1. Chapter 4 Inventory Control Subject to Known Demand
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Breakdown of the Total Investment in Inventories in the U. S
Breakdown of the Total Investment in Inventories in the U.S. Economy (1999)

3. Reasons for Holding Inventories
Economies of Scale
Uncertainty in delivery leadtimes
Speculation. Changing Costs Over Time
Smoothing.
Demand Uncertainty
Costs of Maintaining Control System

4. 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 it’s arrival. Can assume known or unknown.
Review Time - Is system reviewed periodically or is system state known at all times?
Treatment of Excess Demand.
Backorder all Excess Demand
Lose all excess demand
Backorder some and lose some
Inventory that changes over time
perishability
obsolescence

5. Relevant Costs a) Physical Cost of Space (3%)
Holding Costs - Costs proportional to the quantity of inventory held. Includes:
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%)
Note: Since inventory may be changing on a continuous basis, holding cost is proportional to the area under the inventory curve. (See examples.)

6. Inventory as a Function of Time

7. Relevant Costs (continued)
Ordering Cost (or Production Cost).
Includes both fixed and variable components.
slope = c K
C(x) = K + cx for x > 0 and =0 for x = 0.

8. Relevant Costs (continued)
Penalty or Shortage Costs. All costs that accrue when insufficient stock is available to meet demand. These include:
Loss of revenue for lost demand
Costs of bookeeping 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.

9. Simple EOQ Model 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).
4. Order quantity is fixed at Q per cycle. (can be proven optimal.)
5. Cost structure:
a) Fixed and marginal order costs (K + cx)
b) Holding cost at h per unit held per unit time.

10. Inventory Levels for the EOQ Model
Saw structure is typical. First order when inventory is 0. Reordering Q everytime when inventory is 0 must be optimal

11. The EOQ Model: Notation
D = λ 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 C(Q) = K + cQ Holding Cost:
Time (t)
Inventory (I(t))
Assume Constant Demand T Q
Time between orders
slope = -l
Instantaneous
Replenishment
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 G(Q) = average order cost + average holding cost
What is the average annual cost?
G(Q) = average order cost + average holding cost
Average ordering
cost per time T
Average inventory
level at any time

14. Total Costs
What is the average annual cost?

15. The Average Annual Cost Function G(Q)

16. The Average Annual Cost Function G(Q)

17. Minimize Annual Costs YES! Take the derivative of G(Q)
Is this a minimum?
EOQ:
YES!

19. Properties of the EOQ 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. 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

21. Example Q 240 T = = = . 07 years * (250 working days/year) l 3600 =
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? Q
240 T = = = . 07
years *
(250
working
days/year) l
3600 =
17.5
working
days = 18
days

22. Order Point for the EOQ Model
Does it matter if
τ < T or τ > T ?
Keep track of
time left to zero inventory or set automatic reorder at a particular
inventory level, R.
R = λ*τ, if τ < T
R = λ*MOD(τ/T),
if τ > T τ τ
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 τ

23. Sensitivity Analysis
independent
of Q
Let G(Q) be the average annual holding and set-up cost function given by
Holding & Setup costs
and let G* be the optimal average annual holding and setup cost. Then it can be shown that:
Cost penalties are quite small

24. 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 > λ),
arriving continuously.

25. Inventory Levels for Finite Production Rate Model

26. The EPQ Model: Notation
D = λ 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 = 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

27. The EPQ Model: Formula
For EOQ:

28. 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

29. All-Units Discount Order Cost Function

30. All-Units Discount Average Annual Cost Function
G(Q) Q
500
1,000
G0(Q)
G1(Q)
G2(Q)
Gmin(Q)

31. Incremental Discount Order Cost Function

32. Average Annual Cost Function for Incremental Discount Schedule

33. 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 EOQ’s.

34. Example
Supplier of paint to the maintenance department has announced new pricing:
$8 per gallon if order is < 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?

35. Solution Step 1: For Price 1:
Step 2: As Q(1) < 300, EOQ is realizable.
Step 3: Price 2:
Step 4: As Q(2) < 300, EOQ is not realizable.

36. Cost Function Realizable G(Q|p1) G(Q|p2) Not Realizable C(Q) 240 277
300 Q

37. Cost Function Only possible solutions G(Q|p1) G(Q|p2) C(Q) 240 277 300

38. Solution Step 5: Compare costs of possible solutions.
For $8 price, Q=240:
For $6 price, Q=300:
Q=300 is the optimal quantity.

39. 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 

40. 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:

41. 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:

42. 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 (n+1) equations

43. 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 (n+2) equations

44. 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

45. 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

46. Homework: Read Ch. 4 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
Work on appendix 4-A,

47. 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.

48. Reorder Point Calculation for Example 4.1

49. Reorder Point Calculation for Lead Times Exceeding One Cycle

50. 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:

51. EOQ With Finite Production Rate
Suppose that items are produced internally at a rate P > λ. 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- λ/P)

52. Inventory Levels for Finite Production Rate Model

53. 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.

54. All-Units Discount Order Cost Function

55. Incremental Discount Order Cost Function

56. 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 succeeding it. (See Figure 4-11).
For incremental discounts, the optimal will always occur at a realizable EOQ value. Compare costs at all realizable EOQ’s. (See Figure 4-12).

57. All-Units Discount Average Annual Cost Function

58. Average Annual Cost Function for Incremental Discount Schedule

59. 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:
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.

60. 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 we must have that

61. 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.
And the optimal production quantities are given by: