Inventory Control Models

1
Inventory Control Models

• Ch 4 (Known Demands)
• Ch 5 (Uncertainty of Demand)
• R. R. Lindeke
• IE 3265, Production And Operations Management

2
Reasons for Holding Inventories

• Economies of Scale
• Uncertainty in delivery leadtimes
• Speculation. Changing Costs Over Time
• Smoothing seasonality, Bottlenecks
• Demand Uncertainty
• Costs of Maintaining Control System

3
Characteristics of Inventory Systems

• Demand
• May Be Known or Uncertain
• May be Changing or Unchanging in Time
• Lead Times - time that elapses from placement of
  order until its arrival. Can assume known or
  unknown.
• Review Time. Is system reviewed periodically or
  is system state known at all times?

4
Characteristics of Inventory Systems

• Treatment of Excess Demand.
• Backorder all Excess Demand
• Lose all excess demand
• Backorder some and lose some
• Inventory whos quality changes over time
• perishability
• obsolescence

5
Real Inventory Systems ABC ideas

• This was the true basis of Paretos Economic
  Analysis!
• In a typical Inventory System most companies find
  that their inventory items can be generally
  classified as
• A Items (the 10 - 20 of skus) that represent up
  to 80 of the inventory value
• B Items (the 20 30) of the inventory items
  that represent nearly all the remaining worth
• C Items the remaining 50 70 of the inventory
  items skus) stored in small quantities and/or
  worth very little

6
Real Inventory Systems ABC ideas and Control

• A Items must be well studied and controlled to
  minimize expense
• C Items tend to be overstocked to ensure no
  runouts but require only occasional review
• See mhia.org there is an e-lesson on the
  principles of ABC Inventory management check it
  out! do it!

7
Relevant Inverntory Costs

• Holding Costs - Costs proportional to the
  quantity of inventory held. Includes
• Physical Cost of Space (3)
• Taxes and Insurance (2 )
• Breakage Spoilage and Deterioration (1)
• Opportunity Cost of alternative investment. (18)
  
• Holding issues total 24
• Therefore, in inventory systems, the holding cost
  is taken as
• h ? .24Cost of product

8
Lets Try one

• Problem 4, page 193 cost of inventory
• Find h first (yearly and monthly)
• Total holding cost for the given period
• THC 26666.67
• Average Annual Holding Cost
• assumes an average monthly inventory of trucks
  based on onhand data
• 3333

9
Relevant Costs (continued)

• Ordering Cost (or Production Cost).
• Includes both fixed and variable components
• slope c
• K C(x) K cx for x gt 0 0 for x 0.

10
Relevant Costs (continued)

• Penalty or Shortage Costs. All costs that accrue
  when insufficient stock is available to meet
  demand. These include
• Loss of revenue due to lost demand
• Costs of book-keeping for backordered demands
• Loss of goodwill for being unable to satisfy
  demands when they occur.

11
Relevant Costs (continued)

• When computing Penalty or Shortage Costs
  inventory managers generally assume cost is
  proportional to number of units of excess demand
  that will go unfulfilled.

12
The Simple EOQ Model the most fundamental of
all!

• Assumptions
• 1. Demand is fixed at l units per unit time.
• 2. Shortages are not allowed.
• 3. Orders are received instantaneously. (this
  will be relaxed later).

13
Simple EOQ Model (cont.)

• Assumptions (cont.)
• 4. Order quantity is fixed at Q per cycle. (we
  will find this is an optimal value)
• 5. Cost structure
• a) includes fixed and marginal order costs (K
  cx)
• b) includes holding cost at h per unit held
  per unit time.

14
Inventory Levels for the EOQ Model
15
The Average Annual Cost Function G(Q)
16
Modeling Inventory
17
Subbing Q/? for T
18
Finding an Optimal Level of Q the so-called
EOQ

• Take derivative of the G(Q) equation with respect
  to Q
• Set derivative equals Zero
• Now, Solve for Q

19
Properties of the EOQ (optimal) Solution

• Q is increasing with both K and ? and decreasing
  with h
• Q changes as the square root of these quantities
• Q is independent of the proportional order cost,
  c. (except as it relates to the value of h Ic)

20
Try ONE!

• A company sells 145 boxes of BlueMountain
  BobBons/week (a candy)
• Over the past several months, the demand has been
  steady
• The store uses 25 as a holding factor
• Candy costs 8/bx and sells for 12.50/bx
• Cost of making an order is 35
• Determine EOQ (Q) and how often an order should
  be placed

21
Plugging and chugging

• h 8.25 2
• ? 14552 7540

22
But, Orders usually take time to arrive!

• This is a realistic relaxation of the EOQ ideas
  but it doesnt change the model
• This requires the user to know the order Lead
  Time
• And then they trigger an order at a point before
  the delivery is needed to assure no stock outs
• In our example, what if lead time is 1 week?
• We should place an order when we have 145 boxes
  in stock (the one week draw down)
• Note make sure lead time units match units in T!

23
But, Orders usually take time to arrive!

• What happens when order lead times exceed T?
• We proceed just as before (but we compute ?/T)
• ? is the lead time is units that match T
• Here, lets assume ? 6 weeks then
• ?/T 6/3.545 1.69
• Place order 1.69 cycles before we need product
• Trip Point is then 0.69Q .69514 356 boxes
• This trip point is not for the next stock out but
  the one after that (1.69 T from now!) be
  very careful!!!

24
Sensitivity Analysis

• Let G(Q) be the average annual holding and set-up
  cost function given by
• and let G be the optimal average annual cost.
  Then it can be shown that

25
Sensitivity

• We find that this model is quite robust to Q
  errors if holding costs are relatively low
• We find, given a ?Q error in ordering quantity
• that Q ?Q has smaller error than Q - ?Q
• That is, we tend to have a greater penalty cost
  (Error means extra inventory maintenance costs)
  if we order too little than too much

26
EOQ With Finite Production Rate

• Suppose that items are produced internally at a
  rate P (gt ?, the consumption rate). Then the
  optimal production quantity to minimize average
  annual holding and set up costs has the same form
  as the EOQ, namely
• Except that h is defined as h h(1- ?/P)

27
This is based on solving
28
Inventory Levels for Finite Production Rate Model
29
Lets Try one

• We work for Sams Active Suspensions
• They sell after market kits for car Pimpers
• They have an annual demand of 650 units
• Production rate is 4/day (wotking at 250 d/y)
• Setup takes 2 technicians working 45 minutes
  _at_21/hour and requires an expendable tool costing
  25

30
Continuing

• Each kit costs 275
• Sams uses MARR of 18, tax at 3, insurance at
  2 and space cost of 1
• Determine h, Q, H, T and break T down to
• T1 production time in a cycle (Q/P)
• T2 non producing time in a cycle (T T1)

31
Quantity Discount Models

• All Units Discounts the discount is applied to
  ALL of the units in the order. Gives rise to an
  order cost function such as that pictured in
  Figure 4-9
• Incremental Discounts the discount is applied
  only to the number of units above the breakpoint.
  Gives rise to an order cost function such as that
  pictured in Figure 4-10.

32
All-Units Discount Order Cost Function
33
Incremental Discount Order Cost Function
34
Properties of the Optimal Solutions

• For all units discounts, the optimal will occur
  at the bottom of one of the cost curves or at a
  breakpoint. (It is generally at a breakpoint.).
  One compares the cost at the largest realizable
  EOQ and all of the breakpoints beyond it. (See
  Figure 4-11).
• For incremental discounts, the optimal will
  always occur at a realizable EOQ value. Compare
  costs at all realizable EOQs. (See Figure 4-12).

35
All-Units Discount Average Annual Cost Function
36
To Find EOQ in All Units discount case

• Compute Q for each cost level
• Check for Feasibility (the Q computed is
  applicable to the range) Realizable
• Compute G(Q) for each of the realizable Qs and
  the break points.
• Chose Q as the one that has lowest G(Q)

37
Lets Try one

• Product cost is 6.50 in orders lt600, 3.50 above
  600.
• Organizational I is 34
• K is 300 and annual demand is 900

38
Lets Try one

• Both of these are Realizable (the value is in
  range)
• Compute G(Q) for both and breakpoint (600)
• G(Q) c? (?K)/Q (hQ)/2

Order 674 at a time!
39
Average Annual Cost Function for Incremental
Discount Schedule
40
In an Incremental Case

• Cost is strictly a varying function of Q -- It
  varies by interval
• Calculate a C(Q) for the applied schedule
• Divide by Q to convert it to a unit cost
  function
• Build G(Q) equations for each interval
• Find Q from each Equation
• Check if Realizable
• Compute G(Q) for realizable Qs

41
Trying the previous problem (but as Incremental
Case)

• Cost Function Basically states that we pay 6.50
  for each unit up to 600 then 3.50 for each unit
  ordered beyond 601
• C(Q) 6.5(Q), Q lt 600
• C(Q) 3.5(Q 600) 6.5600, Q ? 600
• C(Q)/Q 6.5, Q lt 600 (order up to 600)
• C(Q)/Q 3.5 ((3900 2100)/Q), Q ? 600 3.5
  (1800)/Q (orders beyond 601)

42
Trying the previous but as Incremental Case

• For the First Interval
• Q ?(2300900)/(.346.50) 495 (realizable)
• For order gt 600, find Q by writing a G(Q)
  equation and then optimizing
• G(Q) c? (?K)/Q (hQ)/2

43
Differentiating G2(Q)
Realizable!
44
Now Compute G(Q) for both and cusp

• G(495) 9006.5 (300900)/495
  .34((6.5495)/2) 6942.43
• G(600) 9006.5 (300900/600)
  .34((6.5600)/2) 6963.00
• G(1763) 900(3.5 (1800/1783)) (300900)/1783
  .34(3.5 (1800/1783))(1783/2) 5590.67

Lowest cost purchase 1783 about every 2 years!
45
Properties of the Optimal Solutions

• Lets jump back into our teams and do some!

46
Resource Constrained Multi-Product Systems

• Consider an inventory system of n items in which
  the total amount available to spend is C and
  items cost respectively c1, c2, . . ., cn. Then
  this imposes the following constraint on the
  system

47
Resource Constrained Multi-Product Systems

• When the condition that
• is met, the solution procedure is
  straightforward. If the condition is not met, one
  must use an iterative procedure involving
  Lagrange Multipliers.

48
EOQ Models for Production Planning

• Consider n items with known demand rates,
  production rates, holding costs, and set-up
  costs. The objective is to produce each item once
  in a production cycle. For the problem to be
  feasible the following equation must be true

49
Issues

• We are interested in controlling Family MAKESPAN
  (we wish to produce all products within our
  chosen cycle time)
• Underlying Assumptions
• Setup Cost (times) are not Sequence Dependent
  (this assumption is not always accurate as we
  will later see)
• Plants uses a Rotation Policy that produces a
  single batch of each product each cycle a
  mixed line balance assumption

50
EOQ Models for Production Planning

• The method of solution is to express the average
  annual cost function in terms of the cycle time,
  T. The optimal cycle time has the following
  mathematical form
• We must assure that this time allows for all
  setups and of production times.

51
Working forward

• This last statement means
• ?(sj(Qj/Pj) ? T
• Of course Qj ?jT
• So with substitution ?(sj((?jT )/Pj) ? T
• Or T?(?sj/(1- ?j/Pj) Tmin
• Finally, we must Choose T(actual cycle time)
  MAX(T,Tmin)

52
Lets Try Problem 30
Given 20 days/month and 12 month/year 85/hr
for setup
53
Compute the Following in teams!
54
Lets do a QUICK Exploration of Stochastic
Inventory Control (Ch 5)

• We will examine underlying ideas
• We base our approaches on Probability Density
  Functions (means std. Deviations)
• We are concerned with two competing ideas Q and
  R
• Q (as earlier) an order quantity and R a
  stochastic estimate of reordering time and level
• Finally we are concerned with Servicing ideas
  how often can we supply vs. not supply a demand
  (adds stockout costs to simple EOQ models)

55
The Nature of Uncertainty

• Suppose that we represent demand as
• D Ddeterministic Drandom
• If the random component is small compared to the
  deterministic component, the models of chapter 4
  will be accurate. If not, randomness must be
  explicitly accounted for in the model.
• In this chapter, assume that demand is a random
  variable with cumulative probability distribution
  F(t) and probability density function f(t).

56
The Newsboy Model

• At the start of each day, a newsboy must decide
  on the number of papers to purchase. Daily sales
  cannot be predicted exactly, and are represented
  by the random variable, D.
• Costs co unit cost of overage
• cu unit cost of underage
• It can be shown that the optimal number of papers
  to purchase is the fractile of the demand
  distribution given by F(Q) cu / (cu co).

57
Determination of the Optimal Order Quantity for
Newsboy Example
58
Computing the Critical Fractile

• We wish to minimize competing costs (Co Cu)
• G(Q,D) CoMAX(0, Q-D) CuMAX(0, D-Q) ---- D
  is actual (potential) Demand
• G(Q) E(G(Q,D)) an expected value
• Therefore

59
Applying Leibnizs Rule

• d(G(Q))/dQ CoF(Q) Cu(1 F(Q))
• F(Q) is a cumulative Prob. Density Function (as
  earlier of the quantity ordered)
• Thus G(Q) (Cu)/(Co Cu)
• This is the critical fractile for the order
  variable

60
Lets see about this Prob 5 pg 241

• Observed sales given as a number purchased during
  a week (grouped)
• Lets assume some data was supplied
• Make Cost 1.25
• Selling Price 3.50
• Salvageable Parts 0.80
• Co overage cost 1.25 - 0.80 0.45
• Cu underage cost 3.50 - 1.25 2.25

61
Continuing

• Compute Critical Ratio
• CR Cu/(Co Cu) 2.25/(.45 2.25) .8333
• If we assume a continuous Probability Density
  Function (lets choose a normal distribution)
• Z(CR) ? 0.967 when F(Z) .8333 (from Std. Normal
  Tables!)
• Z (Q - ?)/?)
• From the problem data set, we compute mean 9856
  St.Dev. 4813.5)

62
Continuing

• Q ?Z ? 4813.5.967 9856 14511
• Our best guess economic order quantity is 14511
• (We really should have done it as a Discrete
  problem -- Taking this approach we would find
  that Q is only 12898)

63
Lot Size Reorder Point Systems

• Assumptions
• Inventory levels are reviewed continuously (the
  level of on-hand inventory is known at all times)
• Demand is random but the mean and variance of
  demand are constant. (stationary demand)

64
Lot Size Reorder Point Systems
Assumptions

• There is a positive leadtime, t. This is the time
  that elapses from the time an order is placed
  until it arrives.
• The costs are
• Set-up each time an order is placed at K per
  order
• Unit order cost at C for each unit ordered
• Holding at H per unit held per unit time ( i.
  e., per year)
• Penalty cost of P per unit of unsatisfied demand

65
Describing Demand

• The response time of the system (in this case) is
  the time that elapses from the point an order is
  placed until it arrives. Hence,
• The uncertainty that must be protected against is
  the uncertainty of demand during the lead time.
• We assume that D represents the demand during the
  lead time and has probability distribution F(t).
  Although the theory applies to any form of F(t),
  we assume that it follows a normal distribution
  for calculation purposes.

66
Decision Variables

• For the basic EOQ model discussed in Chapter 4,
  there was only the single decision variable Q.
• The value of the reorder level, R, was determined
  by Q.
• Now we treat Q and R as independent decision
  variables.
• Essentially, R is chosen to protect against
  uncertainty of demand during the lead time, and Q
  is chosen to balance the holding and set-up
  costs. (Refer to Figure 5-5)

67
Changes in Inventory Over Time for
Continuous-Review (Q, R) System
68
The Cost Function

• The average annual cost is given by
• Interpret n(R) as the expected number of
  stockouts per cycle given by the loss integral
  formula see Table A-4 (std. values).
• The optimal values of (Q,R) that minimizes G(Q,R)
  can be shown to be

69
Solution Procedure

• The optimal solution procedure requires iterating
  between the two equations for Q and R until
  convergence occurs (which is generally quite
  fast).
• A cost effective approximation is to set QEOQ
  and find R from the second equation.
• A slightly better approximation is to set Q
  max(EOQ,s)
• where s is the standard deviation of lead time
  demand when demand variance is high.

70
Service Levels in (Q,R) Systems

• In many circumstances, the penalty cost, p, is
  difficult to estimate. For this reason, it is
  common business practice to set inventory levels
  to meet a specified service objective instead.
  The two most common service objectives are
• Type 1 service Choose R so that the probability
  of not stocking out in the lead time is equal to
  a specified value.
• Type 2 service. Choose both Q and R so that the
  proportion of demands satisfied from stock equals
  a specified value.

71
Computations

• For type 1 service, if the desired service level
  is a then one finds R from F(R) a and QEOQ.
• Type 2 service requires a complex interative
  solution procedure to find the best Q and R.
  However, setting QEOQ and finding R to satisfy
  n(R) (1-ß)Q (which requires Table A-4) will
  generally give good results.

72
Comparison of Service Objectives

• Although the calculations are far easier for type
  1 service, type 2 service is generally the
  accepted definition of service.
• Note that type 1 service might be referred to as
  lead time service, and type 2 service is
  generally referred to as the fill rate.
• Refer to the example in section 5-5 to see the
  difference between these objectives in practice
  (on the next slide).

73
Comparison (continued)

• Order Cycle Demand
  Stock-Outs
• 1 180 0
• 2 75 0
• 3 235 45
• 4 140 0
• 5 180 0
• 6 200 10
• 7 150 0
• 8 90 0
• 9 160 0
• 10 40 0
• For a type 1 service objective there are two
  cycles out of ten in which a stockout occurs, so
  the type 1 service level is 80. For type 2
  service, there are a total of 1,450 units demand
  and 55 stockouts (which means that 1,395 demand
  are satisfied). This translates to a 96 fill
  rate.

74
(s, S) Policies

• The (Q,R) policy is appropriate when inventory
  levels are reviewed continuously. In the case of
  periodic review, a slight alteration of this
  policy is required. Define two levels, s lt S, and
  let u be the starting inventory at the beginning
  of a period. Then
• (In general, computing the optimal values of s
  and S is much more difficult than computing Q and
  R.)