Production and Operation Managements

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Production and Operation Managements
Inventory Control Subject to Known Demand
Professor JIANG Zhibin Department of Industrial
Engineering Management Shanghai Jiao Tong
University
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• Inventory Control Subject to Known Demand

• Contents
• Types of Inventories
• Motivation for Holding Inventories
• Characteristics of Inventory System
• Relevant Costs
• The EOQ Model
• EOQ Model with Finite Production Rate
• Quantity Discount Models

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Introduction

• Inventory is the stock of any item or resource
  used in an organization.
• An inventory system is the set of policies and
  controls that monitors levels of inventory or
  determines what levels should be maintained.
• Generally , inventory is being acquired or
  produced to meet the need of customers
• Dependant demand system-the demand of components,
  subassemblies, and assemblies are intersected
  (lower levels depend on higher level)-MRP (
  Material Requirement Planning) system

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Introduction

• The fundamental problem of inventory management
  
• When to place order for replenish the stock ?
• How much to order?
• The complexity of the resulting model depends on
  the assumptions about the various parameters of
  the system
• The major distinction is between models for
  known demand and random demand.

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Introduction

• The current investment in inventories in USA is
  enormous
• It amounted up to 1.37 trillion in the last
  quarter of 1999
• It accounts for 20-25 of the total annual GNP
  (general net product)
• There exists enormous potential for improving the
  efficiency of economy by scientifically
  controlling inventories

Fig. 4-1 Breakdown of the Total Investment in
Inventories in the U.S. Economy (1999)
Inventory model discussed here are most
applicable to manufacturing, wholesale, and
retail sectors, composing 82 of the total .
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Types of Inventories

• A natural classification is by value added from
  manufacturing (the values are added to
  inventories at each level of the manufacturing
  operations, finally all values are cumulated with
  finished goods)
• Raw materials-Resources required in the
  production or processing activity of the firm.
• Components-Includes parts and subassemblies.
• Work-in-process (WIP)-the inventory either
  waiting in the system for processing or being
  processed.
• It may includes component inventories and some
  raw materials
• the level of WIP is taken as a measure of the
  efficiency of a production scheduling system.
• JIT aims at reducing WIP to zero.
• Finished good-also known as end items-the final
  products.

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Why Hold Inventories

• For economies of scale-It may be economical to
  produce a relatively large number of items in
  each production run and store them for future
  use.
• Coping with Uncertainties
• Uncertainty in demand
• Uncertainty in lead time
• Uncertainty in supply
• For speculation-
• Purchase large quantities at current low prices
  and store them for future use.
• Cope with considerable fluctuation in price of
  costly commodities required in large quantities
• Cope with labor strike

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Why Hold Inventories

• For Transportation
• Pipeline inventories is the inventory moving from
  point to point, e.g., materials moving from
  suppliers to a plant, from one operation to the
  next in a plant.
• It exists for purpose of transportation or
  materials handling in a plant
• Smoothing-Producing and storing inventory in
  anticipation of peak demand helps to alleviate
  the disruptions caused by changing production
  rates and workforce level.
• Logistics-To cope with constraints in purchasing,
  production, or distribution of items that may
  causes a system maintain inventory
• Purchase an item in minimum quantities
• Logistics of manufacture-zero inventory is
  impossible in order to keep continuity in
  manufacturing process
• Control costs-More inventory need less and
  simpler control

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Characteristics of Inventory Systems

• Demand patterns and characteristics
• Constant versus variable
• Known versus random
• Lead Time
• Ordered from the outside
• Produced internally
• Review patterns
• Continuous-supermarket
• Periodic-regular stock-taking for a grocery
  store
• Excess demand-demand that cannot be filled
  immediately from stock backordered or lost.
• Changing inventory
• Limited shelf life- perishability
• Become obsolete- obsolescence

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Relevant Costs- Holding cost

• Holding cost (carrying or inventory cost)-the sum
  of costs that are proportional to the amount of
  inventory physically on hand at any point in
  time.
• Some items of holding costs
• Cost of providing the physical space to store the
  items
• Taxes and insurance
• Breakage, spoilage, deterioration, and
  obsolescence
• Opportunity cost of alternative investment

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Relevant Costs- Holding cost

• Use cost of capital to account for the
  opportunity cost
• If we have 10,000RMB on hand, and save it in bank
  for one month at interest rate 0.25/month, then
  we may earn 25RMB/month from bank
• If we use this amount money to by some goods and
  store them in warehouse, then we lose
  25RMB/month.
• Inventory cost fluctuates with time-inventory as
  a function of time

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Relevant Costs- Holding cost
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Relevant Costs- Order cost

• It depend on the amount of inventory that is
  ordered or produced.
• Two components
• The fixed cost K independent of size of order as
  long as it is not zero
• The variable cost c incurred on per-unit basis

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Relevant Costs- Order cost
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Relevant Costs- Penalty Cost

• Also know as shortage cost or stock-out cost-is
  the cost of not having sufficient stock on hand
  to satisfy a demand when it occurs.
• Two interprets
• In back-order case include whatever bookkeeping
  and/or delay costs may be involved
• In lost-sale case include of loss-of-goodwill
  cost, a measure of customer satisfaction.
• Two approaches
• Penalty cost, p, is charged per-unit basis. Each
  time a demand occurs that cannot be satisfied
  immediately, a cost p is incurred independent of
  how long it takes to eventually fill the demand.
• Charge the penalty cost on a per-unit-time basis.

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The EOQ Model-Basic Model

• EOQ-economic order quantity model is the simplest
  and most fundamental of all inventory models.
• The basic assumption
• Known and constant demand rate ? (units/unit
  time, yr)
• No shortage
• No order lead time (will be relaxed)
• Costs include
• Set up cost at K per positive order placed
• Proportional order cost at c per unit ordered
• Holding cost at h per unit held per unit time

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The EOQ Model-Basic Model

• Considerations
• On-hand inventory level at the time zero is zero
• An order must be placed at time zero
• Q is the size of the order (lot size)
• Next order is placed just when the inventory
  level drops to zero
• The order cycle TQ/ ?

Average holding cost is Q/2

• The objective is to choose Q to minimize the
  average cost per unit time (usually, a year)

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The EOQ Model-Basic Model

• Express the average annual cost as a function of
  the lot size
• Order cost in each order cycle C(Q)KcQ
• The average holding cost during one order cycle
  is hQ/2
• The average annual cost (suppose there are n
  cycles in a year)

Average holding cost for one cycle that for one
year.
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The EOQ Model-Basic Model

• Since Qgt0,G(Q) is convex function of Q
• G(Q) is minimized at Q--economic order quantity,
  EOQ

• Notes
• Q is the value of Q where the two curves
  interest
• The constant order cost component, c, does not
  appear explicitly in the expression of Q, since
  ?c is independent from Q in the C(Q)
• Because ?c is constant, it is ignored while
  computing average cost.

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The EOQ Model-Basic Model

• Example 4.1
• Pencils are sold at a fairly steady rate of 60
  per week
• Pencils cost 2 cents each and sell for 15 cents
  each
• Cost 12 to initiate an order, and holding costs
  are based on annual interest rate of 25.
• Determine the optimal number of pencils for the
  book store to purchase each time and the time
  between placement of orders

• Solutions
• Annual demand rate ??60??523,120
• The holding cost is the product of the variable
  cost of the pencil and the annual interest-h0.02
  ?0.250.05

Back
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The EOQ Model-Considering Lead Time

• Since there exits lead time ? (4 moths for
  Example 4.1), order should be placed some time
  ahead of the end of a cycle
• Reorder point R-determines when to place order in
  term of inventory on hand, rather than time.

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The EOQ Model-Considering Lead Time

• Determine the reorder point when the lead time
  exceeds a cycle.

Computing R for placing order 2.31 cycles ahead
is the same as that 0.31 cycle ahead.

• Example
• EOQ25
• ?500/yr
• ?6 wks
• T25/5002.6 wks
• ?/T2.31---2.31 cycles are included in LT.
• Action place every order 2.31 cycles in advance.

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The EOQ Model- Sensitivity

• How sensitive is the annual cost function to
  errors in the calculation of Q?

• Considering Example 4.1. Suppose that the
  bookstore orders pencils in batches of 1,000,
  rather than 3,870 as the optimal solution
  indicates. What additional cost is it incurring
  by using a suboptimal solution?

By substituting Q1,000, we can find the average
annual cost for this lot size.
Which is considerably larger than the optimal
cost of 19.35.
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The EOQ Model- Sensitivity

• Lets obtain a universal solution to the
  sensitivity problem.
• Let G be the average annual holding and setup
  cost at the optimal solution. Then

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The EOQ Model- Sensitivity

• To see how one would use this result, consider
  using a suboptimal lot size in Example 4.1. The
  optimal solution was Q3,870, and we wished to
  evaluate the cost error of using Q1,000. Forming
  the ratio Q/Q gives 3.87. Hence,
  G(Q)/G(0.5)(3.871/3.87)(0.5)(4.128)2.06.
  This says that the average annual holding and
  setup cost with Q1,000 is 2.06 times the optimal
  average holding and setup cost.
• In general, the cost function G(Q) is relative
  insensitive to errors in Q. For example, if Q is
  twice as large as Q, then G/Q1.25 , meaning
  that an error of 100 in Q will generate an error
  of 25 in annual average cost.
• And suppose that the order quantity differed from
  the optimal by ?Q units. A value of QQ ?Q
  would result in a lower average annual cost than
  a value of QQ- ?Q. ---Not symmetric.

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The EOQ Model for Finite Production Rate

• The simple EOQ model is based on assumption that
  the items are obtained from an outside supplier,
  and thus entire lot is delivered at the same
  time
• EOQ model is also effective when units are
  internally produced, based on assumption that
  production rate is infinite
• If the production rate is finite and comparable
  to the rate of demand, the simple EOQ model will
  be ineffective.

• Assumption
• Items are produced at a rate P during a
  production run
• Pgt??? for feasibility
• Let
• Q is the lot size of each production run
• T is the cycle length, the time between
  successive startups. TT1T2 , where T1 is
  production time, while T2 is the downtime (no
  production)
• Note that the maximum level of on-hand inventory
  during a cycle is no longer Q.

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The EOQ Model for Finite Production Rate

• The number of units consumed in each cycle is ?T
• The number of units produced at rate P in a
  production run T is QT1P
• ?TT1P?Q?T1Q/P
• The maximum level of inventory on hand is HT1(P-
  ?)Q(1- ?/P)
• Since average inventory level is H/2, thus the
  annual average inventory cost follows

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The EOQ Model for Finite Production Rate
Example 4.3

• Determine
• Optimized size of a production run Q
• The length of each production run T
• The average annual cost of holding cost and
  setup
• Maximum level of inventory on hand.

• Given
• P10,000 units/yr
• K50
• ??2,500 units/yr
• h2?0.30.6

• Solutions
• hh(1- ? / P)0.6(1- 2,500 / 10,000)0.45
• Q(2K ?/h)1/2745
• TQ/ ?745/2,5000.298 yr
• The production time (uptime) T1Q/P0.0745 yr
• The downtime is T2T-T10.2235 yr
• G(Q)K ? / QhQ/2335.41
• HQ(1- ?/P)559 units

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Quantity Discount Model

• The suppliers may charge less per unit for larger
  orders to encourage the customer to buy their
  products in larger batches.
• Two popular ways of discounts
• All-units discount is applied to all of the
  units in an order
• Incremental only applied to additional units
  beyond the breakpoints

• Example 4.4 Weighty Trash Bag Companys pricing
  schedule for its large trash can liners
• For orders of less than 500 bags, charges 30
  cents per bag
• for orders of 500 or more but fewer than 1,000
  bags, charges 29 cents per bag and
• for orders of 1,000 or more, charges 28 cents per
  bag.

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Quantity Discount Model

• The breakpoints are 500 and 1,000. The discount
  schedule is all-units
• The order cost function C(Q) is defined as

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Quantity Discount Model- Optimal Policy for
All-Units Discount Schedule

• Example 4.4 If Weighty uses trash bags at a
  fairly constant rate of 600 per yr, how to place
  order?
• Suppose that fixed cost of placing an order is
  8, and holding costs are based on 20 annual
  interest rate.
• First compute EOQ values corresponding to each of
  the unit cost.

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Quantity Discount Model- Optimal Policy for
All-Units Discount Schedule

• Each curve is valid only for certain values of Q,
  thus the average annual cost function is given by
  discontinuous curves.

• An EOQ value is realizable, if it falls within
  the interval of EOQ that corresponds to the unit
  cost that has been used to compute it. (Q0 for
  the example)

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Quantity Discount Model- Optimal Policy for
All-Units Discount Schedule

• The goal is to find the minimum of this
  discontinuous curve, which corresponds to the
  EOQ.
• Generally, the optimal solution will be either
  the largest realizable EOQ or one of the
  breakpoints that exceeds it.
• The three candidates are 400, 500, and 1,000.

Conclusion the optimal solution is to place a
standing order for 500 units with Weighty at an
annual cost of 198.10.
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Quantity Discount Model- Incremental Quality
Discounts
The order function follows
The average annual cost function follows
G(Q)?C(Q)/QK ?/QIC(Q)Q/2
G(Q) has been divided into three segments G1(Q),
G2(Q), and G3(Q), each of which is obtained by
using one of the three segments of C(Q) .
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Quantity Discount Model- Incremental Quality
Discounts

• The optimal solution occurs at the minimum of one
  of the three average annual cost curves.
• Procedures
• Compute the three minima of the three curves
• Find the realizable values (the minimum falls
  into correct interval)

(3) Compare G at these realizable values, one
with the smallest G is the optimal
solution. Q0400 (R), Q1519 (R), Q2702