Lecture Introduction QM. Aim of the course

Transcription

1 Valua%on and pricing (November 5, 2013) Lecture 14 Review Session Olivier J. de Jong, LL.M., MM, MBA, CFD, CFFA, AA Aim of the course Discuss indexes (especially those used for business purposes) and concepts related to probability theory, including independent and dependent events. Understand different forecasting models, and how to use these in business settings, and what their limitations are. Understand decision making models and how they can be used to solve business problems. Discuss and use queuing theory for common business situations. Discuss and use linear programming for common business situations. Lecture Introduction QM 1. Describe the quantitative analysis approach 2. Understand the application of quantitative analysis in a real situation 3. Describe the three categories of business analytics 4. Describe the use of modeling in quantitative analysis 5. Discuss possible problems in using quantitative analysis 6. Perform a break-even analysis 1 3

2 Introduction Mathematical tools have been used for thousands of years Quantitative analysis can be applied to a wide variety of problems Not enough to just know the mathematics of a technique Must understand the specific applicability of the technique, its limitations, and assumptions Successful use of quantitative techniques usually results in a solution that is timely, accurate, flexible, economical, reliable, and easy to understand and use 1 4 What is Quantitative Analysis? Quantitative analysis is a scientific approach to managerial decision making in which raw data are processed and manipulated to produce meaningful information Raw Data Quantitative Analysis Meaningful Information 1 5 Business Analytics BUSINESS ANALYTICS CATEGORY Descrip%ve analy%cs QUANTITATIVE ANALYSIS TECHNIQUE (CHAPTER) Sta%s%cal measures such as means and standard devia%ons (Chapter 2) Sta%s%cal quality control (Chapter 15) Predic%ve analy%cs Decision analysis and decision trees (Chapter 3) Regression models (Chapter 4) Forecas%ng (Chapter 5) Project scheduling (Chapter 11) Wai%ng line models (Chapter 12) Simula%on (Chapter 13) Markov analysis (Chapter 14) Prescrip%ve analy%cs Inventory models such as the economic order quan%ty (Chapter 6) Linear programming (Chapters 7, 8) Transporta%on and assignment models (Chapter 9) Integer programming, goal programming, and nonlinear programming (Chapter 10) Network models (Chapter 9) 1 6

3 The Quantitative Analysis Approach FIGURE 1.1 Defining the Problem Developing a Model Acquiring Input Data Developing a Solution Testing the Solution Analyzing the Results Implementing the Results 1 7 How To Develop a Quantitative Analysis Model Profit = Revenue (Fixed cost + Variable cost) Profit = (Selling price per unit)(number of units sold) [Fixed cost + (Variable costs per unit) (Number of units sold)] Profit = sx [f + vx] Profit = sx f vx where s = selling price per unit v = variable cost per unit f = fixed cost X = number of units sold 1 8 How To Develop a Quantitative Analysis Model Profit = Revenue (Fixed The cost parameters + Variable cost) of this Profit = (Selling price per model unit)(number are f, v, of and units s sold) as [Fixed cost + (Variable these are costs the per inputs unit) (Number of units inherent sold)] in the model Profit = sx [f + vx] The decision variable of Profit = sx f vx interest is X where s = selling price per unit v = variable cost per unit f = fixed cost X = number of units sold 1 9

4 Lecture Probability 1. Understand the basic foundations of probability analysis. 2. Describe statistically dependent and independent events. 3. Use Bayes theorem to establish posterior probabilities. 4. Describe and provide examples of both discrete and continuous random variables. 5. Explain the difference between discrete and continuous probability distributions. 6. Calculate expected values and variances and use the normal table Types of Probability Objective Approach Relative frequency approach P (event) = Number of occurrences of the event Classical or logical method P (head) = 1 2 P (spade) = Total number of trials or outcomes Number of ways of getting a head Number of possible outcomes (head or tail) Number of chances of drawing a spade Number of possible outcomes of Revising Probabilities with Bayes Theorem Bayes theorem is used to incorporate additional information and help create posterior probabilities from original or prior probabilities FIGURE 2.3 Prior Probabili?es Bayes Process Posterior Probabili?es New Informa?on 2 12

5 Variance of a Discrete Probability Distribution Standard deviation is the square root of the variance σ = Variance = σ 2 where = square root σ = standard deviation 2 13 The Normal Distribution Symmetrical with the midpoint representing the mean Shifting the mean does not change the shape Values on the X axis measured in the number of standard deviations away from the mean As standard deviation becomes larger, curve flattens As standard deviation becomes smaller, curve becomes steeper 2 14 Lecture Decision Making 1. List the steps of the decision-making process. 2. Describe the types of decision-making environments. 3. Make decisions under uncertainty. 4. Use probability values to make decisions under risk. 5. Develop accurate and useful decision trees. 6. Revise probabilities using Bayesian analysis. 7. Use computers to solve basic decision-making problems. 8. Understand the importance and use of utility theory in decision making. 3 15

6 The Six Steps in Decision Making 1. Clearly define the problem at hand 2. List the possible alternatives 3. Identify the possible outcomes or states of nature 4. List the payoff (typically profit) of each combination of alternatives and outcomes 5. Select one of the mathematical decision theory models 6. Apply the model and make your decision 3 16 Types of Decision-Making Environments Decision making under certainty The decision maker knows with certainty the consequences of every alternative or decision choice Decision making under uncertainty The decision maker does not know the probabilities of the various outcomes Decision making under risk The decision maker knows the probabilities of the various outcomes 3 17 Decision Making Under Uncertainty Criteria for making decisions under uncertainty 1. Maximax (optimistic) 2. Maximin (pessimistic) 3. Criterion of realism (Hurwicz) 4. Equally likely (Laplace) 5. Minimax regret 3 18

7 Decision Making Under Risk When there are several possible states of nature and the probabilities associated with each possible state are known Most popular method choose the alternative with the highest expected monetary value (EMV) EMV(alternative) = X i P(X i ) where X i = payoff for the alternative in state of nature i P(X i ) = probability of achieving payoff X i (i.e., probability of state of nature i) = summation symbol Educa%on, Inc Expected Value of Perfect Information (EVPI) Expanded EVwPI becomes And EVwPI = (best payoff for first state of nature) x (probability of first state of nature) + (best payoff for second state of nature) x (probability of second state of nature) + + (best payoff for last state of nature) x (probability of last state of nature) EVPI = EVwPI Best EMV 3 20 Expected Value of Perfect Information (EVPI) Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable) Additional information will cost $65,000 Should Thompson Lumber purchase the information? 3 21

8 Expected Opportunity Loss Expected opportunity loss (EOL) is the cost of not picking the best solution Construct an opportunity loss table For each alternative, multiply the opportunity loss by the probability of that loss for each possible outcome and add these together Minimum EOL will always result in the same decision as maximum EMV Minimum EOL will always equal EVPI 3 22 Sensitivity Analysis FIGURE 3.1 EMV Values RANGE OF BEST ALTERNATIVE P VALUES Do nothing Less than Construct a small plant Construct a large plant Greater than $300,000 $200,000 Point 2 EMV (large plant) $100,000 Point 1 EMV (small plant) 0 $100, Values of P EMV (do nothing) $200, Decision Trees Any problem that can be presented in a decision table can be graphically represented in a decision tree Most beneficial when a sequence of decisions must be made All decision trees contain decision points/nodes and state-of-nature points/nodes At decision nodes one of several alternatives may be chosen At state-of-nature nodes one state of nature will occur 3 24

9 Structure of Decision Trees Trees start from left to right Trees represent decisions and outcomes in sequential order Squares represent decision nodes Circles represent states of nature nodes Lines or branches connect the decisions nodes and the states of nature 3 25 Expected Value of Sample Information Thompson wants to know the actual value of doing the survey Expected value EVSI = with sample information Expected value of best decision without sample information = (EV with SI + cost) (EV without SI) EVSI = ($49,200 + $10,000) $40,000 = $19, Efficiency of Sample Information Possibly many types of sample information available Different sources can be evaluated Efficiency of sample Market information survey = EVSI is only EVPI 100% 32% as efficient as perfect information For Thompson Efficiency of sample information = 19, % = 32% 60,

10 Lecture Regeression 1. List the steps of the decision-making process. 2. Describe the types of decision-making environments. 3. Make decisions under uncertainty. 4. Use probability values to make decisions under risk. 5. Develop accurate and useful decision trees. 6. Revise probabilities using Bayesian analysis. 7. Use computers to solve basic decision-making problems. 8. Understand the importance and use of utility theory in decision making Calculating Revised Probabilities Calculating posterior probabilities P(B A) P(A) P(A B) = P(B A) P(A)+ P(B A ") P( A ") where A, B = any two events A = complement of A A = favorable market B = positive survey Educa%on, Inc Bayesian Analysis Many ways of getting probability data Management s experience and intuition Historical data Computed from other data using Bayes theorem Bayes theorem incorporates initial estimates and information about the accuracy of the sources Allows the revision of initial estimates based on new information Educa%on, Inc. 3 30

11 Utility Theory Monetary value is not always a true indicator of the overall value of the result of a decision The overall value of a decision is called utility Economists assume that rational people make decisions to maximize their utility 3 31 Utility Curve Typical of a risk avoider Less utility from greater risk Avoids situations where high losses might occur As monetary value increases, utility curve increases at a slower rate A risk seeker gets more utility from greater risk As monetary value increases, the utility curve increases at a faster rate Risk indifferent gives a linear utility curve Educa%on, Inc Utility as a Decision-Making Criteria Once a utility curve has been developed it can be used in making decisions Replaces monetary outcomes with utility values Expected utility is computed instead of the EMV Educa%on, Inc. 3 33

12 Introduction Regression analysis very valuable tool for a manager Understand the relationship between variables Predict the value of one variable based on another variable Simple linear regression models have only two variables Multiple regression models have more than one independent variable 4 34 Introduction Variable to be predicted is called the dependent variable or response variable Value depends on the value of the independent variable(s) Explanatory or predictor variable Dependent variable Independent = + variable Independent variable 4 35 Simple Linear Regression Regression models used to test relationships between variables Random error Y = β 0 + β 1 X +ε where Y = dependent variable (response) X = independent variable (predictor or explanatory) β 0 = intercept (value of Y when X = 0) β 1 = slope of the regression line ε = random error 4 36

13 Simple Linear Regression True values for the slope and intercept are not known Estimated using sample data ˆ Y = b 0 + b 1 X where ^ Y = predicted value of Y b 0 = estimate of β 0, based on sample results b 1 = estimate of β 1, based on sample results 4 37 FIGURE 4.3 Four Values of the Correlation Coefficient Y Y Y (a) Perfect Positive Correlation: r = +1 X Y (b) Positive Correlation: 0 < r < 1 X (c) No Correlation: r = 0 X (d) Perfect Negative Correlation: r = X Assumptions of the Regression Model With certain assumptions about the errors, statistical tests can be performed to determine the model s usefulness 1. Errors are independent 2. Errors are normally distributed 3. Errors have a mean of zero 4. Errors have a constant variance A plot of the residuals (errors) often highlights glaring violations of assumptions 4 39

14 Estimating the Variance Errors are assumed to have a constant variance (σ 2 ), usually unknown Estimated using the mean squared error (MSE), s 2 s 2 = MSE = SSE n k 1 where n = number of observations in the sample k = number of independent variables 4 40 Testing the Model for Significance If there is very little error, MSE would be small and the F statistic would be large model is useful If the F statistic is large, the significance level (p-value) will be low, unlikely would have occurred by chance When the F value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r 2 are meaningful 4 41 Steps in a Hypothesis Test 1. Specify null and alternative hypotheses H 0 : β 1 = 0 H 1 : β Select the level of significance (α) Common values are 0.01 and Calculate the value of the test statistic F = MSR MSE 4 42

15 Reject if F calculated > F α,df1,df 2 Steps in a Hypothesis Test 4. Make a decision using one of the following methods a) Reject the null hypothesis if the test statistic is greater than the F value from the table in Appendix D. Otherwise, do not reject the null hypothesis: df 1 = k df 2 = n k 1 b) Reject the null hypothesis if the observed significance level, or p-value, is less than the level of significance (α). Otherwise, do not reject the null hypothesis: p-value = P(F > calculated test statistic) Reject if p-value < α 4 43 LECTURE Forecasting 1. Understand and know when to use various families of forecasting models. 2. Compare moving averages, exponential smoothing, and other time-series models. 3. Seasonally adjust data. 4. Understand Delphi and other qualitative decisionmaking approaches. 5. Compute a variety of error measures FIGURE 5.1 Forecasting Models Forecasting Techniques Qualitative Models Time-Series Methods Causal Methods Delphi Methods Moving Average Regression Analysis Jury of Executive Opinion Exponential Smoothing Multiple Regression Sales Force Composite Trend Projections Consumer Market Survey Decomposition 5 45

16 Time-Series Models Predict the future based on the past Uses only historical data on one variable Extrapolations of past values of a series Ignores factors such as Economy Competition Selling price 5 Components of a Time Series Sequence of values recorded at successive intervals of time Four possible components Trend (T) Seasonal (S) Cyclical (C) Random (R) 5 47 Time-Series Models Two basic forms Multiplicative Demand = T x S x C x R Additive Demand = T + S + C + R Combinations are possible 5 48

17 Measures of Forecast Accuracy Compare forecasted values with actual values See how well one model works To compare models Forecast error = Actual value Forecast value Measure of accuracy Mean absolute deviation (MAD): forecast error MAD = n 5 49 Forecasting Random Variations No other components are present Averaging techniques smooth out forecasts Moving averages Weighted moving averages Exponential smoothing of Moving Averages Used when demand is relatively steady over time The next forecast is the average of the most recent n data values from the time series Smooths out short-term irregularities in the data series Moving average forecast = Sum of demands in previous n periods n of

18 Moving Averages Mathematically where F t+1 = Y t +Y t Y t n+1 n F t+1 = forecast for time period t + 1 Y t = actual value in time period t n = number of periods to average of Wallace Garden Supply Use a 3-month weighted moving average model to forecast demand Weighting scheme WEIGHTS APPLIED PERIOD 3 Last month 2 Two months ago 1 Three months ago 3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago 6 Sum of the weights of Exponential Smoothing Mathematically where F t+1 = F t +α(y t F t ) F t+1 = new forecast (for time period t + 1) Y t = pervious forecast (for time period t) α = smoothing constant (0 α 1) Y t = pervious period s actual demand The idea is simple the new estimate is the old estimate plus some fraction of the error in the last period of

19 Exponential Smoothing Example In January, February s demand for a certain car model was predicted to be 142 Actual February demand was 153 autos Using a smoothing constant of α = 0.20, what is the forecast for March? New forecast (for March demand) = ( ) = or 144 autos If actual March demand = 136 New forecast (for April demand) = ( ) = or 143 autos of Selecting the Smoothing Constant Selecting the appropriate value for α is key to obtaining a good forecast The objective is always to generate an accurate forecast The general approach is to develop trial forecasts with different values of α and select the α that results in the lowest MAD of Monitoring and Controlling Forecasts Tracking signal measures how well a forecast predicts actual values Running sum of forecast errors (RSFE) divided by the MAD Tracking signal = RSFE MAD (forecast error) = MAD MAD = forecast error n of

20 Monitoring and Controlling Forecasts Positive tracking signals indicate demand is greater than forecast Negative tracking signals indicate demand is less than forecast A good forecast will have about as much positive error as negative error Problems are indicated when the signal trips either the upper or lower predetermined limits Choose reasonable values for the limits of Monitoring and Controlling Forecasts FIGURE 5.7 Plot of Tracking Signals + 0 MADs Upper Control Limit Lower Control Limit Time Signal Tripped Tracking Signal Acceptable Range of Lecture Inventory Control 1. Understand the importance of inventory control and ABC analysis. 2. Use the economic order quantity (EOQ) to determine how much to order. 3. Compute the reorder point (ROP) in determining when to order more inventory. 4. Handle inventory problems that allow quantity discounts or noninstantaneous receipt of

21 5. Understand the use of safety stock. 6. Describe the use of material requirements planning in solving dependent-demand inventory problems. 7. Discuss just-in-time inventory concepts to reduce inventory levels and costs. 8. Discuss enterprise resource planning systems of Introduction FIGURE 6.1 Inventory Planning and Control Feedback metrics to revise plans and forecasts Planning on what to stock and how to get it Controlling inventory levels Forecasting parts/product demand of Importance of Inventory Control Five uses of inventory 1. The decoupling function 2. Storing resources 3. Irregular supply and demand 4. Quantity discounts 5. Avoiding stockouts and shortages of

22 Economic Order Quantity Economic order quantity (EOQ) model One of the oldest and most commonly known inventory control techniques Easy to use A number of important assumptions Objective is to minimize total cost of inventory of Economic Order Quantity Assumptions: Demand is known and constant Lead time is known and constant Receipt of inventory is instantaneous Purchase cost per unit is constant The only variable costs are ordering cost and holding or carrying cost These are constant throughout the year Orders are placed so that stockouts or shortages are avoided completely of Inventory Costs in the EOQ Q Situation = number of pieces to order EOQ = Q* = optimal number of pieces to order D = annual demand in units for the inventory item C o = ordering cost of each order C h = holding or carrying cost per unit per year Annual Number of ordering = orders placed cost per year Annual demand = Number of units in each order Ordering cost per order Ordering cost per order 6 66 = D Q C o 66 of

23 Inventory Costs in the EOQ Q Situation = number of pieces to order EOQ = Q* = optimal number of pieces to order D = annual demand in units for the inventory item C o = ordering cost of each order C h = holding or carrying cost per unit per year Annual holding = Average inventory cost = Carrying cost per unit per year Order quantity (Carrying cost per unit per year) 2 = Q 2 C h of Inventory Costs in the EOQ Situation FIGURE 6.3 Total Cost as a Function of Order Quantity Cost Minimum Total Cost Curve for Total Cost of Carrying and Ordering Optimal Order Quantity Carrying Cost Curve Ordering Cost Curve Order Quantity of Finding the EOQ When the EOQ assumptions are met, total cost is minimized when Annual ordering cost = Annual holding cost D Q C = Q o 2 C h Thus Solving for Q Q 2 C h = 2DC o Q 2 = 2DC EOQ = Q * = o C h Q = 2DC o C h DC o C h 69 of

24 Reorder Point: Determining When To Order Next decision is when to order Time between placing an order and its receipt is called the lead time (L) or delivery time Generally expressed as a reorder point (ROP) Demand ROP = per day = d L Lead time for a new order in days of EOQ Without Instantaneous Receipt When inventory accumulates over time, the instantaneous receipt assumption does not apply Daily demand rate must be taken into account Production run model Inventory Level Maximum Inventory Part of Inventory Cycle During Which Production is Taking Place t FIGURE 6.5 Inventory Control and the Production Process 6 71 There is No Production During This Part of the Inventory Cycle Time 71 of Production Run Model Equation summary Annual holding cost = Q! 2 1 d $ # &C h " p % Annual setup cost = D Q C s Optimal production quantity Q * = C h 2DC s! 1 d $ # & " p % of

25 Quantity Discount Models Quantity discounts are commonly available Basic EOQ model is adjusted by adding in the purchase or materials cost Total cost = Material cost + Ordering cost + Holding cost where Total cost = DC + D Q C o + Q 2 C h D = annual demand in units C o = ordering cost of each order C = cost per unit C h = holding or carrying cost per unit per year of Quantity Discount Models Steps in the process 1. For each discount price (C), compute EOQ = 2DC o IC 2. If EOQ < Minimum for discount, adjust the quantity to Q = Minimum for discount 3. For each EOQ or adjusted Q, compute Total cost = DC + D Q C o + Q 2 C h 4. Choose the lowest-cost quantity of Use of Safety Stock If demand or the lead time are uncertain, the exact ROP will not be known with certainty Safety stock can help prevent stockouts Can be implemented by adjusting the ROP Average demand ROP = + during lead time ROP = where Average demand during lead time SS = safety stock + SS Safety stock of

26 Use of Safety Stock Objective is to choose a safety stock amount the minimizes total holding and stockout costs If variation in demand and holding and stockout costs are known, payoff/cost tables could be used to determine safety stock More general approach is to choose a desired service level based on satisfying customer demand of Use of Safety Stock Set safety stock to achieve a desired service level Service level = 1 Probability of a stockout or Probability of a stockout = 1 Service level of Safety Stock with the Normal Distribution ROP = (Average demand during lead time) + Zσ dlt where Z = number of standard deviations for a given service level σ dlt = standard deviation of demand during the lead time Thus Safety stock = Zσ dlt of

27 Calculating Lead Time Demand and Standard Deviation Three situations to consider Demand is variable but lead time is constant Demand is constant but lead time is variable Both demand and lead time are variable of Calculating Lead Time Demand and Standard Deviation 1. Demand is variable but lead time is constant ROP = dl + Z ( σ d L) where d = average daily demand σ d = standard deviation of daily demand L = lead time in days of Calculating Lead Time Demand and Standard Deviation 2. Demand is constant but lead time is variable where ROP = dl + Z ( dσ L ) L = average lead time σ L = standard deviation of lead time d = daily demand of

28 Calculating Lead Time Demand and Standard Deviation 3. Both demand and lead time are variable ROP = dl + Z Lσ d 2 + d 2 σ L 2 The most general case Can be simplified to the earlier equations of Service Levels, Safety Stock, and Holding Costs As service levels increase Safety stock increases at an increasing rate As safety stock increases Annual holding costs increase of Calculating Annual Holding Cost with Safety Stock Under standard assumptions of EOQ Average inventory = Q/2 Annual holding cost = (Q/2)C h With safety stock Total annual holding cost Holding cost = of regular + inventory THC = Q 2 C h +(SS)C h Holding cost of safety stock where THC = total annual holding cost Q = order quantity C h = holding cost per unit per year SS = safety stock of

29 ABC Analysis The purpose is to divide the inventory into three groups based on the overall inventory value of the items Group A items account for the major portion of inventory costs Typically 70% of the dollar value but only 10% of the quantity of items Forecasting and inventory management must be done carefully Mistakes can be expensive of ABC Analysis Group B items are more moderately priced May represent 20% of the cost and 20% of the quantity Moderate levels of control Group C items are very low cost but high volume It is not cost effective to spend a lot of time managing these items Simple control policies and loose control of Material Structure Tree The first step is to develop a bill of materials (BOM) BOM identifies components, descriptions, and the number required for production of one unit of the final product Material structure tree developed from the BOM Demand for product A is 50 units Each A requires 2 units of B and 3 units of C Each B requires 2 units of D and 3 units of E Each C requires 1 unit of E and 2 units of F of

30 Material Structure Tree Level FIGURE 6.12 Material Structure Tree for Item A B(2) A C(3) D(2) E(3) E(1) F(2) of Material Structure Tree The demand for B, C, D, E, and F is completely dependent on the demand for A The material structure tree has three levels Items above a level are called parents Items below any level are called components The number in parenthesis beside each item shows how many are required for each unit of the parent of LECTURE Linear Program 1. Understand the basic assumptions and properties of linear programming (LP). 2. Graphically solve any LP problem that has only two variables by both the corner point and isoprofit line methods. 3. Understand special issues in LP such as infeasibility, unboundedness, redundancy, and alternative optimal solutions. 4. Understand the role of sensitivity analysis. 5. Use Excel spreadsheets to solve LP problems of

31 Introduction Many management decisions involve making the most effective use of limited resources Linear programming (LP) Widely used mathematical modeling technique Planning and decision making relative to resource allocation Broader field of mathematical programming Here programming refers to modeling and solving a problem mathematically of Requirements of a Linear Programming Problem Four properties in common Seek to maximize or minimize some quantity (the objective function) Restrictions or constraints are present Alternative courses of action are available Linear equations or inequalities of Formulating LP Problems Common LP application product mix problem Two or more products are produced using limited resources Maximize profit based on the profit contribution per unit of each product Determine how many units of each product to produce of

32 Graphical Solution to an LP Problem Easiest way to solve a small LP problems is graphically Only works when there are just two decision variables Not possible to plot a solution for more than two variables Provides valuable insight into how other approaches work Nonnegativity constraints mean that we are always working in the first (or northeast) quadrant of a graph of Graphical Representation of Constraints FIGURE 7.2 Graph of Carpentry Constraint Equation Number of Chairs C (T = 0, C = 80) T Number of Tables (T = 60, C = 0) of Graphical Representation of Constraints FIGURE 7.3 Region that Satisfies the Carpentry Constraint Number of Chairs C (30, 40) 20 (30, 20) T Number of Tables Any point on or below the constraint plot will not violate the restriction Any point above the plot will violate the restriction (70, 40) of

33 Isoprofit Line Solution Method Find the optimal solution from the many possible solutions Speediest method is to use the isoprofit line Starting with a small possible profit value, graph the objective function Move the objective function line in the direction of increasing profit while maintaining the slope The last point it touches in the feasible region is the optimal solution of Corner Point Solution Method The corner point method for solving LP problems Look at the profit at every corner point of the feasible region Mathematical theory is that an optimal solution must lie at one of the corner points or extreme points of Solving Minimization Problems Many LP problems involve minimizing an objective such as cost Minimization problems can be solved graphically Set up the feasible solution region Use either the corner point method or an isocost line approach Find the values of the decision variables (e.g., X 1 and X 2 ) that yield the minimum cost of

34 Lecture Project Man. 1. Understand how to plan, monitor, and control projects with the use of PERT and CPM. 2. Determine earliest start, earliest finish, latest start, latest finish, and slack times for each activity, along with the total project completion time. 3. Reduce total project time at the least total cost by crashing the network using manual or linear programming techniques. 4. Understand the important role of software in project management of Introduction Managing large-scale, complicated projects effectively is a difficult problem and the stakes are high The first step in planning and scheduling a project is to develop the work breakdown structure (WBS) Identify activities that must be performed and their beginning and ending events Identify time, cost, resource requirements, predecessors, and people responsible for each activity A schedule for the project then can be developed of Introduction The program evaluation and review technique (PERT) and the critical path method (CPM) are two popular quantitative analysis techniques for complex projects PERT uses three time estimates to develop a probabilistic estimate of completion time CPM is a more deterministic technique They are so similar they are commonly considered one technique, PERT/CPM of

35 Introduction The program evaluation and review technique (PERT) and the critical path method (CPM) are two popular quantitative analysis techniques for complex projects PERT uses three time estimates to develop a probabilistic estimate of completion time CPM is a more deterministic technique They are so similar they are commonly considered one technique, PERT/CPM of Six Steps of PERT/CPM 1. Define the project and all of its significant activities or tasks. 2. Develop the relationships among the activities. Decide which activities must precede others. 3. Draw the network connecting all of the activities. 4. Assign time and/or cost estimates to each activity. 5. Compute the longest time path through the network; this is called the critical path. 6. Use the network to help plan, schedule, monitor, and control the project of Six Steps of PERT/CPM 1. Define the project and all of its significant activities or tasks. 2. Develop the relationships among the activities. Decide which activities must precede others. 3. Draw the network connecting all of the activities. 4. Assign time and/or cost estimates to each activity. 5. Compute the longest time path through the network; this is called the critical path. 6. Use the network to help plan, schedule, monitor, and control the project of

36 PERT/CPM Questions answered by PERT 1. When will the entire project be completed? 2. What are the critical activities or tasks in the project, that is, the ones that will delay the entire project if they are late? 3. Which are the noncritical activities, that is, the ones that can run late without delaying the entire project s completion? 4. If there are three time estimates, what is the probability that the project will be completed by a specific date? of PERT/CPM Questions answered by PERT 5. At any particular date, is the project on schedule, behind schedule, or ahead of schedule? 6. On any given date, is the money spent equal to, less than, or greater than the budgeted amount? 7. Are there enough resources available to finish the project on time? of Drawing the PERT/CPM Network Two common techniques for drawing PERT networks Activity-on-node (AON) nodes represent activities Activity-on-arc (AOA) arcs represent the activities The AON approach is easier and more commonly found in software packages One node represents the start of the project, one node for the end of the project, and nodes for each of the activities The arcs are used to show the predecessors for each activity of

37 Drawing the PERT/CPM Network FIGURE 11.1 Network for General Foundry A C F Build Internal Construct Install Control Components Collection Stack System E Start Build Burner B D G Modify Roof Pour Concrete Install Pollution and Floor and Install Frame Device H Inspect and Test Finish 109 of Activity Times In some situations, activity times are known with certainty CPM assigns just one time estimate to each activity and this is used to find the critical path In many projects there is uncertainty about activity times PERT employs a probability distribution based on three time estimates for each activity, and a weighted average of these estimates is used for the time estimate and this is used to determine the critical path of Activity Times The time estimates in PERT are Optimistic time (a) = time an activity will take if everything goes as well as possible. There should be only a small probability (say, 1 / 100 ) of this occurring. Pessimistic time (b) = time an activity would take assuming very unfavorable conditions. There should also be only a small probability that the activity will really take this long. Most likely time (m) = most realistic time estimate to complete the activity of

38 Activity Times The time estimates in PERT are Optimistic time (a) = time an activity will take if everything goes as well as possible. There should be only a small probability (say, 1 / 100 ) of this occurring. Pessimistic time (b) = time an activity would take assuming very unfavorable conditions. There should also be only a small probability that the activity will really take this long. Most likely time (m) = most realistic time estimate to complete the activity of Activity Times To find the expected activity time (t), the beta distribution weights the estimates as follows t = a + 4m + b 6 To compute the dispersion or variance of activity completion time! b a$ Variance = # & " 6 % of How to Find the Critical Path To find the critical path, determine the following quantities for each activity 1. Earliest start (ES) time: the earliest time an activity can begin without violation of immediate predecessor requirements 2. Earliest finish (EF) time: the earliest time at which an activity can end 3. Latest start (LS) time: the latest time an activity can begin without delaying the entire project 4. Latest finish (LF) time: the latest time an activity can end without delaying the entire project of

39 How to Find the Critical Path Activity times are represented in the nodes ACTIVITY t ES EF LS LF Earliest times are computed as Earliest finish time = Earliest start time + Expected activity time EF = ES + t Earliest start = Largest of the earliest finish times of immediate predecessors ES = Largest EF of immediate predecessors of How to Find the Critical Path At the start of the project we set the time to zero Thus ES = 0 for both A and B Start A t = 2 ES = 0 EF = = 2 B t = 3 ES = 0 EF = = of How to Find the Critical Path FIGURE 11.4 General Foundry s Earliest Start (ES) and Earliest Finish (EF) Times A 2 C 2 F E 4 H 2 Start B 3 D 4 G Finish 117 of

40 How to Find the Critical Path FIGURE 11.4 General Foundry s Earliest Start (ES) and Earliest Finish (EF) Times A 2 C 2 F E 4 H 2 Start B 3 D 4 G Finish 118 of How to Find the Critical Path Compute latest start (LS) and latest finish (LF) times for each activity by making a backward pass through the network Latest start time = Latest finish time Expected activity time LS = LF t Latest finish time = Smallest of latest start times for following activities LF = Smallest LS of following activities For activity H LS = LF t = 15 2 = 13 weeks of How to Find the Critical Path FIGURE 11.5 General Foundry s Latest Start (LS) and Latest Finish (LF) Times A 2 C 2 F E 4 H 2 Start B 3 D 4 G Finish 120 of

41 How to Find the Critical Path Once ES, LS, EF, and LF have been determined, find the amount of slack time for each activity Slack = LS ES, or Slack = LF EF Activities A, C, E, G, and H have no slack time These are called critical activities and they are said to be on the critical path The total project completion time is 15 weeks Industrial managers call this a boundary timetable of Sensitivity Analysis and Project Management A predecessor activity is one that must be accomplished before the given activity can be started A successor activity is one that can be started only after the given activity is finished A parallel activity is one that does not directly depend on the given activity Once these have been defined, we can explore the impact that an increase (decrease) in an activity time for a critical path activity would have on other activities in the network of PERT/COST PERT is an excellent method of monitoring and controlling project length but it does not consider the very important factor of project cost PERT/Cost is a modification of PERT that allows a manager to plan, schedule, monitor, and control cost as well as time of

42 Four Steps of the Budgeting Process 1. Identify all costs associated with each of the activities. Then add these costs together to get one estimated cost or budget for each activity. 2. If you are dealing with a large project, several activities can be combined into larger work packages. A work package is simply a logical collection of activities. Since the General Foundry project we have been discussing is small, each activity will be a work package of Four Steps of the Budgeting Process 3. Convert the budgeted cost per activity into a cost per time period. To do this, we assume that the cost of completing any activity is spent at a uniform rate over time. Thus, if the budgeted cost for a given activity is $48,000 and the activity s expected time is four weeks, the budgeted cost per week is $12,000 (=$48,000/4 weeks). 4. Using the earliest and latest start times, find out how much money should be spent during each week or month to finish the project by the date desired of Monitoring and Controlling Project Costs The value of work completed, or the cost to date for any activity Value of work completed The activity difference = (Percentage of work complete) x (Total activity budget) Activity difference = Actual cost Value of work completed of

43 Project Crashing Projects will sometimes have deadlines that are impossible to meet using normal procedures By using exceptional methods it may be possible to finish the project in less time Costs usually increase Reducing a project s completion time is called crashing of Four Steps to Project Crashing 1. Find the normal critical path and identify the critical activities. 2. Compute the crash cost per week (or other time period) for all activities in the network using the formula. Crash cost/time period = Crash cost Normal cost Normal time Crash time of Four Steps to Project Crashing 3. Select the activity on the critical path with the smallest crash cost per week. Crash this activity to the maximum extent possible or to the point at which your desired deadline has been reached. 4. Check to be sure that the critical path you were crashing is still critical. Often, a reduction in activity time along the critical path causes a noncritical path or paths to become critical. If the critical path is still the longest path through the network, return to step 3. If not, find the new critical path and return to step of

44 LECTURE Waiting Lines 1. Describe the trade-off curves for cost-of-waiting time and cost of service. 2. Understand the three parts of a queuing system: the calling population, the queue itself, and the service facility. 3. Describe the basic queuing system configurations. 4. Understand the assumptions of the common models dealt with in this chapter. 5. Analyze a variety of operating characteristics of waiting lines of Introduction Queuing theory is the study of waiting lines One of the oldest and most widely used quantitative analysis techniques The three basic components of a queuing process Arrivals Service facilities The actual waiting line Analytical models of waiting lines can help managers evaluate the cost and effectiveness of service systems of Waiting Line Costs Most waiting line problems are focused on finding the ideal level of service a firm should provide Generally service level is something management can control Often try to find the balance between two extremes A large staff and many service facilities High levels of service but high costs The minimum number of service facilities Service cost is lower but may result in dissatisfied customers Service facilities are evaluated on their total expected cost which is the sum of service costs and waiting costs Find the service level that minimizes the total expected cost of

45 Characteristics of a Queuing System FIGURE 12.2 Four Basic Queuing System Configurations Arrivals Arrivals Queue Queue Single-Channel, Single-Phase System Type 1 Service Facility Single-Channel, Multiphase System Service facility Type 2 Service Facility Departures After Service Departures After Service 133 of Characteristics of a Queuing System FIGURE 12.2 Four Basic Queuing System Configurations Arrivals Queue Multichannel, Single-Phase System Service Facility 1 Service Facility 2 Service Facility Departures After Service 134 of Characteristics of a Queuing System FIGURE 12.2 Four Basic Queuing System Configurations Arrivals Queue Type 1 Service Facility 1 Type 1 Service Facility 2 Multichannel, Multiphase System Type 2 Service Facility 1 Type 2 Service Facility Departures After Service 135 of

46 Identifying Models Using Kendall Notation A notation for queuing models that specifies the pattern of arrival, the service time distribution, and the number of channels Basic three-symbol Kendall notation has the form Arrival distribution Service time distribution Number of service channels open Specific letters used to represent probability distributions M = Poisson distribution for number of occurrences D = constant (deterministic) rate G = general distribution with known mean and variance Identifying Models Using Kendall Notation A single-channel model with Poisson arrivals and exponential service times would be represented by M/M/1 When a second channel is added M/M/2 A three-channel system with Poisson arrivals and constant service time would be M/D/3 A four-channel system with Poisson arrivals and normally distributed service times would be M/G/ Single-Channel Model, Poisson Arrivals, Exponential Service Times (M/M/1) Assumptions of the model 1. Arrivals are served on a FIFO basis 2. There is no balking or reneging 3. Arrivals are independent of each other but the arrival rate is constant over time 4. Arrivals follow a Poisson distribution 5. Service times are variable and independent but the average is known 6. Service times follow a negative exponential distribution 7. Average service rate is greater than the average arrival rate

47 Single-Channel Model, Poisson Arrivals, Exponential Service Times (M/M/1) Assumptions of the When model these assumptions are met, 1. Arrivals are served we can on develop a FIFO basis a series of equations that define the queue s 2. There is no balking or reneging operating characteristics 3. Arrivals are independent of each other but the arrival rate is constant over time 4. Arrivals follow a Poisson distribution 5. Service times are variable and independent but the average is known 6. Service times follow a negative exponential distribution 7. Average service rate is greater than the average arrival rate Remember Remember this overview is just a selection of some of the issues we discussed during lecture and tutorial, and needs to be viewed with your notes, MIP, questions and slides provided during the course of the program. Any questions please olivier.edu@gmail.com Quick WeChat -> olivier.edu and make notes as you do so, in whatever way works best for you in terms of remembering information (your performance on this course is only assessed by exam). Lots of success tomorrow of