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Quantitative Review I Spring 2013 Vicky Gu 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) P2 P1 Growth Rate 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? (150 10) units P1 0.70 units / hour 200 hours (180 3) units P2 0.77 units / hour 230 hours P2 P1 0.77 0.70 Growth Rate 0. 1 P1 0.70 A 10% increase 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 in productivity? 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 200 units $12 / unit $2400 1.41 200units $6.50 / unit 40 hours $10 / hour $1700 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 45 Number of reservations 46 47 48 49 50 0.1 51 50 0.18 0.25 0.15 0.22 0.1 51 0.06 0.13 0.13 0.28 0.28 0.02 52 0.06 0.125 0.175 0.2 0.1 How many passengers should they book? 52 0.35 0.05 0.02 0.02 # of passengers actually showed up 45 # of seats booked Reservations 46 47 48 49 50 0.1 51 50 0.18 0.25 0.15 0.22 0.1 51 0.06 0.13 0.13 0.28 0.28 0.02 52 0.06 0.125 0.175 0.2 0.1 52 0.35 0.05 0.02 0.02 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 Your first job is in hotel management and recently you were promoted to Hotel Manager for a large convention hotel in downtown New Orleans. Answer the following questions given the information below for one day. What is the total contribution to profit and overhead? What is your hotel effectiveness percentage? Characteristic/Variable Business Hotel Customers (B) Convention Association Hotel Customers (C) 260 room nights rented (DB) 400 room nights rented (DC) $125 (PB) $85 (PC) Variable cost/room night (VC) $25 $25 Maximum price/room night (called the rack rate) $150 $110 300 room nights available 700 room nights available Customers for this day (D) Average price/room night(P) Maximum number rooms available for sale this day Contribution to profit and overhead ($) = (PB - VC)*DB+(PC -VC)*DC = ($125 - $25)*260 + ($85- $25)*400 = $50000 Hotel Management Effectiveness (%) = Actual hotel revenue 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 150 *300+110* 700 = 125 * 260 +85* 400 54.5% = Key Concept: Forecasting 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 Important forecasting methods to project the demand 1) 2) 3) 4) 5) Moving Average (Simple vs. Weighted) Exponential Smoothing Seasonality forecasting Linear Regression Tracking signal 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. Week Sales 415 + 458 +460 = 444.3 1 432 3 2 396 3 415 4 458 5 460 6 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? January 120 February 95 March 100 April 75 May 100 June 50 50*40% +100*30%+75*30% = 72.5 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) Ft 1 At 1 Ft – – – – 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). Week Actual Demand in Patients Exponential Smoothing Forecast 1 430 430 2 234 3 506 4 470 5 468 6 365 Absolute Deviation Week Actual Demand in Patients Exponential Smoothing Forecast 1 430 430 2 234 430 234-430 = 196 3 506 391 506-391 =115 4 470 414 470-414 = 56 5 468 425 468-425 = 43 6 365 434 365- 434 = 69 Mean absolute deviation(MAD) for wk 2~6 Absolute Deviation 430-430 =0 =(196+115+56+43+69)/5 = 95.8 Ft+1 = αAt+(1-α)Ft Week 2 forecast F2 = .2 (430)+.8(430) = 430 Week 3 forecast F3 = .2 (234)+.8(430) = 391 Week 4 forecast F4 = .2 (506)+.8(391) = 414 Week 5 forecast F5 = .2 (470)+.8(414) = 425 Week 6 forecast F6 = .2 (468)+.8(425) = 434 • 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): 1 n MAD = Ai Fi n i =1 – A good measure of the actual error in a forecast (See the previous exponential smoothing example) • Tracking Signal (TS) - Exposes forecast bias (positive or negative) - Positive tracking signal =under-forecasting - Negative = over-forecasting Cumulative error actual- forecast TS MAD Example: Given the actual demand and forecast from Jan. to Apr. what will be the MAD and TS? Month Actual Demand (A) Forecast (F) Jan 60 68 Feb 50 52 Mar 65 55 Apr 35 40 A-F Absolute Deviation Total -8 8 -2 2 10 10 -5 5 -5 25 MAD =25/4 =6.25 TS actual- forecast MAD TS 5/ 6.25 .8 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. Historical Data 1999 2000 2001 90 110 200 120 420 500 300 600 650 380 450 510 Historical Data The given data 1999 2000 2001 Spring 90 110 200 Summer Fall 120 420 500 300 600 650 Winter 380 890 450 1580 510 1860 2000 890/4=222.5 1580/4=395 1860/4= 465 2000/4=500 Total 1. Calculate the average for each year Current Year Average 2002 Seasonal Index 1999 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 2000 2001 90/222.5= .40 110/395 = .28 200/465 = .43 120/222.5=.54 420/395 =1.06 500/465 = 1.08 300/222.5=1.35 600/395 = 1.52 650/465 =1.40 380/222.5=1.71 450/395 =1.14 510/465 =1.10 Average index Forecast (1999-2001) 2002 (.40+.28+.43)/3 = .37 .37*500= 185 (.54+1.06+1.08)/3 = .89 .89 *500=446 (1.35+1.52+1.40)/3=1.42 1.42*500=711 (1.71+1.14+1.10).3=1.31 1.31*500=657 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 Tx y a bX Tx = 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 • Identify dependent (y) and independent (x) variables • Solve for the slope of the line XY n X Y b X n((X) ) 2 2 • Solve for the y intercept a Y bX • Develop your equation for the trend line Tx or y =a + bX Example: 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? XY n XY b X n((X) ) 2 a Y bX 2 Rainfall "X" Sales "Y" X*Y X2 City A 35 $2800 98000 1225 City B 30 $2000 60000 900 City C 15 $800 12000 225 Total 80 $5600 170000 2350 26.67 $1866.67 Average b (170000 3 * 26.67 * 1866.67) /[(1225 900 225) 3 * (26.67 ^ 2)] 95.38 a 1866.67 95.38 * 26.67 677 Y = a +bX = -677 +95.4*40= $3138 Key Concept: Break-Even Analysis A way of finding the point, in dollars and units, at which costs equal revenues (Supplement 7 p. 292) 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 Q SP VC 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? FC Q SP VC 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 the BE 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 actual output (100%) Utilization effective = effective capacity 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? Utilization effective Utilization design actual output 28 (100%) (100%) 140% effective capacity 20 actual output 28 (100%) (100%) 93% theoretical capacity 30 • 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? Utilization design Solve for X: actual output X (100%) (100%) 83% theoretical capacity 900 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 Expand small Expand large High demand Expand Don’t expand Low demand High demand Step 2. Add our possible states of probabilities, and potential revenue $40,000 0.35 Expand small Expand 0.65 1 $60,000 2 Don’t expand Expand large 0.35 $40,000 $20,000 X 0.65 $110,000 It is obvious that not to expand is not a good choice Step 3. Determine the expected value of each decision 0.35 Expand small $40,000 Expand 0.65 1 $60,000 2 Do nothing Expand large 0.35 0.65 $40,000 $20,000 $110,000 EVsmall = (0.35)*40,000 +0.65*60,000 = $53000 EVlarge = (0.35)*40,000+(0.65)*110,000 = $85500 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 Capacity: (60 min/hr) /2 (min/unit) = 30 units/hr B 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 order Make patties Cook burgers 30 sec/unit 60 sec/unit Add Veggie & cheese 10 sec/unit Wrap 20 sec/unit 45 sec/unit Make patties 30 sec/unit Cook burgers Add Veggie & cheese 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