Mr. X is considering three universities for admission and must select based on location, academic reputation, and political violence. He analyzes each university using a decision tree that weights reputation most heavily. The analysis shows University B has the best overall score. Decision analysis allows ranking alternatives based on weighted criteria to select the optimal choice.
This document is a project report on an inventory model submitted by three students. It includes:
1. An introduction to inventory, defining it as stock held for future production or sales in raw material, semi-finished, and finished forms. The objective is to minimize total costs or maximize profits.
2. A section on economic order quantity (EOQ) that describes costs of holding too much or too little inventory and assumptions like uniform demand. It includes a table calculating EOQ for different order quantities.
3. The EOQ formula and an example calculation of EOQ for Frooti beverages based on given demand and costs.
4. A re-order level calculation for Pad
This document provides an overview of decision theory and various decision-making environments. It discusses the six steps in decision theory as applying to a case study about a lumber company expanding its product line. The types of decision-making environments covered are decisions under certainty, risk, and uncertainty. Decision-making under uncertainty further explores criteria for decisions like maximax, maximin, weighted average, equally likely, and minimax regret. Game theory and its applications to strategic decision-making between competitors are also briefly introduced.
This document summarizes a presentation on decision theory given by Bhushan Vijay Phirke to the MBA program at Rajarshi Shahu College of Engineering on April 17, 2020. It defines decision theory, outlines the six steps in the decision theory process, and describes key concepts like problem formulation, payoff tables, decision environments, and optimization criteria like optimism, pessimism, and regret. It concludes that decision theory provides a logical framework to help managers determine the most beneficial course of action when facing uncertainty.
The document discusses decision theory and decision trees. It introduces decision making under certainty, risk, and uncertainty. It defines elements related to decisions like goals, courses of action, states of nature, and payoffs. It also discusses concepts like expected monetary value, expected profit with perfect information, expected value of perfect information, and expected opportunity loss. Examples are provided to demonstrate calculating these metrics. Finally, it provides an overview of how to construct a decision tree, including defining the different node types and how to calculate values within the tree.
Inroduction to Decision Theory and Decision Making Under CertaintyAbhi23396
This document introduces decision theory and decision-making under certainty. It defines decision theory as a descriptive and prescriptive approach to classify levels of knowledge when making decisions. Under certainty, a decision maker has perfect information about outcomes for each alternative, allowing them to choose the best option. An example is provided where a manufacturer must choose between two machines, M1 and M2, to process an order of 1000 units. All costs are known for each machine's setup time, tooling costs, and machining time per unit. Calculations show the total cost is lower to use machine M2, so it is the best choice under the certain conditions given.
This document discusses inventory models for independent demand, including the basic economic order quantity (EOQ) model and production order quantity model. It provides information on their objectives, assumptions, variables, and equations. It also defines reorder point and provides examples of calculating optimal order quantity, number of orders, time between orders, total annual costs, and reorder point based on given demand, costs, lead time, and other parameters.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
This document is a project report on an inventory model submitted by three students. It includes:
1. An introduction to inventory, defining it as stock held for future production or sales in raw material, semi-finished, and finished forms. The objective is to minimize total costs or maximize profits.
2. A section on economic order quantity (EOQ) that describes costs of holding too much or too little inventory and assumptions like uniform demand. It includes a table calculating EOQ for different order quantities.
3. The EOQ formula and an example calculation of EOQ for Frooti beverages based on given demand and costs.
4. A re-order level calculation for Pad
This document provides an overview of decision theory and various decision-making environments. It discusses the six steps in decision theory as applying to a case study about a lumber company expanding its product line. The types of decision-making environments covered are decisions under certainty, risk, and uncertainty. Decision-making under uncertainty further explores criteria for decisions like maximax, maximin, weighted average, equally likely, and minimax regret. Game theory and its applications to strategic decision-making between competitors are also briefly introduced.
This document summarizes a presentation on decision theory given by Bhushan Vijay Phirke to the MBA program at Rajarshi Shahu College of Engineering on April 17, 2020. It defines decision theory, outlines the six steps in the decision theory process, and describes key concepts like problem formulation, payoff tables, decision environments, and optimization criteria like optimism, pessimism, and regret. It concludes that decision theory provides a logical framework to help managers determine the most beneficial course of action when facing uncertainty.
The document discusses decision theory and decision trees. It introduces decision making under certainty, risk, and uncertainty. It defines elements related to decisions like goals, courses of action, states of nature, and payoffs. It also discusses concepts like expected monetary value, expected profit with perfect information, expected value of perfect information, and expected opportunity loss. Examples are provided to demonstrate calculating these metrics. Finally, it provides an overview of how to construct a decision tree, including defining the different node types and how to calculate values within the tree.
Inroduction to Decision Theory and Decision Making Under CertaintyAbhi23396
This document introduces decision theory and decision-making under certainty. It defines decision theory as a descriptive and prescriptive approach to classify levels of knowledge when making decisions. Under certainty, a decision maker has perfect information about outcomes for each alternative, allowing them to choose the best option. An example is provided where a manufacturer must choose between two machines, M1 and M2, to process an order of 1000 units. All costs are known for each machine's setup time, tooling costs, and machining time per unit. Calculations show the total cost is lower to use machine M2, so it is the best choice under the certain conditions given.
This document discusses inventory models for independent demand, including the basic economic order quantity (EOQ) model and production order quantity model. It provides information on their objectives, assumptions, variables, and equations. It also defines reorder point and provides examples of calculating optimal order quantity, number of orders, time between orders, total annual costs, and reorder point based on given demand, costs, lead time, and other parameters.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
1) The document discusses several criteria for decision making under uncertainty including maximax, maximin, minimax, minimin, and Laplace.
2) The maximax criterion is optimistic and chooses the alternative with the highest possible payoff. Maximin is pessimistic and chooses the alternative with the highest minimum payoff.
3) Minimax regret considers the maximum regret for each alternative and chooses the one with the minimum maximum regret. This accounts for opportunity loss across states of nature.
This document provides an overview of game theory concepts. It defines game theory as analyzing situations of conflict and competition involving decision making by two or more participants. Some key points:
- Game theory was developed in the 20th century, with a seminal 1944 book discussing its application to business strategy.
- Basic concepts include players, pure and mixed strategies, zero-sum vs. non-zero-sum games, and payoff matrices to represent outcomes.
- Solutions include finding equilibrium points using minimax and maximin principles for pure strategies or solving systems of equations for mixed strategies when no equilibrium exists.
- Dominance rules can reduce game matrices, and graphical or algebraic methods solve for mixed strategies without saddles
Inventory models with two supply modelsMOHAMMED ASIF
We develop dynamic programming models for periodic inventory systems that allow for both regular and emergency orders to be placed periodically. There are two key cases - whether a fixed cost exists for emergency orders. If emergency ordering is possible, there is a critical inventory level such that emergency orders are placed if inventory falls below this level at review times. We also provide simple procedures to compute optimal policy parameters - the optimal order-up-to level solves a myopic cost function. Thus, the optimal policies are easy to implement.
This document discusses simulation as a technique used in operations research to analyze the behavior of systems. It provides examples of how simulation works by initializing a system, generating inputs, observing outputs, and collecting statistics. Some key uses of simulation mentioned include testing policy decisions, conducting experiments without disrupting real systems, and obtaining operating characteristics estimates faster than working with actual systems. The document also outlines some advantages and limitations of the simulation approach. It includes two examples demonstrating how to simulate daily demand for a bakery and daily production for a moped manufacturer using random numbers.
Linear programming is a mathematical modeling technique used to determine optimal resource allocation to achieve objectives. It involves converting a problem into a linear mathematical model with decision variables, constraints, and an objective function. The optimal solution is found by systematically increasing the objective function value until infeasibility is reached. For example, a linear programming model was used to determine the optimal production mix and levels of two drug combinations to maximize profit given resource constraints. The optimal solution was found to be 320 dozen of drug X1 and 360 dozen of drug X2, utilizing all available resources and achieving $4,360 in weekly profit.
The document discusses decision theory and decision-making under uncertainty. It defines key concepts in decision theory including decision maker, courses of action, states of nature, payoff, and expected monetary value. It describes three types of decision-making environments: certainty, risk, and uncertainty. Under risk, decisions are made using probability assessments and expected monetary value calculations. Several steps and concepts in decision making under risk are outlined, including constructing payoff matrices, calculating expected values, and opportunity loss analysis.
Solving Degenaracy in Transportation Problemmkmanik
- The document discusses solving degeneracy in transportation problems using the example of a transportation problem with 4 sources and 5 destinations.
- An initial basic feasible solution is found using the least cost method, but it results in a degenerate solution since the number of allocated cells is less than m + n - 1.
- To solve the degeneracy, an unallocated cell is selected and allocated a value to satisfy the condition. Here, an unallocated cell value of 5 is selected and assigned the value ε.
- The solution is then optimized using the U-V method by calculating Uj + Vi = Cij for allocated cells and penalties Pij for unallocated cells until all penalties are less than
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
The document discusses transportation and assignment models in operations research. The transportation model aims to minimize the cost of distributing a product from multiple sources to multiple destinations, while satisfying supply and demand constraints. The assignment model finds optimal one-to-one matching between sources and destinations to minimize costs. Some solution methods for transportation problems include the northwest corner method, row minima method, column minima method, and least cost method. The Hungarian method is commonly used to solve assignment problems by finding the minimum cost matching.
1) The document discusses the Hungarian method for solving assignment problems. It involves minimizing the total cost or maximizing the total profit of assigning resources like employees or machines to activities like jobs.
2) The method includes steps like developing a cost matrix, finding the opportunity cost table, making assignments to zeros in the table, and revising the table until an optimal solution is reached.
3) There are examples showing the application of these steps to problems with unique and multiple optimal solutions, as well as an unbalanced problem with more resources than activities.
This document provides an overview of queuing theory, which is used to model waiting lines. It discusses key concepts like arrival processes, service systems, queuing models and their characteristics. Some examples where queuing theory is applied include telecommunications, traffic control, and manufacturing layout. Common elements of queuing systems are customers, servers and queues. The document also presents examples of single and multiple channel queuing models.
Decision making under condition of risk and uncertaintysapna moodautia
The document discusses decision making under conditions of certainty, risk, and uncertainty. It defines each condition and explains how the degree of knowledge affects the decision. Certainty exists when all alternatives and outcomes are known. Risk exists when probabilities can be estimated but information is imperfect. Uncertainty exists when probabilities cannot be estimated due to limited information. Modern approaches to decision making under uncertainty include risk analysis, decision trees, and preference theory.
Operational research (OR) uses analytical techniques to improve decision-making and efficiency. It encompasses problem-solving methods applied to optimize performance. OR analyzes systems through mathematical modeling, simulation, and other techniques. It aims to make the best use of resources by carefully planning and analyzing processes. Examples of OR applications include scheduling, facility planning, forecasting, yield management, and defense logistics. The field originated from military planning in World War II and has since expanded to business, industry, and public policy problems.
The document provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
Decision theory deals with determining the optimal course of action when alternatives have uncertain consequences. There are several key concepts: decision alternatives are available options; states of nature are uncontrollable events; and payoff is the numerical outcome of alternatives and states. The decision process involves defining the problem, listing states, identifying alternatives, expressing payoffs, and applying a model to select the optimal alternative based on criteria. Decision making can occur under certainty, risk, or uncertainty depending on what is known about states and payoffs. Different techniques are used depending on the environment.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
This document contains definitions of accounting and finance terms. It includes over 100 terms defined in 1-2 sentences each, organized alphabetically from A-C. Some key terms defined include accounting period, accrual, amortization, balance sheet, breakeven analysis, capital, capital asset, and capital budgeting. The definitions provide brief explanations of concepts and terminology commonly used in accounting and corporate finance.
The document defines operation research as the application of scientific methods to optimize solutions to problems involving complex systems. It provides three definitions of OR from different authors. The document then gives an example problem of determining the optimal product mix for interior and exterior paint production. It proceeds to outline the key steps of formulating an OR problem: defining decision variables and constraints, determining the objective, developing a mathematical model, and finding a feasible solution. Finally, it discusses the major components involved in formulating an OR problem: the environment, decision maker, objectives, and alternative actions/constraints.
1) The document discusses several criteria for decision making under uncertainty including maximax, maximin, minimax, minimin, and Laplace.
2) The maximax criterion is optimistic and chooses the alternative with the highest possible payoff. Maximin is pessimistic and chooses the alternative with the highest minimum payoff.
3) Minimax regret considers the maximum regret for each alternative and chooses the one with the minimum maximum regret. This accounts for opportunity loss across states of nature.
This document provides an overview of game theory concepts. It defines game theory as analyzing situations of conflict and competition involving decision making by two or more participants. Some key points:
- Game theory was developed in the 20th century, with a seminal 1944 book discussing its application to business strategy.
- Basic concepts include players, pure and mixed strategies, zero-sum vs. non-zero-sum games, and payoff matrices to represent outcomes.
- Solutions include finding equilibrium points using minimax and maximin principles for pure strategies or solving systems of equations for mixed strategies when no equilibrium exists.
- Dominance rules can reduce game matrices, and graphical or algebraic methods solve for mixed strategies without saddles
Inventory models with two supply modelsMOHAMMED ASIF
We develop dynamic programming models for periodic inventory systems that allow for both regular and emergency orders to be placed periodically. There are two key cases - whether a fixed cost exists for emergency orders. If emergency ordering is possible, there is a critical inventory level such that emergency orders are placed if inventory falls below this level at review times. We also provide simple procedures to compute optimal policy parameters - the optimal order-up-to level solves a myopic cost function. Thus, the optimal policies are easy to implement.
This document discusses simulation as a technique used in operations research to analyze the behavior of systems. It provides examples of how simulation works by initializing a system, generating inputs, observing outputs, and collecting statistics. Some key uses of simulation mentioned include testing policy decisions, conducting experiments without disrupting real systems, and obtaining operating characteristics estimates faster than working with actual systems. The document also outlines some advantages and limitations of the simulation approach. It includes two examples demonstrating how to simulate daily demand for a bakery and daily production for a moped manufacturer using random numbers.
Linear programming is a mathematical modeling technique used to determine optimal resource allocation to achieve objectives. It involves converting a problem into a linear mathematical model with decision variables, constraints, and an objective function. The optimal solution is found by systematically increasing the objective function value until infeasibility is reached. For example, a linear programming model was used to determine the optimal production mix and levels of two drug combinations to maximize profit given resource constraints. The optimal solution was found to be 320 dozen of drug X1 and 360 dozen of drug X2, utilizing all available resources and achieving $4,360 in weekly profit.
The document discusses decision theory and decision-making under uncertainty. It defines key concepts in decision theory including decision maker, courses of action, states of nature, payoff, and expected monetary value. It describes three types of decision-making environments: certainty, risk, and uncertainty. Under risk, decisions are made using probability assessments and expected monetary value calculations. Several steps and concepts in decision making under risk are outlined, including constructing payoff matrices, calculating expected values, and opportunity loss analysis.
Solving Degenaracy in Transportation Problemmkmanik
- The document discusses solving degeneracy in transportation problems using the example of a transportation problem with 4 sources and 5 destinations.
- An initial basic feasible solution is found using the least cost method, but it results in a degenerate solution since the number of allocated cells is less than m + n - 1.
- To solve the degeneracy, an unallocated cell is selected and allocated a value to satisfy the condition. Here, an unallocated cell value of 5 is selected and assigned the value ε.
- The solution is then optimized using the U-V method by calculating Uj + Vi = Cij for allocated cells and penalties Pij for unallocated cells until all penalties are less than
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
The document discusses transportation and assignment models in operations research. The transportation model aims to minimize the cost of distributing a product from multiple sources to multiple destinations, while satisfying supply and demand constraints. The assignment model finds optimal one-to-one matching between sources and destinations to minimize costs. Some solution methods for transportation problems include the northwest corner method, row minima method, column minima method, and least cost method. The Hungarian method is commonly used to solve assignment problems by finding the minimum cost matching.
1) The document discusses the Hungarian method for solving assignment problems. It involves minimizing the total cost or maximizing the total profit of assigning resources like employees or machines to activities like jobs.
2) The method includes steps like developing a cost matrix, finding the opportunity cost table, making assignments to zeros in the table, and revising the table until an optimal solution is reached.
3) There are examples showing the application of these steps to problems with unique and multiple optimal solutions, as well as an unbalanced problem with more resources than activities.
This document provides an overview of queuing theory, which is used to model waiting lines. It discusses key concepts like arrival processes, service systems, queuing models and their characteristics. Some examples where queuing theory is applied include telecommunications, traffic control, and manufacturing layout. Common elements of queuing systems are customers, servers and queues. The document also presents examples of single and multiple channel queuing models.
Decision making under condition of risk and uncertaintysapna moodautia
The document discusses decision making under conditions of certainty, risk, and uncertainty. It defines each condition and explains how the degree of knowledge affects the decision. Certainty exists when all alternatives and outcomes are known. Risk exists when probabilities can be estimated but information is imperfect. Uncertainty exists when probabilities cannot be estimated due to limited information. Modern approaches to decision making under uncertainty include risk analysis, decision trees, and preference theory.
Operational research (OR) uses analytical techniques to improve decision-making and efficiency. It encompasses problem-solving methods applied to optimize performance. OR analyzes systems through mathematical modeling, simulation, and other techniques. It aims to make the best use of resources by carefully planning and analyzing processes. Examples of OR applications include scheduling, facility planning, forecasting, yield management, and defense logistics. The field originated from military planning in World War II and has since expanded to business, industry, and public policy problems.
The document provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
Decision theory deals with determining the optimal course of action when alternatives have uncertain consequences. There are several key concepts: decision alternatives are available options; states of nature are uncontrollable events; and payoff is the numerical outcome of alternatives and states. The decision process involves defining the problem, listing states, identifying alternatives, expressing payoffs, and applying a model to select the optimal alternative based on criteria. Decision making can occur under certainty, risk, or uncertainty depending on what is known about states and payoffs. Different techniques are used depending on the environment.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
This document contains definitions of accounting and finance terms. It includes over 100 terms defined in 1-2 sentences each, organized alphabetically from A-C. Some key terms defined include accounting period, accrual, amortization, balance sheet, breakeven analysis, capital, capital asset, and capital budgeting. The definitions provide brief explanations of concepts and terminology commonly used in accounting and corporate finance.
The document defines operation research as the application of scientific methods to optimize solutions to problems involving complex systems. It provides three definitions of OR from different authors. The document then gives an example problem of determining the optimal product mix for interior and exterior paint production. It proceeds to outline the key steps of formulating an OR problem: defining decision variables and constraints, determining the objective, developing a mathematical model, and finding a feasible solution. Finally, it discusses the major components involved in formulating an OR problem: the environment, decision maker, objectives, and alternative actions/constraints.
This document discusses micro insurance as a tool for poverty alleviation and managing vulnerability in Bangladesh. It defines key terms like microfinance, micro insurance, poverty and vulnerability. It describes the history and necessity of micro insurance and the major risks faced by micro insurance companies and the poor, such as agricultural, health and life risks. It discusses different models for delivering micro insurance and the challenges. It explains how micro insurance can help reduce poverty and vulnerability by providing protection from risks. Finally, it outlines some common micro insurance products offered.
This document outlines rules related to Value Added Tax (VAT) in Bangladesh. It discusses several key points:
1) It defines important terms related to VAT such as "tax", "registered person", and "divisional officer".
2) It establishes procedures for registered persons/businesses to declare the value of goods and services for tax assessment purposes using specified forms. It also allows for reassessment of declared values.
3) It provides rules for businesses with annual turnover under 20 lakh Taka to pay a flat 4% "turnover tax" by enlisting with tax authorities and filing periodic returns. Higher penalties are outlined for noncompliance.
4) It delegates powers to VAT officers to
The document discusses linear programming problems. It defines linear programming as finding non-negative values of variables to satisfy linear constraints and optimize a linear objective function. It provides examples of transportation problems and diet problems formulated as linear programs. It describes graphical and algebraic methods for solving linear programs, including introducing slack and surplus variables to transform inequalities to equations.
This document provides an overview and table of contents for a study material on Value Added Tax (VAT) in Bangladesh. It includes definitions of VAT, key features of the VAT system in Bangladesh such as applicable rates, scope, and mechanisms. It also lists examples that will be used to illustrate VAT calculations. The document aims to help readers understand the theory and application of VAT in Bangladesh.
This document is the Value Added Tax Act of Bangladesh from 1991. Some key points:
- It establishes a 15% value added tax on most goods imported into or produced in Bangladesh, as well as services rendered in Bangladesh, with some exemptions.
- Zero-rated tax applies to exported goods and services.
- VAT is paid by importers on imported goods, producers/manufacturers on goods produced locally, service providers for services, and suppliers in other cases.
- It defines various terms related to VAT like input tax, output tax, taxable goods and services, and establishes rules around determination of the tax base and timing of tax payments.
The document discusses the concept of duality in linear programming problems. There are five steps to formulate the dual problem from the primal problem: 1) objective functions switch between maximization and minimization, 2) right hand sides of primal constraints become coefficients in the dual objective, 3) primal objective coefficients become right hand side values in the dual constraints, 4) transpose the primal constraint coefficients for the dual constraints, and 5) switch inequality signs. The dual problem maximizes the right hand side values subject to constraints with the primal objective coefficients and reversed inequality signs. The primal and dual problems are symmetric and related through their coefficients, constraints, and objective functions.
1. The document provides an overview of key income tax concepts in Bangladesh including definitions, tax rates, and types of taxes. It covers income tax authority, types of taxes, important definitions like income, tax, assessee, person, income year, assessment year, and tax day.
2. It distinguishes between resident and non-resident taxpayers and provides the criteria for each. For individuals, it is 182 days presence in Bangladesh or 90 days presence plus presence in Bangladesh for 365 days during the last 4 years. For companies and firms, residence is determined based on place of control and management.
3. The document outlines various tax rates applicable to different types of income and taxpayers. These include rates for salary
Panera Bread has expanded from 602 units in 2003 to 1,270 units today through strategic positioning and execution. It was founded in 1981 and acquired Saint Louis Bread Company in the 1990s. Panera developed the concept of "fast casual", serving high quality, healthy food quickly in a relaxing environment. This unique positioning allowed Panera to attract customers from both the fast food and casual dining segments. While competition is growing, Panera's strengths like signature foods, atmosphere, and first-mover advantage create barriers for competitors.
The document contains 18 code snippets demonstrating solutions to common programming problems. The code snippets include programs to: 1) convert temperature between Celsius and Fahrenheit, 2) find the larger of two numbers, 3) determine if a number is even or odd, and 4) calculate the square of a number. The programs demonstrate a variety of programming concepts like if/else statements, for loops, functions, and more.
This document discusses various techniques for decision making under uncertainty. It defines subjective and objective probabilities, and consistency requirements for probabilities. It describes the key elements of decision problems including alternatives, states of nature, payoff tables, and uncertainty. It defines three types of decision making environments: certainty, risk, and uncertainty. For decision making under risk, it discusses expected monetary value, expected opportunity loss, and expected value of perfect information as decision criteria. It provides examples to illustrate these concepts and techniques. Finally, it outlines several criteria that can be used for decision making under uncertainty including maximax, maximin, and minimax.
This document discusses key concepts in decision theory and decision making under uncertainty. It begins by defining decision theory and describing the degree of certainty in decision making problems. It then outlines elements of decision analysis like states of nature, chance occurrences governed by probabilities, and payoff matrices. An example involving production decisions for a dairy product is provided. The document also discusses criteria for decision making under uncertainty like Laplace, maximin, maximax, Hurwicz, and regret. It concludes by covering expected monetary value, expected opportunity loss, expected value of perfect information, and decision trees as approaches to decision making under risk.
The document defines and provides examples of decision trees. It explains that decision trees use a branching graph structure to illustrate all possible outcomes of a decision. They are used in operations research and decision analysis to help identify the most likely strategy to reach a goal. The document provides an example of an email management decision tree and explains how decision tree analysis involves forecasting outcomes and assigning probabilities. It outlines some common applications of decision trees and provides an example decision tree for weekend activity choices based on weather, finances, and parental visits.
This document introduces decision tree analysis for addressing complex decisions involving uncertainty. It provides examples to illustrate key concepts.
1) A decision tree is used to represent a company's choice between developing two new products, accounting for the costs, potential revenues, and risks of failure for each option. 2) Probabilities of success or failure can be added to the tree, but do not necessarily make the best choice clear. 3) The concept of expected value is introduced as a decision-making criterion that considers outcomes and their probabilities.
The document discusses decision trees, which are diagrams that illustrate decisions and their potential consequences. It provides examples of decision trees used by two companies - Manly Plastics and Vine Desserts - to analyze decisions about new product development and business location selection. It also discusses key concepts in decision trees, including decision nodes, chance nodes, expected value calculations, and how decision trees can be used for regression and survival analysis involving continuous or time-to-event outcomes.
The document introduces decision tree analysis for addressing complex decisions involving uncertainty. It provides an example where a company is deciding whether to develop a temperature sensor or pressure sensor. A decision tree is constructed to represent the possible outcomes and their probabilities. Expected value is introduced as a decision-making criterion that considers both outcomes and their probabilities. An example uses expected value to analyze whether one should accept a bet involving the roll of a die.
The document introduces decision tree analysis for addressing complex decisions involving uncertainty. It provides an example where a company is deciding whether to develop a temperature sensor or pressure sensor. A decision tree is constructed to represent the possible outcomes and their probabilities. Expected value is introduced as a decision-making criterion that considers both outcomes and their probabilities. An example uses expected value to analyze whether one should accept a bet involving the roll of a die.
The document discusses decision theory and decision-making under conditions of certainty, risk, and uncertainty. It provides steps in decision theory including listing states of nature, identifying alternatives, expressing payoffs, and choosing an alternative. Decision-making can occur under certainty, risk, or uncertainty depending on what is known about consequences and probabilities. The document also provides an example of a company considering strategies with different payoffs based on possible sales outcomes and states of nature.
The document discusses concepts in risk mathematics including Eugene Fama's efficient market hypothesis, random walks, Pascal's triangle for calculating probabilities of coin flips and dice rolls, standard deviation as a measure of risk, and the Capital Asset Pricing Model. It explains how standard deviation measures how returns vary from the mean and is thus a measure of risk. Higher standard deviation means higher risk. The Capital Asset Pricing Model uses beta to quantify an asset's risk relative to the market and determines the discount rate used in net present value calculations.
1) The Pittsburgh Development Corporation is deciding on the size of a new luxury condominium complex with three options - small (30 units), medium (60 units), or large (90 units).
2) There is uncertainty around the demand for the condominiums which could be strong or weak. This is represented as two states of nature - strong demand or weak demand.
3) Three decision making approaches are discussed to make a decision under uncertainty when probabilities of states of nature are unknown: optimistic, conservative, and minimax regret. The optimistic approach selects the option with the highest potential payoff, conservative takes the option with the highest minimum payoff, and minimax regret minimizes maximum regret.
Mean and Variance of Discrete Random Variable.pptxMarkJayAquillo
The document discusses computing the mean, variance, and standard deviation of discrete random variables. It provides examples of calculating the mean of different probability distributions by taking the sum of each value multiplied by its probability. It also gives examples of finding variance by taking the sum of the squared differences between each value and the mean multiplied by its probability, and defines standard deviation as the square root of variance. The document aims to help readers understand how to calculate and interpret these statistical measures for discrete random variables.
1. Modification of Problem 9.3 from the book Jean Clark i.docxjackiewalcutt
1. Modification of Problem 9.3 from the book:
Jean Clark is the manager of the Midtown Saveway Grocery Store. She now needs to replenish her
supply of strawberries. Her regular supplier can provide as many cases as she wants. However, because
these strawberries already are very ripe, she will need to sell them tomorrow and then discard any that
remain unsold. Jean estimates that she will be able to sell 10, 11, 12, or 13 cases tomorrow. She can
purchase the strawberries for $3 per case and sell them for $8 per case. Jean now needs to decide how
many cases to purchase.
Jean has checked the store’s records on daily sales of strawberries. On this basis, she estimates that the
prior probabilities are 0.2, 0.4, 0.3, and 0.1 for being able to sell 10, 11, 12, and 13 cases of strawberries
tomorrow, respectively.
a) Develop a decision analysis formulation of this problem by identifying the decision
alternatives, the states of nature, and the payoff table. (Build a table similar to the
Table 9.3 in the textbook or the table on Slide 9 of Lecture Notes 11 – Decision
Analysis).
State of Nature
Alternative Sell 10 cases Sell 11 cases Sell 12 cases Sell 13 cases
Buy 10 cases $50 $50 $50 $50
Buy 11 cases $47 $55 $55 $55
Buy 12 cases $44 $52 $60 $60
Buy 13 cases $41 $49 $57 $65
Prior Probability 0.2 0.4 0.3 0.1
b) If Jean is dubious about the accuracy of these prior probabilities and so chooses to
ignore them and use the maximax criterion, how many cases of strawberries should
she purchase? Show how you reach to your answer using the table you have in part
a).
Max(Buy 10) = $50,
Max(Buy 11) = $55,
Max(Buy 12) = $60,
Max(Buy 13) = $65.
Maximax = $65 with buying 13 cases.
State of Nature
Alternative Sell 10 cases Sell 11 cases Sell 12 cases Sell 13 cases
Buy 10 cases $50 $50 $50 $50
Buy 11 cases $47 $55 $55 $55
Buy 12 cases $44 $52 $60 $60
Buy 13 cases $41 $49 $57 $65
c) How many cases should be purchased if she uses the maximin criterion? Show how
you reach to your answer using the table you have in part a).
Min(Buy 10) = $50,
Min(Buy 11) = $47,
Min(Buy 12) = $44,
Min(Buy 13) = $41.
Maximin = $50 with buying 10 cases.
State of Nature
Alternative Sell 10 cases Sell 11 cases Sell 12 cases Sell 13 cases
Buy 10 cases $50 $50 $50 $50
Buy 11 cases $47 $55 $55 $55
Buy 12 cases $44 $52 $60 $60
Buy 13 cases $41 $49 $57 $65
d) How many cases should be purchased if she uses the maximum likelihood criterion?
Show how you reach to your answer using the table you have in part a).
The most likely state of nature is to sell 11 cases. Under this state, she should buy 11
cases with a payoff of $55.
State of Nature
Alternative Sell 10 cases Sell 11 cases Sell 12 cases Sell 13 cases
Buy 10 cases $50 $50 $50 $50
Buy 11 cases $47 $55 $55 $55
Buy 12 cases $44 $52 $60 $60
Buy 13 cases $41 $49 $57 $65
Prior Probability 0.2 0.4 0.3 0.1
e) ...
This document discusses risk and return related to various investment alternatives. It provides background information on risk, defines risk and expected return, and discusses different types of risk including stand-alone risk and market risk. It then presents a case study where the reader is asked to analyze the risk and return characteristics of different investment options and construct a portfolio based on the information provided, including calculating expected returns, standard deviations, betas, and required rates of return. The goal is to choose the best investment alternative given the risk-return tradeoff.
This document discusses decision making environments and techniques. It describes three types of decision making environments: decision making under certainty, uncertainty, and risk. It also discusses decision trees, Bayesian analysis, and utility theory as tools for decision making under uncertainty and risk. The key techniques covered are expected value analysis, maximax/maximin criteria, and expected opportunity loss criterion for decision making under risk.
This document provides an overview of a course on decision analysis and dynamic programming. It introduces key concepts like decision trees, probabilities, expected values, and dynamic decision problems. Examples discussed include assigning tables in a restaurant, playing Tetris, and managing a portfolio. The document outlines approaches for structuring decision problems, assessing probabilities, and determining the best course of action.
A decision tree is a diagram that visually represents decisions, uncertainties, and outcomes of a complex decision-making process. It breaks down a complex problem into a step-by-step process. The document provides examples of how to construct a decision tree by defining decision points and possible outcomes as branches. It also explains how to evaluate a decision tree by assigning values and probabilities to outcomes and working backwards to calculate expected values at decision points in order to determine the optimal decision.
1. In the construction of decision trees, which of the following s.docxhyacinthshackley2629
1. In the construction of decision trees, which of the following shapes represents a state of nature node? (Points : 1)
square
circle
diamond
triangle
Question 2.2. In the construction of decision trees, which of the following shapes represents a decision node? (Points : 1)
square
circle
diamond
triangle
Question 3.3. A market research study is being conducted to determine if a product modification will be well received by the public. A total of 1,000 consumers are questioned regarding this product.
The table below provides information regarding this sample.
Positive
Reaction
Neutral
Reaction
Negative
Reaction
Male
240
60
100
Female
260
220
120
What is the probability that a randomly selected person would be a female who had a positive reaction? (Points : 1)
0.250
0.260
0.455
0.840
Question 4.4. The probability that a typical tomato seed will germinate is 60%. A seed company has developed a hybrid tomato that they claim has an 85% probability of germination. If a gardener plants the new hybrid tomato in batches of 12, what is the probability that 10 or more seeds will germinate in a batch? (Points : 1)
0.064
0.083
0.264
0.736 <-- Not sure
Question 5.5. Historical data indicates that only 20% of cable customers are willing to switch companies. If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that no more than 3 customers would be willing to switch their cable? (Points : 1)
0.85
0.15
0.20
0.411
Question 6.6. Lock combinations are made using 3 digits followed by 2 letters. How many different lock combinations can be made if repetition of digits is allowed? (Points : 1)
6
260
6,760
676,000
Question 7.7. In 2012 the stock market took some big swings up and down. One thousand investors were asked how often they tracked their investments. The table below shows their responses. What is the probability that an investor tracks the portfolio weekly?
How often tracked?
Response
Daily
235
Weekly
278
Monthly
292
Few times a year
136
Do not track
59
(Points : 1)
0.235
0.278
0.513
0.722
Question 8.8. In hypothesis testing, the null and the alternative hypotheses are ________. (Points : 1)
not mutually exclusive
mutually exclusive
always false
always true
Question 9.9. If we fail to reject the null hypothesis, ________. (Points : 1)
we have found evidence to support the alternative hypothesis
the null hypothesis is proved to be true
we have only failed to find evidence to support the alternative hypothesis
the hypothesis test is inconclusive
Question 10.10. The probability of a Type I error can be specified by the investigator. The probability of a Type II error is ________. (Points : 1)
one minus the probability of Type I erro.
Quantitative Analysis for Managers Notes.pdfMahdyHasan6
This document discusses quantitative analysis and decision-making under conditions of uncertainty, risk, and certainty. It provides examples of how expected value, opportunity loss, and incremental analysis can be used to determine the optimal production level for a bread manufacturing company given demand probabilities. Under absolute uncertainty with no demand information, the optimal decision balances risk and potential loss. With some demand information, expected value analysis indicates producing 42 units. And with full certainty of demands, opportunity loss is minimized at 42 units as well. Quantitative methods can help decision-making even if outcomes may initially differ from reality in the short-term.
This document provides an overview of decision theory and decision making under uncertainty. It discusses structuring decision problems using decision trees and different types of decision making environments including uncertainty, risk, and certainty. It then covers various decision making criteria for uncertainty including optimistic, conservative, minimax regret, equally likely, and criterion of realism approaches. Expected values of perfect and sample information are also introduced. Examples are provided to illustrate key concepts such as calculating expected values and values of information.
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(2) Guidelines are given on calculating the rate for technical service fees, which is the number of foreign employees multiplied by a certain percentage of their salary. Registration of agreements related to technical service fees must be done according to the Registration Act.
(3) Examples are given of technical services that can be provided and the types of technical fees that can be charged for those services.
This document summarizes amendments made to several laws in Bangladesh related to banking and financial regulations. Key points:
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The document provides details of the specific sections amended and the new/amended definitions. In
The document summarizes key amendments made to the Patents and Designs Act, 1911 through the Patents and Designs (Amendment) Act, 2023. Some of the key changes include expanding the scope of patentable inventions, establishing patent offices and providing guidelines for granting patents. It also discusses procedures for filing and reviewing patents and establishing infringement and dispute resolution mechanisms.
This document establishes the formation of a new organization called the Digital Bangladesh Technology Park Authority (DBTPA) through an act of parliament. Some key points:
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2. It contains notices regarding registration of various documents under the Registration Act, 1908 as well as levy of stamp duty on certain instruments under the Stamp Act, 1899.
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1. 1
Decision Theory
Definition of Decision Analysis: Decision analysis is defined as the use of a rational process/technique
for selecting the best of several alternatives. The “goodness” of a selected alternative depends on the
quality of the data used in describing decision situation.
Example: Mr X is a bright student, has received full academic scholarships from three institutions: U of
A, U of B and U of C. To select a university Mr. X lists two three main criteria: location, academic
reputation and political violence. Being the excellent student he is, he judges academic reputation to be
five times as important as location and less political violence, which gives a weight of approximation
80% to reputation, 15% to location and 5% to political violence. He then use a systematic analysis to
rank the three universities from the standard point of location, reputation and political violence. The
analysis produces the following estimates
Location: U of A (13%) U of B (28%) U of C (59%)
Reputation U of A (54%) U of B (27%) U of C (19%)
Political Violence U of A (25%) U of B 45% U of C (30%)
The structure of the decision problem is summarized below in Figure.
Select a
University
Location
(0.15)
Reputation
(0.80)
Political
Violence 0.05
U of A
0.13
U of B
0.28
U of C
0.59
U of A
0.54
U of B
0.27
U of C
0.19
U of A
0.25
U Of B
0.45
U of C
0.30
Sazzad Hossain
128
2. 2
To select the university here we use the technique of decision analysis.
Classification of Decision Situation
Decision making process can be classified into the following three broad categories
(1) Decision making under certainty in which the data are known deterministically
(2) Decision making under risk in which the data can be described by probability distributions.
3. Decision making under uncertainty in which the data cannot be assigned relative weights that
represent their degree of relevance in the decision process.
Decision Making Under Certainty:
Linear programming and transportation models provide an example of decision making under
certainty. Here only one state of nature exists, there is complete certainty about the future. The
decision maker simply finds the best payoff in that one column and chooses the associated
alternative. Few complex managerial decision making problems, however ever enjoy the luxury of
having complete information about the future and hence decision making under uncertainty is of
little consequential interest. This section presents a different approaches for the situation in which
ideas, feelings and emotions are quantified to provide a numeric scale for prioritizing decision
alternatives. The approach is known as the
Decision Making Under Risk
Here, more than one states of nature exist and decision maker has sufficient information to assign
probabilities to each of these states. For this reason decision making under risk is usually based on
the expected value criteria in which decision alternatives are compared based on the maximization of
expected profits or the minimization of expected cost. This approach has limitations in the sense that
it may not be applicable to certain situations
Expected Value Criteria:
It is also known as the expected monetary value (EMV) criteria, it consists of the following steps.
(i) Construct a payoff table listing the alternative decisions and the various states of nature. Enter the
conditional profit for each decision-event combination along with the associated probabilities. In
general, a decision problem may include n states of nature and m alternatives. If jp is the
probability of occurrence for state of nature j and let ija is the payoff of alternative i
given state of nature j (i= 1, 2,……,m; j = 1, 2,……,n)
3. 3
State j
Probability( jp )
Alternatives
1, 2, ……………. i………. ….. m
1
2
.
.
.
j
.
.
n
1p
2p
.
.
.
jp
.
.
np
11a 21a ………………. i1a ……………. m1a
12a 22a ………………… i2a …………… m2a
. ………………………………………..
. ………………………………………..
. ………………………………………..
1ja 2ja ………………….. ija ……………… mja
. …………………………………………….
. ………………………………………..
1na 2na …………………… ina ……………… mna
(ii) Calculate the EMV for each decision alternative by multiplying the conditional profits by the
assigned probabilities and adding the resulting conditional values. The expected monetary value
or expected payoff for alternative I is computed as
i ij j
1
EMV = a p ; i = 1, 2,.....,m
n
j
(iii) The best alternative is the one associated with
*
i i iEMV =max [EMV ] or
*
i i iEMV =min [EMV ]
depending, respectively, on whether the payoff of the problem represents profit (income) or loss
(expense)
Problem: A newspaper boy has the following probabilities of selling a sports magazine
No. of Copies Sold Probabilities
10 0.01
11 0.15
12 0.20
13 0.25
14 0.30
Cost of a copy is TK 30 and sale price is TK 50. He cannot return unsold copies. How many
copies should ne order?
Solution: The numbers of copies for purchases and for sales are 10, 11, 12, 13 and 14. These are
his sales magnitudes. There is no reason for him to buy less than 10 or more than 14 copies.
Stocking 10 copies each day will always result in profit TK200 irrespective of demand. For
instance, even if the demand on some day is more than 10 copies, he can sell only 10 and hence
his conditional profit is TK200. When he stocks 11 copies, his conditional profits is TK220 on
days buyers request 11, 12, 13 or 14 copies. But on days when he has 11 copies on stock and
buyers buy only 10 copies, his profit will be TK 170. The profit of TK 200 on the 10 copies sold
4. 4
must be reduced by TK 30, the cost of one copy left unsold. The same will be true when he
stocks 12, 13, or 14 copies. The conditional profit is given below;
Payoff = (20 copies sold-30 copies unsold ) TK
The payoff table is constructed below;
Table 1: Payoff Table: Conditional profit in TK
Possible Demand
No. of Copies
Prob. Possible stock action
10 Copies 11 Copies 12 Copies 13 Copies 14 Copies
10
11
12
13
14
0.10
0.15
0.20
0.25
0.30
200
200
200
200
200
170
220
220
220
220
140
190
240
240
240
110
160
210
260
260
80
130
180
230
280
Now the expected value of each decision alternative is obtained by adding the multiplying its
conditional profit and the associated probability. The is shown in Table below;
Table: Expected profit
Possible Demand
No. of Copies
Prob. Expected profit from stocking in TK
10 Copies 11 Copies 12 Copies 13 Copies 14 Copies
10
11
12
13
14
0.10
0.15
0.20
0.25
0.30
20
30
40
50
60
17
33
44
55
66
14
28.5
48
60
72
11
24
42
65
78
8
19.5
36
57.5
84
Total Expected Profit 200 215 222.5 220 205
Thus the newsboy must order 12 copies to earn highest possible average daily profit. This stocking
will maximize the total profits over a period of time. Of course there is no guarantee that he will make
a profit of TK222.5 tomorrow. However, if he stocks 12 copies each day under the condition given, he
will have average profit of TK 222.5 per day. This is the best he can do because choice of any of the
other four possible stock actions result in a lower daily profit.
Expected Opportunity Loss (EOL) Criterion
This approach is an alternative to EMV. EOL or expected value of regrets is the amount by which
maximum possible profit will be reduced under various possible actions. The course of action that
minimize these losses is the optimal decision alternative. It consists of the following steps:
Step 1: Construct the conditional profit table for each decision-event combination and write the
associated probabilities
5. 5
Step 2: For each event, calculate the conditional opportunity loss (COL) by subtracting the payoff
from the maximum payoff for the event.
Step 3: Calculate the expected opportunity loss (EOL) for each decision alternative by
multiplying the COL’s by the associated probabilities and then adding the values.
Step 4. Select the alternative that yields the lowest EOL
Problem: A newspaper boy has the following probabilities of selling a sports magazine
No. of Copies Sold Probabilities
10 0.01
11 0.15
12 0.20
13 0.25
14 0.30
Cost of a copy is TK 30 and sale price is TK 50. He cannot return unsold copies. How many
copies should ne order?
Solution: The best alternative for demand 10 copies is to order 10 copies, resulting in optimal
profit of TK 200. The conditional opportunity loss for each stock action is obtained just by
subtraction the respective conditional profit from TK 200. Likewise, for demand of 11, 12, 13
and 14 copies we subtract the conditional payoff values for each of these rows from the
maximum of that row. The resulting conditional opportunity loss table is shown below;
Table 1: Conditional Loss Table in TK
Possible Demand
No. of Copies
Prob. Possible stock action
10 Copies 11 Copies 12 Copies 13 Copies 14 Copies
10
11
12
13
14
0.10
0.15
0.20
0.25
0.30
0
20
40
60
80
30
0
20
40
60
60
30
0
20
40
90
60
30
0
20
120
90
60
30
0
Now the EOL value of each decision alternative is obtained by adding the multiplying its conditional
opportunity loss and the associated probability. The is shown in Table below;
Table: Expected profit
Possible Demand
No. of Copies
Prob. Expected profit from stocking in TK
10 Copies 11 Copies 12 Copies 13 Copies 14 Copies
10
11
12
13
14
0.10
0.15
0.20
0.25
0.30
0
3
8
15
24
3
0
4
10
8
6
4.5
0
5
12
9
9
6
0
6
12
13.5
12
7.5
0
Total Expected Profit 50 35 27.5 30 45
6. 6
The optimal stock action is the one which will minimize expected opportunity losses. If the
newsboy stock 12 copies each day, the expected loss will be minimum.
Note : It may be pointed out that EMV and EOL decision criteria are completely consistent and
yield the same optimal decision alternative. However, while the decisions alternative will be
always based on finding the minimum EOL value under EOL criterion irrespective of whether the
problem is of maximization expected profit or minimizing expected loss.
Expected Value of Perfect Information (EVPI): Complete and accurate information about
the future demand referred to as perfect information, would remove all uncertainty from the
problem. With this perfect information, the decision maker would know in advance exactly about
the future demand. EVPI represents the maximum amount he would pay to get this additional
information on which may be based the decision alternative.
Problem: A dairy firm wants to determine the quantity of butter it should produce to meet the
demand. The past records have shown the following demand patterns;
Quantity Demanded (in kg) No. of Days Demand Occurred
15
20
25
30
35
40
50
6
14
20
80
40
30
10
The stock levels are restricted to the range of 15 kg to 50 kg and the butter left unsold at the end
of the day must be disposed of due to inadequate storing facilities. Butter costs TK 80 and sold
TK 110 per kg.
(i) Construct a conditional profit table
(ii) Determine the action alternative associated with the maximization of expected profit
(iii) Determine the EVPI
Solution: The dairy firm would not produce butter less than 15 kg and more than 50 kg. Form the
given data we can calculate the conditional profit for each stock action and event (demand
combination. If CP is the conditional profit, S is the quantity in stock and D is the quantity
demanded, then CP is given by;
30S; when D S
CP=
110D-80S; when D<S
The probability of demand of 15 kg is = 6/200 = 0.03. The probabilities associated with other
demand levels are given below win payoff table below:
7. 7
Possible
Demand
(Event in
kg) (D)
Prob. Possible stock action (Alternative ) (in kg) (S)
15 20 25 30 35 40 50
15
20
25
30
35
40
50
0.03
0.07
0.10
0.40
0.20
0.15
0.05
450
450
450
450
450
450
450
50
600
600
600
600
600
600
-350
200
750
750
750
750
750
-750
-200
350
900
900
900
900
-1150
-600
-50
500
1050
1050
1050
-1550
-1000
-450
100
650
1200
1200
-2350
-1800
-1250
-700
-150
400
1500
The expected profit for each alternative is given below
Table: Expected profit
Possible
Demand
(Event in
kg) (D)
Prob. Possible stock action (Alternative ) (in kg) (S)
15 20 25 30 35 40 50
15
20
25
30
35
40
50
0.03
0.07
0.10
0.40
0.20
0.15
0.05
13.5
31.5
45.0
180
90
67.5
22.5
1.5
42
60
240
120
90
30
-10.5
14
75
300
150
112.5
37.5
-22.5
-14
35
360
180
135
45
-34.5
-42
-5
200
210
157.5
52.5
-46.5
-70
-45
40
130
180
60
-70.5
-126
-125
-280
-30
60
75
EMV 450.0 583.5 678.5 718.5 538.5 248.5 -496.5
Since the maximum EMV is TK 718.5 for stock of 30 kg of butter, thus the dairy firm may
produce 30 kg of butter and can expect average daily profit of TK 718.5
Now the expected profit with perfect information is given below in Table ()
Table : Expected profit with perfect information
Market Demand
(in kg)
Probabilities Conditional Profit under
certainty
Expected profit
with perfect
information
15
20
25
30
35
40
50
0.03
0.07
0.10
0.40
0.20
0.15
0.05
450
600
750
900
1050
1200
1500
13.5
42
75
360
210
180
75
EPPI 955.5
8. 8
The expected value of perfect information is given by
EVPI = EPPI- max(EMV)
= 955.5-718.5
= 237 TK
Use of Incremental (Marginal ) Analysis.
Any additional unit purchased will be either sold or remain unsold. If p denotes the probability of
selling one additional unit, then (1-p) must be the probability of not selling it. If the additional
unit sold, the conditional profit will increase as a result of the profit earned from this additional
unit. This is termed as the incremental profit or marginal profit. Let the define the marginal profit
is MP. If the additional unit is not sold, the conditional profit is reduced and the amount of
reduction is called the incremental loss or marginal loss. Let us define the marginal loss is ML.
Thus the expected incremental profit will be pMP while the expected incremental loss will be
(1-p) ML. Thus the units should be stocked up to the point such that
pMp (1-p)ML
or at least p(MP+ML) = ML
or at least p=ML/(MP+ML)
Thus additional unit should be stocked so long as the probability of selling at least one additional
unit is greater than p.
Problem: A milkman buys milk at Tk 30 and sells for TK 40 per litre. Unsold milk has to be thrown
away. The daily demand has the following probability distribution
Demand (Liter) Probability
46
48
50
52
54
56
58
60
62
64
0.01
0.03
0.06
0.10
0.20
0.25
0.15
0.10
0.05
0.05
If each day’s demand is independent of previous days’ demand, how many liters should he
ordered every day.
Solution: The marginal profit is given as
MP = 10 Tk
And the marginal loss is given as
9. 9
ML = TK 30
The milkman should stock additional liters of milk so long as the probability of selling at least an
additional liter of milk is greater than p where
ML
p=
MP+ML
30
0.75
10 30
The value of 0.75 for p implies that in order to justify the stocking of an additional unit, there
must be at least 0.75 cumulative probability of selling that unit. The cumulative probability of
sales are computed below;
Demand (Liter) Probability Cumulative probability that sales
will be at this level or higher
46
48
50
52
54
56
58
60
62
64
0.01
0.03
0.06
0.10
0.20
0.25
0.15
0.10
0.05
0.05
1.00
0.99
0.96
0.90
0.80
0.60
0.35
0.20
0.10
0.05
The optimum number of liters of milk to be stocked is 54 liters. If the number increased to 56
liters, the cumulative probability will become 0.60 which is less than the required probability
0.75.
The expected incremental profit is given by;
EIP = pMP=0.75 10=7.5 TK
The expected incremental loss is given by;
EIL = (1-p)ML = 0.25 0.30=7.5
If the milkman stocks 56 liters of milk, the expected incremental loss will be more than expected
incremental gain. The approach give you the same decision as provided by the EMV and EOL
approaches. However, the computational effort required in this approach is much less.
Problem: A vegetable seller buys a box of tomatoes for TK 300 and sells them for TK 450 a box.
If the box is not sold on the first selling day, it is worth of TK 200 as salvage. The past records
indicate that the demand is normally distributed with a mean of 30 boxes daily and standard
10. 10
deviation is 9. How many boxes should be stock?
Decision Trees Analysis:
A decision tree is a graphical representation of the decision process indicating decision
alternatives, states of nature, probabilities attached to the state of nature and conditional benefits
and losses. It consists of a network of nodes and branches. Two types of nodes are used: (i)
decision node represented by a square and state of nature (chance or event) node represented by a
circle. Alternative courses of action (strategies) originate from the decision nodes as main
branches (decision branches). At the end of each decision branch, there is a state of nature node
from which emanate chance events in the form of sub-branches (chance branches). The
respective payoff and the probabilities associated with alternative courses and the chance events
are shown alongside these branches. At the terminal of the chance branches are shown the
expected value of the outcome.
The general approach used in decision tree analysis is to work backward through the tree from
right to left, computing the expected value of each chance node. We then choose the particular
branch leaving a decision node which leads to the chance node with the highest expected value.
This approach is known as roll back or fold back process.
Example: A client asks an estate agent to sell three properties A, B, and C for him and agrees to
pay him 5% commission on each sale. He specifies certain conditions. The estate agent must sell
property A first, and this he must do within 60 days. If and when A is sold the agent receives his
5% commission on that sale. He can then either back out at this stage or nominate and try to sell
one of the remaining two properties within 60 days. If he does not succeed in selling the
nominated property in that period, he is not given the opportunity to sell the third property on the
same conditions. The prices, selling costs( incurred by the estate agent whenever a sale is made)
and the estate agent’s estimated probabilities of making
a sale are given below;
Property Price of Property Selling Cost Probability of Sale
A 125000 5000 0.70
B 175000 4000 0.60
C 225000 6000 0.50
(i) Draw up an appropriate decision tree for the state agent
(ii) What is the estate agent’s best strategy under EMV approach?
Solution: The state agent’s gets 5% commission if he sells the properties and satisfies the
specified conditions. The amount he receives as commission on sale of properties A, B and C will
be TK 6250, TK 8750, and TK 11250 respectively. Since the selling costs incurred by his are TK
5000, TK 4000 and TK 6000 respectively and his conditional profits are TK 1250, TK 4750 and TK
11. 11
5250 respectively. The decision tree for the problem is shown below;
EMV of Node D = TK [(5250 0.50)+(0 0.50)]=TK2625
EMV of Node E = TK [(4750 0.60)+(0 0.40)]=TK2850
EMV of Node 3 = Max of TK [ 2625, 0] = TK2625
EMV of Node 4 = Max of TK [ 2850, 0] = TK 2850
EMV of Node B = TK [(4750+2625) 0.60)+(0 0.40)]=TK4425
EMV of Node C = TK [(5250+2850) 0.50)+(0 0.50)]=TK4050
EMV of Node 2 = Max of [ 4425, 4050] = TK 4425
EMV of Node A = TK [(1250+4425) 0.70)+(0 0.30)]=TK3972.5
EMV of Node 1 = TK 3972.5
Thus the optimal strategy for the estate agent is to sell A, if he sells A, then try to sell B and if he sells
B and then try to sell C to get an optimum amount of TK 3972.5
Sells C, 5250. 0.5
Takes c
Stops, TK 0 Sells Does Not C, TK 0
4750, 0.60 Stop, TK0 4750, 0.6
Sells A, Takes B Does Not Sell B Sells B
TK 0, 0.40 Takes B Does
Accepts TK1250, 0.70 Take C Sells, 5250, 0.50 Not Sell
Does not Sell, TK 0 Does Not Sell C Stops, TK0 TK0
0.30 0.50, TK 0
1 A
2
B
C
3
4
D
E