Inventory Control

1
Inventory Control

• Mahmut Ali GÖKÇE
• Industrial Systems Engineering Dept.
• Izmir University of Economics

2
Some Basic Definitions

• An inventory is an accumulation of a commodity
  that will be used to satisfy some future demand.
• Inventories of the following form
• - Raw material
• - Components (subassemblies)
• - Work-in-process
• - Finished goods
• - Spare parts
• - Purchased products in retailing

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Functional Classification of Inventories

• Anticipation Stock You have to keep some
  inventory to satisfy the expected demand of the
  customer.
• Customer comes to the store and requires
  immediate purchase of what s/he needs.
• You have to keep some items ready to satisfy such
  immediate requests based on your anticipation of
  the average demand
• These inventories are referred to as anticipation
  stock

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Functional Classification of Inventories

• Cycle Inventories produce or buy in larger
  quantities than needed.
• Economies of scale
• Quantity discounts
• Restrictions (technological,transportation,)

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Functional Classification of Inventories

• Safety Stock Provides protection against
  irregularities and uncertainties, in order to
  avoid stockouts.

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Functional Classification of Inventories

• Production Smoothing Consider, low demand in one
  part of the year, hence build up stock for the
  high demand season.

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Functional Classification of Inventories

• Hedge inventories expect changes in the
  conditions (price, strike, supply, etc.)
• Pipeline (or work-in-process) inventories goods
  in transit, between levels of a supply chain,
  between work stations.

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Characteristics of Inventory Systems

• Demand
• Constant versus Variable
• Known versus Random
• Lead Time
• Review Time
• Periodic or Continuous
• Excess Demand
• Backorder allowed - Lost Partial - Impatience
• Changing Inventory
• Perishable or Not

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Framework for Inventory Control

• Large number of items
• Large manufacturer 500,000 items
• Retailer 100,000
• Items show different characteristics
• Demand can occur in many ways
• Unit by unit, in cases, by the dozen, etc.

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Goals Reduce Cost, Improve Service

• By effectively managing inventory
• Xerox eliminated 700 million inventory from its
  supply chain
• Wal-Mart became the largest retail company
  utilizing efficient inventory management
• GM has reduced parts inventory and transportation
  costs by 26 annually

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Goals Reduce Cost, Improve Service

• By not managing inventory successfully
• In 1994, IBM continues to struggle with
  shortages in their ThinkPad line (WSJ, Oct 7,
  1994)
• In 1993, Liz Claiborne said its unexpected
  earning decline is the consequence of higher than
  anticipated excess inventory (WSJ, July 15,
  1993)
• In 1993, Dell Computers predicts a loss Stock
  plunges. Dell acknowledged that the company was
  sharply off in its forecast of demand, resulting
  in inventory write downs (WSJ, August 1993)

12
Understanding Inventory

• The inventory policy is affected by
• Demand Characteristics
• Lead Time
• Number of Products
• Objectives
• Service level
• Minimize costs
• Cost Structure

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Cost Structure

• Order costs
• Fixed (Set-up cost, K)
• Variable (c)
• Holding Costs (h)
• Insurance
• Maintenance and Handling
• Taxes
• Opportunity Costs
• Obsolescence (risk of loosing some of its value)
• Stock out Cost (p)
• Customer goodwill
• Lost sales

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3 Basic Questions

• Decision making in production and inventory
  management involves dealing with large number of
  items, with very diverse characteristics and with
  external factors.
• We want to resolve
• How often the inventory status (of an item)
  should be determined ?
• When a replenishment order should be placed ?
• How large the replenishment order should be ?

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Why EOQ ? Economic Order Quantity

• Easy to compute
• Does not require data that is hard to obtain
• Policies are surprisingly robust
• Assumptions can be relaxed
• Gives a good overall idea
• Can be starting point for more complicated models

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Assumptions Leading to EOQ

• Demand rate, D, is constant and deterministic
  over time (units/day, units/year, etc.)
• The order quantities are fixed at Q items per
  order, need not be discrete, and there are no
  minimum or maximum restrictions
• Unit variable cost, v, does not depend on Q (no
  discounts)
• A fixed cost K is incurred every time an order is
  placed
• Cost factors do not change over time
• Single item (no interaction with others)

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EOQ Assumptions (Continued)

• Replenishment lead time is negligible
• No shortages are allowed
• Entire order quantity is delivered at the same
  time
• The planning horizon is infinite
• The initial inventory is 0
• As we will see, these assumptions can be relaxed.

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Economic Order Quantity Model

• Our aim is to determine the best replenishment
  strategy (recall when and how much to order)
  under the criterion that the relevant costs will
  be minimized over time (i.e., minimize cost per
  unit time)
• It is reasonable to consider the following
  optimization problem

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EOQ Model - Intuition

• When we should place a new replenishment order ?
• Demand is deterministic and at a constant rate
• lead time is negligible
• no backorders are allowed
• Hence, a replenishment order should be placed
  each time inventory level drops to zero

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EOQ Model - Intuition

• How much we have to order each time ?
• Parameters do not change over time !
• There is no reason for ordering different
  quantities. So, each time a replenishment order
  is placed we order the same quantity Q

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EOQ - Notation

• Recall the notation we introduced earlier
• Q replenishment order quantity
• K fixed cost incurred with each replenishment
• c unit variable cost of an item /unit
• h the holding cost
• D demand rate, units/unit time

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Economic Lot Size Model
Costs K order cost h inventory holding cost Q
order size (decision var.)
Inventory
T
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Deriving EOQ

• Note from the figure that the same picture occurs
  over and over again (why ? Everything is
  stationary over time!) . Any one of the triangles
  will be called a cycle. Then,

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Deriving EOQ

• Cycle time T Q/D (slope, -D, -Q/T)
• Since Q/D is the time between two replenishments
  (and the cycle time), D/Q is the number of
  replenishments per unit time (4 months between
  two replenishments will give 0.25 replenishments
  per month).
• In each replenishment we pay, KQc (at each
  cycle)
• Replenishment costs/unit time

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Deriving EOQ

• In each replenishment cycle the total inventory
  carried is the area of one triangle. Why ? It can
  be computed as
• Then inventory carried per unit time (average
  inventory) is Q/2 (also by intuition !).
• Inventory holding cost/unit time Q h /2
• Inventory holding cost at each cycle Q hT /2

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EOQ Total Cost is Convex
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EOQ Total Cost is Convex

• Lets verify For Q gt 0
• TRC(Q) is convex

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Deriving EOQ

• Average inventory level in a cycle is Q/2
• Average inventory cost at each cycle is hTQ/2
• Total cost at every cycle is C(Q) K hTQ/2
• Cycle time T Q/D
• Cost per unit time is KD/Q hQ/2

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EOQ Costs
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Turnover Ratio

• Turnover Ratio A measure of effective inventory
  control D/(Average Inventory)

At optimal Q value
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EOQ An Example

• Number 2 pencils at the bookstore are sold at a
  fairly steady rate of 60 per week. The pencils
  cost the book store 2 cents each and sell for 15
  cents each. It costs the book store 12 to
  initiate an order, and holding costs are based on
  an annual interest rate of 25. Determine the
  optimum number of pencils for the bookstore to
  purchase and the time between placements of
  orders. What are the yearly holding and setup
  costs for the item?

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EOQ An Example

• h Ic0.250.020.005 per unit per year
• Order Quantity that Minimizes Average Total Cost
• (2KD/h)1/2 (2123120/.005)1/2
• 3870 units
• T cycle time Q/D3870/3120 1.24 years
• Average annual holding cost hQ/2.0053870/29.
  675
• Average Annual setup cost KD/Q
  123120/38709.675

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EOQ lead time
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EOQ Sensitivity

• If we cannot order Q units?
• G(Q) KD/Q hQ/2
• G(Q) KD/ Q h Q/2
• For any other Q

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EOQ Sensitivity

• Let Q3870. What if we order Q1000
• G(Q) /G(Q) 0.5 (3.87 1/3.87) 2.06
• We can conclude that G(Q) is relatively
  insensitive to errors in Q
• Hence one can order 4000 pencils if space
  available rather than 3870!

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Relaxing EOQ Production
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EOQProduction - Example

• A local company produces programmable EPROM for
  several industrial clients. It has experienced a
  relatively flat demand of 2,500 units per year
  for the product. The EPROM is produced at a rate
  of 10,000 units per year. The accounting
  department has estimated that it costs 50 to
  initiate a production run, each unit costs the
  company 2 to manufacture, and the cost of
  holding is based on a 30 percent annual interest
  rate. Determine the optimal size of a production
  run, the length of each production run, and the
  average cost of holding and setup. What is the
  maximum level of on-hand inventory of EPROMs?
  (Nahmias, p. 213)

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Relaxing EOQ Quantity Discounts
G(Q)
G0(Q)
G1(Q)
G2(Q)
Q
500
1000
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EOQQuantity Discounts - Example
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Multiproduct Systems A-B-C Analysis

• There is a trade off between the cost of
  controlling the system and the potential benefits
  from that control.
• Vilfredo Pareto studied the distribution of
  wealth in 19th century and noted that large
  portion of the wealth is owned by small segment
  of the population.
• Typically the top 20 of the items account for
  the 80 of the annual dollar value of sales, the
  next 30 percent for the next 15.
• To use ABC
• select criterion for ranking
• rank items on basis of criterion
• calculate percentages
• set up classes around break points

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Example A-B-C Analysis
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Example A-B-C Analysis Sorted by Usage
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Example A-B-C Analysis Cumulatives

• 5 24.22064 (39340/162426)
• 10 45.02026
• 15 62.86979
• 20 79.98034
• 25 89.1277
• 30 92.82754
• 35 96.23547
• 40 99.04374
• 45 99.46436
• 50 99.65564

55 99.73275 60 99.80628 65 99.86058 70 99.91023
75 99.94351 80 99.96693 85 99.98471 90 99.9923
1 95 99.9982 100 100
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Example A-B-C Analysis Cumulatives Graph
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A-B-C Analysis

• Since A items account for the lions share of the
  yearly revenue, these items should be watched
  closely and inventory levels for A items should
  be the monitored continuously.
• More sophisticated forecasting techniques might
  be used
• More care would be taken in the estimation of
  the various cost parameters required in
  calculating optimal policies.