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DISTRIBUTION INVENTORY SYSTEMS

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DISTRIBUTION INVENTORY SYSTEMS Powered By Docstoc
					Distribution Inventory
       Systems
Dr. Everette S. Gardner, Jr.
Competing interests in inventory
management
                      Controller:
                  Inventory investment




   Marketing manager:                Operations manager:
    Customer service                 Stock replenishment
                                          workload
                         Inventory                         2
Average inventory behavior with
stable demand
        12
        11
        10                                     On hand
        9
        8
Stock




        7
        6
        5                                                Avg. inv.
        4
        3
        2                                                     ROP
        1
        0
             0   4   8   12      16       20       24    28
                              Day
                              Inventory                              3
Average inventory behavior with stable
demand (cont.)

 Demand = 1 unit per day
 Leadtime = 2 days
 Leadtime demand (LTD) = 2 units

 Reorder point (ROP) = LTD = 2 units
 Order quantity (Q) = 10 units
 Maximum inventory = Q = 10 units
 Avg. investment = Q/2 = 5 units




                           Inventory     4
The economic order quantity
Objective
      Minimize total variable costs (TVC)

Ordering costs (OC)
      OC = Cost per order x Nbr. of orders
      Nbr. of orders = Demand/Order qty.

Holding costs (HC)
      HC = Holding cost per unit per year x Avg. inv. balance
      Avg. inv. balance = Order qty./2

Total variable costs
      TVC = OC + HC

                            Inventory                           5
The economic order quantity (cont.)

    $

            Total costs




                          Ordering costs



               EOQ


                    Inventory              6
Economic order quantity (cont.)

EOQ in units of stock

   QU = ((2 * demand in units * cost per order) / holding cost per unit per year)1/2


EOQ in dollars of stock

   Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2


where the holding rate is a fraction of inventory value




                                      Inventory
                                                                  EOQ.xls              7
Economic order quantity (cont.)
Example
annual demand              =   100 units
cost per order             =   $10
holding cost per unit      =   $5
unit price                 =   $25
holding rate               =   .20

  QU = ((2 * 100 * 10) / 5)1/2             = 20 units

  Q$ = ((2 * 25 * 100 * 10) / .20)1/2      = $500

                         Inventory                      8
EOQ in perspective

• Ordering and holding costs should be marginal
  (out of pocket) costs.

• Accounting systems generate average costs.

• In reality, ordering costs are semifixed.




                       Inventory                  9
 EOQ in perspective (cont.)
                         Assumption:           Reality:

    Total
    ordering
    costs

                         Nbr. of orders       Nbr. of orders



• In reality, holding costs depend on executive judgments on
  the cost of capital.


                                 Inventory                     10
 Using the EOQ when costs are
 unknown

Costs can be taken out of EOQ formulas and used as
  policy variables to achieve management goals for
  workload and average cycle stock investment.

Formula for EOQ in dollars

  Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2




                               Inventory                              11
 Using the EOQ when costs are
 unknown (cont.)
Remove all constants and unknowns

  K = ((2 * cost per order) / holding rate)1/2

K is called the EOQ constant

Simplified EOQ formula

  Q$ = K (demand in $)1/2


                            Inventory            12
 Calculations with the EOQ cost
 constant
Let unit price = $10, annual demand = 100 units

Low investment, high workload policy
       K=1
       Q = K (demand in $)1/2
         = 1 ($10 * 100)1/2 = $31.62

Avg. investment = order qty./2
                = $31.62/2 = $15.81

Workload        = demand/order qty.
                = $1000/$31.62 = 31.62 orders
                              Inventory           13
 Calculations with the EOQ cost
 constant (cont.)
High investment, low workload policy
       K=6
       Q = K (demand in $)1/2
         = 6 ($10 * 100)1/2 = $189.74

Avg. investment = order qty./2
                = $189.74 / 2 = $94.87

Workload       = demand/order qty.
               = $1000 / $189.74 = 5.3 orders


                             Inventory          14
Tradeoffs between investment and
workload
        Avg. invest. =          Workload =
K       order qty./2            demand/order qty.
1       $15.81                  31.62 orders
2        31.62                  15.8
3        47.43                  10.5
4        63.24                   7.9
5        79.04                   6.3
6        94.86                   5.3
6.32    100.00                   5.0
8       126.48                   4.0
10      158.10                   3.5
20      316.20                   1.6
                    Inventory                       15
Tradeoffs between investment and
workload (cont.)
                  $

                      300

                      250           The optimal
Avg. investment




                                    policy curve
                      200

                      150

                      100


                       50



                            0   5   10    15       20      25   30
                                         Workload
                                               Inventory             16
 Optimal policies for multi-item
 inventories
Given only the sum of square roots of demand in $, you can
   compute aggregate workload and investment.

Read Σ as “the sum of”:

Investment formula

Single-item                           Multi-item

Q$ = K (demand in $)1/2                Σ Q$ = Σ K (demand in $)1/2
                                       Σ Q$ = K Σ (demand in $)1/2
Q$ / 2 = (K/2) * (demand in $)1/2      Σ Q$ / 2 = K/2 Σ (demand in $)1/2

                                Inventory                                  17
Optimal policies for multi-item
inventories (cont.)
Workload (F) formulas

Single-item                    Multi-item

F = (demand in $) / Q$          Σ F = Σ (demand in $) / Q$

F = (1/K) * (demand in $)1/2    Σ F = 1/K Σ (demand in $)1/2




                           Inventory                           18
Multi-item example

5,000 line-item inventory

Σ (demand in dollars)1/2 = $250,000

K     K/2     Investment                1/K    Workload
1     0.5        $125,000                1.0   250,000 orders
2     1.0         250,000                0.5   125,000
5     2.5         625,000                0.2    50,000
10    5.0       1,250,000                0.1    25,000



                            Inventory                           19
 Multi-item example (cont.)
For K = 5:

avg. investment = Σ Q$/2

Σ Q$/2         = (K/2) Σ (demand in $)1/2
               = 2.5 * 250,000
               = 625,000

workload       = 1/K Σ (demand in $)1/2
               = 0.2 * $250,000
               = 50,000 orders


                              Inventory     20
Achieving management goals for
investment
Goal = Σ Q$/2

Goal = (K/2) * Σ (demand in $)1/2

Solving for K yields:

K = (2 * goal) / Σ (demand in $)1/2

This value of K meets the investment goal exactly.

The workload for that K is:

Σ F = (1/K) * Σ (demand in $)1/2
   = (Σ (demand in $)1/2)2 / (2 * goal)
                             Inventory               21
Average inventory behavior with
uncertain demand
        12                                     On hand
        11
        10
        9
        8
Stock




        7                                                     Avg. inv.
        6
        5
        4                                                     ROP
        3
        2                                                     SS
        1
        0
             0   4   8   12      16       20      24     28
                              Day
                              Inventory                                   22
Average inventory behavior with
uncertain demand (cont.)
Demand = 1 unit per day
Leadtime = 2 days
Leadtime demand (LTD) = 2 units
Safety stock (SS) = 2 units
Reorder point (ROP) = LTD + SS = 4 units
Order quantity (Q) = 10 units
Maximum inventory = Q + SS = 12 units
Avg. investment = Q/2 + SS = 7 units



                        Inventory          23
 Reorder point with uncertain
 demand
Assumption
      Length of leadtime is constant

Concepts
      Reorder point = mean demand         +   safety
                     during leadtime          stock

                                     standard deviation
      Safety stock = safety factor * of forecast errors
                                      during leadtime



                              Inventory                   24
 Reorder point with uncertain
 demand (cont.)

The standard deviation is a measure of variability of the
  forecast errors.

With a perfect forecast, the standard deviation is zero.

As forecast accuracy gets worse, the standard deviation gets
   larger.

The larger the safety factor, the smaller the risk of running out
  of stock.


                              Inventory                         25
Safety factor and probability of shortage
 z       P(z)       z          P(z)
0.00   0.50000     2.30      0.01072
0.10   0.46017     2.40      0.00820
0.20   0.42074     2.50      0.00621
0.30   0.38209     2.60      0.00466
0.40   0.34458     2.70      0.00347
0.50   0.30854     2.80      0.00256
0.60   0.27425     2.90      0.00187
0.70   0.24196     3.00      0.00135
0.80   0.21186     3.10      0.00097
0.90   0.18406     3.20      0.00069
1.00   0.15866     3.30      0.00048
1.10   0.13567     3.40      0.00034
1.20   0.11507     3.50      0.00023
1.30   0.09680     3.60      0.00016
1.40   0.08076     3.70      0.00011
1.50   0.06681     3.80      0.00007
1.60   0.05480     3.90      0.00005
1.70   0.04457     4.00      0.00003
1.80   0.03593     4.10      0.00002
1.90   0.02872     4.20      0.00001
2.00   0.02275     4.30      0.00001
2.10   0.01786     4.40      0.00001
2.20   0.01390     4.50      0.00000

                          Inventory         26
 Probability of shortage on one order cycle

P(Z) = Probability demand will exceed Z standard deviations
       of safety stock on one order cycle

Example:

  Mean demand during LT = 100 units
  Std. dev. = 20 units
  Safety factor (Z) = 1.5
  Reorder point = mean demand + Z (std. dev.)
                      during LT
                  = 100 + 1.5 (20)
                  = 130 units

  From table, P(Z) = .06681
                              Inventory
                                                ROP.xls       27
Probability of shortage on one order
cycle (cont.)
From table, P(Z) = .06681




                       .50         .43319

      Z
                             0              1.5
      X
                             100            130




                             Inventory            28
 Number of annual shortage
 occurrences (SO)
The probability of shortage on one order cycle is misleading
  since the ordering rate can vary widely across the
  inventory. A better measure of customer service is the
  number of annual shortage occurrences.

          Probability of                 Number of
SO    =   shortage on one    *           annual
          order cycle                    order cycles




                             Inventory                         29
 Number of annual shortage
 occurrences (SO) (cont.)
Example:
  Safety factor = 1.5

  Probability = .06681

  Annual demand = 1000 units

  Order quantity = 50 units

  Number of annual order cycles = 1000/50 = 20

  SO = .06681 * 20 = 1.34



                               Inventory         30
 Units or dollars backordered as a
 service measure

E(Z)   =   Expected units backordered for a distribution
           with mean = 0 and standard deviation = 1 on
           one order cycle

E(Z)σ =    Expected units backordered for a distribution
           with mean = X and standard deviation = σ
           on one order cycle




                         Inventory                         31
Units or dollars backordered as a
service measure (cont.)
Example:
  Annual demand = 1000 units
  Order quantity = 50 units
  X = 25 units
  σ=5
  Z = 1.2
  From table, E(Z) = .05610
  Reorder point = 25 + 5 (1.2) = 31
  Units short per cycle = .05610 (5) = .2805
  Annual order cycles = 1000/50 = 20
  Units short per year = 20 (.2805) = 5.61
                              Inventory        32
 Quiz #1: Computing customer
 service measures
Given:
         Annual demand = $2,000
         Order quantity = $100
         Mean demand during leadtime = $80
         Standard deviation = $60

Suppose we want the probability of shortage on one order cycle to be
  .09680. Compute the following:
         Safety stock
         Reorder point
         Number of annual shortage occurrences
         Dollars backordered during one order cycle
         Dollars backordered per year
         Average cycle stock investment
         Average inventory investment
                                   Inventory                       33
 Quiz #2: Computing customer
 service measures

For the same data as the previous problem, what reorder
  point will yield:

  a. 3 shortage occurrences per year?

  b. 5% of annual sales backordered?




                          Inventory                       34
 Inventory tradeoff curves
A variety of different workload and investment combinations
   yield exactly the same customer service.

To develop a tradeoff curve for dollars backordered:

1. Compute and plot the optimal policy curve showing tradeoffs
   between cycle stock investment and workload.

2. Select a percentage goal for dollars backordered.

3. For each workload, compute the safety stock needed to meet the
   goal.

4. Add cycle stock to safety stock to obtain total investment.


                                Inventory                           35
Inventory tradeoff curves (cont.)

                $                       “Isoservice curve” -- each
                                         point yields the same
                                         dollars backordered
   Investment




                                                    Safety
                                                    stock
                    Optimal
                    policy or
                    cycle stock curve

                                   Workload



                                        Inventory                    36
U.S. Navy application of tradeoff
curves
Inventory system
     8 Naval supply centers
     Average inventory statistics at each center
           80,000 line items
           $25 million investment

Budget constraint
     Average investment limited to 2.5 months of
     stock

                        Inventory                  37
U.S. Navy application of tradeoff
curves (cont.)
Investment allocation strategies
                               Old                    New
Safety stock                   1.5 months             1.0
   months
Cycle stock                    1.0 months             1.5
   months
Total                          2.5                    2.5

Results of new allocation
        Reordering workload cut from 840,000 to 670,000 per year
        $2 million annual savings in manpower

                             Inventory                             38
Workload/service tradeoffs
                                (Total investment fixed at 2.5 months)
                        90%
                                         New policy         Previous policy
     Customer service


                        85%




                        80%



                        75%
                              0.0       0.5       1.0           1.5       2.0
                                        Safety stock (months)

                              112      121        184           240       289

                                    Workload (thousands of orders)

                                                Inventory                       39
Strategic problems in managing
distribution inventories
1. Controlling inventory growth as sales increase

2. Controlling inventory growth as new locations are
   added

3. Push vs. pull decision rules

4. Continuous review of stock balances vs. periodic review

5. Choosing a customer service measure




                            Inventory                        40
Inventory growth
Inventories should grow at slower rate than sales.
Why? Order quantities are proportional to the square root of sales.

Example:
        One inventory item
        K=1
        Q$ = K (demand in $)1/2

                Sales                      Average      Investment
Sales           Growth            Q$       Investment   Growth

100               ---             10       10/2 = 5        ---
200             100%              14       14/2 = 7       40%

                               Inventory                              41
 Effects of adding inventory locations
Inventories must increase as new locations are added.

One reason is that forecasting is easier when customer demands are
  consolidated. Thus forecast errors are smaller and less safety stock
  is required.

Another reason stems from the EOQ:
        One inventory item
        Sales of $100
        K=1

With one location:
        Q$ = K (demand in $)1/2
        Q$ = 1 (100)1/2 = 10
        Average investment = 10/2 = 5
                               Inventory                             42
 Effects of adding inventory locations
 (cont.)
With two locations:
      Location 1: Q$ = 1 (50)1/2 = 7.07
      Location 2: Q$ = 1 (50)1/2 = 7.07
      Average investment = (7.07 + 7.07) / 2 = 7.07

      Investment increase = 7.05 – 5 = 2.05
      Percentage increase = 2.05 / 5 = 41%




                             Inventory                43
Continuous review vs. periodic review
systems
Continuous review
       Check stock balance after each transaction

       If stock on hand below reorder point, place new order in a fixed
       amount

Periodic review
       Check stock balance on a periodic schedule

       If stock on hand below reorder point, place new order:
                 in a fixed amount, or
                 in a variable amount (maximum level – on hand)

Investment requirements
       Periodic review always requires more investment than continuous
       review to meet any customer service goal.
                                  Inventory                               44
 Push vs. pull control systems
Push or centralized system
      Central authority forecasts demand, sets stock levels,
      and pushes stock to each location.

Pull or decentralized system
      Each location forecasts its own demand and sets its own
      stock levels.

Investment requirements
      A pull system always requires more investment than a
      push system to meet any customer service goal.

                            Inventory                           45
 Comparison of shortage values

Inventory statistics
       5,790 line items
       $45 million annual sales

Shortage values at investment constraint of $5 million

Shortage value                    Shortage       Dollars
minimized                         occurrences    backordered

shortage occurrences              1,120          $3.46 million
dollars backordered               2,812           1.48




                                  Inventory                      46

				
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