ECLT 5930/SEEM 5740: Engineering Economics 2014–15 Second Term Master of Science in ECLT & SEEM Instructors: Dr. Anthony Man–Cho So Dr. Man Hong Keith Wong Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong January 22, 2015 Recap: Cost Concepts and Analysis 1. Various cost terminologies and their characteristics 2. Cost–volume relationships; breakeven points 3. Cost–driven design optimization ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 1 Application: To Produce or Not to Produce? • Department A of a manufacturing plant occupies 100 square meters and produce, among other things, 576 pieces of product X per day. • The average daily production costs for product X are summarized as follows: Direct labor 1 operator working 4 hours per day at $22.50 per hour; part–time manager at $30 per day Direct material Overhead $120.00 $86.40 at $0.82 per square meter Total cost per day $82.00 $288.40 • One can also outsource the production of X to another company at a cost of $0.35 per piece. This results in a total purchase cost of 576 × $0.35 = $201.60. Question: Should the plant shut down the production line for X and purchase it from the other company? ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 2 This Lecture: Cost Estimation Techniques • Cost is an integral element in evaluating and analyzing feasible alternatives. – – – – provide information for setting prices determine whether a proposed product can be made and distributed at a profit evaluate how much capital can be justified for improvements establish benchmarks for productivity improvement programs • Previous discussions assume that they are exactly known. • In reality, various cost parameters must be estimated or forecasted. • A decision based on these cost estimated are economically sound only to the extent that those estimates are representative of what would occur. – Garbage in, garbage out! ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 3 Fundamental Cost Estimation Approaches • Top–Down – use historical data from similar projects and adjust them to current levels – get a rough estimate, best used early in the estimation process • Bottom–Up – break down a project into small, manageable units and estimate their economic consequences – get a detailed estimate, best used when details concerning the desired output have been defined ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 4 Integrated Approach for Cost Estimation • Work Breakdown Structure (WBS): explicitly defining the elements of a project and their interrelationships at successive levels of detail • Cost and Revenue Structure: delineation of cost and revenue categories for the purpose of cash flow estimation at each level of WBS • Estimating Techniques and Models: use selected mathematical models to estimate the cost and revenue elements ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 5 Work Breakdown Structure (WBS) • WBS is a framework for – – – – defining all project work elements and their interrelationships, collecting and organizing information, developing relevant cost and revenue data, and integrating project management activities. • It is developed from top–down in successive levels of detail. • Both functional (e.g., logistical support, marketing, etc.) and physical (e.g., labor, material, etc.) elements are included in the WBS. • A WBS can include both recurring (e.g., maintenance) and non–recurring (e.g., initial construction) work elements. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 6 Work Breakdown Structure: Example Commercial Building Project Physical • Site Work and Foundation – Site Gardening – Excavation – ... • Exterior – Framing – Siding – ... • Interior – Doors – Flooring/Stairways – ... ECLT 5930/SEEM 5740 (2014–15 Second Term) Functional • Sales – Leasing/Asset Sales – Adminstrative Support – ... • Project Management – Technical Management – Legal – ... • Engineering Services – Design – Consulting – ... January 22, 2015 7 Cost and Revenue Structure • This structure is used to identify and categorize the costs and revenues that need to be included in the analysis. • Some categories include (but are not limited to) capital investment material costs property taxes/insurance disposal costs quality costs ECLT 5930/SEEM 5740 (2014–15 Second Term) labor costs maintenance costs overhead costs sales revenues market/salvage values January 22, 2015 8 Cost Estimation • The purpose of estimation is not to produce exact data about future cash flows, but rather a reasonably accurate projection. • Different levels of accuracy – order–of–magnitude estimates: used in the planning or initial evaluation stage – budget estimates: used in the preliminary or conceptual design stage – definitive estimates: used in the detailed engineering stage • Sources of Data – accounting records: good source of historical data, but have their limitations – sources within a firm: e.g., people/data from engineering, sales, production, quality, purchasing, personnel, etc. – sources outside a firm: e.g., government statistics, marketing research, personal contact, etc. – research and development (R&D): could be a large undertaking and need rigorous techniques ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 9 Cost Estimation (Cont’d) • How estimates are accomplished? – conference: people with good information are gathered to estimate the quantities in question – comparison: use similar situations or designs to extrapolate relevant estimates – quantitative: use of mathematical techniques to derive estimates • Question: What are the limitations of each of the above approaches? We shall focus on quantitative techniques in this lecture. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 10 Use of Indices • An index is a dimensionless number that indicates how cost or price has changed with time with respect to a base year. • Mathematically, an estimate of the cost of an item in year n, Cn (typically current), with respect to that of year k, Ck (which is in the past), can be obtained by In Cn = Ck × , Ik where – – – – – – k = reference year (e.g., 2000) for which cost or price is known; n = year for which cost or price is to be estimated (note that n > k); Ck = cost or price of item in year k; Cn = estimated cost or price of item in year n; Ik = index value in year k; In = index value in year n. • Indices capture the ratio of price changes. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 11 Use of Indices: Example • A company installed a boiler for $525,000 in 2000, when the index was 468. • The company wants to install another boiler of the same type now in 2012, when the index is 542. • Question: What is the approximate cost of the new boiler? – Using the index equation, the approximate cost is given by C2012 = C2000 × ECLT 5930/SEEM 5740 (2014–15 Second Term) I2012 542 = $608, 013. = 525, 000 × I2000 468 January 22, 2015 12 Use of Indices (Cont’d) • Question: What are the advantages and disadvantages regarding the use of indices? ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 13 Computation of Indices • An index can be compiled for a single or multiple items. Examples – Consumer Price Index (CPI): tracks the prices of consumer goods and services (such as food and beverage, transport, entertainment) purchased by households – Commodity Price Index: tracks the prices of agricultural and industrial commodities (such as oil, metal, corn) consumed by industries • Suppose that we want to incorporate M items in an index. Let – Ckm = cost or price of item m in year k; – Cnm = estimated cost or price of item m in year n; – Wm = weight (relative importance) assigned to item m. • Then, we can form the following weighted (or composite) index: In = W1(Cn1/Ck1) + W2(Cn2/Ck2) + · · · + WM (CnM /CkM ) × Ik . W1 + W2 + · · · + WM ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 14 Composite Index: Example • Consider the composite price for gasoline (in cents per gallon) in year 1992, 2006 and 2011: Premium Unleaded plus Regular unleaded 1992 114 103 93 2006 240 230 221 2011 320 303 291 • The weight on regular unleaded is three times that of the other two, based on the amount sold. • Suppose that the index in 1992 was 100. Then, the index value in 2011 is I2011 = (320/114) + (303/103) + 3(291/93) × 100 ≈ 302.7172. 1+1+3 • Now, suppose that the index in 2012 is estimated to be I2012 = 327. Then, the price of gasoline in 2012 can be estimated using the composite index: cost(premium)2012 = cost(premium)2011 × ECLT 5930/SEEM 5740 (2014–15 Second Term) I2012 ≈ 345.67 cents per gallon. I2011 January 22, 2015 15 Unit and Factor Technique • The unit technique involves using a per unit factor that can be estimated effectively. Examples capital cost of plant per kilowatt capacity revenue per mile material cost per unit square revenue per customer served • The factor technique extends the unit technique by taking into account multiple cost components and forming a single cost estimate. Mathematically, the total cost C estimated by this technique is given by X X fmUm, C= Cd + d m where – Cd = cost of component d that is estimated directly; – fm = cost per unit of component m; – Um = number of units of component m. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 16 Unit and Factor Technique: Example • Suppose that your company has a commerical building for lease. The annual revenue can be estimated as 3 X where – – – – – – – – Sj × uj × dj , R = 12 × P × rp + 12 × B × rb + | {z } direct estimation {z } |j=1 factors P = number of parking spaces; B = number of billboards; rp = monthly rate of parking spaces; rb = monthly rate of billboards; j = type of building space; Sj = available type–j space (in square feet); uj = utilization rate of space type j; dj = rate (per square foot utilized per year) of space type j. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 17 Parametric Cost Estimating • Basic Idea: use both historical data and statistical techniques to predict future costs. • Statistical techniques are used to produce Cost Estimating Relationships (CER), which take the form cost (or price) of an item = f (cost driver1, cost driver2, . . .). Here, f is a function that we specify, which may include parameters that need to be estimated from historical data. • Two immediate questions: – What are the cost drivers? – What kind of f should we use? ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 18 Cost Drivers • Cost drivers are design variables that account for large portion of total cost. • They depend on the product/service being considered. • Examples Product software turbine engine housing cellphone plan set menu Cost Drivers number of lines of codes, target audience maximum thrust, fuel consumption size, neighborhood, interest rate number of features, coverage quality of food, ambience, rent • Caution: One can always come up with a large number of cost drivers, but not all of them are necessarily significant (more on this later). ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 19 CER I: Power–Sizing Technique • Also known as the exponential model. • Commonly used to estimate capital investment on industrial plant or equipment. • Idea: cost varies as some power of the change in capacity or size. Mathematically, we have X X SA SA CA = ⇐⇒ CA = CB , CB SB SB where – – – – – CA = cost of plant A; CB = cost of plant B; SA = size of plant A; SB = size of plant B; X = cost–capacity factor to reflect economies of scale. • Note: The costs must be determined at the same point in time for which the estimate is desired. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 20 Power–Sizing Technique: Example • A power plant with capacity 200–MW cost $100M 20 years ago. The cost index was 400. • Now in 2012, consider a power plant with capacity of 600 MW, and the cost index is 1200. • Suppose that the cost–capacity factor is 0.79. • Question: Estimate the cost of building such a plant. – To apply the power–sizing technique, we first need to find the current cost of a 200–MW plant: 1200 = $300 million. cost(200 MW)2012 = cost(200 MW)1992 × 400 – Then, by the power–sizing technique, we have cost(600 MW)2012 = cost(200 MW)2012 × ECLT 5930/SEEM 5740 (2014–15 Second Term) 600 200 0.79 January 22, 2015 ≈ $714 million. 21 CER II: Learning Curve • In practice, it is observed that workers’ efficiency and organizational performance improve with repetitive production of a good or service. • To formalize this observation, we can use the learning curve, which measures the percentage reduction occurs in, say, labor hours, as the number of units produced is doubled. Example – Suppose that there is a 10% reduction in labor hours when the number of bicycles produced is doubled. – Suppose that it takes 10 hours to assemble the first bicycle. – Then, it takes 10 × 0.9 = 9 hours to assemble the second bicycle, 10 × 0.9 × 0.9 = 8.1 hours to assemble the fourth bicycle, etc. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 22 CER II: Learning Curve (Cont’d) • Let – Zu = number of input resource units needed to produce the u–th unit; – K = number of input resource units needed to produce the first unit; – s = learning curve slope parameter (e.g., s = 0.9 means a 90% learning curve). • Then, the definition of learning curve says Z 2n = K × sn . • Letting n = log2 u, we have the following equivalent form: Zu = K × slog2 u. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 23 Learning Curve: Example • Suppose that the time needed to assemble the first car is 100 hours, and the learning curve is 80%. • Questions: – What is the time needed to assemble the 10th car? – The total time required to assemble all 10 cars? ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 24 Learning Curve: Example • Suppose that the time needed to assemble the first car is 100 hours, and the learning curve is 80%. • Questions: – What is the time needed to assemble the 10th car? Z10 = 100 × (0.8)log2 10 ≈ 47.651 hours. – What is the total time required to assemble all 10 cars? total time = Z1 + Z2 + · · · + Z10 ≈ 631 hours. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 25 CER III: Functional Regression • Sometimes we may suspect that the cost and the cost drivers follow certain functional relationships. Examples Type Linear Power Logarithmic Exponential Functional Form cost = b0 + b1x1 + b2x2 + · · · + bk xk cost = b0 + b1xb111 xb212 + · · · cost = b0 + b1 log(x1) + b2 log(x2) + · · · + bk log(xk ) cost = b0 + b1 exp(b11x1 ) + b2 exp(b22x2 ) + · · · + bk exp(bkk xk ) • To use a particular functional form to produce cost estimates, we need to estimate the coefficients (i.e., b0, b1, . . .) from historical data. – Problem: The historical data may not fit the chosen functional form exactly. (Why?) ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 26 Functional Regression: Schematic View ǫi xi f f (xi ) + (unknown) yi = f (xi ) + ǫi Figure 1: Schematic view of functional regression ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 27 Coefficient Estimation in Linear Regression • Suppose we decide that there is one cost driver (x) for our product, and the cost (y) is related to this cost driver via y = b0 + b1x. • To determine b0 and b1, we collect past data. We observe that when the cost driver is set to xi, the cost is given by yi, where i = 1, . . . , n, and n is the total number of observations. • As mentioned before, these points may not fall exactly on a line. • To find the “best” line, we may try to find b0, b1 that minimize the following penalty function: n n X X e2i . (yi − (b0 + b1xi))2 = i=1 i=1 This is the well–known least squares approach. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 28 Coefficient Estimation in Linear Regression (Cont’d) y (x6 , y6 ) e6 (x1 , y1 ) e1 0 (x3 , y3 ) (x5 , y5 ) e3 e4 e2 y = b0 + b1 x (x4 , y4 ) (x2 , y2 ) x Figure 2: Linear regression and the relevant parameters ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 29 Coefficient Estimation in Linear Regression (Cont’d) • The solution is given by b1 = b0 = n 1 n Pn Pn Pn i=1 xi yi − ( i=1 xi ) ( i=1 yi ) , Pn Pn 2 2 n i=1 xi − ( i=1 xi) n X i=1 yi − b1 n X i=1 ! xi . • Questions: – How do we know if this solution is good? – Are there other approaches besides least squares? ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 30 Model Validation y (x3 , y3 ) (x2 , y2 ) (x4 , y4 ) (x5 , y5 ) (x1 , y1 ) (x6 , y6 ) x 0 Figure 3: Mis–specification of the model ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 31 Model Validation (Cont’d) • To decide whether we have a good fit, we need some statistical tools. • Two widely used measures include standard error and correlation coefficient. – Standard error measures the deviation of the actual costs from the predicted costs. – Correlation coeffcient measures whether there is in fact a linear relationship between the cost and the cost driver. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 32 Presence of Outliers y (x3 , y3 ) (x5 , y5 ) (x2 , y2 ) (x4 , y4 ) (x1 , y1 ) x 0 Figure 4: Outliers in the data ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 33 Presence of Outliers (Cont’d) • In general, it is hard to tell which data points are outliers. • To mitigate the effect of outliers, one can minimize other penalty functions. For example, consider minimizing the following: n X i=1 |yi − (b0 + b1xi)| = n X |ei|. i=1 This is known as the least absolute error approach. • This approach does not admit a closed–form solution. Nevertheless, the solution can be found using linear programming. ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 34 What’s Next? • Assignment: Read Chapter 3 of the course textbook. • Next: The time value of money (Chapter 4 of the course textbook) ECLT 5930/SEEM 5740 (2014–15 Second Term) January 22, 2015 35
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