Download Single-Period Models (Discrete Demand)

LESSON 16: INVENTORY MODELS
(STOCHASTIC)
Outline
• Single-Period Models
– Discrete Demand
– Continuous Demand, Uniform Distribution
– Continuous Demand, Normal Distribution
• Multi-Period Models
– Given a Q, R Policy, Find Cost
– Optimal Q, R Policy without Service Constraint
– Optimal Q, R Policy with Type 1 Service Constraint
– Optimal Q, R Policy with Type 2 Service Constraint
Stochastic Inventory Control Models
Inventory Control with Uncertain Demand
• In Lessons 16-20 we shall discuss the stochastic
inventory control models assuming that the exact
demand is not known. However, some demand
characteristics such as mean, standard deviation and
the distribution of demand are assumed to be known.
• Penalty cost, p: Shortages occur when the demand
exceeds the amount of inventory on hand. For each
unit of unfulfilled demand, a penalty cost of p is
charged. One source of penalty cost is the loss of
profit. For example, if an item is purchased at $1.50
and sold at $3.00, the loss of profit is $3.00-1.50 =
$1.50 for each unit of demand not fulfilled.
Stochastic Inventory Control Models
Inventory Control with Uncertain Demand
• More on penalty cost, p
– Penalty cost is estimated differently in different situation.
There are two cases:
1. Backorder - if the excess demand is backlogged and
fulfilled in a future period, a backorder cost is
charged. Backorder cost is estimated from
bookkeeping, delay costs, goodwill etc.
2. Lost sales - if the excess demand is lost because the
customer goes elsewhere, the lost sales is charged.
The lost sales include goodwill and loss of profit
margin. So, penalty cost = selling price - unit variable
cost + goodwill, if there exists any goodwill.
Single- and Multi- Period Models
• Stochastic models are classified into single- and multiperiod models.
• In a single-period model, items are received in the
beginning of a period and sold during the same period.
The unsold items are not carried over to the next
period.
• The unsold items may be a total waste, or sold at a
reduced price, or returned to the producer at some
price less than the original purchase price.
• The revenue generated by the unsold items is called
the salvage value.
Single- and Multi- Period Models
• Following are some products for which a single-period
model may be appropriate:
– Computer that will be obsolete before the next order
– Perishable products
– Seasonal products such as bathing suits, winter
coats, etc.
– Newspaper and magazine
• In the single-period model, there remains only one
question to answer: how much to order.
Single- and Multi- Period Models
• In a multi-period model, all the items unsold at the
end of one period are available in the next period.
• If in a multi-period model orders are placed at regular
intervals e.g., once a week, once a month, etc, then
there is only one question to answer: how much to
order.
• However, we discuss Q, R models in which it is
assumed that an order may be placed anytime. So,
as usual, there are two questions: how much to order
and when to order.
Single-Period Models
• In the single-period model, there remains only one
question to answer: how much to order.
• An intuitive idea behind the solution procedure will
now be given: Consider two items A and B.
– Item A
• Selling price $900
• Purchase price $500
• Salvage value $400
– Item B
• Selling price $600
• Purchase price $500
• Salvage value $100
Single-Period Models
• Item A
– Loss resulting from unsold items = 500-400=$100/unit
– Profit resulting from items sold = $900-500=$400/unit
• Item B
– Loss resulting from unsold items = 500-100=$400/unit
– Profit resulting from items sold = $600-500=$100/unit
• If the demand forecast is the same for both the items, one
would like to order more A and less B.
• In the next few slides, a solution procedure is discussed
that is consistent with this intuitive reasoning.
Single-Period Models
• First define two terms:
• Loss resulting from the items unsold (overage cost)
co= Purchase price - Salvage value
• Profit resulting from the items sold (underage cost)
cu = Selling price - Purchase price
• The Question
Given costs of overestimating/underestimating
demand and the probabilities of various demand
sizes how many units will be ordered?
Single-Period Models (Discrete Demand)
Decision Rule
• Demand may be discrete or continuous. The demand
of computer, newspaper, etc. is usually an integer.
Such a demand is discrete. On the other hand, the
demand of gasoline is not restricted to integers. Such
a demand is continuous. Often, the demand of
perishable food items such as fish or meat may also
be continuous.
• Consider an order quantity Q
• Let p
= probability (demand<Q)
= probability of not selling the Qth item.
• So, (1-p) = probability of selling the Qth item.
Single-Period Models (Discrete Demand)
Decision Rule
• Expected loss from the Qth item = pco
• Expected profit from the Qth item = 1  p cu
• So, the Qth item should be ordered if pco  (1  p )cu
cu
or , p 
co  cu
• Decision Rule (Discrete Demand):
– Order maximum quantity Q such that
cu
p
co  cu
where p = probability (demand<Q)
Single-Period Models (Discrete Demand)
Example 1: Demand for cookies:
Demand
Probability of Demand
1,800 dozen
0.05
2,000
0.10
Note: The demand is
2,200
0.20
discrete. So, the demand
2,400
0.30
cannot be other numbers
2,600
0.20
e.g., 1900 dozens, etc.
2,800
0.10
3,000
0,05
Selling price=$0.69, cost=$0.49, salvage value=$0.29
a. Construct a table showing the profits or losses for each
possible quantity (Self study)
b. What is the optimal number of cookies to make?
c. Solve the problem by marginal analysis.
Self Study
Single-Period Models (Discrete Demand)
Demand
Prob
(dozen) (Demand)
1800
2000
2200
2400
2600
2800
3000
Prob
Expected Revenue Revenue Total
(Selling
Number
From
From Revenue
all the units) Sold
Sold
Unsold
Items
Items
Cost
Profit
0.05
0.1
0.2
0.3
0.2
0.1
0.05
a. Sample computation for order quantity = 2200:
Expected number sold = 1800Prob(demand=1800)
+2000Prob(demand=2000) + 2200Prob(demand2200)
= 1800(0.05)+2000(0.1)+2200(0.2+0.3+0.2+0.1+0.05)
= 1800(0.05)+2000(0.1)+2200(0.85) = 2160
Revenue from sold items=2160(0.69)=$1490.4
Revenue from unsold items=(2200-2160)(0.29)=$11.6
Self Study
Single-Period Models (Discrete Demand)
Demand
Prob
(dozen) (Demand)
1800
2000
2200
2400
2600
2800
3000
0.05
0.1
0.2
0.3
0.2
0.1
0.05
Prob
Expected Revenue Revenue Total
(Selling
Number
From
From Revenue
all the units) Sold
Sold
Unsold
Items
Items
1
1800 1242.0
0.0
1242
0.95
1990 1373.1
2.9
1376
0.85
2160 1490.4
11.6
1502
0.65
2290 1580.1
31.9
1612
0.35
2360 1628.4
69.6
1698
0.15
2390 1649.1
118.9
1768
0.05
2400 1656.0
174.0
1830
Cost
882
980
1078
1176
1274
1372
1470
Profit
360
396
424
436
424
396
360
a. Sample computation for order quantity = 2200:
Total revenue=1490.4+11.6=$1502
Cost=2200(0.49)=$1078
Profit=1502-1078=$424
b. From the above table the expected profit is maximized for
an order size of 2,400 units. So, order 2,400 units.
Single-Period Models (Discrete Demand)
c. Solution by marginal analysis:
cu 
co 
Order maximum quantity, Q such that
cu
p  Probabilitydemand  Q  

cu  c0
Demand, Q Probability(demand) Probability(demand<Q), p
1,800 dozen
0.05
2,000
0.10
2,200
0.20
2,400
0.30
2,600
0.20
2,800
0.10
3,000
0,05
Single-Period Models (Discrete Demand)
• In the previous slide, one may draw a horizontal line that
separates the demand values in two groups. Above the line,
the values in the last column are not more than
cu

cu  c0
The optimal solution is the last demand value above the line.
So, optimal solution is 2,400 units.
Continuous Distribution
• Often the demand is continuous. Even when the
demand is not continuous, continuous distribution may
be used because the discrete distribution may be
inconvenient.
• For example, suppose that the demand of calendar can
vary between 150 to 850 units. If demand varies so
widely, a continuous approximation is more convenient
because discrete distribution will involve a large number
of computation without any significant increase in
accuracy.
• We shall discuss two distributions:
– Uniform distribution
– Continuous distribution
Continuous Distribution
• First, an example on continuous approximation.
• Suppose that the historical sales data shows:
Quantity
No. Days sold
Quantity
No. Days sold
14
1
21
11
15
2
22
9
16
3
23
6
17
6
24
3
18
9
25
2
19
11
26
1
20
12
Continuous Distribution
• A histogram is constructed with the above data and shown
in the next slide. The data shows a good fit with the normal
distribution with mean = 20 and standard deviation = 2.49.
• There are some statistical tests, e.g., Chi-Square test, that
can determine whether a given frequency distribution has
a good fit with a theoretical distribution such as normal
distribution, uniform distribution, etc. There are some
software, e.g., Bestfit, that can search through a large
number of theoretical distributions and choose a good one,
if there exists any. This topic is not included in this course.
Continuous Distribution
Mean = 20
Standard deviation = 2.49
Continuous Distribution
The figure below shows
an example of uniform
distribution.
Single-Period Models (Continuous Demand)
Decision Rule
• The decision rule for continuous demand is similar to
the decision rule for the discrete demand
• Decision Rule (Continuous Demand):
– Order quantity Q such that
cu
p
co  cu
where p = probability (demandQ)
– Note the following difference
• The word “maximum”
• The “= “ in the formula for p
• The “ “ in the definition of p
Single-Period Models (Continuous Demand)
Decision Rule
• There is a nice pictorial interpretation of the decision
rule. Although we shall discuss the interpretation in
terms of normal and uniform distributions, the
interpretation is similar for all the other continuous
distributions.
• The area under the curve is 1.00. Identify the vertical
line that splits the area into two parts with areas
cu
p
on the left and
co  cu
co
1  p  
on the right
co  cu
The order quantity corresponding to the vertical line
is optimal.
Single-Period Models (Continuous Demand)
Decision Rule
Probability
• Pictorial interpretation of the decision rule for uniform
distribution (see Examples 2, 3 and 4 for application
of the rule):
Area
p
 1  p 
cu

co  cu
co

co  cu
Area
150
Q*
Demand
850
Single-Period Models (Continuous Demand)
Decision Rule
Probability
• Pictorial interpretation of the decision rule for normal
distribution (see Examples 3 and 4 for application of
the rule):
Area
Area
p

 1  p 
cu
co  cu

Q*
Demand
co
co  cu
Single-Period Models (Continuous Demand)
Example 2: The J&B Card Shop sells calendars. The oncea-year order for each year’s calendar arrives in
September. The calendars cost $1.50 and J&B sells them
for $3 each. At the end of July, J&B reduces the calendar
price to $1 and can sell all the surplus calendars at this
price. How many calendars should J&B order if the
September-to-July demand can be approximated by
a. uniform distribution between 150 and 850
Single-Period Models (Continuous Demand)
Solution to Example 2:
Overage cost
co = Purchase price - Salvage value =
Underage cost
cu = Selling price - Purchase price =
Single-Period Models (Continuous Demand)
p
cu
co  cu
=
Now, find the Q so that p = probability(demand<Q) =
Probability
Q* = a+p(b-a) =
Area
=
Area =
150
Demand
Q*
850
Single-Period Models (Continuous Demand)
Example 3: The J&B Card Shop sells calendars. The oncea-year order for each year’s calendar arrives in
September. The calendars cost $1.50 and J&B sells them
for $3 each. At the end of July, J&B reduces the calendar
price to $1 and can sell all the surplus calendars at this
price. How many calendars should J&B order if the
September-to-July demand can be approximated by
b. normal distribution with  = 500 and =120.
Single-Period Models (Continuous Demand)
Solution to Example 3: co =$0.50, cu =$1.50 (see Example 2)
p
cu
1.50
=
= 0.75
co  cu
1.50  0.50
Single-Period Models (Continuous Demand)
Now, find the Q so that p = 0.75
Self Study
Single-Period Models (Continuous Demand)
Example 4: A retail outlet sells a seasonal product for $10
per unit. The cost of the product is $8 per unit. All units not
sold during the regular season are sold for half the retail
price in an end-of-season clearance sale. Assume that the
demand for the product is normally distributed with  =
500 and  = 100.
a. What is the recommended order quantity?
b. What is the probability of a stockout?
c. To keep customers happy and returning to the store
later, the owner feels that stockouts should be avoided if
at all possible. What is your recommended quantity if the
owner is willing to tolerate a 0.15 probability of stockout?
d. Using your answer to part (c), what is the goodwill cost
you are assigning to a stockout?
Self Study
Single-Period Models (Continuous Demand)
Solution to Example 4:
a. Selling price=$10,
Purchase price=$8
Salvage value=10/2=$5
cu =10 - 8 = $2, co = 8-10/2 = $3
Now, find the Q so that
p = 0.4
or, area (1) = 0.4
Look up Table A-1 for
Area (2) = 0.5-0.4=0.10
Area=0.10
Probability
cu
2
p
=
= 0.4
co  cu 2  3
 100
Area=0.40
(2)
(1)
(3)
z = 0.255 =500
Demand
Self Study
Single-Period Models (Continuous Demand)
z = 0.25 for area = 0.0987
z = 0.26 for area = 0.1025
So, z = 0.255 (take -ve, as p = 0.4 <0.5) for area = 0.10
So, Q*=+ z =500+(-0.255)(100)=474.5 units.
b. P(stockout)=P(demandQ)=1-P(demand<Q)=1-p=1-0.4=0.6
Self Study
c. P(stockout)=Area(3)=0.15
Look up Table A-1 for
Area (2) = 0.5-0.15=0.35
z = 1.03 for area = 0.3485
z = 1.04 for area = 0.3508
Probability
Single-Period Models (Continuous Demand)
Area=0.35
 100
Area=0.15
(2)
(1)
=500
Demand
So, z = 1.035 for area = 0.35
So, Q*=+ z =500+(1.035)(100)=603.5 units.
(3)
z = 1.035
Self Study
Single-Period Models (Continuous Demand)
d. p=P(demand<Q)=1-P(demandQ)
=1-P(stockout)=1-0.15=0.85
For a goodwill cost of g
cu =10 - 8+g = 2+g, co = 8-10/2 = $3
cu
2 g
Now, solve g in p =
=
=0.85
co  cu
(2  g )  3
Hence, g=$15.
READING AND EXERCISES
Lesson 16
Reading:
Section 5.1 - 5.3 , pp. 245-254 (4th Ed.), pp. 232-245
(5th Ed.)
Exercise:
8a, 12a, and 12b, pp. 256-258 (4th Ed.), pp. 248-250
(5th Ed.)