Operation Research (O.R.) is a systematic study aimed at optimizing decision-making within organizations by employing scientific methods and interdisciplinary approaches. It encompasses various domains such as resource allocation, production planning, procurement, marketing, finance, and personnel management, utilizing quantitative analysis for problem-solving. Key decision-making criteria discussed include Maximax, Maximin, Minimax, Hurwicz, and Laplace, applicable under conditions of certainty, uncertainty, and risk.

1. OPERATION
RESEARCH

2. OPERATION RESEARCH
of
An operation may be defined as the set of acts
required for the achievements of a desired
outcomes.

3. DEFNITIONS OF O.R.
 I
OPERATION RESEARCH IS SYSTEMATIC, METHOD ORIENTED
STUDY OF THE BASIC STRUCTURE, CHARACTERISTICS,
FUNCTION & RELATIONSHIP OF AN ORGANISATION TO PROIDE
THE EXECUTIVE WITH A SOUND, SCIENTIFIC AND QUANITATIVE
BASIS FOR THE DECESION MAKING.
------------------------------BY E.L.ARNOFF &
M.J.NETZORG
O.R. IS AN AID FOR THE EXECUTIVE IN MAKING HIS DECISIONS
BY PROVIDING HIM WITH THE NEEDED QUANTITATIVE
INFORMATION BASED ON THE SCIENTIFIC METHOD OF
ANALYSIS ---------------BY C. KITTEL
O.R. IS THE APPLICATION OF SCIENTIFIC METHODS TO THE
PROBLEM ARISING FROM OPERATIONS INVOLVING
INTEGRATED SYSTEMS OF MEN, MACHINE AND MATERIALS. IT
NORMALLY UTILIZES THE KNOWLEDGE AND SKILLS OF
INTERDISCIPLINARY RESEARCH TEAM TO PROVIDE THE
MANAGERS OF SUCH SYSTEMS WITH OPTIMUM OPERATING
SOLUTIONS. --------------------BY FABRYCKY & TORGERSEN

4. CHARACTERISTICS OF OR:
 ITS SYSTEM ORIENTED
 USE OF INTERDISCIPLINARY TEAM
 APPLICATION OF SCIENTIFIC METHODS
 UNCOVERING OF NEW PROBLEMS
 IMPROVE THE QUALITY OF DECESIONS
 USE OF COMPUTERS
 QUANTITATIVE SOLUTION
 HUMAN FACTORS

5. SCOPE OF OPERATION RESEARCH:
I. ALLOCATION AND DISTRIBUTION:
Optimal allocation of limited resources such as
men, machine, and material.
Location and size of warehouses, distribution
centre, retail depot etc.
Distribution policy.
II. PRODUCTION AND FACILITY PLANNING:
Selection, location and design of production
plant.
Project scheduling & allocation of resources.
Forecasting.
Maintenance policy.
Scheduling & sequencing.

6. III. PROCUREMENT:
What, when and how to purchase at minimum
procurement cost
Bidding and replacement policies.
IV. MARKETING:
Product selection, timing & competitive action
Selection of advertising media.
Demand forecast and stock level.
Customer’s preference for size, colour &
packaging of various products.
V. FINANCE:
Capital requirement, cash flow analysis.
Credit policies, credit risks etc.
Profit plan of the company.
Determination of optimum replacement policies.

7. VI. PERSONNEL:
Selection of personnel, determination of
retirement age and skills
Recruitment of policies & assignments of jobs.
VII. RESEARCH AND DEVELOPMENT:
Determination of areas of Research and
Development
Reliability & control of development of projects.
Selection of projects & preparation of their
budgets.

8. METHODOLOGY OF OR:
FORMULATE THE PROBLEM
CONSTRUCT A MATHEMATICAL MODEL
SOLVE THE MODEL
TEST THE MODEL
ANALYSE THE RESULT
IMPLEMENTATION OF SELECTED STRATEGY

9. DIFFICULTY IN O.R.
PROBLEM FORMULATION
DATA COLLECTION
STUDY BASED ON OBSERVATION OR OLD
LAWS
TIME FACTOR
HUMAN FACTOR

10. DECESION THEORY:
DECESION THEORY PROVIDES A RATIONAL
APPROACH IN DEALING PROBLEMS CONFRONTED
WITH THE PARTIAL , IMPERFECT OR UNCERTAIN
FUTURE CONDITION

11. STEPS IN DECESION THEORY
APPROACH:
LIST ALL THE VIABLE
ALTERNATIVES
IDENTIFY THE EXPECTED
FUTURE EVENTS
CONSTRUCT A PAY-OFF TABLE
SELECT OPTIMUM DECESION
CRITERION

12. DECESION MAKING ENVIRONMENT:
DECESIONS ARE MADE UNDER THREE TYPES OF
ENVIRONMENT:
D.M.E.
CERTAINITY UNCERTAINITY RISK
IN THIS , ONLY
ONE STATE OF
NATURE EXISTS
i.e. THERE IS
COMPLETE
CERTAINITY
ABOUT THE
FUTURE
HERE MORE THAN
ONE S.O.N. EXISTS
BUT D. MAKER
LACKS
SUFFICIENT
KNOWLEDGE TO
ALLOW HIM
ASSIGN PROB TO
VARIOUS S.O.N.
HERE ALSO MORE
THAN ONE S.O.N.
EXISTS BUT THE D.
MAKER HAS
SUFFICIENT INFO
TO ALLOW HIM
ASSIGN PROB TO
EACH OF THESE
STATES

13. D.M. UNDER UNCERTAINITY:
Under condition of uncertainty, the decision maker has
knowledge about states of nature that happens but
lacks the knowledge about the probabilities of their
occurrence.
Under conditions of uncertainty, a few decision
criterions are available which could be of help to the
decision maker.
D.M. UNDER UNCERTAINITY
Maximax
Criterion
or
Criterion of
optimism
Maximin
Criterion
or
Criterion of
pessimism
Minimax
Criterion
or
Regret
Criterion
Hurwicz
Criterion
or
Criterion of
Realism
Laplace
Criterion
or
Criterion of
Rationality

14. ILLUSTRATION: CONSIDERING A MANUFACTURING
COMPANY THAT IS THINKING OF VARIOUS
ALTERNATIVES TO INCREASE ITS PRODUCTION TO
MEET THE INCREASING MARKET DEMAND.
WHICH STRATEGY OR ALTERNATIVE WILL THE CO.
EMPLOY ON THE BASIS OF VARIOUS METHODS.
ALTERNATIVES
STATE OF NATURE (PRODUCT DEMAND)
HIGH MODERATE LOW NIL
EXPAND 50,000 25,000 - 25,000 - 45,000
CONSTRUCT 70,000 30,000 - 40,000 - 80,000
SUBCONTRACT 30,000 15,000 - 1,000 - 10,000

15. (I) MAXIMAX CRITERION OR CRITERION
OF OPTIMISM:
 This criterion provides the decision maker with optimistic
criterion. The working method is summarizing as follow.
Locate the maximum payoff values corresponding to each
alternative (or course of action or strategy), then
Select an alternative with maximum payoff value.
THUS THE MAXIMAX PAYOFF IS Rs. 70,000 CORRESPONDING TO
THE ALTERNATIVE “CONSTRUCT”.
ALTERNA
TIVES
STATE OF NATURE (PRODUCT DEMAND)
HIGH MODERA
TE
LOW NIL
EXPAND 50,000 25,000 - 25,000 - 45,000
CONSTRU
CT
70,000 30,000 - 40,000 - 80,000
SUBCONT
RACT
30,000 15,000 - 1,000 - 10,000
MAXIMUM
OF
ROW
50,000
70,000
30,000

16. (II) MAXIMIN CRITERION OR CRITERION
OF PESSIMISM:
This criterion provides the decision maker with pessimistic
criterion. The working method is summarizing as follow.
Locate the minimum payoff values corresponding to each
alternative (or course of action or strategy), then
Select an alternative with maximum payoff value.
THUS THE MINIMAX PAYOFF IS Rs. – 10,000 CORRESPONDING TO THE
ALTERNATIVE - “SUBCONTRACT”
ALTERNA
TIVES
STATE OF NATURE (PRODUCT DEMAND)
HIGH MODERAT
E
LOW NIL
EXPAND 50,000 25,000 - 25,000 - 45,000
CONSTRU
CT
70,000 30,000 - 40,000 - 80,000
SUBCONT
RACT
30,000 15,000 - 1,000 - 10,000
MINIMUM
OF
ROW
-45,000
-80,000
-10,000

17. (III) MINIMAX CRITERION OR MINIMUM
REGRET CRITERION:
This criterion is also known as opportunity loss
decision criterion or minimax regret criterion. The
working method is summarizing as follow.
Determine the amount of regret corresponding to
each alternative for each state of nature. The regret
for jth
event corresponding to ith
alternative is given by
ith
regret = (maximum payoff – ith
payoff) for the jth
event
Determine the maximum regret amount for each
alternative.
Choose the alternative which corresponds to the
minimum of the maximum regrets.

18. ALTERNATIVE
S
STATE OF NATURE (PRODUCT DEMAND)
HIGH MODERATE LOW NIL
EXPAND 50,000 25,000 - 25,000 - 45,000
CONSTRUCT 70,000 30,000 - 40,000 - 80,000
SUBCONTRACT 30,000 15,000 - 1,000 - 10,000
ALTERNATI
VES
CALCULATION OF REGRET
HIGH MODERAT
E
LOW NIL
EXPAND 20000 5000 24000 35000
CONSTRUC
T
0 0 39000 70000
SUBCONTR
ACT
40000 15,000 0 0
MAXIMUM
OF
ROW
35,000
70,000
40,000
THIS TABLE SHOWS THAT THE COMPANY WILL MINIMIZE ITS
REGRET TO RS 35,000 BY SELECTING ALTERNATIVE-
“EXPANSION”

19. (IV) HURWICZ CRITERION OR CRITERION OF REALISM:
Also called weighted average criterion, it is a compromise
between the maximax (optimistic) and minimax
(pessimistic) decision criterion. This concept allows the
decision maker to take into account both maximum and
minimum for each alternative and assign them weights
according to his degree of optimism (or pessimism). The
working method is summarizing as follow:
Choose an appropriate degree of optimism, α so that (1-α)
represents degree of pessimism.
Determine the maximum as well as minimum of each
alternative and obtain
P = α. Maximum + (1-α). Minimum
for each alternative.
Choose the alternative that yields the maximum value of
P.

20. HERE LET α = 0.8
WORKING NOTES:
H1 = 0.8 * 50000 + 0.2 * -45000 = 31000
H2 = 0.8 * 70000 + 0.2 * -80000 = 40000
H3 = 0.8 * 30000 + 0.2 * -10000 = 22000
THUS ACCORDING TO HURWICZ CRITERION , COMPANY WILL
CHOOSE ALTERNATIVE – “CONSTRUCT”
ALTER
NATIV
ES
STATE OF NATURE
HIGH MOD LOW NIL
EXP 50,000 25,000 - 25,000 - 45,000
CONST 70,000 30,000 - 40,000 - 80,000
SUBCO
NTRACT
30,000 15,000 - 1,000 - 10,000
H
-80000
-10000
40000
MAX
OF
ROW
-45000
70,000
31000
MIN
OF
ROW
50000
30000 22000

21. (V) LAPLACE CRITERION OR CRITERION OF
RATIONALITY:
Also known as equal probabilities criterion or
criterion of rationality. Since the probability of states
of nature are not known, it is assumed that all states
of nature will occur with equal probability, i.e. assign
an equal probability. The working method is
summarizing as follow:
Determine expected value for each alternative; if n
denotes the number of events and P’s denote the
payoffs, then expected value is given by 1n[P1+P2+….
+Pn]
Choose the alternative that yields the maximum value
of P.

22. ALTERNA
TIVES
STATE OF NATURE EXPECTE
D PAYOFF
HIGH MODERAT
E
LOW NIL
EXPAND 50,000 25,000 - 25,000 - 45,000
CONSTRU
CT
70,000 30,000 - 40,000 - 80,000
SUBCONT
RACT
30,000 15,000 - 1,000 - 10,000
WORKING NOTES:
(E.P.)1 = ¼(50000 + 25000 - 25000 – 45000) = 1250
(E.P.)2 = ¼(70000 + 30000 – 40000 – 80000)= - 5000
(E.P.)3 = ¼(30000 + 15000 – 1000 – 10000 ) = 8500
THUS ACCORDING TO LAPLACE CRITERION , COMPANY WILL
CHOOSE ALTERNATIVE – “SUBCONTRACT”
- 5000
1250
8500

23. ILLUSTRATION: THE FOLLOWING MATRIX GIVES
THE PAYOFF OF DIFFERENT STRATEGIES S1, S2, S3
AGAINST CONDITIONS N1, N2, N3 AND N4.
INDICATE THE DECESION TAKEN UNDER THE
FOLLOWING APPROACH:
OPTIMISTIC
PESSIMISTIC
REGRET
HURWICZ, THE DEGREE OF OPTIMISM BEING 0.7
EQUAL PROBABILITY
STRATEGY CONDITIONS
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 4000 - 100 6000 18000
S2 20000 5000 400 0
S3 20000 15000 - 2000 1000

24. SOLUTION
(I) OPTIMISTIC CRITERION:
SO, ACCORDING TO O.C., MAXIMAX PAYOFF IS Rs. 20000
CORRESPONDING TO THE STRATEGY – “S2” AND “S3” .
(II) PESSIMISTIC CRITERION:
SO, ACCORDING TO P.C., MAXIMIN PAYOFF IS Rs. 0
CORRESPONDING TO THE STRATEGY – “S2”
STRATEGY
CONDITIONS
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 4000 - 100 6000 18000
S2 20000 5000 400 0
S3 20000 15000 - 2000 1000
MAX
OF
ROW
20000
20000
18000
STRATEGY
CONDITIONS
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 4000 - 100 6000 18000
S2 20000 5000 400 0
S3 20000 15000 - 2000 1000
MIN
OF
ROW
- 2000
0
- 100

25. (III) REGRET (SAVAGE CRITERION):
THE BEST PAY OFFS FOR EACH STATE OF NATURE
N1, N2, N3 & N4 ARE Rs. 20000, Rs. 15000, Rs. 6000 & RS.
18000 RESPECTIVELY.
SUBSTRACTING FROM THESE THE PAYOFFS OF
CORRESPONDING COLUMN WE GET
THE MINIMAX REGRET CORRESPONDS TO
STRATEGY “S1”.
STRATEGY
CONDITION
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 16000 15100 0 0
S2 0 10000 5600 18000
S3 0 0 8000 17000
MAX
OF
ROW
17000
18000
16000

26. (IV) HURWICZ CRITERION:
STRAT
EGY
CONDITION H=
.max +
(1-)
min
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 4000 - 100 6000 18000
S2 20000 5000 400 0
S3 20000 15000 - 2000 1000
MAX
OF
ROW
20000
20000
18000
MIN
OF
ROW
- 2000
0
- 100
13400
14000
12570
HERE  = 0.7
WORKING NOTES:
H1 = 0.7 *18000 + 0.3 * (- 100) = 12570
H2 = 0.7 * 20000+ 0.3 * 0 = 14000
H3 = 0.7 * 20000 + 0.3 * (- 2000) = 13400
THE MAXIMUM VALUE OF H = Rs. 14000 WHICH CORRESPONDS TO
STRATEGY “S2”.

27. (V) EQUAL PROBABILITY CRITERION:
WORKING NOTES:
(E.P.)1 = ¼(4000 - 100 + 6000 + 18000) = 6975.
(E.P.)2 = ¼(20000 + 5000 + 400 + 0 ) = 6350.
(E.P.)3 = ¼(20000 + 15000 – 2000 + 1000 ) = 8500.
THE MAXIMUM PAYOFF IS Rs. 8500 WHICH
CORRESPONDS TO THE STRATEGY – “S3”.
STRATEGY
CONDITION
N1
(Rs)
N2
(Rs)
N3
(Rs)
N4
(Rs)
S1 4000 - 100 6000 18000
S2 20000 5000 400 0
S3 20000 15000 - 2000 1000
EXPECTE
D
PAYOFF
8500
6975
6350

28. DECESION MAKING UNDER
RISK:
Here more than one state of nature exists and
the decision maker has sufficient information
to assign probabilities to each of these states.
These probabilities could be obtained from the
past records or simply the subjective judgment
of the decision maker.
Under conditions of risk, knowing the
probability distribution of the state of nature,
the best decision is to select the course of
action which has the largest expected pay off
value.

29. DECESION MAKING UNDER
RISK:
Expected Value
Criterion
or
Expected Monetary
Value Criterion
Expected
Opportunity Loss
Criterion
or
Expected Value of
Regret
Expected Value
for Perfect
Information
Conditional Profit
Table
Expected Profit Table
Conditional Profit
Table
Conditional Loss
table
Expected Loss Table
Conditional Profit
Table with P.I.
Expected Profit Table
with P.I.

30. ILLUSTRATION
A newspaper boy has the following probabilities of
selling a magazine:
No. of copies sold Probability
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
Cost of the copy is 30 paisa and sale price is 50 paisa. He
cannot return the unsold copies. How many should he
order?

31. EXPECTED VALUE CRITERION:
The expected monetary value for a given course of
action is the weighted sum of possible payoffs for each
alternative. It is obtained by summing the payoffs for
each course of action multiplied by the probabilities
associated with state of nature. It consists of following
steps:
Construct a payoff table listing the alternative decisions
and the various state of nature. Enter the conditional profit
for each decision event combination along with the
associated probabilities. (Construct Conditional profit
table).
Calculate the EMV for each decision alternative by
multiplying the conditional profits by assigned
probabilities and adding the resulting conditional values.
(Construct expected profit table).
Select the alternative that yields the highest EMV.

32. SOLUTION
Cost Price = 30 paisa.
Selling Price = 50 paisa.
Profit = Selling price – Cost price = 20 paisa.
STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
Conditional
Profit =
Profit * S.P. = 20 S.P. ;When D ≥ S
S.P. * D – C.P. * S = 50D – 30S; When D < S
200
200 220
240
260
280
200
200
200 220
220
220
240
240 260
110 80
130
160
190
210 180
230
140
170

33. STEP II: CONSTRUCT EXPECTED PROFIT TABLE:
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
20 17 14 11 8
30
40
33 28.5 24 19.5
44 48 42 36
50 55 60 65 57.5
60 66 72 78 84
200 222.5 220 205
TOTAL EXPECTED
PROFIT
215
THE NEWS BOY MUST, THEREFORE, ORDER 12
COPIES TO EARN THE HIGHEST POSSIBLE
AVERAGE DAILY PROFIT OF 222.5 PAISE

34. EXPECTED OPPORTUNITY LOSS CRITERION:
EOL represents the amount by which maximum possible
profit will be reduced under various possible stock actions.
The course of action that minimizes these losses or
reductions is the optimal decision alternative. The
procedure to calculate expected opportunity losses is as
follows:
Prepare the conditional profit table for each decision-event
combination and write associated probabilities.
(Construct Conditional profit table).
For each event, determine the conditional opportunity loss
(COL) by subtracting the payoff from the maximum payoff
for that event. (Construct Conditional loss table).
Calculate the expected opportunity loss for each decision
alternative by multiplying the COL’s by the associated
probabilities and then adding the values. (Construct
Expected loss table).
Select the alternative that yields the lowest EOL.

35. SOLUTION
Cost Price = 30 paisa.
Selling Price = 50 paisa.
Profit = Selling price – Cost price = 20 paisa.
STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE
Conditional
Profit =
Profit * S.P. = 20 S.P. ;When D ≥ S
S.P. * D – C.P. * S = 50D – 30S; When D < S
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
200
200 220
240
260
280
200
200
200 220
220
220
240
240 260
110 80
130
160
190
210
230
140
170
180

36. STEP II: CONSTRUCT CONDITIONAL LOSS TABLE
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
0
20 0
0
0
0
60
80
40 20
40
60
20
40 20
90 120
90
60
30
30
30
60
30
60
ROW-WISE SUBSTRACTION
ROW MAX – OTHER ELEMENTS OF ROW

37. STEP III: CONSTRUCT EXPECTED LOSS TABLE:
THE OPTIMUM STOCK ACTION IS THE ONE WHICH
WILL MINIMIZE EXPECTED OPPORTUNITY LOSS;
THIS ACTION CALLS FOR THE STOCKING OF 12
COPIES EACH DAY AT WHICH POINT THERE IS
MINIMUM EXPECTED LOSS OF 27.5 PAISE.
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
0
3 0
0
0
0
15
24
8 4
10
18
5
12 6
9 12
13.5
9
4.5
6
7.5
6
3
12
E.O.L. 50 35 27.5 30 45

38. EXPECTED VALUE FOR PERFECT INFORMATION:
Perfect Information means complete and accurate
information about the future demand and that remove
all the uncertainty for future.
EVPI represents the maximum amount of money the
decision maker has to pay to get this additional
information about the occurrence of various state of
nature before a decision has to be made. The
procedure to calculate expected value of perfect
information is as follows:
Construct conditional profit table with perfect
information.
Construct expected profit table with perfect
information.
Determine EVPI from relation;
EVPI = EPPI – max EMV

39. SOLUTION
Cost Price = 30 paisa.
Selling Price = 50 paisa.
Profit = Selling price – Cost price = 20 paisa.
STEP I : CONSTRUCT CONDITIONAL PROFIT
TABLE
Possible
Demand
(No. of
Copies)
Probability
Possible Stock Action
10
Copies
11
Copies
12
Copies
13
Copies
14
Copies
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
200
220
240
260
280

40. STEP II: CONSTRUCT EXPECTED PROFIT
TABLE WITH PERFECT INFORMATION:
DEMAND
(No. Of Copies)
Probability Conditional Profit
Under Certainty
Expected Profit With
Perfect Information
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30
200
220
240
260
280
20
33
48
65
84
EPPI = 250
STEP III: The expected value of perfect information is
given by
EVPI = EPPI – max EMV
= 250 – 222.5
= 27.5 Paise
Thus this is the maximum amount which the newsboy
willing to pay, per day, for a perfect information.
= min E.O.L.

41. ILLUSTRATION
Under an employment promotion program, it is proposed to
allow sale of newspapers on the buses during off peak hours.
The vendor can purchase the newspaper at a special
concessional rate of 25 paise per copy against the selling
price of 40 paise. Any unsold copies are, however a dead loss.
A vendor has estimated the following probability distribution
for the no. of copies demanded:
a) How many copies should he ordered so that his expected
profit will be maximum?
b) Compute EPPI
c) The vendor is thinking of spending on a small market
survey to obtain additional information regarding the
demand levels. How much should he be willing to spend on
such a survey?
No. of copies 15 16 17 18 19 20
Probability 0.04 0.19 0.33 0.26 0.11 0.07

42. SOLUTION
(a) CALCULATION OF EXPECTED PROFIT:
Cost Price = 25 paisa.
Selling Price = 40 paisa.
Profit = Selling price – Cost price = 40 – 25 = 15 paisa.
STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE:
Demand
(No. of
Copies) Prob
Possible Stock Action
15
Copies
16
Copies
17
Copies
18
Copies
19
Copies
20
Copies
15 0.04
16 0.19
17 0.33
18 0.26
19 0.11
20 0.07
225
225 240
255
270
285
225
225
225 240
240
240
255
255 270
150 125
165
190
215
230 205
245
175
200
260
100
140
180
220
285
225 240 255 270 300