Additional Lecture Notes

Transcription

1 Additional Lecture Notes Lecture 1: Introduction Overview Te purposes of tis lecture are (i) to take care of introductions; (ii) to transact course business; (iii) to consider te value of models; (iv) to conduct te world s fastest course in calculus; and (v) to introduce decision trees. Te last of tese is te main focus of te lecture. Notes 1. Introduction (a) Introduce course (b) Introduce self i. Want you to succeed see me or gsi. ii. Office is F675, pone , ermalin@aas.berkeley.edu (c) Introduce Simon Wakeman 2. Course business (a) Class reps. (b) Seating cart (c) Cold calling (d) Minimal use of PowerPoint 3. Value of Models (a) From Sylvie and Bruno Concluded by Lewis Carroll (1893): Tat s anoter ting we ve learned from your Nation, said Mein Herr, map-making. But we ve carried it muc furter tan you. Wat do you consider te largest map tat would be really useful? About six inces to te mile. Only six inces! exclaimed Mein Herr. We very soon got to six yards to te mile. Ten we tried a undred yards to te mile. And ten came te grandest idea of all! We actually made a map of te country, on te scale of a mile to te mile! Copyrigt c 2004 Benjamin E. Hermalin. All rigts reserved.

2 Have you used it muc? I enquired. It as never been spread out, yet, said Mein Herr: te farmers objected: tey said it would cover te wole country, and sut out te sunligt! So we now use te country itself, as its own map, and I assure you it does nearly as well. (b) Wat does economics offer: A framework from wic to better understand te information, objectives, and circumstances faced in business. A framework from wic to make better decisions. A systematic approac to solving business problems more efficiently and successfully. (c) Two types of analysis: Normative: Wat you sould do to accomplis your objectives. Positive: Wat oters are likely to do. (d) Models are te tools used A model is an abstraction from reality. Its goal is, as Serlock Holmes put it, to recognize out of a number of facts wic are incidental and wic vital. Tose tat are incidental are ignored, tose tat are vital are kept. Good models and bad models one goal of tis course is to elp you tell tem apart. (e) Te ideas of economists..., bot wen tey are rigt and wen tey are wrong, are more powerful tan is commonly understood. Indeed, te world is ruled by little else. Practical men [and women], wo believe temselves to be quite exempt from any intellectual influences, are usually te slaves of some defunct economist. Jon Maynard Keynes 4. Matematics World s Fastest Calculus Course (a) A wee bit of calculus fortunately calculus is really easy 2

3 value/unit v MF(x) F(x 1 ) x 1 x 1 + units (x) Figure 1: TerateofcangeinF ( ) at x 1 is MF(x 1 ). Calculus Calculus can be seen as means of determining areas under curves and determining te rate at wic tese areas are increasing (or decreasing). Consider Figure 1. It sows a function, called MF( ), wic is a flat (orizontal) line at eigt v. Tatis,forallx, MF(x) =v. Let te area under te MF( ) curve from 0 to x be denoted F (x). Using te formula for te area of a rectangle, eigt widt, we see tat F (x) =vx. One question we migt ask is te rate at wic F ( ) increases as we increase its argument. For instance, if we tougt of x as time (e.g., ours) and v as speed (e.g., km/our), ten vx would be distance traveled (e.g., kilometers). Recall tat vx = F (x). Recall too tat speed is te rate at wic distance increases. Hence, te rate at wic F ( ) increases is v, wic is to say te rate of cange of F ( ) atanyx is MF(x). Te M in MF( ) stands for marginal. Marginal means rate of cange. In matematics, MF( ) would be referred to as te derivative of F ( ). Observe tat we could ave arrived at tis conclusion anoter way. One way to calculate te rate is to consider te difference in F ( ) at two points normalized by te difference between te two points. Tat is, calculate F (x 1 + ) F (x 1 ) (x 1 + ) x 1. Te numerator is te difference in F ( ) evaluated at two points x 1 and x 1 + (see Figure 1). Te denominator is te difference between te points. We divide te numerator by te denominator because te rate needs to be expressed on a per-wole-unit basis (e.g., we talk about km./r., not km. per alf our). In matematics, we migt, terefore, write df (x)/dx = MF(x) orf (x) =MF(x). 3

4 value/unit MG(x) G(x 1 ) x 1 x 1 + units (x) Figure 2: TerateofcangeinG( ) at x 1 is MG(x 1 ). Notice we can rewrite te above fraction as F (x 1 + ) F (x 1 ) = v (x 1 + ) v (x 1 ) (x 1 + ) x 1 = v = v. Figure 2 sows a somewat more complex situation. Here we ve graped te function MG( ). We ve defined te function G( ) so tat G(x) is te area under MG( ) from0tox. We would like to know te rate of cange of G( ) at x 1. To calculate tat rate of cange, observe tat we would know ow to calculate tat rate of cange were MG( ) a flat function like MF( ) in Figure 1. Were it flat, ten te rate of cange would just be te eigt of te rectangle (i.e., MG(x 1 )). But observe, from Figure 2, tat for a small cange, from x 1 to x 1 +, were is small, te cange in G( ) is approximately equal to te cross-atced rectangle wose eigt is MG(x 1 ) and wose widt is. And we know te rate of cange of area in tis rectangle it s MG(x 1 ). As gets smaller, te rectangle becomes a better and better approximation of te cange in G( ). Hence, we can conclude tat MG(x 1 ) is te rate of cange in G( ) at x 1. Note tat te argument just given approximates te same algebra we de- 4

5 ployed wit respect to F ( ) andmf( ): Rate of cange in G( ) atx 1 G(x 1 + ) G(x 1 ) (x 1 + ) x 1 = G(x 1 + ) G(x 1 ). Tis approximation gets better te smaller is. Indeed, if we let srink all te way to zero, ten tis approximation will be exact. For example, if G(x) =ax 2 + bx + c, were a c are constants, ten we ave ( a(x1 + ) 2 + b(x 1 + )+c ) ( ax bx 1 + c ) Rate of cange in G( ) atx 1 = ax2 1 +2ax 1 + a 2 + bx 1 + b + c ax 2 1 bx 1 c = 2ax 1 + a 2 + b =2ax 1 + a + b. If we srink all te way to zero, tis becomes 2ax 1 + b. In oter words, if G(x) =ax 2 + bx + c, ten MG(x) =2ax 1 + b. Finally, observe tat we ve also sown tat te area under a function, say t( ), from 0 to x is equal to te function, call it T ( ), suc tat marginal T ( ) equals t( ). In oter words, if we reverse te process of calculating te marginal, ten we get te area under te curve. For example, suppose tat t(x) =2ax+b. Reversing te process of taking te marginal, we ave T (x) =ax 2 + bx + c because we just saw tat te marginal of tat function is 2ax + b. One problem, toug, is we don t know wat c is any constant c would work. Fortunately, a region of zero widt as zero area. So T (0) te area from 0 to 0 is 0. Terefore, 0=a0 2 + b0+c = c. Te constant c =0. To summarize, te marginal (derivative) of a function is te rate at wic te area te function represents is increasing or decreasing. Te area under a curve (te integral) is te function wose marginal is te curve itself. For tose of you familiar wit calculus notation, T (x) = x 0 t(z)dz, werez is a dummy of integration. 5

6 5. Decision Trees (a) Problem Solving i. Key ingredients By systematic Know te question. Focus on wat s relevant for answering te question. Focus on analyses relevant for answering te question. Always ask yourself, is wat I m doing getting me closer to answering te question? Use available frameworks, metods, formulæ. But don t expect tese metods to substitute for tinking tey re an aid to tinking. ii. J.M. Juran: Analyze te symptoms Teorize as to causes Test te teories Establis te cause(s) Test te remedy under operating conditions Establis controls to old te gain iii. Sigeru Mizuno: Seek out problem points List possible causes Identify te primary causes Devise measures to correct te problem Implement te corrective measures Ceck te results Institutionalize te new measures iv. Fisbone analysis v. See Sterling pump failure team (page 7). (b) Decision trees i. decision nodes ii. payoffs iii. backward induction iv. arrowing and intermediate values (c) Cance nodes i. expected value calculation ii. intermediate values 6

7 Sterling Pump Failure Team Tis note is drawn from Science, Specific Knowledge, and Total Quality Management by Karen H. Wruck and Micael C. Jensen (Journal of Accounting and Economics, Vol. 18 (1994), pp ). A complete copy is available electronically troug te Long Library. Pump failures in te ptalic anydride production process cost Sterling at least an average of $149,000 per year. In 1988, a team of engineers and manufacturing supervisors was formed to address te pump failure problem. Tis team immediately concluded tat eac of te 22 pump failures tat occurred over te last year was due to a special or one-of-a-kind cause. In mid-1989 a quality facilitator was assigned to te pump failure problem. A second team was formed. Tis new team consisted primarily of ourly personnel: tree operators from te production unit, four macinists, one mecanical engineer, and a tecnical services engineer. Te first action of te new team was to get pump failure data. Tey went all te way back to Tey found tat 22 failures per year was not far out of te ordinary. Te average failure rate since 1977 was 14.9 per year. Te first team ad been wrong in concluding tat te failures were due to special causes. Te failures were due to te establised metods for installing, starting and operating te pumps. Once te team recognized tat it was a systems problem, tey moved to identify sources of common cause variation. Tey identified inconsistencies in bot operating and maintenance procedures. Standardizing startup and repair procedures were gimmies tey took tese inconsistencies out of te system. But tey alone would not eliminate te failures. Based on te team members knowledge and te data tey ad collected, te team brainstormed and listed 57 teories tat potentially explained te ig pump failure rates. Te team reviewed and edited te brainstorming list, testing eac teory against te data. Troug tis process, tey reduced te brainstorming list to four potential causes of failure: i) te pump seal installation procedure; ii) pump suction pressure; iii) excessive pump vibration; and iv) missing or broken equipment upstream from te pump. Tey ten experimented to determine wic of tese causes were important determinants of pump failure. Testing te pump suction teory rejected it as a cause of failure. Te broken or missing equipment teory was eliminated troug inspection. Since testing te two remaining teories required making canges and observing te results of tose canges upstream from te pump over time, te team developed recommendations to address bot te pump seal installation procedures and excessive pump vibration. Te recommendations were implemented in early Since ten tere ave been no pump failures. 7