Chapter 4 Inventory Control Subject to Deterministic Demand

1
Chapter 4Inventory Control Subject to
Deterministic Demand
2
Operations Management

• Inventory control
• Scheduling
• Capacity planning
• Forecasting
• Quality control
• Equipment maintenance

• Finished product
• Work-In-Progress
• Components
• Rough materials

3
Reasons for Holding Inventories

• Economies of Scale
• cheaper to produce/order in large batches
• Uncertainties
• demand lead times delivery of supply
• Speculation
• finished inventory held in anticipation of a rise
  in their value or costs
• Smoothing
• irregularities of demand pattern
• Transportation
• inventory in transit from one location to other
• Costs of Maintaining Control System
• can be cheaper to order once a year, than
  continuously monitor orders and deliveries

4
Relevant Costs

• Holding Costs - Costs proportional to the
  quantity of inventory held. Includes, for
  example
• a) Physical Cost of Space (3 )
• b) Taxes and Insurance (2
  )
• c) Breakage Spoilage and Deterioration
  (1 )
• d) Opportunity Cost of alternative investment
  (18 )
• (Total 24 )

5
Relevant Costs (continued)

• Penalty or Shortage Costs. Including
• Loss of revenue for lost demand
• Costs of book keeping for backordered demands
• Loss of goodwill for being unable to satisfy
  demands when they occur.
• Generally assume cost is proportional to number
  of units of excess demand.

6
Relevant Costs (continued)

• Ordering Cost (or Production Cost)
• Includes both fixed and variable components
• K

7
Example MedEquip dilemma Harris 1913

• Small manufacturer of operating-room monitoring
  and diagnostic equipment
• Production process mounting electronic
  components in standard metal racks
• The racks are purchased from a local metalworking
  shop, which must set up its equipment (presses,
  machining stations, welding stations) every time
  it produces run of racks ? setup cost
• As result , the metalworking shop does not want
  to produce small batches each time, but MedEquip
  does not want to buy in excessive quantities ?
  inventory cost
• How many parts to make/order at once?

8
Inventory Levels for the EOQ Model
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The EOQ Model Assumptions

• Production is instantaneous
• There is no capacity constraint, and entire lot
  is produced simultaneously
• Delivery is immediate
• There is no time lag between production and
  availability to satisfy demand
• Demand is deterministic
• There is no uncertainty about the quantity or
  timing of demand
• Demand is constant over time
• It can be presented as a straight line, so that
  if annual demand is 365 units, this translates to
  a daily demand of one unit
• A production run incurs a fixed setup cost
• Regardless of the size of the lot or the status
  of the factory, the setup cost is the same
• Products can be analyzed individually
• Either there is only a single product or there
  are no interactions (e.g. shared equipment)
  between products

10
Inventory Levels for the EOQ Model
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The EOQ Model Notation

• D ? is the demand rate (in units per year)
• c unit production cost, not counting setup or
  inventory costs (in dollars per unit)
• K setup costs (per placed order) in dollars
• h holding cost (in dollars per unit per year),
  if the holding cost consists entirely of interest
  on money tied up in inventory,
• h ic, where i is an annual interest rate
• Q lot size (order size) in units
• T time between orders (cycle length)
• G(Q) average annual cost

12
Relationships

• Ordering Costs (Order amount Q)
• C(Q) K cQ
• Holding Cost
• h Ic (Interest Rate)(Cost of Inv.)
• Average Inventory Size?
• Under constant demand Q/2
• Time Between Orders
• l Q/T
• T Q/l

Rate of consumption l
Q
T
13
Total Costs

• What is the average annual cost?
• G(Q) average order cost average holding cost

14
Total Costs

• What is the average annual cost?

15
The Average Annual Cost Function G(Q)
16
Minimize Annual Costs

• Take the derivative of G(Q)
• Is this a minimum?
• EOQ

YES!
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Properties of the EOQ Solution

• This formula is well-known economic order
  quantity, is also known as economic lot size
• This is a tradeoff between lot size and inventory
• Garbage in, garbage out - usefulness of the
  EOQ formula for computational purposes depends on
  the realism of input data
• Estimating setup cost is not easily reduced to a
  single invariant cost K

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Example

• Uvic requires 3600 gallons of paint annually for
  scheduled maintenance of buildings. Cost of
  placing an order is 16 and the interest rate
  (annual) is 25. Price of paint is 8 per
  gallon.
• How much paint should be ordered, and how often?

240
Q




days/year)

working
(250



years

07
.
T
l
3600
days

18



days

working
17.5


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Order Point for the EOQ Model

• Does it matter if
• t lt T or t gt T ?
• Keep track of
• time left to zero inventory or set
  automatic reorder at a particular
• inventory level, R.
• R ?t, if t lt T
• R ?MOD(t/T),
• if t gt T

Assumption Delivery is immediate There is no
time lag between production and availability to
satisfy demand Relax this assumption! Let the
order lead time to be equal to t
20
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 holding
and setup cost. Then it can be shown that
Cost penalties are quite small
21
Finite Replenishment Rate Economic
Production Quantity (EPQ)

• Assumptions for EOQ
• Production is instantaneous
• There is no capacity constraint, and entire lot
  is produced simultaneously
• Delivery is immediate
• There is no time lag between production and
  availability to satisfy demand
• Example
• Parts produced at the same factory
• production rate is P (P gt ?),
• arriving continuously.

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Inventory Levels for Finite Production Rate Model
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The EPQ Model Notation

• D ? is the demand rate (in units per year)
• c unit production cost, not counting setup or
  inventory costs (in dollars per unit)
• K setup costs (per placed order) in dollars
• h holding cost (in dollars per unit per year),
  if the holding cost consists entirely of
  interest on money tied up in inventory, hic,
  where i is an annual interest rate
• Q size of each production run (order) in units
• T time between initiation of orders arrival
  (cycle length)
• T1 production (replenishment) time
• T2 downtime
• H maximum on-hand inventory
• G(Q) average annual setup holding cost

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The EPQ Model Formula
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Quantity Discount Models

• One of the most severe assumptions the unit
  variable cost c did not depend on the
  replenishment quantity
• In practice quantity discounts exist based on
  the purchase price or transportation costs take
  advantage of these can result in substantial
  savings
• 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 in Ch. 4.7
• 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

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All-Units Discount Order Cost Function
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The Average Annual Cost Function G(Q)
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All-Units Discount Average Annual Cost Function
G0(Q)
G1(Q)
G2(Q)
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Incremental Discount Order Cost Function
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Average Annual Cost Function for Incremental
Discount Schedule
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Properties of the Optimal Solutions

• For all units discounts, the optimal will occur
  at the minimum point of one of the cost curves or
  at a discontinuity point
• One compares the cost at the largest realizable
  EOQ and all of the breakpoints succeeding it
• For incremental discounts, the optimal will
  always occur at a realizable EOQ value.
• Compare costs at all realizable EOQs.

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Example

• Supplier of paint to the maintenance department
  has announced new pricing
• 8 per gallon if order is lt 300 gallons
• 6 per gallon if order is 300 gallons
• Data remains as before
• K 16, I 25, l 3600
• Is this a case of all units or incremental
  discount?

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Solution

• Step 1 For Price 1
• Step 2 As Q(1) lt 300, EOQ is realizable.
• Step 3 Price 2
• Step 4 As Q(2) lt 300, EOQ is not realizable.

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Cost Function
Realizable
G(Qp1)
C(Q)
G(Qp2)
Not Realizable
240
277
Q
300
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Cost Function
Only possible solutions
G(Qp1)
C(Q)
G(Qp2)
240
277
Q
300
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Solution

• Step 5 Compare costs of possible solutions.
• For 8 price, Q240
• For 6 price, Q300
• Q300 is the optimal quantity.

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Resource Constrained Multi-Product Systems

• Classic EOQ model is for a single item. Setup
  plan for n items.
• Option A Treat one system with multiple items
  as multiple systems with one item
• Works if There are no interactions among items,
  such as sharing common resources budget,
  storage capacity, or both
• Option B Modify classic EOQ to insure no
  violation of the resource constraints
• Works if Have not made any mistakes and know how
  to use Lagrange multipliers ?

38
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 budget constraint on
  the system
• , where Qi is the order size for product i
• , where wi is the volume occupied by product
  i
• Minimize
• s.t. and

For EOQ
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Resource Constrained Multi-Product Systems

• Minimize
• s.t.

Budget constraint Space constraint
Lagrange multipliers method relax one or more
constraints Minimize by solving necessary
conditions
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Resource Constrained Multi-Product Systems Steps
to Find Optimal Solution

• Single constraint
• Solve the unconstrained problem. If constraint is
  satisfied, this solution is the optimal one.
• If the constraint is violated, rewrite objective
  function using Lagrange multipliers
• Obtain optimal Qi by solving (n1) equations

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Resource Constrained Multi-Product Systems Steps
to Find Optimal Solution

• Double constraints
• Solve the unconstrained problem. If both
  constraints are satisfied, this solution is the
  optimal one.
• Otherwise rewrite objective function using
  Lagrange multipliers by including one of the
  constraints, say budget, and solve one-constraint
  problem to find optimal solution. If the space
  constraint is satisfied, this solution is the
  optimal one.
• Otherwise repeat the process for the only space
  constraint.
• If both single-constraint solutions do not yield
  the optimal solution, then both constraints are
  active, and the Lagrange equation with both
  constraints must be solved.
• Obtain optimal Qi by solving (n2) equations

42
EOQ Models for Production Planning

• Problem determine optimal procedure for
  producing n products on a single machine
• Consider n items with known demand rates ,
  production rates , holding costs , and
  set-up costs . The objective is to minimize
  the cost of holding and setups, and to have no
  stock-outs. For the problem to be feasible we
  must have that
• Assumption rotation cycle policy exactly one
  setup for each product in each cycle production
  sequence stays the same in each next cycle

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The method of solution is to express the average
annual cost function in terms of the cycle time,
T to assure no stock-outs. The optimal cycle time
has the following mathematical form, where sj is
a setup time And the optimal production
quantities are given by where T max T,
Tmin, see pp.216-217
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Recommended problems

• Look over Ch. 4.1 4.4
• Read Ch. 4.5 4.9
• Problems
• 4.5, 4.12, 4.15, 4.16
• 4.17, 4.18, 4.22, 4.24, 4.25
• 4.26, 4.27, 4.28, 4.30

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References

• Presentations by McGraw-Hill/Irwin and by
  Wilson,G.R.
• Production Operations Analysis by S.Nahmias
• Factory Physics by W.J.Hopp, M.L.Spearman
• Inventory Management and Production Planning and
  Scheduling by E.A. Silver, D.F. Pyke, R.
  Peterson
• Production Planning, Control, and Integration
  by D. Sipper and R.L. Bulfin Jr.