11/16/12. Introduction to Quantitative Analysis. Developing a Model. Acquiring Input Data. Course Overview. The Quantitative Analysis Approach

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1 Course Overview Itroductio to uatitative Aalysis Bus--M Lecture 4 Course review To accompay uatitative Aalysis for Maagemet, Eleveth Editio, by Reder, Stair, ad Haa Lecture TITLE Break Eve Aalysis (Chapter ) Bayes Theorem Regressio Aalysis (Chapter, ot Eglish) 3, 4, 5, 6 Decisio Aalysis (Chapter 3) 7, 8 Forecastig (Chapter 5) 9 Ivetory Cotrol (Chapter 6) 0, Liear Programmig (Chapter 7) Project Maagemet (Chapter 3) 3 Waitig Lies (Chapter 4) - The uatitative Aalysis Approach Defiig the Problem Developig a Model Acquirig Iput Data Developig a Solutio Testig the Solutio Developig a Model uatitative aalysis models are realistic, solvable, ad uderstadable mathematical represetatios of a situatio. $ Sales Y b 0 + b X $ Advertisig There are differet types of models: Figure. Aalyzig the Results Implemetig the Results -3 Scale models Schematic models -4 Acquirig Iput Data How To Develop a uatitative Aalysis Model Iput data must be accurate Garbage I Process Garbage Out A mathematical model of profit: Profit Reveue Expeses Data may come from a variety of sources such as compay reports, compay documets, iterviews, o-site direct measuremet, or statistical samplig. -5-6

2 How To Develop a uatitative Aalysis Model Expeses ca be represeted as the sum of fixed ad variable costs. Variable costs are the product of uit costs times the umber of uits. Profit Reveue (Fixed cost + Variable cost) Profit (Sellig price per uit)(umber of uits sold) [Fixed cost + (Variable costs per uit)(number of uits sold)] Profit sx [f + vx] Profit sx f vx s sellig price per uit v variable cost per uit f fixed cost X umber of uits sold -7 How To Develop a uatitative Aalysis Model Expeses ca be represeted as the sum of fixed ad variable costs ad variable The parameters costs are the of this product model of uit costs times the umber are f, v, of ad uits s as these are the Profit Reveue (Fixed iputs cost iheret + Variable i the cost) model Profit (Sellig price The per decisio uit)(umber variable of of uits sold) [Fixed iterest cost + is (Variable X costs per uit)(number of uits sold)] Profit sx [f + vx] Profit sx f vx s sellig price per uit v variable cost per uit f fixed cost X umber of uits sold -8 Break Eve Poit Compaies are ofte iterested i the break-eve poit (BEP). The BEP is the umber of uits sold that will result i $0 profit. 0 sx f vx, or 0 (s v)x f Solvig for X, we have f (s v)x BEP X f s v Fixed cost (Sellig price per uit) (Variable cost per uit) -9 Types of Probability Determiig!objec've!probability):)! Rela.ve)frequecy)! Typically)based)o)historical)data) P (evet) Number of occurreces of the evet Total umber of trials or outcomes! Classical or logical method! Logically determie probabilities without trials P (head) Number of ways of gettig a head Number of possible outcomes (head or tail) -0 Types of Probability Subjec've!probability)is)based)o)the) experiece)ad)judgmet)of)the)perso) makig)the)es.mate.)! Opiio)polls)! Judgmet)of)experts)! Delphi)method) Mutually Exclusive Evets Evets)are)said)to)be)mutually!exclusive)if) oly)oe)of)the)evets)ca)occur)o)ay)oe) trial.)! Tossig a coi will result i either a head or a tail.! Rollig a die will result i oly oe of six possible outcomes. - -

3 Collectively Exhaustive Evets Evets)are)said)to)be)collec.vely)exhaus.ve)if) the)list)of)outcomes)icludes)every)possible) outcome.)! Both)heads)ad)) tails)as)possible)) outcomes)of)) coi)flips.)! All)six)possible)) outcomes)) of)the)roll)) of)a)die.) OUTCOME OF ROLL PROBABILITY / 6 / 6 3 / 6 4 / 6 5 / 6 6 / 6 Total -3 Three Types of Probabilities! Margial)(or)simple))probability)is)just)the)probability)of)a) sigle)evet)occurrig.) P)(A)! Joit probability is the probability of two or more evets occurrig ad is equal to the product of their margial probabilities for idepedet evets. P (AB) P (A) x P (B)! Coditioal probability is the probability of evet B give that evet A has occurred. P (B A) P (B)! Or the probability of evet A give that evet B has occurred P (A B) P (A) -4 Revisig Probabilities with Bayes Theorem Bayes theorem is used to icorporate additioal iformatio ad help create posterior probabilities. Prior Probabilities Geeral Form of Bayes Theorem We ca compute revised probabilities more directly by usig: B A) A) A B) B A) A) + B A! ) A! ) New Iformatio Bayes Process Posterior Probabilities A! the complemet of the evet A; for example, if A is the evet fair die, the A! is loaded die. Figure Radom Variables A radom variable assigs a real umber to every possible outcome or evet i a experimet. X umber of refrigerators sold durig the day Discrete!radom!variables)ca)assume)oly)a)fiite) or)limited)set)of)values.)) Co'uous!radom!variables)ca)assume)ay)oe) of)a)ifiite)set)of)values.) Probability Distributio of a Discrete Radom Variable For discrete radom variables a probability is assiged to each evet. The studets i Pat Shao s statistics class have just completed a quiz of five algebra problems. The distributio of correct scores is give i the followig table:

4 RANDOM VARIABLE (X Score) Probability Distributio of a Discrete Radom Variable NUMBER RESPONDING PROBABILITY P (X) / / / / /00 Total /00 Table.6 The Probability Distributio follows all three rules:. Evets are mutually exclusive ad collectively exhaustive.. Idividual probability values are betwee 0 ad. 3. Total of all probability values equals. -9 Expected Value of a Discrete Probability Distributio The expected value is a measure of the cetral tedecy of the distributio ad is a weighted average of the values of the radom variable. P E( X ) X i X i ) i X X ) + X X ) X X ) X i radom variable s possible values ( X i ) probability of each of the radom variable s possible values summatio sig idicatig we are addig all i possible values ) expected value or mea of the radom sample -0 E(X Variace of a Discrete Probability Distributio For a discrete probability distributio the variace ca be computed by E(X ) [ X i E( X σ Variace [ X E( X )] X i i i X i X i radom variable s possible values expected value of the radom variable )] differece betwee each value of the radom variable ad the expected mea ) probability of each possible value of the radom variable ) Variace of a Discrete Probability Distributio A related measure of dispersio is the stadard deviatio. σ Variace σ square root stadard deviatio σ - - Probability Distributio of a Cotiuous Radom Variable Probability Distributio of a Cotiuous Radom Variable Sice radom variables ca take o a ifiite umber of values, the fudametal rules for cotiuous radom variables must be modified.! The sum of the probability values must still equal.! The probability of each idividual value of the radom variable occurrig must equal 0 or the sum would be ifiitely large. The probability distributio is defied by a cotiuous mathematical fuctio called the probability desity fuctio or just the probability fuctio.! This is represeted by f (X). -3 Figure.6 Probability Weight (grams) -4 4

5 The Normal Distributio The)ormal!distribu'o)is)the)oe)of)the)most) popular)ad)useful)co.uous)probability) distribu.os.)))! The)formula)for)the)probability)desity)fuc.o)is) rather)complex:) ( x µ ) σ f ( X ) e σ π! The ormal distributio is specified completely whe we kow the mea, µ, ad the stadard deviatio, σ. -5 The Normal Distributio " The ormal distributio is symmetrical, with the midpoit represetig the mea. " Shiftig the mea does ot chage the shape of the distributio. " Values o the X axis are measured i the umber of stadard deviatios away from the mea. " As the stadard deviatio becomes larger, the curve flattes. " As the stadard deviatio becomes smaller, the curve becomes steeper. -6 The Normal Distributio The Normal Distributio 40 µ Smaller µ, same σ Same µ, smaller σ µ Larger µ, same σ Same µ, larger σ Figure µ 60 Figure.9 µ -7-8 Usig the Stadard Normal Table Usig the Stadard Normal Table Step Covert the ormal distributio ito a stadard ormal distributio.! A stadard ormal distributio has a mea of 0 ad a stadard deviatio of! The ew stadard radom variable is Z Z X µ σ X value of the radom variable we wat to measure µ mea of the distributio σ stadard deviatio of the distributio Z umber of stadard deviatios from X to the mea, µ -9 For example, µ 00, σ 5, ad we wat to fid the probability that X is less tha 30. X µ Z σ 5 30 std dev 5 X < 30) Figure.0 µ 00 σ 5 X I Z X µ σ -30 5

6 Usig the Stadard Normal Table Step Look up the probability from a table of ormal curve areas.! Use Appedix A or Table.9 (portio below).! The colum o the left has Z values.! The row at the top has secod decimal places for the Z values. AREA UNDER THE NORMAL CURVE Z Table.9 (partial) X < 30) Z <.00) The Expoetial Distributio! The)expoe'al!distribu'o)(also)called)the) ega've!expoe'al!distribu'o))is)a)co.uous) distribu.o)oie)used)i)queuig)models)to) describe)the).me)required)to)service)a)customer.)) Its)probability)fuc.o)is)give)by:) x f ( X ) µ e µ X radom variable (service times) µ average umber of uits the service facility ca hadle i a specific period of time e.78 (the base of atural logarithms) -3 The Expoetial Distributio The Six Steps i Decisio Makig Expected value Average service time µ Variace µ f(x) X. Clearly defie the problem at had.. List the possible alteratives. 3. Idetify the possible outcomes or states of ature. 4. List the payoff (typically profit) of each combiatio of alteratives ad outcomes. 5. Select oe of the mathematical decisio theory models. 6. Apply the model ad make your decisio. Figure Types of Decisio-Makig Eviromets Decisio Makig Uder Ucertaity Type): )Decisio)makig)uder)certaity)! The)decisio)maker)kows!with!certaity)the) cosequeces)of)every)altera.ve)or)decisio) choice.) Type): )Decisio)makig)uder)ucertaity)! The)decisio)maker)does!ot!kow)the) probabili.es)of)the)various)outcomes.) Type)3: )Decisio)makig)uder)risk)! The)decisio)maker)kows!the!probabili'es)of) the)various)outcomes.) There are several criteria for makig decisios uder ucertaity:. Maximax (optimistic). Maximi (pessimistic) 3. Criterio of realism (Hurwicz) 4. Equally likely (Laplace) 5. Miimax regret

7 Maximax Used to fid the alterative that maximizes the maximum payoff.! Locate the maximum payoff for each alterative.! Select the alterative with the maximum umber. ALTERNATIVE Costruct a large plat Costruct a small plat STATE OF NATURE FAVORABLE MARKET ($) 3-37 UNFAVORABLE MARKET ($) MAXIMUM IN A ROW ($) 00,000 80,000 00,000 Maximax 00,000 0,000 00,000 Do othig Table 3. Criterio of Realism (Hurwicz) This is a weighted average compromise betwee optimism ad pessimism.! Select a coefficiet of realism α, with 0 α.! A value of is perfectly optimistic, while a value of 0 is perfectly pessimistic.! Compute the weighted averages for each alterative.! Select the alterative with the highest value. Weighted average α(maximum i row) + ( α)(miimum i row) 3-38 Miimax Regret Decisio Makig Uder Risk Based)o)opportuity!loss)or)regret,)this)is)the) differece)betwee)the)op.mal)profit)ad)actual) payoff)for)a)decisio.)! Create)a)opportuity)loss)table)by)determiig)the) opportuity)loss)from)ot)choosig)the)best)altera.ve.)! Opportuity)loss)is)calculated)by)subtrac.g)each)payoff) i)the)colum)from)the)best)payoff)i)the)colum.)! Fid)the)maximum)opportuity)loss)for)each)altera.ve) ad)pick)the)altera.ve)with)the)miimum)umber.) 3-39! This)is)decisio)makig)whe)there)are)several)possible) states)of)ature,)ad)the)probabili.es)associated)with) each)possible)state)are)kow.)! The)most)popular)method)is)to)choose)the)altera.ve) with)the)highest)expected!moetary!value!(emv).!! This)is)very)similar)to)the)expected!value!calculated)i)the)last) chapter.) EMV (alterative i) (payoff of first state of ature) x (probability of first state of ature) + (payoff of secod state of ature) x (probability of secod state of ature) + + (payoff of last state of ature) x (probability of last state of ature) 3-40 Expected Value of Perfect Iformatio (EVPI) EVPI places a upper boud o what you should pay for additioal iformatio. EVPI EVwPI Maximum EMV EVwPI is the log ru average retur if we have perfect iformatio before a decisio is made. EVwPI (best payoff for first state of ature) x (probability of first state of ature) + (best payoff for secod state of ature) x (probability of secod state of ature) + + (best payoff for last state of ature) x (probability of last state of ature) 3-4 Expected Opportuity Loss! Expected!opportuity!loss)(EOL))is)the)cost)of)ot) pickig)the)best)solu.o.)! First)costruct)a)opportuity)loss)table.)! For)each)altera.ve,)mul.ply)the)opportuity)loss) by)the)probability)of)that)loss)for)each)possible) outcome)ad)add)these)together.)! Miimum)EOL)will)always)result)i)the)same) decisio)as)maximum)emv.! Miimum)EOL)will)always)equal)EVPI

8 Sesitivity Aalysis! Sesitivity aalysis examies how the decisio might chage with differet iput data.! For the Thompso Lumber example: P probability of a favorable market ( P) probability of a ufavorable market Five Steps of Decisio Tree Aalysis. Defie the problem.. Structure or draw the decisio tree. 3. Assig probabilities to the states of ature. 4. Estimate payoffs for each possible combiatio of alteratives ad states of ature. 5. Solve the problem by computig expected moetary values (EMVs) for each state of ature ode Structure of Decisio Trees Trees start from left to right. Trees represet decisios ad outcomes i sequetial order. Squares represet decisio odes. Circles represet states of ature odes. Lies or braches coect the decisios odes ad the states of ature. Expected Value of Sample Iformatio Expected value with sample EVSI iformatio, assumig o cost to gather it Expected value of best decisio without sample iformatio (EV with sample iformatio + cost) (EV without sample iformatio) Calculatig Revised Probabilities! Recall Bayes theorem: B A) A) A B) B A) A) + B A! ) A! ) A, B ay two evets A! complemet of A Utility Theory! Moetary)value)is)ot)always)a)true)idicator) of)the)overall)value)of)the)result)of)a)decisio.)! The)overall)value)of)a)decisio)is)called)u'lity.!! Ecoomists)assume)that)ra.oal)people)make) decisios)to)maximize)their)u.lity.) For this example, A will represet a favorable market ad B will represet a positive survey

9 Forecastig Models ualitative Models ualitative Models Delphi Methods Jury of Executive Opiio Sales Force Composite Cosumer Market Survey Forecastig Techiques Time-Series Methods Movig Average Expoetial Smoothig Tred Projectios Decompositio Causal Methods Regressio Aalysis Multiple Regressio Figure 5.! ualita've!models)icorporate)judgmetal)or) subjec.ve)factors.)! These)are)useful)whe)subjec.ve)factors)are) thought)to)be)importat)or)whe)accurate) qua.ta.ve)data)is)difficult)to)obtai.)! Commo)qualita.ve)techiques)are:)! Delphi)method.)! Jury)of)execu.ve)opiio.)! Sales)force)composite.)! Cosumer)market)surveys.) ualitative Models Time-Series Models Delphi Method This is a iterative group process (possibly geographically dispersed) respodets provide iput to decisio makers. Jury of Executive Opiio This method collects opiios of a small group of high-level maagers, possibly usig statistical models for aalysis. Sales Force Composite This allows idividual salespersos estimate the sales i their regio ad the data is compiled at a district or atioal level. Cosumer Market Survey Iput is solicited from customers or potetial customers regardig their purchasig plas. 5-5 Time-series models attempt to predict the future based o the past. Commo time-series models are: Movig average. Expoetial smoothig. Tred projectios. Decompositio. Regressio aalysis is used i tred projectios ad oe type of decompositio model. 5-5 Causal Models Measures of Forecast Accuracy! Causal!models)use)variables)or)factors)that) might)ifluece)the)qua.ty)beig)forecasted.)! The)objec.ve)is)to)build)a)model)with)the)best) sta.s.cal)rela.oship)betwee)the)variable) beig)forecast)ad)the)idepedet)variables.)! Regressio)aalysis)is)the)most)commo) techique)used)i)causal)modelig.) We compare forecasted values with actual values to see how well oe model works or to compare models. Forecast error Actual value Forecast value! Oe measure of accuracy is the mea absolute deviatio (MAD): MAD forecast error

10 Measures of Forecast Accuracy There are other popular measures of forecast accuracy. The mea squared error: MSE (error)! The mea absolute percet error: error actual MAPE 00%! Ad bias is the average error. Time-Series Forecastig Models A time series is a sequece of evely spaced evets. Time-series forecasts predict the future based solely o the past values of the variable, ad other variables are igored Compoets of a Time-Series A time series typically has four compoets:. Tred (T) is the gradual upward or dowward movemet of the data over time.. Seasoality (S) is a patter of demad fluctuatios above or below the tred lie that repeats at regular itervals. 3. Cycles (C) are patters i aual data that occur every several years. 4. Radom variatios (R) are blips i the data caused by chace or uusual situatios, ad follow o discerible patter. Movig Averages! Movig averages ca be used whe demad is relatively steady over time.! The ext forecast is the average of the most recet data values from the time series.! This methods teds to smooth out shortterm irregularities i the data series. Sum of demadsiprevious periods Movig average forecast ! Mathematically: Movig Averages F t+ Yt + Yt Yt + Where: F forecast for time period t + t+ Y t actual value i time period t umber of periods to average Weighted Movig Averages! Weighted movig averages use weights to put more emphasis o previous periods.! This is ofte used whe a tred or other patter is emergig. F t +! Mathematically: ( Weight iperiod i)( Actual value i period) F w Y + w Y ( Weights) t t t + t+ w + w w w Y w i weight for the i th observatio 0

11 Expoetial Smoothig Expoetial Smoothig! Expoe'al!smoothig)is)a)type)of)movig)average) that)is)easy)to)use)ad)requires)liwle)record)keepig) of)data.) New forecast Last period s forecast + α(last period s actual demad Last period s forecast) Here α is a weight (or smoothig costat) i which 0 α. Mathematically: Where: F t+ Ft + α( t t Y F ) F t+ ew forecast (for time period t + ) F t pervious forecast (for time period t) α smoothig costat (0 α ) Y t pervious period s actual demad The idea is simple the ew estimate is the old estimate plus some fractio of the error i the last period Expoetial Smoothig with Tred Adjustmet Like all averagig techiques, expoetial smoothig does ot respod to treds. A more complex model ca be used that adjusts for treds. The basic approach is to develop a expoetial smoothig forecast, ad the adjust it for the tred. Forecast icludig tred (FIT t+ ) Smoothed forecast (F t+ ) + Smoothed Tred (T t+ ) Expoetial Smoothig with Tred Adjustmet The equatio for the tred correctio uses a ew smoothig costat β. T t must be give or estimated. T t+ is computed by: T t+ ( β) Tt + β( Ft + t FIT ) T t smoothed tred for time period t F t smoothed forecast for time period t FIT t forecast icludig tred for time period t α smoothig costat for forecasts β smoothig costat for tred Selectig a Smoothig Costat Tred Projectios! As with expoetial smoothig, a high value of β makes the forecast more resposive to chages i tred.! A low value of β gives less weight to the recet tred ad teds to smooth out the tred.! Values are geerally selected usig a trial-aderror approach based o the value of the MAD for differet values of β.! Tred projectio fits a tred lie to a series of historical data poits.! The lie is projected ito the future for medium- to log-rage forecasts.! Several tred equatios ca be developed based o expoetial or quadratic models.! The simplest is a liear model developed usig regressio aalysis

12 Tred Projectio The mathematical form is Yˆ b + 0 bx Where Ŷ predicted value b 0 itercept b slope of the lie X time period (i.e., X,, 3,, ) Seasoal Variatios with Tred! Whe)both)tred)ad)seasoal)compoets)are)preset,) the)forecas.g)task)is)more)complex.)! Seasoal)idices)should)be)computed)usig)a)cetered! movig!average)(cma))approach.)! There)are)four)steps)i)compu.g)CMAs:). Compute)the)CMA)for)each)observa.o)() possible).). Compute)the)seasoal)ra.o))Observa.o/CMA)for) that)observa.o.) 3. Average)seasoal)ra.os)to)get)seasoal)idices.) 4. If)seasoal)idices)do)ot)add)to)the)umber)of) seasos,)mul.ply)each)idex)by)(number)of)seasos)/ (Sum)of)idices).) The Decompositio Method of Forecastig with Tred ad Seasoal Compoets! Decomposi'o)is)the)process)of)isola.g)liear)tred)ad) seasoal)factors)to)develop)more)accurate)forecasts.)! There)are)five)steps)to)decomposi.o:). Compute)seasoal)idices)usig)CMAs.). Deseasoalize)the)data)by)dividig)each)umber)by)its) seasoal)idex.) 3. Fid)the)equa.o)of)a)tred)lie)usig)the) deseasoalized)data.) 4. Forecast)for)future)periods)usig)the)tred)lie.) 5. Mul.ply)the)tred)lie)forecast)by)the)appropriate) seasoal)idex.) Usig Regressio with Tred ad Seasoal Compoets! Mul'ple!regressio)ca)be)used)to)forecast)both)tred) ad)seasoal)compoets)i)a).me)series.)! Oe)idepedet)variable)is).me.)! Dummy)idepedet)variables)are)used)to)represet)the)seasos.)! The)model)is)a)addi.ve)decomposi.o)model:) Y ˆ a + b + X + b X + b3 X 3 b4 X 4 X time period X if quarter, 0 otherwise X 3 if quarter 3, 0 otherwise X 4 if quarter 4, 0 otherwise Moitorig ad Cotrollig Forecasts! Trackig sigals ca be used to moitor the performace of a forecast.! A trackig sigal is computed as the ruig sum of the forecast errors (RSFE), ad is computed usig the followig equatio: RSFE Trackig sigal MAD MAD 5-7 forecast error Adaptive Smoothig! Adap've!smoothig)is)the)computer)moitorig)of) trackig)sigals)ad)self^adjustmet)if)a)limit)is) tripped.)! I)expoe.al)smoothig,)the)values)of)α)ad)β)are) adjusted)whe)the)computer)detects)a)excessive) amout)of)varia.o.! 5-7

13 Importace of Ivetory Cotrol Importace of Ivetory Cotrol Five uses of ivetory: The decouplig fuctio Storig resources Irregular supply ad demad uatity discouts Avoidig stockouts ad shortages Decouple maufacturig processes. Ivetory is used as a buffer betwee stages i a maufacturig process. This reduces delays ad improves efficiecy Storig resources. Seasoal products may be stored to satisfy offseaso demad. Materials ca be stored as raw materials, work-iprocess, or fiished goods. Labor ca be stored as a compoet of partially completed subassemblies. Compesate for irregular supply ad demad. Demad ad supply may ot be costat over time. Ivetory ca be used to buffer the variability Ecoomic Order uatity! The)ecoomic!order!qua'ty)(EO))model)is)oe) of)the)oldest)ad)most)commoly)kow) ivetory)cotrol)techiques.)! It)is)easy)to)use)but)has)a)umber)of)importat) assump.os.)! Objec.ve)is)to)miimize)total)cost)of)ivetory.) 6-75 Ecoomic Order uatity Assump.os:). Demad)is)kow)ad)costat.). Lead).me)is)kow)ad)costat.) 3. Receipt)of)ivetory)is)istataeous.) 4. Purchase)cost)per)uit)is)costat)throughout)the) year.) 5. The)oly)variable)costs)are)the)cost)of)placig)a) order,)orderig!cost,)ad)the)cost)of)holdig)or) storig)ivetory)over).me,)holdig)or)carryig!cost,) ad)these)are)costat)throughout)the)year.) 6. Orders)are)placed)so)that)stockouts)or)shortages)are) avoided)completely.) 6-76 Ivetory Costs i the EO Situatio Mathematical equatios ca be developed usig: umber of pieces to order EO * optimal umber of pieces to order D aual demad i uits for the ivetory item C o orderig cost of each order C h holdig or carryig cost per uit per year Ivetory Costs i the EO Situatio Mathematical equatios ca be developed usig: umber of pieces to order EO * optimal umber of pieces to order D aual demad i uits for the ivetory item C o orderig cost of each order C h holdig or carryig cost per uit per year Number of Aual orderig cost orders placed per year D C o Orderig cost per order Average Aual holdig cost ivetory C h Carryig cost per uit per year

14 Ecoomic Order uatity (EO) Model Purchase Cost of Ivetory Items Summary of equatios: Aual orderig cost Aual holdig cost EO * 6-79 DC C h D o C o C h Total ivetory cost ca be writte to iclude the cost of purchased items. Give the EO assumptios, the aual purchase cost is costat at D C o matter the order policy, C is the purchase cost per uit. D is the aual demad i uits. At times it may be useful to kow the average dollar level of ivetory: (C) Average dollar level 6-80 Purchase Cost of Ivetory Items Ivetory carryig cost is ofte expressed as a aual percetage of the uit cost or price of the ivetory. This requires a ew variable. I Aual ivetory holdig charge as a percetage of uit price or cost! The cost of storig ivetory for oe year is the thus, C h IC * DC IC 6-8 o Sesitivity Aalysis with the EO Model! The)EO)model)assumes)all)values)are)kow)ad)fixed) over).me.)! Geerally,)however,)some)values)are)es.mated)or)may) chage.)! Determiig)the)effects)of)these)chages)is)called) sesi'vity!aalysis.)! Because)of)the)square)root)i)the)formula,)chages)i)the) iputs)result)i)rela.vely)small)chages)i)the)order) qua.ty.! DCo EO C 6-8 h Reorder Poit: Determiig Whe To Order! Oce)the)order)qua.ty)is)determied,)the)ext) decisio)is)whe!to!order.!! The).me)betwee)placig)a)order)ad)its)receipt) is)called)the)lead!'me)(l))or)delivery!'me.)! Whe)to)order)is)geerally)expressed)as)a)reorder! poit)(rop).) EO Without The Istataeous Receipt Assumptio Whe ivetory accumulates over time, the istataeous receipt assumptio does ot apply. Daily demad rate must be take ito accout. The revised model is ofte called the productio ru model. Demad ROP per day Lead time for a ew order i days d L Figure 6.5 4

15 Aual Carryig Cost for Productio Ru Model! I)produc.o)rus,)setup!cost)replaces)orderig)cost.)! The)model)uses)the)followig)variables:) umber of pieces per order, or productio ru C s setup cost C h holdig or carryig cost per uit per year p daily productio rate d daily demad rate t legth of productio ru i days 6-85 Aual Carryig Cost for Productio Ru Model Maximum ivetory level (Total produced durig the productio ru) (Total used durig the productio ru) (Daily productio rate)(number of days productio) (Daily demad)(number of days productio) (pt) (dt) sice we kow Maximum ivetory level Total produced pt t p & d # pt dt p d $! p p % p " 6-86 Aual Carryig Cost for Productio Ru Model Sice the average ivetory is oe-half the maximum: & d # Average ivetory $! % p " Aual Setup Cost for Productio Ru Model Setup cost replaces orderig cost whe a product is produced over time. Aual setup cost D C s ad & d # Aualholdig cost $! % p " C h replaces Aual orderig cost D C o Determiig the Optimal Productio uatity By settig setup costs equal to holdig costs, we ca solve for the optimal order quatity Aual holdig cost Aual setup cost Solvig for, we get & d # D $! C % p " * 6-89 h C s DC s & d # Ch$! % p " Productio Ru Model Summary of equatios & d # Aualholdig cost $! % p " Aual setup cost Optimal productio quatity 6-90 D * C s C h DCs & d # Ch$! % p " 5