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## Quantitative Review I Spring 2013 Vicky Gu

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**Key Concepts: 1. Productivity**2.Productivity Change Ch.1, p.14 Productivity is the ratio of outputs (goods and services) divided by the inputs (resources, such as labor and capital) Productivity (P) = = Productivity Change (Productivity index)is used to compare a process’ productivity at a given time (P2) to the same process’ productivity at an earlier time (P1)**Example**• Last week a company produced 150 units using 200 hours of labor, and found to have 10 defective units • This week, the same company produced 180 units with 3 defective units using 230 hours of labor • What is the change in productivity?**Productivity of last week**Productivity of this week Productivity change**If inputs increase by 30% and outputs decrease by 15%, what**is the percentage change inproductivity? P1= outputs/inputs = 1/1 P2= (1- 0.15)/ (1+0.3) =0.654 Productivity change = (P2-P1)/ P1 = 0.654-1= -0.3462 1**Key Concept:Multifactor Productivity**It measures productivity using ratio of outputs to several inputs such as labor, material, energy…… Ch 1. p.15 • Convert all inputs & outputs to $ value • Example: • 200 units produced sell for $12.00 each • Materials cost $6.50 per unit • 40 hours of labor were required at $10 an hour Calculate the multifactor productivity**Revenue Management Systems**(also called Yield Management) Ch.2 • Airline booking • Overbooking–accepting more reservations than capacity • available, assuming that a certain percentage of customers • will not show up or will cancel prior to using the service**Example: A regional airline that operates a 50-seat jet**prices the ticket for one popular business flight at $250. If the airline overbooks the reservations, overbooked passengers receive a $450 travel business flight voucher. The airline is considering overbooking by up to 2 seats, and the demand for the flight always exceeds the number of reservations it might accept. The probabilities of the number of passengers who show up is given for each booking scenario in the following table: Number of passengers showing up Number of reservations How many passengers should they book?**# of passengers actually showed up**# of seats booked Reservations Expected Profit 50 =250*(45*0.18+46*0.25+47*0.15+48*0.22+49*0.1+50*0.1) =$11777.5 51 =250*(45*0.06+46*0.13+47*0.13+48*0.1+49*0.28+50*0.28) -450*0.02 =$11818.5 52 =250*(45*0.06+46*0.125+47*0.175+48*0.2+49*0.35+50*0.05)-450*(0.02+0.02) =$11463.2 They should book 51 passengers**Hotel Management**• -Contribution to profit and overhead • -Hotel Management Effectiveness**Contribution to profit and overhead ($)**= (PB - VC)*DB+(PC -VC)*DC = ($125 - $25)*260 + ($85- $25)*400 = $50000 Hotel Management Actual hotel revenue Effectiveness (%) Maximum possible hotel revenue = (Actual prices for each room night)*(Actual number of room nights rented) = Maximum price for each room night)*(Maximum number of room nights available 125 * 260 +85* 400 = = 54.5% 150 *300+110* 700**Key Concept:Forecasting**Important forecasting methods to project the demand • Moving Average (Simple vs. Weighted) • Exponential Smoothing • Seasonality forecasting • Linear Regression • Tracking signal Forecasting is the art and science of predicting future events. Quantitative forecasting involves taking historical data and project them Into the future with mathematical models. Ch. 4. p.104 Time Series Models Casual Model Used to monitor forecast accuracy**Simple Moving Average – Uses an average of the n most**recent periods of data to forecast the next period (Ch 4. p.109) (when we assume that market demands will stay fairly steady over time) Example:Lauren's Beauty Boutique has experienced the following weekly sales. Calculate a 3 period moving average for Week 6. 415 + 458 +460 = 444.3 3 Week Sales 1 2 3 4 5 6 432 396 415 458 460**Weighted Moving Average – use weights to place more**emphasis on recent values (Ch 4. p. 110) (This is used when a detectable trend or pattern is present) Example: A firm has the following order history over the last 6 months. What would be a 3-month weighted moving average forecast for July, using weights of 40% for the most recent month, 30% for the month preceding the most recent month, and 30% for the month preceding that one? 50*40% +100*30%+75*30% = 72.5 January 120 February 95 March 100 April 75 May 100 June 50**Exponential Smoothing – Uses a weighted average of past**time-series values to forecast the value of the time series in the next period (Ch 4. p. 112) • Last period’s forecast (Ft) • Last periods actual value (At) • Select value of smoothing coefficient α, between 0 and 1.0 • The forecast “smoothes out” the irregular fluctuations in the time series • Forecast quality is dependent on selection of alpha • (Typical values for α are in the range of 0.1-0.5, larger values of α place • more emphasis on recent data, if the time series is very volatile and • contains substantial random variability, a small value of the smoothing • constant is preferred.)**Example: The manager of a small health clinic would like to**use exponential smoothing to forecast demand for emergency services in their facility. If she uses an alpha value of 0.2, what is the mean absolute deviation of her forecasts from Weeks 2 Through 6? (Assume that the forecast for Week 1 is 430).**Mean absolute deviation (MAD) – A measure of the overall**forecast error for a model (Ch 4. p. 113) MAD = N: number of periods of data**Tracking Signal – It is used to measure of how well a(TS)**forecast is predicting actual values Ch. 4, p. 132 • Mean Absolute Deviation (MAD): • A good measure of the actual error in a forecast • Tracking Signal (TS) • - Exposes forecast bias (positive or negative) • - Positive tracking signal =under-forecasting • - Negative = over-forecasting (See the previous exponential smoothing example) Cumulative error**Example: Given the actual demand and forecast from Jan.**to Apr. what will be the MAD and TS? MAD =25/4 =6.25**Seasonal Forecasting –forecast method used to project**seasonal demand based on seasonal variation in historical data (regular up-and-down movements in a time series that relate to recurring events such as weather or holidays) (Ch.4, p. 121)Example: Joe’s Equipment Distributors sells “Raider Power” brand lawn mowers. The demand forecast for 2002 is 2000 units. Given the historical sales figures listed below derive a forecast for each quarter in 2002.**The given data**Spring Summer Fall Winter 1. Calculate the average for each year 2. Calculate the seasonal index for each quarter in each year 3-year spring average index 3. Calculate the average index for each season, then calculate the forecast of each season 3-year winter average index**Regression analysis – A method for building a statistical**model that defines a relationship between a single dependent variable and one or more independent variables (Ch 4. p.126) The Regression Equation or Trend Forecast = trend forecast or y variable a = estimate of Y-axis intercept where x = 0 b = estimate of slope of the demand line X = period number or independent variable**Linear Regression**• Identifydependent (y) and independent (x) variables • Solve for the slope of the line • Solve for the y intercept • Develop your equation for the trend line • Tx or y =a + bX**Cover Me, Inc. sells umbrellas in three cities. Management**assumes that annual rainfall is the primary determinant of umbrella sales, and it wants to generate a linear regression equation to estimate potential sales in other cities. Given the data, what is the estimated amount of sales for 40 inches of rain utilizing a linear regression equation? Example:**(Supplement 7 p. 292)**Key Concept: Break-Even Analysis A way of finding the point, in dollars and units, at which costs equal revenues Total cost = FC +VC*Q Total revenue = SP *Q At break-even point FC +VC*Q= SP*Q Solve for Q: Q (SP-VC) =FC FC : Fixed Cost VC: Variable Cost SP: Selling Price Q: Number of units produced**Example: Blaster Radio Company is trying to decide whether**or not to introduce a new model. If they introduce it, there will be additional fixed costs of $400,000 per year. The variable costs have been estimated to be $20 per radio. If Blaster sells the new radio model for $30 per radio, how many must they sell to break even? Q = $400,000/ ($30-$20) Q = 40,000 The company has to sell 40,000 radios to break even**Example: If Blast radio company can’t sell 40,000 radios**in the first year, instead, their sales forecast is as follows: Year 1: Sell 25,000 Year 2: Sell 42,000 Year 3: Sell 60,000 At which year will the company achieve break even? Answer: To achieve break even in each year (i.e. to cover both the FC & VC), Sales need to reach 40,000 unit per year from what we just found out Year 1: 25,000 – 40,000 = -15,000 (short of 15,000 radios) Year 2: 42,000 – 40,000 = 2,000 (over 2000 radios) Year 3: Need 40,000 + (15000-2000)= 53,000 to break even 53,000/60,000 =0.88 0.88*12 months = 10.6, 10.6 months in year 3 or by November theBE will be reached**Key Concept: Manufacture capacity utilization and efficiency**Supplement 7, p. 283 Capacity - The maximum output rate of production or service facility or units of resource availability Theoretical capacity - Also called ideal capacity, designed capacity, (best operating level) Maximum output rate under idea conditions e.g. A bakery can make 30 custom cakes per day when pushed at holiday time**Effective capacity - Also called realistic capacity**It is the maximum output rate under normal conditions e.g. On the average this bakery can make 20 custom cakes per day Capacity Utilization - measures how much of the available capacity is actually being used Utilization effective= Utilization design =**Example: A bakery can make 30 custom cakes per day when**pushed at holiday time (or the design capacity is 30 custom cakes per day), but under normal condition, it makes 20 custom cakes per day on average. Currently the bakery is producing 28 cakes per day. What is the bakery’s capacity utilization relative to both theoretical and effective capacity? • The current utilization is only slightly below its theoretical capacity and considerably above its effective capacity • The bakery can only operate at this level for a short period of time**Example: A clinic has been set up to give flu shots to the**elderly in a large city. The theoretical capacity is 50 seniors per hour, and the effective capacity is 44 seniors per hour. Yesterday the clinic was open for ten hours and gave flu shots to 330 seniors.(a) What is the theoretical utilization?(b) What is the effective utilization? Yesterday the clinic was open for ten hours and gave flu shots to 330 seniors So the actual output is 330 senior / ten hours 33 senior / hour We know the theoretical capacity is 50 senior / hour We also know the effective capacity is 44 senior / hour Utilization theoretical = 33/50 =66% Utilization effective = 33/44 = 75%**Example: A manufacturer of printed circuit boards has a**theoretical capacity of 900 boards per day. The theoretical capacity utilization is 83% currently, what is the current production? Solve for X: 83% * 900 =74 7**Key Concept: Decision Trees for Capacity Planning Decisions**• Build from the present to the future: • Distinguish between decisions (under your control) & chance events (out of your control, but can be estimated to a given probability) • Solve from the future to the present: • Generate an expected value for each decision point based on probable outcomes of subsequent events**Example: The owners of Sweet-Tooth Bakery have determined**that they need to expand their facility in order to meet their increased demand for baked goods. The decision is whether to expand now with a large facility or expand small with the possibility of having to expand again in 5 years. The owners have estimated the following chances for demand: • The likelihood of demand being high is 0.65. · • The likelihood of demand being low is 0.35. • for each alternative have been estimated as follows: • Large expansion has an estimated profitability of either $110,000 or $40,000, depending on whether demand turns out to be high or low. • Small expansion has a profitability of $40,000, assuming demand is low. • Small expansion with an occurrence of high demand would require considering whether to expand further. If the bakery expands at this point, the profitability is to be $60,000, if not, $20,000. • What decision should the bakery make, and what is the expected value of that decision?**Step 1. We start by drawing the decision trees**Low demand Low demand Expand Expand small High demand High demand Don’t expand Expand large**Step 2. Add our possible states of probabilities, and**potential revenue $40,000 $60,000 0.35 $20,000X 0.35 $40,000 Expand Expand small 0.65 1 0.65 2 Don’t expand $110,000 Expand large It is obvious that not to expand is not a good choice**Step 3. Determine the expected value of each decision**$40,000 $60,000 $40,000 0.35 $20,000 0.35 Expand Expand small 0.65 $110,000 1 0.65 2 Do nothing EVsmall = (0.35)*40,000 +0.65*60,000 = $53000 EVlarge = (0.35)*40,000+(0.65)*110,000 = $85500 Expand large Expanding large generates the greatest expected profit, so our choice is to expand large, and the expected value for this decision is $85500**Interpretation**• At decision point 2, we chose to expand to maximize profits ($60,000 > $20,000) • Calculate expected value of small expansion: • EVsmall = 0.35($40,000) + 0.65($60,000) = $53000 • Calculate expected value of large expansion: • EVlarge = 0.35($40,000) + 0.65($110,000) = $85500 • At decision point 1, compare alternatives & choose the large expansion to maximize the expected profit: • $85500 > $53000 • Choose large expansion despite the fact that there is a 35% chance it’s the worst decision: • Take the calculated risk!**Key Concepts:**• Bottleneck • - The limiting factor or constraint in a system. • Process time of a station • -The time to produce units at a single workstation. • Process time of a system • The time of the longest (slowest) process; the • bottleneck. • Process cycle time • - The time it takes for a product to go through the production process with no waiting.**Three-Station Assembly Line**There are 60 minutes in each hour B Capacity: (60 min/hr) /2 (min/unit) = 30 units/hr Capacity: 60 (min/hr) /3 (min/unit) = 20 units/hr A C 4 min/unit 2 min/unit Capacity: 60 (min/hr) /4 (min/unit) = 15 units/hr 3 min/unit Process time for each station: 2 minutes, 4 minutes, 3 minutes Process time for the system: 4 minutes (the bottleneck) Process cycle time: 2+4+3 =9 minutes (the time to produce one finished product)**Capacity Analysis with Simultaneous Process**Cook burgers Add Veggie & cheese Make patties 30 sec/unit 60 sec/unit 10 sec/unit order Wrap 20 sec/unit 45 sec/unit Cook burgers Add Veggie & cheese Make patties 30 sec/unit 60 sec/unit 10 sec/unit The process time of each assembly line is 60 second The process time of the combined assembly line operations is 60 sec per two burgers, or 30 sec per burger. Thus, the wrapping becomes the bottleneck for the entire operation which is 45 sec per burger. Capacity: within each hour which is 3600 second, 80 burgers are made (3600 /45=80) If productivity needs to be increased, then the bottleneck station should be the first to start