per day, d = 3 units per day, LT = 2 days; OI = 7 days
Amount on hand = 71 units, Desired serve level = 99%.


Solution: z = 2.33 for 99% service level.
Amount to order = 30 (7 + 2) + 2.33 (3) (7 + 2) - 71 = 270 + 20.97 - 71 = 220




                           Introduction to Operations Management
                                                                                 68

				
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