Decision Theory in Operational Research By Faziii
- 1. Decision Theory Rameen Fatima 22011598-149 Hasnain Masood 22011598-150
- 2. Decision Making A decision may be defined as the selection of the action by the decision maker which is considered to be the best according to some predetermined standard from amongst the available options
- 3. Steps in Decision Making STEP 1 STEP 2 STEP 3 STEP 4 Identification of all possible outcomes called as state of nature or events Identification of all courses of action called as acts Determination of Payoff Function Choosing from among the alternatives, the best possible action on the basis of some criterion
- 4. Terminologies The Decision Maker The decision maker may be an individual or a group of individuals who is responsible for making the selection of the course of action from a set of possible courses of action State of Nature They are also known as events. States of Nature are various possible outcomes or occurrences which are outside the decision makers control and they determine the level of success for a given act Course of Action or Acts The acts are alternative course of action or strategies that are available to the decision maker. The decision maker has control over the choices of these acts.
- 5. Terminologies Payoff It measures the net benefit resulting from each combination of a state of nature and a course of action . The payoff values are also known as conditional profit values or conditional economic consequences. Payoff Table A tabular arrangement of conditional payoff values is known as payoff matrix Suppose there are m possible states of nature or events 𝑆1, 𝑆2, 𝑆3, … , 𝑆𝑚 And n course of action or acts or strategies 𝐴1, 𝐴2, 𝐴3, … , 𝐴𝑛 𝑃𝑖𝑗 = the payoff corresponding to the strategy 𝐴𝑗 of the decision maker under the state of nature 𝑆𝑖 Totally there are mn payoff values
- 6. Terminologies State of Nature Acts 𝐴1 𝐴2 𝐴3 --- ---- 𝐴𝑛 𝑆1 𝑃11 𝑃12 𝑃13 𝑃1𝑛 𝑆2 𝑃21 𝑃22 𝑃23 𝑃2𝑛 𝑆3 𝑃31 𝑃32 𝑃33 𝑃3𝑛 ----- 𝑆𝑚 𝑃𝑚1 𝑃𝑚2 𝑃𝑚3 𝑃𝑚𝑛
- 7. Types of Decision Making Environments 1 2 3 Decision making under certainty Decision making under uncertainty Decision making under risk
- 8. Decision Making Under Certainty In this case the decision maker has the complete knowledge of the consequences of every alternative or decision choice. Here the decision maker presumes hat only one state of nature is relevant for this purpose. He identifies this state of nature , takes it for granted and presumes complete knowledge as to its occurrence.
- 9. Decision Making Under Uncertainty In this case the decision maker faces multiple state of nature but has no means to arrive at the probability values to the likelihood of occurrences of these state of nature .
- 10. Decision Making Under Risk In this case the decision maker has to faces several state of nature. But he has some knowledge or experience to assign probabilities value to the likely occurrences of these states of nature. The objective is to maximize the expected profit or minimize the expected opportunity loss.
- 12. Decision Making Under Uncertainty Decision under uncertainty means making a choice when you don’t know all the outcomes or their likelihoods. You have limited or no information about how things will turn out, which makes the decision challenging. Example: Imagine you’re planning an outdoor event but don’t know if it will rain. You have to decide between booking an expensive indoor venue or taking a risk with the outdoor option. Since you don’t know the weather for sure, you’re making a decision under uncertainty. .
- 13. Criterion MAXIMAX MAXIMIN MINIMIN LAPLACE MINIMAX REGRET
- 14. Maximax This is the OPTIMISTIC type of criterion In this criterion the decision maker does not want to miss the opportunity to achieve the largest possible profit. Procedure 1) Locate the maximum payoff values corresponding to each act 2) From among the maximums choose the highest value 3) The act corresponding to the highest value will be the decision
- 15. Example 1) maximum payoff values corresponding to act 𝐴1=120 i.e. maximum in 1st column maximum payoff values corresponding to act 𝐴2=80 i.e. maximum in 2nd column maximum payoff values corresponding to act 𝐴3=100 i.e. maximum in 3rd column 2) From among the maximums choose the highest value maximum payoff values out of (120,80,100)=120 3) The act corresponding to the highest value 120 is 𝐴1 will be the decision State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 40 80 100 𝑆2 70 30 80 𝑆3 120 10 20
- 16. Maximin This is the PESSIMISTIC type of criterion. This is a conservative approach. Here the decision maker attempts in maximizing the minimum possible profit Procedure 1) Locate the minimum payoff values corresponding to each act 2) From among the minimum choose the maximum value 3) The act corresponding to the maximum value will be the decision
- 17. Example 1) minimum payoff values corresponding to act 𝐴1=40 i.e. maximum in 1st column minimum payoff values corresponding to act 𝐴2=10 i.e. maximum in 2nd column minimum payoff values corresponding to act 𝐴3=20 i.e. maximum in 3rd column 2) From among the minimum choose the maximum value maximum payoff values out of (40,10,20)=40 3) The act corresponding to the highest value 40 is 𝐴1 will be the decision State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 40 80 100 𝑆2 70 30 80 𝑆3 120 10 20
- 18. Minimin This is the PESSIMISTIC type of criterion. Procedure 1) Locate the minimum payoff values corresponding to each act 2) From among the minimum choose the minimum value 3) The act corresponding to the minimum value will be the decision
- 19. Example 1) minimum payoff values corresponding to act 𝐴1=40 i.e. maximum in 1st column minimum payoff values corresponding to act 𝐴2=10 i.e. maximum in 2nd column minimum payoff values corresponding to act 𝐴3=20 i.e. maximum in 3rd column 2) From among the minimum choose the minimum value minimum payoff values out of (40,10,20)=10 3) The act corresponding to the minimum value 10 is 𝐴2 will be the decision State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 40 80 100 𝑆2 70 30 80 𝑆3 120 10 20
- 20. Laplace In this criteria it is assumed that all states of nature will occur with equal probability. Here the decision maker finds the average outcome for each act and picks up the act corresponding to maximum value Procedure 1) Find the average payoff values corresponding to each act 2) From among the averages choose the maximum value 3) The act corresponding to the maximum value will be the decision
- 21. Example 1) average payoff values corresponding to act 𝐴1 = 40+70+120 3 = 230 3 = 76.67 i.e. average in 1st column average payoff values corresponding to act 𝐴2 = 80+30+10 3 = 120 3 = 40 i.e. average in 2nd column average payoff values corresponding to act 𝐴3 = 100+80+20 3 = 200 3 = 66.67 i.e. average in 3rd column 2) From among the averages choose the maximum value minimum payoff values out of (76.67,40,66.67)=76.67 3) The act corresponding to the maximum value 76.67 is 𝐴1 will be the decision State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 40 80 100 𝑆2 70 30 80 𝑆3 120 10 20
- 22. Minimax Regret In this criteria the decision maker identifies the maximum regret for each act and selects the act for which maximum regret is minimum. Procedure 1) Convert conditional profit matrix into regret matrix i.e. opportunity loss table 2) Find the maximum values corresponding to each act 2) From among the averages choose the minimum value 3) The act corresponding to the minimum value will be the decision
- 23. Opportunity Loss Table The Opportunity Loss is defined as the difference between the highest possible profit for a state of nature and the actual profit obtained for the particular action taken. Maximum pay of 𝑆1=Max{𝑃11, 𝑃12, 𝑃13, … 𝑃1𝑛} = Max{S1} State of Nature Acts 𝐴1 𝐴2 𝐴3 --- 𝐴𝑛 𝑆1 𝑃11 𝑃12 𝑃13 𝑃1𝑛 𝑆2 𝑃21 𝑃22 𝑃23 𝑃2𝑛 𝑆3 𝑃31 𝑃32 𝑃33 𝑃3𝑛 ---- 𝑆𝑚 𝑃𝑚1 𝑃𝑚2 𝑃𝑚3 𝑃𝑚𝑛 Conditional Profit Matrix State of Nature Acts 𝐴1 𝐴2 𝐴3 --- 𝐴𝑛 𝑆1 Max S1 − 𝑃11 Max S1 − 𝑃12 Max S1 − 𝑃13 Max S1 − 𝑃1𝑛 𝑆2 Max S2 − 𝑃21 Max S2 − 𝑃22 Max S2 − 𝑃23 Max S2 − 𝑃2𝑛 𝑆3 Max S3 − 𝑃31 Max S3 − 𝑃32 Max S3 − 𝑃33 Max S3 − 𝑃3𝑛 ----- 𝑆𝑚 Max S𝑚 − 𝑃𝑚1 Max S𝑚 − 𝑃𝑚2 Max S𝑚 − 𝑃𝑚3 Max S𝑚 − 𝑃𝑚𝑛 Opportunity Loss Table or Regret Table
- 24. Example Maximum of 𝑆1=Max{𝑆1}=Max{40,80,100}=100 Maximum of 𝑆2=Max{𝑆2}=Max{70,30,80}=80 Maximum of 𝑆3=Max{𝑆3}=Max{120,10,20}=120 State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 40 80 100 𝑆2 70 30 80 𝑆3 120 10 20 Conditional Payoff Matrix State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 100-40 100-80 100-100 𝑆2 80-70 80-30 80-80 𝑆3 120-120 120-10 120-20 Opportunity Loss Table State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 60 20 0 𝑆2 10 50 0 𝑆3 0 110 100 Opportunity Loss Table
- 25. Example Step 2) Maximum of 𝐴1=Max{60,10,0}=60 Maximum of 𝐴2=Max{20,50,110}=110 Maximum of 𝐴3=Max{0,0,100}=100 Step 3) From among the averages choose the minimum value i.e. min{60,110,100}=60 Step 4) The act corresponding to the minimum value 60 will be the decision i.e. 𝐴1 State of Nature Acts 𝐴1 𝐴2 𝐴3 𝑆1 60 20 0 𝑆2 10 50 0 𝑆3 0 110 100 Opportunity Loss Table