University of South Florida Probability Questions & Decision Tree

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Engineering Decision Making Under Uncertainty Aimen Sudhir, RUID: 197007651 To make the recommendation, the following decision tree was made. As seen from the decision tree, it is profitable for Ectron Fragrance Corporation to conduct the market survey. Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Introduction and Basic Concepts Melike Baykal-Gürsoy ISE - Rutgers University gursoy@soe.rutgers.edu gursoy.rutgers.edu September 2, 2020 Dr. Baykal-Gürsoy gursoy.rutgers.edu Prob. Review Melike Baykal-Gürsoy Prof. in Dept. of I&SE Director of Lab. for Stochastic Systems GRIST-Game Research for Infrastructure SecuriTyLab Introduction I PhD - Univ. of Pennsylvania Systems Eng. I BS & MS - Bogazici Univ. Turkiye - EE I Areas of interest: stochastic modeling, queueing, Markov decision processes, stochastic games, and applications in inventory, transportation, communication networks. I Courses: Eng. Decision Making, Stochastic Modeling, Inventory Control, Process Modeling and Control, Queueing Basic Definitions and Relationships Learning Objectives Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Introduction Basic Definitions and Relationships Table of Contents Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Introduction Basic Definitions and Relationships Prob. Review Initial Definitions Melike Baykal-Gürsoy Consider and experiment whose outcome is not predictable in advance. I Sample Space: set of all possible outcomes of an experiment. We will denote it by S I Event: Any subset S of the sample space S. Event A Sample Space Event B Introduction Basic Definitions and Relationships Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I Example: A family with two cars is selected. For both the older and the newer car, we note whether the car was manufactured in Asia, Europe, or America. I In this case, the sample space can be written as {(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ), (As , Am ), (As , E ), (As , As )}. I The total number of possible outcomes (cardinality of the sample space) is 32 = 9. Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I Example: A family with two cars is selected. For both the older and the newer car, we note whether the car was manufactured in Asia, Europe, or America. I In this case, the sample space can be written as {(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ), (As , Am ), (As , E ), (As , As )}. I The total number of possible outcomes (cardinality of the sample space) is 32 = 9. Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I Example: A family with two cars is selected. For both the older and the newer car, we note whether the car was manufactured in Asia, Europe, or America. I In this case, the sample space can be written as {(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ), (As , Am ), (As , E ), (As , As )}. I The total number of possible outcomes (cardinality of the sample space) is 32 = 9. Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let us consider three events: I Event 1: One car is American and the other is foreign. E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )} I Event 2:At least one car is foreign. E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ), (E , E ), (E , As ), (As , As ), (As , E )} I Event 3: Both cars are American. E3 = {(Am , Am )} Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let us consider three events: I Event 1: One car is American and the other is foreign. E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )} I Event 2:At least one car is foreign. E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ), (E , E ), (E , As ), (As , As ), (As , E )} I Event 3: Both cars are American. E3 = {(Am , Am )} Motivating example Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let us consider three events: I Event 1: One car is American and the other is foreign. E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )} I Event 2:At least one car is foreign. E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ), (E , E ), (E , As ), (As , As ), (As , E )} I Event 3: Both cars are American. E3 = {(Am , Am )} Notation Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let A and B be events: I Union of A and B is the event that at least A or B occurs. A ∪ B; I Intersection of A and B is the event that A and B occur. A ∩ B or simply AB; I Complement of A is the probability that A does not occur. Ac = S \ A. Notation Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let A and B be events: I Union of A and B is the event that at least A or B occurs. A ∪ B; I Intersection of A and B is the event that A and B occur. A ∩ B or simply AB; I Complement of A is the probability that A does not occur. Ac = S \ A. Notation Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let A and B be events: I Union of A and B is the event that at least A or B occurs. A ∪ B; I Intersection of A and B is the event that A and B occur. A ∩ B or simply AB; I Complement of A is the probability that A does not occur. Ac = S \ A. Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Prob. Review Example Melike Baykal-Gürsoy I I I I I I E1 ∪ E2 = E2 (because E1 is a subset of E2 ) E2 ∪ E3 = S (includes all possible outcomes) E 1 ∩ E2 = E1 E 1 ∩ E3 = ∅ E 2 ∩ E3 = ∅ E2c = S \ E2 = E3 Sample Space E1 E2 E3 Introduction Basic Definitions and Relationships Initial Definitions Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I The probability that the event A occurs is written P{A} I The certain event S always occurs, P{S} = 1. I The impossible event ∅ never occurs, P{∅} = 0. I Events A and B are said to be disjoint if A ∩ B = ∅ (They cannot both occur) I E1 and E3 , and E2 and E3 are two examples of disjoint events. I Events A and B are said to be mutually exclusive if A ∩ B = ∅ and A ∪ B = S. I E2 and E3 are mutually exclusive. Initial Definitions Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I The probability that the event A occurs is written P{A} I The certain event S always occurs, P{S} = 1. I The impossible event ∅ never occurs, P{∅} = 0. I Events A and B are said to be disjoint if A ∩ B = ∅ (They cannot both occur) I E1 and E3 , and E2 and E3 are two examples of disjoint events. I Events A and B are said to be mutually exclusive if A ∩ B = ∅ and A ∪ B = S. I E2 and E3 are mutually exclusive. Initial Definitions Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I The probability that the event A occurs is written P{A} I The certain event S always occurs, P{S} = 1. I The impossible event ∅ never occurs, P{∅} = 0. I Events A and B are said to be disjoint if A ∩ B = ∅ (They cannot both occur) I E1 and E3 , and E2 and E3 are two examples of disjoint events. I Events A and B are said to be mutually exclusive if A ∩ B = ∅ and A ∪ B = S. I E2 and E3 are mutually exclusive. Initial Definitions Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I The probability that the event A occurs is written P{A} I The certain event S always occurs, P{S} = 1. I The impossible event ∅ never occurs, P{∅} = 0. I Events A and B are said to be disjoint if A ∩ B = ∅ (They cannot both occur) I E1 and E3 , and E2 and E3 are two examples of disjoint events. I Events A and B are said to be mutually exclusive if A ∩ B = ∅ and A ∪ B = S. I E2 and E3 are mutually exclusive. Initial Definitions Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I The probability that the event A occurs is written P{A} I The certain event S always occurs, P{S} = 1. I The impossible event ∅ never occurs, P{∅} = 0. I Events A and B are said to be disjoint if A ∩ B = ∅ (They cannot both occur) I E1 and E3 , and E2 and E3 are two examples of disjoint events. I Events A and B are said to be mutually exclusive if A ∩ B = ∅ and A ∪ B = S. I E2 and E3 are mutually exclusive. Initial definitions (cont.) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I For disjoint events A, B: P{A ∪ B} = P{A} + P{B}. I In general, P{A ∪ B} = P{A} + P{B} − P{A ∩ B}. Initial definitions (cont.) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I For disjoint events A, B: P{A ∪ B} = P{A} + P{B}. I In general, P{A ∪ B} = P{A} + P{B} − P{A ∩ B}. Prob. Review Law of Total Probability Melike Baykal-Gürsoy Let A1 , A2 , . . . , An , . . . be events with Ai and Aj disjoint whenever i 6= j. Then: P{∪ni=1 Ai } = n X P{Ai } i=1 If S = A1 ∪ A2 ∪ A3 ∪ . . ., we have P{S} = 1 = ∞ X P{Ai } (Law of Total Probability) i=1 Law of Total Probability asserts that, for any event B, P{B} = ∞ X i=1 P{B ∩ Ai }. Introduction Basic Definitions and Relationships Example (cont.) Prob. Review Melike Baykal-Gürsoy Introduction Example: Assume we also note the color of the two cars, which can be white or black. The sample space (state space) is then: S = {(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ), (As , Am ), (As , E ), (As , As ), (W , W ), (W , B), (B, W ), (B, B)}. Let us also note event 4, E4 the event in which at least one car is black. E4 = {(B, B), (W , B), (B, W )} Basic Definitions and Relationships Independence Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships I Events A and B are independent if P{A ∩ B} = P{A}P{B} . I E2 and E4 are independent events. Practice (Pairwise Independent Events that are not Independent) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let a ball be drawn from an urn containing 4 balls, numbered 1, 2, 3, 4. Each of the four outcomes is equally likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}. I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2 (independent). I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2 (independent). I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2 (independent). I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } = 1/2 · 1/2 · 1/2 (not independent). Practice (Pairwise Independent Events that are not Independent) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let a ball be drawn from an urn containing 4 balls, numbered 1, 2, 3, 4. Each of the four outcomes is equally likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}. I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2 (independent). I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2 (independent). I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2 (independent). I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } = 1/2 · 1/2 · 1/2 (not independent). Practice (Pairwise Independent Events that are not Independent) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let a ball be drawn from an urn containing 4 balls, numbered 1, 2, 3, 4. Each of the four outcomes is equally likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}. I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2 (independent). I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2 (independent). I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2 (independent). I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } = 1/2 · 1/2 · 1/2 (not independent). Practice (Pairwise Independent Events that are not Independent) Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Let a ball be drawn from an urn containing 4 balls, numbered 1, 2, 3, 4. Each of the four outcomes is equally likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}. I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2 (independent). I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2 (independent). I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2 (independent). I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } = 1/2 · 1/2 · 1/2 (not independent). Conditional Probability Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships The conditional probability of A given B is written as P{A | B} = P{A ∩ B} P{A, B} = P{B} P{B} Prob. Review Example Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships What is the probability that the first die is 6 given that the sum of two dice is 7? P{1st is 6 and sum = 7} P{sum = 7} P{1st is 6 and 2nd = 1} = P{sum = 7} 1/6 · 1/6 = = 1/6 6/36 P{1st is 6 | sum = 7} = Prob. Review Example Melike Baykal-Gürsoy Introduction Assume that each child born is equally likely to be a boy or a girl. If a family has 2 children, what is the probability that both are girls, given that a) the oldest is a girl, P{both girls, oldest is girl} P{oldest is girl} P{both are girls} = P{oldest is girl} 1/2 · 1/2 = = 1/2 1/2 P{both girls | oldest is girl} = Basic Definitions and Relationships Prob. Review Example (cont.) Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships and b) at least one is a girl. State Space = {(G , G ), (B, G ), (G , B), (B, B)} P{both girls} P{at least one is girl} 1/2 · 1/2 = 1/3 = 3/4 P{both girls | at least one is girl} = Prob. Review Melike Baykal-Gürsoy Introduction Basic Definitions and Relationships Q&A Decision Making Under Uncertainty A. Altay and M. Baykal-Gürsoy 1 1 Industrial & Systems Engineering Department Rutgers University September 1, 2021 Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 1 / 71 Dr. Baykal-Gürsoy gursoy.rutgers.edu Prof. in Dept. of I&SE Director of Lab. for Stochastic Systems GRIST-Game Research for Infrastructure SecuriTy- Lab PhD - Univ. of Pennsylvania - Systems Eng. BS & MS - Bogazici Univ. Turkiye - EE Areas of interest: stochastic modeling, queueing, Markov decision processes, stochastic games, and applications in inventory, transportation, communication networks. Courses: Eng. Decision Making, Stochastic Modeling, Inventory Control, Process Modeling and Control, Process Modeling and Control, Queueing Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 2 / 71 Decision Making Introduction 1 Introduction 1 2 2 Single criterion decision making under Uncertainty and Risk 1 2 3 3 Elementary methods under uncertainty Decision making under risk (Expected value and expected regret) Decision trees (Type I) Utility theory 1 2 3 4 Decision environments Value of information Deal or no deal game Utility calculation Risk premiums Materials management under uncertainty 1 Single period inventory uncertainties Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 3 / 71 Introduction Introduction Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 4 / 71 Introduction Introduction Decision analysis is an analytical and systematic way to tackle problems. A good decision is based on logic: rational decision-maker (DM) We will start with: One-dimensional (single criterion) decision making. Multi-criteria decision making Value of information: Value of perfect information Value of sample information Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 5 / 71 Introduction Decision Making Environments Type 1 : DM knows nothing! Type 2 : DM knows the payoff information with certainty, does not know the uncertainties associated with the payoffs. Type 3 : DM knows every value involved in the problem with certainty. A decision problem has: an objective, alternatives (options), possible states of nature, payoff values (cost or reward). Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 6 / 71 Single Criterion Decision Making Single Criterion Decision Making Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 7 / 71 Single Criterion Decision Making A Decision Table A decision (payoff) table presenting profits for giving a concert: Indoor venue Outdoor venue Good weather $ 7000 $ 15000 Bad weather $ 8000 $ -5000 A decision (payoff) table presenting expanding a business: Construct large plant Construct small plants Do nothing Altay & Baykal-Gürsoy Favorable market $ 200,000 $ 100,000 $0 Decision Making Under Uncertainty Unfavorable market $ ´180,000 $ ´20,000 $0 September 1, 2021 8 / 71 Solution Methods Maximax Maximin Criterion of Realism Minimax Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 9 / 71 Maximax Also known as the optimistic method, it proposes to choose the alternative with the maximum reward. Indoor venue Outdoor venue Good weather $ 7000 $ 15000 Bad weather $ 8000 $ -5000 Maximum Output Favorable market Unfavorable market Maximum output $ 200,000 $ ´180,000 $ 100,000 $ ´20,000 $0 $0 Construct large plant Construct small plant Do nothing Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 10 / 71 Maximin Also known as the pessimistic method, it proposes to choose the most secure alternative. Indoor venue Outdoor venue Construct large plant Construct small plant Do nothing Altay & Baykal-Gürsoy Good weather $ 7000 $ 15000 Bad weather $ 8000 $ -5000 Favorable market Unfavorable market $ 200,000 $ ´180,000 $ 100,000 $ ´20,000 $0 $0 Decision Making Under Uncertainty Worst case Worst case September 1, 2021 11 / 71 Criterion of Realism (CoR) Let’s balance optimism-pessimism with a coefficient α: The most optimistic value has the emphasis of α, The most pessimistic value has the emphasis of 1 ´ α. The concert example: ”The concert is in July. How bad can the weather be, anyway? The weather is good 90% of the time” or ”You know what? The concert is in fall when it rains most days.” or How would your decision change? Indoor venue Outdoor venue Altay & Baykal-Gürsoy Good weather $ 7000 $ 15000 Decision Making Under Uncertainty Bad weather $ 8000 $ -5000 September 1, 2021 12 / 71 Criterion of Realism (CoR) Let’s see mathematically whether your intuition holds. Case 1. ”The concert is in July. How bad can the weather be, anyway? The weather is good almost all the time” Indoor venue Outdoor venue Altay & Baykal-Gürsoy Good weather $ 7000 $ 15000 Decision Making Under Uncertainty Bad weather $ 8000 $ -5000 September 1, 2021 13 / 71 Criterion of Realism (CoR) Case 2. ”You know what? The concert is in fall when it rains most days.” Indoor venue Outdoor venue Altay & Baykal-Gürsoy Good weather $ 7000 $ 15000 Decision Making Under Uncertainty Bad weather $ 8000 $ -5000 September 1, 2021 14 / 71 Criterion of Realism (CoR) Case 3. ”I really have no idea how the weather is going to be. It can go either way.” Indoor venue Outdoor venue Altay & Baykal-Gürsoy Good weather $ 7000 $ 15000 Decision Making Under Uncertainty Bad weather $ 8000 $ -5000 September 1, 2021 15 / 71 The Expansion Example with CoR We will do this one together. The new Amazon fulfillment center is opening in Bensalem, PA. Should we open a large center or a small center? Or should I not open this center at all? Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy High utilization Medium utilization Low utilization $ 200,000 $ -10,000 $ -180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Decision Making Under Uncertainty September 1, 2021 16 / 71 Minimax What about the opportunity loss? Minimax promotes the alternative with the minimum regret. Indoor venue Outdoor venue Good weather $ 7000 $ 15000 Bad weather $ 8000 $ -5000 Regret table: Good weather Bad weather Maximum regret Indoor venue Outdoor venue Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 17 / 71 Minimax High utilization Medium utilization Low utilization $ 200,000 $ -10,000 $ -180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Construct a large center Construct a small center Do not open a center High utilization Medium utilization Low utilization Maximum regret Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 18 / 71 Results of the examples Results of all methods are Maximax Maximin Criterion of Realism Minimax Concert example Amazon example Outdoor venue Construct a large center Indoor venue Do not open a center Depends on α Indoor venue Construct a small center Which one do you think the DM should choose? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 19 / 71 Decision Making Under Risk Decision Making Under Risk Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 20 / 71 Decision Making Under Risk Decision Making Under Risk Probability: a numerical statement about the likelihood that an event will occur. The probability, P, of any event occurring is greater than or equal to 0 and less than or equal to 1. The sum of the simple probabilities for all possible outcomes of an activity must equal 1. Objective probability: Determined by observations and experiments. Subjective probability: Determined by belief, experience, judgment, or intuition. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 21 / 71 Decision Making Under Risk When probabilities are given... The concert example: Probabilities Indoor venue Outdoor venue Good weather 0.60 $ 7000 $ 15000 Bad weather 0.40 $ 8000 $ -5000 The Amazon example: Probabilities Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy High utilization 0.3 Medium utilization 0.5 Low utilization 0.2 $ 200,000 $ ´10,000 $ ´180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Decision Making Under Uncertainty September 1, 2021 22 / 71 Decision Making Under Risk Expected Monetary Value Favors the alternative with the maximum weighted row average. Probabilities Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy Highly utilization 0.3 Medium utilization 0.5 Low utilization 0.2 200000 -10000 -180000 100000 10000 - 20000 0 0 0 Decision Making Under Uncertainty Expected value September 1, 2021 23 / 71 Decision Making Under Risk Expected Opportunity Loss Favors the alternative with the minimum weighted row average of the regret matrix Probabilities Indoor venue Outdoor venue Good weather 0.60 $ 7000 $ 15000 Bad weather 0.40 $ 8000 $ -5000 Regret table: Good weather Bad weather Expected regret Indoor venue Outdoor venue Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 24 / 71 Decision Making Under Risk Value of Information Assume an oracle (or a consultant) that will give you the results of the market analysis that will give the information about the true state? What is the value of such an analysis? How much should the DM pay? Expected value of Perfect Information = EV with PI - Maximum EV under risk Probabilities Construct a large center Construct a small center Do not open a center High utilization 0.3 Medium utilization 0.5 Low utilization 0.2 $ 200,000 $ -10,000 $ -180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Maximum EV under risk = $ 31000 as we found on the previous slide EV with PI = 200000 ¨ 0.3 ` 10000 ¨ 0.5 ` 0 ¨ 0.2 “ $ 65000 Expected value of Perfect Information = 65000 - 31000 = 34000 Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 25 / 71 Decision Making Under Risk Expected Opportunity Loss Probabilities Construct a large center Construct a small center Do not open a center High utilization 0.3 Medium utilization 0.5 Low utilization 0.2 $ 200,000 $ -10,000 $ -180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Frequently utilized Somehow utilized Not used much Expected regret Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 26 / 71 Decision Trees Decision Trees Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 27 / 71 Decision Trees Decision Trees A decision tree is a diagram consisting of decision nodes (squares) chance nodes (circles) decision branches (alternatives) chance branches (state of natures), and terminal nodes (payoffs or rewards) Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 28 / 71 Decision Trees Decision Tree for the Amazon Example The decision: construct a large/small center, or do not construct a new. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 29 / 71 Decision Trees Decision Tree for the Amazon Example The decision: construct a large/small center, or do not construct a new. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 30 / 71 Decision Trees Decision Tree for the Amazon Example For the first option we have three different outcomes Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 31 / 71 Decision Trees Decision Tree for the Amazon Example For the second option we have three different outcomes. For the last option, we only have one outcome. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 32 / 71 Decision Trees Decision Tree for the Amazon Example The outcomes are added to the tree. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 33 / 71 Decision Trees Decision Tree for the Amazon Example The probabilities are added to the tree. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 34 / 71 Decision Trees Decision Tree for the Amazon Example We work our way backwards and find a value for every chance and decision node. We have the chance nodes first. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 35 / 71 Decision Trees Decision Tree for the Amazon Example At every decision node, we select the highest value. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 36 / 71 Decision Trees Another Decision Tree Example For the warehouse location problem, let’s assume a market research is available for a price of 10,000. This research will not provide the perfect information such that: If there will be frequent utilization, there is a 90% chance that the market research will predict it. It will predict a medium utilization with a probability of %10. If the utilization will be medium, there is a 70% chance that the research will predict medium utilization. With a 20% chance, it will predict a frequent utilization and with a 10% chance, it will predict a low utilization. If the utilization will be low, there is a 80% chance that it will predict a low utilization and a 20% chance that it will predict a medium utilization. Should we conduct the market research? Should we open a center? If so, what should be the size? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 37 / 71 Decision Trees Example 2 (contd.) First decision: Should we conduct the market research? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 38 / 71 Decision Trees Example 2 (contd.) There are three possible outcomes. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 39 / 71 Decision Trees Example 2 (contd.) For any outcome, we have to choose from the three alternatives. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 40 / 71 Decision Trees Example 2 (contd.) Let’s solve each branch! First branch: Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 41 / 71 Decision Trees Example 2 (contd.) If we solve all last branches Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 42 / 71 Decision Trees Example 2 (contd.) What about market research output probabilities? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 43 / 71 Utility Theory Utility Theory Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 44 / 71 Utility Theory Utility Theory - Deal or No Deal The contestant then begins choosing cases that are to be removed from play. The amount inside each chosen case is immediately revealed Throughout the game, after a predetermined number of cases have been opened, the banker offers the contestant an amount of money to quit the game The offer based roughly on the amounts remaining in play and the contestant’s demeanor, the bank tries to ’buy’ the contestant’s case for a lower price than what’s inside the case. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 45 / 71 Utility Theory Utility Theory - Deal or No Deal The player then answers ”Deal or no deal” question, choosing: ”Deal”, accepting the offer and ending the game ”No Deal”, rejecting the offer and continuing This process of removing cases and receiving offers continues, until either the player accepts an offer to ’deal’, or all offers have been rejected and player wins the cash inside her/his case. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 46 / 71 Utility Theory Utility Theory - Deal or No Deal Deal or no deal? What about $ 200,000? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 47 / 71 Utility Theory Utility Theory Utility assessment may assign the worst payoff a utility of 0 and the best payoff a utility of 1. A standard gamble is used to determine utility values: When DM is indifferent between two alternatives, the utility values of them are equal. Choose the alternative with the maximum expected utility Back to our example: Probabilities Construct a large center Construct a small center Do not open a center Altay & Baykal-Gürsoy High utilization 0.3 Medium utilization 0.5 Low utilization 0.2 $ 200,000 $ ´10,000 $ ´180,000 $ 100,000 $ 10,000 $ ´20,000 $0 $0 $0 Decision Making Under Uncertainty September 1, 2021 48 / 71 Utility Theory Utility Theory - Example (contd.) Output $200,000 $100,000 $10,000 $0 -$10,000 -$20,000 -$180,000 Altay & Baykal-Gürsoy Utility 1 0 Decision Making Under Uncertainty September 1, 2021 49 / 71 Utility Theory Utility Theory - Example (contd.) Let’s finish it together! Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 50 / 71 Utility Theory Utility Theory - Example (contd.) Output $200,000 $100,000 $10,000 $0 -$10,000 -$20,000 -$180,000 Altay & Baykal-Gürsoy Utility 1 0 Decision Making Under Uncertainty September 1, 2021 51 / 71 Utility Theory Utility Theory - Example (contd.) Then, the decision tree becomes Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 52 / 71 Utility Theory Risk Premium Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 53 / 71 Utility Theory Risk Premium If the DM is risk averse (avoids risk), RP ¿ 0 They prefer to receive a sum of money equal to the expected value of a lottery than to enter the lottery itself. If the DM is risk prone (seeks risk), RP ¡ 0 They prefer to enter a lottery instead of receiving a sum of money equal to its expected value. If the DM is risk neutral, RP = 0 They are indifferent between entering any lottery and receiving a sum of money equal to its expected value. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 54 / 71 Utility Theory Q&A Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 55 / 71 Utility Theory Overview Uncertainties in Materials Management Single-Period Inventory Uncertainties Multi-Period Inventory Uncertainties Continuous Review Inventory Models Periodic Review Inventory Models Markov Chains Markov Decision Processes Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 56 / 71 Uncertainties in Materials Management Uncertainties in Materials Management Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 57 / 71 Uncertainties in Materials Management Uncertainties in Materials Management and Lot Sizing Uncertainty means that demand is a random variable. It is defined by a probability distribution. This probability distribution is estimated from past history of demands. The cost function is a random variable, as well. The general assumption is to minimize the expected cost. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 58 / 71 Uncertainties in Materials Management Overview of Some Probability Distributions Normal Distribution Let D denote the demand. For normal distribution is denoted by D „ Npµ, σ 2 q. µ is the mean of the distribution. σ is the standard deviation. A normal distribution is a continuous probability distribution for a random variable. The graph of a normal distribution is called the normal curve. φpD “ xq “ Altay & Baykal-Gürsoy Decision Making Under Uncertainty 2 2 1 ? e ´px´µq {2σ σ 2π September 1, 2021 59 / 71 Uncertainties in Materials Management The Normal Distribution 1) Which distribution has greater mean? 2) Which distribution has greater variance / standard variation? The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any value can be transformed into a z-score by using the formula z“ Altay & Baykal-Gürsoy Value ´ Mean x ´µ “ StandardDeviation σ Decision Making Under Uncertainty September 1, 2021 60 / 71 Uncertainties in Materials Management The Normal Distribution Standard Normal Distribution: Using Standard Normal Distribution for Demands: Assume that D „ Np100, 102 q. The probability that the demand is less than 115? zp115q “ 115 ´ 100 “ 1.5 10 We denote this by Φp1.5q. Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 61 / 71 Uncertainties in Materials Management Standard Normal Table Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 62 / 71 Uncertainties in Materials Management Standard Normal Distribution The probability that the demand is less than 95? zp95q “ 95 ´ 100 “ ´0.5 10 Φp´0.5q “? Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 63 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties Single-Period Demand Uncertainties Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 64 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties Single-Period Demand Uncertainties Notation: co : unit holding cost (the cost of per unit of inventory remaining at the end of the period) cu : unit underage cost (the cost of per unit unmet demand at the end of the period) D: demand (random variable) Q: the decision variable (how much we will order at the beginning of the period) C pQ, Dq: total overage and underage cost incurred at the end of the period Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 65 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties Single-Period Demand Uncertainties Notation: co : unit holding cost (the cost of per unit of inventory remaining at the end of the period) cu : unit underage cost (the cost of per unit unmet demand at the end of the period) D: demand (random variable) Q: the decision variable (how much we will order at the beginning of the period) C pQ, Dq: total overage and underage cost incurred at the end of the period Remaining Inventory “ Altay & Baykal-Gürsoy # Q ´D 0 DďQ “ maxp0, Q ´ Dq DąQ Decision Making Under Uncertainty September 1, 2021 65 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties Single-Period Demand Uncertainties Notation: co : unit holding cost (the cost of per unit of inventory remaining at the end of the period) cu : unit underage cost (the cost of per unit unmet demand at the end of the period) D: demand (random variable) Q: the decision variable (how much we will order at the beginning of the period) C pQ, Dq: total overage and underage cost incurred at the end of the period Remaining Inventory “ Unmet Demand “ Altay & Baykal-Gürsoy # Q ´D 0 # 0 D ´Q DďQ “ maxp0, Q ´ Dq DąQ DďQ “ maxp0, D ´ Qq DąQ Decision Making Under Uncertainty September 1, 2021 65 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 66 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq For the optimal amount ΦpD ˚ q “ Altay & Baykal-Gürsoy cu co ` cu Decision Making Under Uncertainty September 1, 2021 66 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq For the optimal amount cu co ` cu Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. ΦpD ˚ q “ co : Cost of an unsold item cu : Cost of an unmet demand Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 66 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq For the optimal amount cu co ` cu Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. ΦpD ˚ q “ co : Cost of an unsold item cu : Cost of an unmet demand 3 “ 0.375 5`3 ´ D ˚ ´ 100 ¯ Φ “ 0.375 10 D ˚ ´ 100 “ ´0.32 10 ˚ D “ 96.8 « 97 Φpz ˚ q “ Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 66 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties Standard Normal Table Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 67 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. Any unsold item is returned for $1. co : Cost of an unsold item cu : Cost of an unmet demand Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 68 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. Any unsold item is returned for $1. co : Cost of an unsold item cu : Cost of an unmet demand 3 “ 0.428 4`3 ´ D ˚ ´ 100 ¯ Φ “ 0.428 10 D ˚ ´ 100 “ ´0.21 10 ˚ D “ 97.9 « 98 Φpz ˚ q “ Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 68 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problem Expected shortage (lost sales) “ σLpzq ˜ ¸ ´Q ´ µ¯ Q ´ µ ´Q ´ µ¯ Lpzq “ φ ´ 1´Φ σ σ σ Expected sales “ µ ´ Expected shortage Expected overage “ Q ´ Expected sales Expected fill rate “ Altay & Baykal-Gürsoy Expected sales µ Decision Making Under Uncertainty September 1, 2021 69 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problems Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. Any unsold item is returned for $1. An unmet demand and an unhappy customer yields 4 more customers to churn. co : Cost of an unsold item cu : Cost of an unmet demand Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 70 / 71 Uncertainties in Materials Management Single-Period Demand Uncertainties The Newsvendor Problems Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8. Any unsold item is returned for $1. An unmet demand and an unhappy customer yields 4 more customers to churn. co : Cost of an unsold item cu : Cost of an unmet demand 15 “ 0.833 15 ` 3 ´ D ˚ ´ 100 ¯ Φ “ 0.833 10 ˚ D ´ 100 “ 1.01 10 ˚ D “ 110.1 « 110 Φpz ˚ q “ Altay & Baykal-Gürsoy Decision Making Under Uncertainty September 1, 2021 70 / 71 Engineering Decision Making Under Uncertainty Aimen Sudhir, RUID: 197007651 To make the recommendation, the following decision tree was made. As seen from the decision tree, it is profitable for Ectron Fragrance Corporation to conduct the market survey.
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1

Decision tree – Assignment 1

First and Last Name
The University of ………………….
Course Code: Name of Course
Instructor Name
Due Date

2
Answer:

Probability Number of units

Return

Details

value
P(A1)

0.2

100,000

$6 per unit

Without Mega

P(A2)

0.1

3,000

$6 per unit

Software Enterprises

P(A3)

0.7

10,000

$6 per unit

P(B1)

0.3

100,000

$3 per unit

With Mega Software

P(B2)

0.1
...


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