ࡱ> ikhY 9bjbjWW f==L]d<|0VRRR A0C0C0C0C0C0C0$21&3g0 U N g0wRVwww XlR:A0 A0ww?*Z,50 *,WQ+ |/73-305 Statistical Quality Control Computer Lab 2 Economic Design of  EMBED Equation.3 Control Chart using Excel Solver Summer 2002 (A) Exercise on Probability Distribution Binomial: find probability of c successes from n trials if P(success) = p P(c successes) = BINOMDIST(c,n,p,FALSE) Binomial: find probability of c or fewer successes from n trials if P(success) = p P(c or fewer successes) = BINOMDIST(c,n,p,TRUE) Poisson: find probability of c arrivals if mean number of arrivals = ( P(c arrivals) = POISSON(c,(,FALSE) Poisson: find probability of c or fewer arrivals if mean number of arrivals = ( P(c or fewer arrivals) = POISSON (c,(,FALSE) Normal: find area from normal distribution variate z Area on the left of z, ((z) = NORMSDIST(z) Normal: find normal distribution variate z from area, ((z) z = NORMSINV(Area on the left of z) Normal: find area from normal distribution variable x Area on the left of x = NORMDIST(x,(,(,TRUE) Normal: find normal distribution variable x from area x = NORMINV(Area on the left of x,(,() Question A Find probability(exactly 1 success) if p=0.4, n=3 probability(0 or 1 success) if p=0.4, n=3 probability(exactly 1 arrival) if (=2 probability(0 or 1 arrival) if (=2 area (under normal curve) on the left of z = 2.5 z corresponding to the area (under normal curve) on the left = 0.4 the area (under normal curve) on the left if x=600, (=300 and (=120 x if the area (under normal curve) on the left =0.4, (=300 and (=120  Exercise on finding cost per period for an  EMBED Equation.3  control chart Given input data  EMBED Equation.3 and trial value of control chart parameters  EMBED Equation.3   EMBED Equation.3 =2*NORMSDIST(-k)  EMBED Equation.3  = NORMSDIST(k-(*SQRT(n))-NORMSDIST(-k -(*SQRT(n))  EMBED Equation.3 ,  EMBED Equation.3 , EMBED Equation.3  Sampling cost per cycle = EMBED Equation.3 , Search cost per cycle = EMBED Equation.3 , Operating cost per cycle =  EMBED Equation.3  Total cost per period =  EMBED Equation.3  Question B Using the data shown below, find the total cost per period if an  EMBED Equation.3 control chart has  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   For each trial value of  EMBED Equation.3  in Question 1, find an optimal  EMBED Equation.3 that minimizes total cost per period. Find the optimal EMBED Equation.3 .  Excel Solver Add-in Solver, if needed: Click on the Tools menu. If the Solver is not one of the items of the Tools menu, choose the Add-Ins. Click on the box adjacent to Solver Add-In (a check mark will appear) and click OK. Define objective: From the Tools menu, choose Solver. You get the Solver Parameters window. Click on the little arrow adjacent to the box Set Target Cell. Click on the Cell containing total cost and click again on the arrow. Click on the circle adjacent to Min. Decision variable: The cell address for the decision variable is entered in the box below By Changing Cells:. The decision variable is the cell containing k. Non-negativity: From the Solver Parameters window click on Options. You get the Solver Options window. Click on a box adjacent to Assume Non-Negativity. Click on OK. Solve: Finally, from the Solver Parameters window click on Solve. A Solver Results window pops up. Click on OK. Supplement: Economic Design of  EMBED Equation.3 Control Chart using Excel Solver Input a1cost of sampling each unita2Expected cost of each searcha3per period cost of operating in an out-of-control state(Probability that the process shifts from an in-control state to an out-of-control state in one period(Average number of standard deviations by which the mean shifts whenever the process is out-of-control. In other words, the mean shifts from ( to ((( whenever the process is out-of-control Output (Decision Variables) Determine n and k, where n is the sample size and k provides the upper and lower control limits as Objective Minimize the average cost per period. Relationships Type I error, ( =  EMBED Equation.3  Type II error, ( =  EMBED Equation.3  ((z) is the cumulative standard normal distribution function and may be obtained from the Excel function NORMSDIST:  EMBED Equation.3 = NORMSDIST(-k),  EMBED Equation.3 =NORMSDIST EMBED Equation.3 ,  EMBED Equation.3 =NORMSDIST EMBED Equation.3  For a given pair of n and k the average per period cost is Setting up the Spreadsheet A spreadsheet is set up to compute the average per period cost from a given pair of n and k Input: Known input values are entered in each of the cells below: D4: a1 D5: a2 D6: a3 D7: ( D8: ( Output (Decision Variables): For given values of n, the Excel solver will output optimal values of k in cells B10, B11, B12, For example, for n=1, the optimal value of k is obtained from B10. Relationships Formulae are entered for relationships explained below: Type I error, ( C10 =2*NORMSDIST(-B10) (copied to C11, C12, ) Type II error, ( D10 = NORMSDIST(B10-$D$8*SQRT(A10)) -NORMSDIST(-B10-$D$8*SQRT(A10)) (copied to D11, D12,) The average per period cost E10 =$D$4*A10+($D$5*(1-D10)*($D$7+C10*(1-$D$7))+$D$6*$D$7)/ (1-D10*(1-$D$7)) (copied to E11, E12, ) In the following, I explain the use of Excel solver to get a minimum cost value of k for n=1. Similarly, other minimum cost values of k for other values of n may be obtained. Comparing the costs resulting from all the pairs of n and k, we can get the best pair of n and k that yields a minimum average cost per period. The Excel Solver Add-in Solver, if needed: Click on the Tools menu. If the Solver is not one of the items of the Tools menu, choose the Add-Ins. Click on the box adjacent to Solver Add-In (a check mark will appear) and click OK. Define objective: From the Tools menu, choose Solver. The Solver Parameters Window pops up. Click on the arrow adjacent to the box Set Target Cell. Click on the Cell E10, that stores the average cost per peirod and click again on the arrow. Click on the circle adjacent to Min. Decision variable: The cell addresses for the decision variables are entered in the box below By Changing Cells:. You may again use the little arrow to enter the decision variable $B$10. Non-negativity: From the Solver Parameters window click on Options. The Solver Options window pops up. Click on a box adjacent to Assume Non-Negativity. Click on OK.  Solve: Finally, from the Solver Parameters window click on Solve. A Solver Results window pops up. Click on OK.  The Solution: See the changes to your spreadsheet. The cell for k (i.e., B10) is expected to contain the optimal solution and its cost in the cell for the average cost per period.  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