Lecture 01 s11 431 Stochastic Inventory

8/3/2019 Lecture 01 s11 431 Stochastic Inventory

1/37

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

LESSON 1: INVENTORY MODELS

(STOCHASTIC)


2/37

Stochastic Inventory Control ModelsInventory Control with Uncertain Demand

In Lessons 16-20 we shall discuss the stochasticinventory control models assuming that the exactdemand is not known. However, some demand

characteristics such as mean, standard deviation andthe distribution of demand are assumed to be known.

Penalty cost, : Shortages occur when the demandexceeds the amount of inventory on hand. For each

unit of unfulfilled demand, a penalty cost of ischarged. One source of penalty cost is the loss ofprofit. For example, if an item is purchased at $1.50and sold at $3.00, the loss of profit is $3.00-1.50 =$1.50 for each unit of demand not fulfilled.

p

p


3/37

More on penalty cost,

Penalty cost is estimated differently in different situation.There are two cases:

1. Backorder - if the excess demand is backlogged andfulfilled in a future period, a backorder cost ischarged. Backorder cost is estimated frombookkeeping, delay costs, goodwill etc.

2. Lost sales - if the excess demand is lost because thecustomer goes elsewhere, the lost sales is charged.The lost sales include goodwill and loss of profitmargin. So, penalty cost = selling price - unit variablecost + goodwill, if there exists any goodwill.

p

Stochastic Inventory Control ModelsInventory Control with Uncertain Demand


4/37

Single- and Multi- Period Models

Stochastic models are classified into single- and multi-period models.

In a single-period model, items are received in thebeginning of a period and sold during the same period.The unsold items are not carried over to the nextperiod.

The unsold items may be a total waste, or sold at areduced price, or returned to the producer at someprice less than the original purchase price.

The revenue generated by the unsold items is calledthe salvage value.


5/37


Following are some products for which a single-periodmodel may be appropriate:

Computer that will be obsolete before the next order

Perishable products

Seasonal products such as bathing suits, wintercoats, etc.

Newspaper and magazine

In the single-period model, there remains only one

question to answer: how much to order.


6/37


In a multi-period model, all the items unsold at theend of one period are available in the next period.

If in a multi-period model orders are placed at regularintervals e.g., once a week, once a month, etc, thenthere is only one question to answer: how much toorder.

However, we discuss Q,R models in which it isassumed that an order may be placed anytime. So,as usual, there are two questions: how much to order

and when to order.


7/37


In the single-period model, there remains only onequestion to answer: how much to order.

An intuitive idea behind the solution procedure willnow 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


8/37


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, onewould like to order more A and less B.

In the next few slides, a solution procedure is discussedthat is consistent with this intuitive reasoning.


9/37


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 QuestionGiven costs of overestimating/underestimatingdemand and the probabilities of various demandsizes how many units will be ordered?


10/37

Demand may be discrete or continuous. The demandof computer, newspaper, etc. is usually an integer.Such a demand is discrete. On the other hand, thedemand of gasoline is not restricted to integers. Sucha demand is continuous. Often, the demand ofperishable food items such as fish or meat may alsobe continuous.

Consider an order quantity Q

Letp = probability (demand


11/37

Expected loss from the Qth

item = Expected profit from the Qth item =

So, the Qth item should be ordered if

Decision Rule (Discrete Demand):

Order maximum quantity Q such that

wherep = probability (demand


12/37

Example 1: Demand for cookies:

Demand Probability of Demand1,800 dozen 0.052,000 0.102,200 0.20

2,400 0.302,600 0.202,800 0.103,000 0,05

Selling price=$0.69, cost=$0.49, salvage value=$0.29

a. Construct a table showing the profits or losses for eachpossible quantity (Self study)

b. What is the optimal number of cookies to make?

c. Solve the problem by marginal analysis.

Single-Period Models (Discrete Demand)

Note: The demand is

discrete. So, the demand

cannot be other numberse.g., 1900 dozens, etc.


13/37

Demand Prob Prob Expected Revenue Revenue Total Cost Profit

(dozen) (Demand) (Selling Number From From Revenue

all the units) Sold Sold UnsoldItems Items

1800 0.05

2000 0.1

2200 0.2

2400 0.3

2600 0.22800 0.1

3000 0.05

a. Sample computation for order quantity = 2200:Expected number sold = 1800Prob(demand=1800)

+2000Prob(demand=2000) + 2200Prob(demand2200)= 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) = 2160Revenue from sold items=2160(0.69)=$1490.4

Revenue from unsold items=(2200-2160)(0.29)=$11.6

Single-Period Models (Discrete Demand)Self Study


14/37

Demand Prob Prob Expected Revenue Revenue Total Cost Profit

(dozen) (Demand) (Selling Number From From Revenue

all the units) Sold Sold UnsoldItems Items

1800 0.05 1 1800 1242.0 0.0 1242 882 360

2000 0.1 0.95 1990 1373.1 2.9 1376 980 396

2200 0.2 0.85 2160 1490.4 11.6 1502 1078 424

2400 0.3 0.65 2290 1580.1 31.9 1612 1176 436

2600 0.2 0.35 2360 1628.4 69.6 1698 1274 4242800 0.1 0.15 2390 1649.1 118.9 1768 1372 396

3000 0.05 0.05 2400 1656.0 174.0 1830 1470 360


a. Sample computation for order quantity = 2200:Total revenue=1490.4+11.6=$1502

Cost=2200(0.49)=$1078Profit=1502-1078=$424

b. From the above table the expected profit is maximized foran order size of 2,400 units. So, order 2,400 units.

Self Study


15/37

c. Solution by marginal analysis:

Order maximum quantity, Q such that

Demand, Q Probability(demand) Probability(demand


16/37

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

The optimal solution is the last demand value above the line.So, optimal solution is 2,400 units.


0cc

c

u

u


17/37

Continuous Distribution

Often the demand is continuous. Even when thedemand is not continuous, continuous distribution maybe used because the discrete distribution may beinconvenient.

For example, suppose that the demand of calendar can

vary between 150 to 850 units. If demand varies sowidely, a continuous approximation is more convenientbecause discrete distribution will involve a large numberof computation without any significant increase in

accuracy. We shall discuss two distributions:

Uniform distribution

Continuous distribution


18/37


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 22 916 3 23 6

17 6 24 3

18 9 25 2

19 11 26 1

20 12


19/37


A histogram is constructed with the above data and shownin the next slide. The data shows a good fit with the normaldistribution with mean = 20 and standard deviation = 2.49.

There are some statistical tests, e.g., Chi-Square test, thatcan determine whether a given frequency distribution has

a good fit with a theoretical distribution such as normaldistribution, uniform distribution, etc. There are somesoftware, e.g., Bestfit, that can search through a largenumber of theoretical distributions and choose a good one,

if there exists any. This topic is not included in this course.


20/37


Mean = 20

Standard deviation = 2.49


21/37


The figure below shows

an example of uniformdistribution.


22/37


23/37

There is a nice pictorial interpretation of the decisionrule. Although we shall discuss the interpretation interms of normal and uniform distributions, theinterpretation is similar for all the other continuousdistributions.

The area under the curve is 1.00. Identify the verticalline that splits the area into two parts with areas

The order quantity corresponding to the vertical line

is optimal.

Single-Period Models (Continuous Demand)Decision Rule

righttheon

andlefttheon

uo

o

uo

u

cc

cp

cc

cp

1


24/37

Pictorial interpretation of the decision rule for uniformdistribution (see Examples 2, 3 and 4 for applicationof the rule):


850150

Demand

Probability

Area

uo

u

cc

c

p

uo

o

ccc

p

1

Area

*Q


25/37

Pictorial interpretation of the decision rule for normaldistribution (see Examples 3 and 4 for application ofthe rule):


Probability

Demand

Area

uo

u

cc

c

p

*Q

uo

o

cc

c

p

1

Area


26/37

Example 2: The J&B Card Shop sells calendars. The once-

a-year order for each years calendar arrives inSeptember. The calendars cost $1.50 and J&B sells themfor $3 each. At the end of July, J&B reduces the calendarprice to $1 and can sell all the surplus calendars at thisprice. How many calendars should J&B order if theSeptember-to-July demand can be approximated by

a. uniform distribution between 150 and 850

Single-Period Models (Continuous Demand)


27/37

Solution to Example 2:

Overage cost

co= Purchase price - Salvage value =

Underage cost

cu= Selling price - Purchase price =



28/37

p =

Now, find the Q so thatp = probability(demand


29/37

Example 3: The J&B Card Shop sells calendars. The once-a-year order for each years calendar arrives inSeptember. The calendars cost $1.50 and J&B sells themfor $3 each. At the end of July, J&B reduces the calendarprice to $1 and can sell all the surplus calendars at thisprice. How many calendars should J&B order if the

September-to-July demand can be approximated byb. normal distribution with = 500 and =120.



30/37

Solution to Example 3: co=$0.50, cu=$1.50 (see Example 2)

p = = 0.750.501.50

1.50

uo

u

cc

c



31/37


Now, find the Q so that p= 0.75

Self Study


32/37

Example 4: A retail outlet sells a seasonal product for $10

per unit. The cost of the product is $8 per unit. All units notsold during the regular season are sold for half the retailprice in an end-of-season clearance sale. Assume that thedemand 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 ifat all possible. What is your recommended quantity if theowner is willing to tolerate a 0.15 probability of stockout?

d. Using your answer to part (c), what is the goodwill costyou are assigning to a stockout?

Single-Period Models (Continuous Demand)Self Study

Self Study


33/37

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

p = = 0.4

Now, find the Qso that

p = 0.4

or, area (1) = 0.4

Look up Table A-1 for

Area (2) = 0.5-0.4=0.10

32

2

uo

u

cc

c

Probability

Demand

=500

Area=0.40

z = 0.255

100

Area=0.10

(1)

(2)

(3)


Self Study


34/37

z = 0.25 for area = 0.0987z = 0.26 for area = 0.1025

So,z = 0.255 (take -ve, asp = 0.4


35/37

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

So,z = 1.035 for area = 0.35So, Q*=+z =500+(1.035)(100)=603.5 units.

Probability

Demand

=500

Area=0.35

z = 1.035

100

Area=0.15

(1)

(2)

(3)


Self Study


36/37

d.p=P(demand


37/37

READING AND EXERCISES

Lesson 1

Reading:

Section 5.1 - 5.3 , pp. 245-254 (4th Ed.), pp. 232-245(5th Ed.), pp. 248-261 (6th Ed.)

Exercise:

8a, 12a, and 12b, pp. 256-258 (4th Ed.), pp. 248-250(5th Ed.), pp. 264-266 (6th Ed.)

Documents

Lecture 01 s11 431 Stochastic Inventory