1. Chapter 8
Inventory Models

2. 8.1 Overview of Inventory Issues
Proper control of inventory is crucial to the success of an enterprise.
Typical inventory problems include:
Basic inventory – Planned shortage
Quantity discount – Periodic review
Production lot size – Single period
Inventory models are often used to develop an optimal inventory policy, consisting of:
An order quantity, denoted Q.
A reorder point, denoted R.

3. Type of Costs in Inventory Models
Inventory analyses can be thought of as cost-control techniques.
Categories of costs in inventory models:
Holding (carrying costs)
Order/ Setup costs
Customer satisfaction costs
Procurement/Manufacturing costs

4. Type of Costs in Inventory Models
Holding Costs (Carrying costs): These costs depend on the order size
Cost of capital
Storage space rental cost
Costs of utilities
Labor
Insurance
Security
Theft and breakage
Deterioration or Obsolescence
Ch = Annual holding cost per unit
in inventory
H = Annual holding cost rate
C = Unit cost of an item
Ch = H * C

5. Type of Costs in Inventory Models
Order/Setup Costs
These costs are independent of the order size.
Order costs are incurred when purchasing a good from a supplier. They include costs such as
Telephone
Order checking
Labor
Transportation
Setup costs are incurred when producing goods for sale to others. They can include costs of
Cleaning machines
Calibrating equipment
Training staff
Co = Order cost or setup cost

6. Type of Costs in Inventory Models
Customer Satisfaction Costs
Measure the degree to which a customer is satisfied.
Unsatisfied customers may:
Switch to the competition (lost sales).
Wait until an order is supplied.
When customers are willing to wait there are two types of costs incurred:
Cb = Fixed administrative costs of an out of stock item ($/stockout unit).
Cs = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year).

7. Type of Costs in Inventory Models
Procurement/Manufacturing Cost
Represents the unit purchase cost (including transportation) in case of a purchase.
Unit production cost in case of in-house manufacturing.
C = Unit purchase or manufacturing cost.

8. Demand in Inventory Models
Demand is a key component affecting an inventory policy.
Projected demand patterns determine how an inventory problem is modeled.
Typical demand patterns are:
Constant over time (deterministic inventory models)
Changing but known over time (dynamic models)
Variable (randomly) over time (probabilistic models)
D = Demand rate (usually per year)

9. Inventory Classifications
Inventory can be classified in various ways:
Used typically by accountants at manufacturing firms.
Enables management to track the production process.
Items are classified by their relative importance
in terms of the firm’s capital needs.
Management of items with short shelf life and long shelf life is very different

10. Review Systems Two types of review systems are used:
Continuous review systems.
The system is continuously monitored.
A new order is placed when the inventory reaches a critical point.
Periodic review systems.
The inventory position is investigated on a regular basis.
An order is placed only at these times.

11. 8.2 Economic Order Quantity Model - Assumptions
Demand occurs at a known and reasonably constant rate.
The item has a sufficiently long shelf life.
The item is monitored using a continuous review system.
All the cost parameters remain constant forever (over an infinite time horizon).
A complete order is received in one batch.

12. The EOQ Model – Inventory profile
The constant environment described by the EOQ assumptions leads to the following observation:
The optimal EOQ policy consists of same-size orders. Q
This observation results in the following inventory profile :

13. Cost Equation for the EOQ Model
Total Annual
Inventory Costs
Total Annual
Holding Costs
Total Annual
ordering Costs
Total Annual
procurement Costs = + +
TC(Q) =
(Q/2)Ch +
(D/Q)Co + DC Q Ch
The optimal order Size
Q* =

14. Note: at the optimal order size total holding costs and ordering costs
TV(Q) and Q*
TV(Q)
Constructing the total annual variable cost curve
Add the two curves to one another
Total Holding Costs
Total annual holding and
ordering costs * *
Note: at the optimal order size
total holding costs and ordering costs
are equal * * o *
Total ordering costs Q Q*
The optimal order size

15. Sensitivity Analysis in EOQ models
The curve is reasonably flat around Q*.
Deviations from the optimal order size
cause only small increase in the total cost. Q*

16. Cycle Time
The cycle time, T, represents the time that elapses between the placement of orders.
Note, if the cycle time is greater than the shelf life, items will go bad, and the model must be modified.
T = Q/D

17. Number of Orders per Year
To find the number of orders per years take the reciprocal of the cycle time
N = D/Q
Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy.
How many orders are placed per year? N = 1000/250 = 4 orders.
How often orders need to be placed (what is the cycle time)? T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.

18. Lead Time and the Reorder Point
In reality lead time always exists, and must be accounted for when deciding when to place an order.
The reorder point, R, is the inventory position when an order is placed.
R is calculated by
L and D must be expressed in the same time unit.
R = L D

19. Lead Time and the Reorder Point – Graphical demonstration: Short Lead Time
Inventory position
Place the order
now L
R = Inventory at hand at the beginning of lead time

20. Lead Time and the Reorder Point – Graphical demonstration: Long Lead Time
Outstanding order
R = inventory at hand
at the beginning of lead time +
one outstanding order
= demand during lead time
= LD
Inventory at
hand L
Place the order now

21. Safety stock Safety stocks act as buffers to handle:
Higher than average lead time demand.
Longer than expected lead time.
With the inclusion of safety stock (SS), R is calculated by
The size of the safety stock is based on having a desired service level.
R = LD + SS

22. Safety stock Reorder Point L Place the order now R = LD Planned
situation
Actual
situation
Reorder
Point
Place the order
now L
R = LD

23. Safety stock Reorder Point ? L Place the order now R = LD + SS Actual
situation ?
Reorder
Point LD
SS=Safety stock
Place the order
now L
The safety stock
prevents excessive shortages.
R = LD
+ SS

24. Inventory Costs Including safety stock
Total Annual
Inventory Costs
Total Annual
Holding Costs
Total Annual
ordering Costs
Total Annual
procurement Costs = + +
TC(Q) =
(Q/2)Ch +
(D/Q)Co +
DC + ChSS
Safety stock holding cost

25. ALLEN APPLIANCE COMPANY (AAC)
AAC wholesales small appliances.
AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units.
Management wishes to determine an optimal ordering policy for the Citron brand juicer

26. ALLEN APPLIANCE COMPANY (AAC)
Data
Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr)
Ch = $1.40 [HC = (14%)($10)]
C = $10.
H = 14% (10% ann. interest rate) + (4% miscellaneous)
D = demand information of the last 10 weeks was collected:

27. ALLEN APPLIANCE COMPANY (AAC)
Data
The constant demand rate seems to be a good assumption.
Annual demand = (120/week)(52weeks) = 6240 juicers.

28. AAC – Solution: EOQ and Total Variable Cost
Current ordering policy calls for Q = 600 juicers.
TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80
The EOQ policy calls for orders of size
Savings of 16% Ö
2(6240)(12)
1.40 =
327 = Q*
TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

29. AAC – Solution: Reorder Point and Total Cost
Under the current ordering policy AAC holds 13 units safety stock (how come? Observe):
AAC is open 5 day a week.
The average daily demand = 120/week)/5 = 24 juicers.
Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers.
Reorder point without Safety stock = LD = 192.
Current policy: R = 205.
Safety stock = 205 – 192 = 13.
For safety stock of 13 juicers the total cost is
TC(327) = ($10) + (13)($1.40) = $62,876.09
TV(327) + Procurement Safety stock cost holding cost

30. AAC – Solution: Sensitivity of the EOQ Results
Changing the order size
Suppose juicers must be ordered in increments of 100 (order 300 or 400)
AAC will order Q = 300 juicers in each order.
There will be a total variable cost increase of $1.71.
This is less than 0.5% increase in variable costs.
Changes in input parameters
Suppose there is a 20% increase in demand. D=7500 juicers.
The new optimal order quantity is Q* = 359.
The new variable total cost = TV(359) = $502
If AAC still orders Q = 327, its total variable costs becomes
Only 0.4%
increase
TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13

31. AAC – Solution: Cycle Time
For an order size of 327 juicers we have:
T = (327/ 6240) = year.
= (52)(5) = 14 days.
This is useful information because:
Shelf life may be a problem.
Coordinating orders with other items might be desirable.
working days per week

32. AAC – Excel Spreadsheet
=SQRT(2*$B$10*$B$14/$B$13)
=1/E11
Copy to cell H12
=$B$15*$B$10+$B$16-INT(($B$15*$B$10+$B$16)/E10)*E10
Copy to cell H13
=E10/B10
Copy to cell H11
=(E10/2)*$B$13+($B$10/E10)*$B$14
Copy to cell H14
=$B$10*$B$11+E14+$B$13*B16
Copy to Cell H15

33. Service Levels and
Safety Stocks

34. 8.3 Determining Safety Stock Levels
Businesses incorporate safety stock requirements when determining reorder points.
A possible approach to determining safety stock levels is by specifying desired service level .

35. Two Types of Service Level
Service levels can be viewed in two ways.
The cycle service level
The probability of not incurring a stockout during an inventory cycle.
Applied when the likelihood of a stockout, and not its magnitude, is important for the firm.
The unit service level
The percentage of demands that are filled without incurring any delay.
Applied when the percentage of unsatisfied demand should be under control.

36. The Cycle Service Level Approach
In many cases short run demand is variable even though long run demand is assumed constant.
Therefore, stockout events during lead time may occur unexpectedly in each cycle.
Stockouts occur only if demand during lead time is greater than the reorder point.

37. The Cycle Service Level Approach
To determine the reorder point we need to know:
The lead time demand distribution.
The required service level.
In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by
R = mL + zasL
1 –a = service level

38. The Cycle Service Level Approach
P(DL<R) = 1 – a
m=192
P(DL>R) = a R
P(DL> R) = P(Z > (R – mL)/sL) = a. Since P(Z > Za) = a, we have Za = (R – mL)/sL, which gives…
R = mL + zasL

39. AAC - Cycle Service Level Approach
Assume that lead time demand is normally distributed.
Estimation of the normal distribution parameters:
Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week.
Estimation of the variance of the weekly demand = Sample variance = juicers2.

40. AAC - Cycle Service Level Approach
To find mLand sL the parameters m (per week) and s (per week) must be adjusted since the lead time is longer than one week.
Lead time is 8 days =(8/5) weeks = 1.6 weeks.
Estimates for the lead time mean demand and variance of demand mL » (1.6)(120) = 192; s2L » (1.6)(83.33) =

41. AAC - Service Level for a given Reorder Point
Let us use the current reorder point of 205 juicers.
205 = z (11.55) z = 1.13
From the normal distribution table we have that a reorder point of 205 juicers results in an 87% cycle service level.

42. AAC – Reorder Point for a given Service Level
Management wants to improve the cycle service level to 99%.
The z value corresponding to 1% right hand tail is 2.33.
R = (11.55) = 219 juicers.

43. AAC – Acceptable Number of Stockouts per Year
AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers.
What is the equivalent service level for this strategy?

44. AAC – Acceptable Number of Stockouts per Year
There will be an average of /327 = lead times per year.
The likelihood of stockouts = 1/19 =
This translates into a service level of 94.76%

45. The Unit Service Level Approach
When lead time demand follows a normal distribution
service level can be calculated as follows:
Determine the value of z that satisfy the equation
L(z) = aQ* / sL
Solve for R using the equation
R = mL + zsL

46. AAC – Cycle Service Level (Excel spreadsheet)
=NORMINV(B7,B5,B6)
=NORMDIST(B8,B5,B6,TRUE)

47. 8.4 EOQ Models with Quantity Discounts
Quantity Discounts are Common Practice in Business
By offering discounts buyers are encouraged to increase their order sizes, thus reducing the seller’s holding costs.
Quantity discounts reflect the savings inherent in large orders.
With quantity discounts sellers can reward their biggest customers without violating the Robinson - Patman Act.

48. 8.4 EOQ Models with Quantity Discounts
Quantity Discount Schedule
This is a list of per unit discounts and their corresponding purchase volumes.
Normally, the price per unit declines as the order quantity increases.
The order quantity at which the unit price changes is called a break point.
There are two main discount plans:
All unit schedules - the price paid for all the units purchased is based on the total purchase.
Incremental schedules - The price discount is based only on the additional units ordered beyond each break point.

49. All Units Discount Schedule
To determine the optimal order quantity, the total purchase cost must be included
TC(Q) = (Q/2)Ch + (D/Q)Co + DCi + ChSS
Ci represents the unit cost at the ith pricing level.

50. AAC - All Units Quantity Discounts
AAC is offering all units quantity discounts to its customers.
Data

51. Should AAC increase its regular order of
327 juicers, to take advantage of the discount?

52. AAC – All units discount procedure
Step 1: Find the optimal order Qi* for each discount level “i”. Use the formula
Step 2: For each discount level “i” modify Q i* as follows
If Qi* is lower than the smallest quantity that qualifies for the i th discount, increase Qi* to that level.
If Qi* is greater than the largest quantity that qualifies for the ith discount, eliminate this level from further consideration.
Step 3: Substitute the modified Q*i value in the total cost formula TC(Q*i ).
Step 4: Select the Q i * that minimizes TC(Q i*)

53. AAC – All units discount procedure
Step 1: Find the optimal order quantity Qi* for each discount level “i” based on the EOQ formula

54. AAC – All Units Discount Procedure
Step 2 : Modify Q i *
$10/unit
$9.75/unit
$9.50
Q1*
Q2*
Q3*
599
336
300
331
999
600

55. AAC – All Units Discount Procedure
Step 2 : Modify Q i *
$10/unit
Q3*
Q3*
Q3*
Q3*
Q3*
$9.50
Q3*
Q1*
Q2*
Q3*
Q3*
336
300
331
999
600

56. AAC – All Units Discount Procedure
Step 3: Substitute Q I * in the total cost function
Step 4
AAC should order 5000 juicers

57. AAC – All Units Discount Excel Worksheet

58. 8.4 Production Lot Size Model - Assumptions
Demand rate is constant.
Production rate is larger than demand rate.
The production lot is not received instantaneously (at an infinite rate), because production rate is finite.
There is only one product to be scheduled.
The rest of the EOQ assumptions stay in place.

59. Production Lot Size Model – Inventory profile
The optimal production lot size policy orders the same amount each time.
This observation results in the inventory profile below:

60. Production Lot Size Model – Understanding the inventory profile
The production increases the
inventory at a rate of P.
Demand accumulation
during production run = DT1
Production time T1
Maximum inventory
Maximum inventory = (P – D)T1 = (P – D)(Q/P) = Q(1 – D/P)
The inventory increases at a net rate of P - D
Production
Lot Size = Q = PT1
The demand decreases the
inventory at a rate of D.
Demand accumulation
during production run

61. Production Lot Size Model – Total Variable Cost
The parameters of the total variable costs function are similar to those used in the EOQ model.
Instead of ordering cost, we have here a fixed setup cost per production run (Co).
In addition, we need to incorporate the annual production rate (P) in the model.

62. Production Lot Size Model – Total Variable Cost
TV(Q) =
(Q/2)(1 - D/P)Ch +
(D/Q)Co
P is the annual production rate
The average inventory
Ch(1-D/P)
The Optimal Order Size
Q* =
2DCo

63. Production Lot Size Model – Useful relationships
Cycle time T = Q / D.
Length of a production run T1 = Q / P.
Time when machines are not busy producing the product T2 = T - T1 = Q(1/D - 1/P).
Average inventory = (Q/2)(1-D/P).

64. FARAH COSMETICS COMPANY
Farah needs to determine optimal production lot size for its most popular shade of lipstick.
Data
The factory operates 7 days a week, 24 hours a day.
Production rate is 1000 tubes per hour.
It takes 30 minutes to prepare the machinery for production.
It costs $150 to setup the line.
Demand is 980 dozen tubes per week.
Unit production cost is $.50
Annual holding cost rate is 40%.

65. FARAH COSMETICS COMPANY – Solution
Dozens
Input for the total variable cost function
D = 613,200 per year [(980 dozen/week (12)/ 7](365)
Ch = 0.4(0.5) = $0.20 per tube per year.
Co = $150
P = (1000)(24)(365) = 8,760,000 per year.

66. FARAH COSMETICS COMPANY – Solution
Current Policy
Currently, Farah produces in lots of 84,000 tubes.
T = (84,000 tubes per run)/(613,200 tubes per year)= years (about 50 days).
T1 = (84,000 tubes per lot)/(8,760,000 tubes per year)= years (about 3.5 days).
T2 = = years (about 46.5 days).
TV(Q = 84,000) = (84,000/2) {1-(613,200/8,760,000)}(0.2) ,200/84,000)(150) = $8907.

67. FARAH COSMETICS COMPANY – Solution
The Optimal Policy
Using the input data we find
TV(Q* = 31,499) = (31,499/2) [1-(613,200/8,760,000)](0.2) (613,200/31,499)(150) = $5,850.
The optimal order size
(0.2)(1-613,200/8760,000)
Q* =
2(613,200)(150)
= 31,499

68. FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)

69. 8.5 Planned Shortage Model
When an item is out of stock, customers may:
Go somewhere else (lost sales).
Place their order and wait (backordering).
In this model we consider the backordering case.
All the other EOQ assumptions are in place.

70. Planned Shortage Model – the Total Variable Cost Equation
The parameters of the total variable costs function are similar to those used in the EOQ model.
In addition, we need to incorporate the shortage costs in the model.
Backorder cost per unit per year (loss of goodwill cost) - Cs.
Reflects future reduction in profitability.
Can be estimated from market surveys and focus groups.
Backorder administrative cost per unit - Cb.
Reflects additional work needed to take care of the backorder.

71. Planned Shortage Model – the Total Variable Cost Equation
The Annual holding cost = Ch[T1/T](Average inventory) =
Ch[T1/T] (Q-S)/2
The Annual shortage cost = Cb(number of backorders per year) + Cs(T2/T)(Average number of backorders).
To calculate the annual holding cost and shortage cost we need to find
The proportion of time inventory is carried, (T1/T)
The proportion of time demand is backordered, (T2/T). T1 T2 T

72. Finding T1/ T and T2/ T Q Average inventory = (Q - S) / 2
Proportion of time
inventory exists
= T1/T
Q - S Q
Q - S
= (Q - S) / Q Q T2 T1 T1
Proportion of time
shortage exists = T2/T T S T S
= S / Q
Average shortage = S / 2

73. Planned Shortage Model – The Total Variable Cost Equation
Annual holding cost: Ch[T1/T](Q-S)/2 = Ch[(Q-S) /Q](Q-S)/2
= Ch(Q-S)2/2Q
Annual shortage cost: Cb(Units in short per year) + Cs[T2/T](Average number of backorders) = Cb(S)(D/Q) + CsS2/2Q

74. Planned Shortage Model – The Total Variable Cost Equation
The total annual variable cost equation
The optimal solution to this problem is obtained under the following conditions
Cs > 0 ;
Cb < \/ 2CoCh / D
(Q -S)2 D Q S2 2Q
Ch +
(Co + SCb) +
TV(Q,S) = CS 2Q
Holding
costs
Ordering
costs
Time independent
backorder costs
Time dependent
backorder costs

75. Planned Shortage Model – The Optimal Inventory Policy
The Optimal Order Size -
Q* =
Ch + Cs Cs x
2DCo
(DCb)2
ChCs Ch
The Optimal Backorder level
S*=
Q* Ch - DCb
Ch + Cs
Reorder Point
R = L D - S*

76. SCANLON PLUMBING CORPORATION
Scanlon distributes a portable sauna from Sweden.
Data
A sauna costs Scanlon $2400.
Annual holding cost per unit $525.
Fixed ordering cost $1250 (fairly high, due to costly transportation).
Lead time is 4 weeks.
Demand is 15 saunas per week on the average.

77. SCANLON PLUMBING CORPORATION
Scanlon estimates a $20 goodwill cost for each week a customer who orders a sauna has to wait for delivery.
Administrative backordrer cost is $10.
Management wishes to know:
The optimal order quantity.
The optimal number of backorders.
Backorder costs

78. SCANLON PLUMBING – Solution
Input for the total variable cost function
D = 780 saunas [(15)(52)]
Co = $1,250
Ch = $525
Cs = $1,040
Cb = $10

79. SCANLON PLUMBING – Solution
The optimal policy x
(780)(10)2
(525)(1040)
525
2(780)(1250)
1040
Q* = 74 - _
S*=
(74)(525)
(780)(10) 20
R = (4 / 52)(780) = 40

80. SCANLON PLUMBING – Spreadsheet Solution

81. 8.7 Review Systems – Continuous Review
(R, Q) Policies
The EOQ, production lot size, and planned shortage models assume that
inventory levels are continuously monitored
Items are sold one at a time.

82. 8.7 Review Systems – Continuous Review
(R, Q) Policies
The above models call for order point (R) order quantity (Q) inventory policies.
Such policies can be implemented by
A point-of-sale computerized system.
The two-bin system.

83. Continuous Review Systems
(R, M) policies
When items are not necessarily sold one at a time, the reorder point might be missed, and out of stock situations might occur more frequently.
The order to level (R, M) policy may be implemented in this situation.

84. Continuous Review Systems
(R,M) policies
The R, M policy replenishes inventory up to a pre-determined level M.
Order Q = Q* + (R – I) = (M – SS) + (R – I) each time the inventory falls to the reorder point R or below. (Order size may vary from one cycle to another).

85. Periodic Review Systems
It may be difficult or impossible to adopt a continuous review system, because of:
The high price of a computerized system.
Lack of space to adopt the two-bin system.
Operations inefficiency when ordering different items from the same vendor separately.
The periodic review system may be found more suitable for these situations.

86. Periodic Review Systems
Under this system the inventory position for each item is observed periodically.
Orders for different items can be better coordinated periodically.

87. Periodic Review Systems
(T,M) Policies
In a replenishment cycle policy (T, M), the inventory position is reviewed every T time units.
An order is placed to bring the inventory level back up to a maximum inventory level M.
M is determined by
Forecasting the number of units demanded during the review period T.
Adding the desired safety stock to the forecasted demand.

88. Periodic Review Systems
Calculation of the replenishment level and order size
Q = M + LD – I
M = TD + SS
T =Review period
L = Lead time
SS= Safety stock
Q = Inventory position
D = Annual demand
I = Inventory position

89. AAC operates a (T, M) policy
Every three weeks AAC receives deliveries of different products from Citron.
Lead time is eight days for ordering Citron’s juicers.
AAC is now reviewing its juicer inventory and finds 210 in stock.
How many juicers should AAC order for a safety stock of 30 juicers?

90. AAC operates a (T, M) policy – Solution
Data
Review period T = 3 weeks = 3/52 = years,
Lead time = L = 8 days = 8/260 = years,
Demand D = 6240 juicers per year,
Safety stock SS = 30 juicers,
Inventory position I = 210 juicers
AAC operates 260 days a year.
(5)(52) = 260.

91. AAC operates a (T, M) policy – Solution
Review period demand = TD = ( 3/52)(6240) = 360 juicers,
M = TD + SS = = 390 juicers,
Q = M + LD – I = (6240) = 372 juicers.

92. AAC operates a (T, M) policy – Solution
Replenishment level
Order
Order
M = maximum inventory
Inventory position
Inventory position SS SS SS L L
Review point
Review point T
Notice: I + Q is designed to satisfy the demand within an interval of T + L To obtain the replenishment level add SS to I + Q.

93. 8.8 Single Period Inventory Model - Assumptions
Shelf life of the item is limited.
Inventory is saleable only within a single time period.
Inventory is delivered only once during a time period.
Demand is stochastic with a known distribution.
At the end of each period, unsold inventory is disposed of for some salvage.
The salvage value is less than the cost per item.
Unsatisfied demand may result in shortage costs.

94. The Expected Profit Function
To find an optimal order quantity we need to balance the expected cost of over-ordering and under ordering.
Expected Profit = S(Profit when Demand=X)Prob(Demand=X) x
The expected profit is a function of the order size, the random demand, and the various costs.

95. The Expected Profit Function
Developing an expression for EP(Q)
Notation
p = per unit selling price of the good.
c = per unit cost of the good.
s = per unit salvage value of unsold good.
K = fixed purchasing costs
Q = order quantity.
EP(Q) = Expected Profit if Q units are ordered.
Scenarios
Demand X is less than the order quantity (X < Q).
Demand X is greater than or equal to the order quantity (X ³Q.

96. The Expected Profit Function
Scenario 1: Demand X is less than the units stocked, Q.
Scenario 2: Demand X is greater than or equal to the units stocked.
Profit = pX + s(Q - X) - cQ - K
Profit = pQ - g(X - Q) - cQ - K
EP(Q) = [pX+s(Q - X) - cQ - K]P(X) [pQ - g(X - Q) - cQ - K]P(X)

97. The Optimal Solution To maximize the expected profit order Q*
For the discrete demand case take the smallest value of Q*
that satisfies the condition
P(D £ Q*) ³ (p - c + g)/(p - s + g)
For the continuous demand case find the Q* that solves
F(Q*) = (p - c + g) /(p - s + g)

98. THE SENTINEL NEWSPAPER
Management at Sentinel wishes to know how many newspapers to put in a new vending machine.
Data
Unit selling price is $0.30
Unit production cost is $0.38.
Advertising revenue is $0.18 per newspaper.
Unsold newspaper can be recycled and net $0.01.
Unsatisfied demand costs $0.10 per newspaper.
Filling a vending machine costs $1.20.
Demand distribution is
discrete uniform between
30 and 49 newspapers.

99. SENTINEL - Solution Input to the optimal order quantity formula
c = [ ]
s = 0.01
g = 0.10
K = 1.20
p+ g - c
p+ g - s
The probability of the optimal service level =
= 0.513 =

100. SENTINEL – Solution Finding the optimal order quantity Q*
1.0
P(D £ 39) = 0.50
P(D £ 40) = 0.55
0.513
0.55
0.50
Q* = 40 30 39 40 49

101. SENTINEL – Spreadsheet Solution
=(B5+B8-B6)/(B5+B8-B7)
=ROUNDUP(B10+E5*(B11-10),0)
=(E6-B10+1)/(B11-B10+1)

102. WENDELL’S BAKERY
Management in Wendell’s wishes to determine the number of donuts to prepare for sale, on weekday evenings
Data
Unit cost is $0.15.
Unit selling price is $0.35.
Unsold donuts are donated to charity for a tax credit of $0.05 per donut.
Customer goodwill cost is $0.25.
Operating costs are $15 per evening.
Demand is normally distributed
with a mean of 120, and a
standard deviation of 20 donuts.

103. WENDELL’S BAKERY - Solution
Input to the optimal order quantity formula
p = $0.35
c = $0.15
s = $0.05
g = $0.25
K = $15.00
The optimal service level =
p+ g - c
p+ g - s = =

104. WENDELL’S BAKERY - Solution Finding the optimal order quantity
From the relationship F(Q*) = we find the corresponding z value.
From the standard normal table we have z =
The optimal order quantity is calculated by
Q* = m + zs
For Wendell’s Q* =
.8182
m=120 Q*

105. WENDELL’S BAKERY - Solution Calculating the expected profit
For the normal distribution
L [(Q* - m ) /s] is obtained from the partial expected value table.
For Wendell’s
EP(138) = ( )(120) - ( )(138) - ( )x(20)L[( ) / 20] - 15 = $6.10
EP(Q*) = (p - s) m - (c - s)Q* - (p + g - s) (s)L[(Q* - m ) /s] - K
L(0.9) =

106. WENDELL’S BAKERY - Spreadsheet Solution
=(B5+B8-B6)/(B5+B8-B7)
=NORMINV(E5,B10,B11)
=(B5-B7)*B10-(B6-B7)*E6-(B5+B8-B7)*B11*(EXP(-(((E6-B10)/B11)^2)/2)/((2*PI())^0.5)-((E6-B10)/B11)*(1-NORMSDIST((E6-B10)/B11)))-B9

107. WENDELL’S – The commission strategy
When commission replaces fixed wages…
Compare the maximum expected profit of two strategies:
$0.13 commission paid per donut sold,
$15 fixed wage per evening (calculated before).
Calculate first the optimal quantity for the alternative policy.
Check the expected difference in pay for the operator.

108. WENDELL’S – The commission strategy - Solution
The unit selling price changes to
c = = $0.22
The optimal order: F(Q*) = ( ) / ( )=
Z = .71 Q* = m + zs = (0.71)(20) » 134 donuts.

109. WENDELL’S – The commission strategy - Solution
Will the bakery’s expected profit increase?
EP(134) = ( )(20) - ( )(134) - ( )x(20)L[( ) / 20] = $5.80 < 6.10
The bakery should not proceed with the alternative plan.

110. WENDELL’S – The commission strategy - Solution
Comments
The operator expected compensation will increase, but not as much as the bakery’s expected loss.
An increase in the mean sales is probable when the commission compensation plan is implemented. This may change the analysis results.

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