1. Blueprint Problem II: Method of Least Squares, Multiple Regression, Learning Curve

   Applying the Concepts

   McCourt Company had the following 10 months of data on materials handling cost and number of moves:

   Month	Materials Handling Cost	Number of Moves
   January	$5,600	475
   February	3,090	125
   March	2,780	175
   April	8,000	600
   May	1,990	200
   June	5,300	300
   July	4,300	250
   August	6,300	400
   September	2,000	100
   October	6,240	425

   Required:

   1. Using the above data on material handling, using regression software such as regression function within Microsoft Excel, and fill in the missing data in the table below (round answers to two decimal places).

   McCourt Company SUMMARY OUTPUT
   Regression Statistics
   Multiple R	0.94
   R Square
   Adjusted R Square
   Standard Error	746.14
   Observations	10

   ANOVA
   	df	SS	MS	F	Significance F
   Regression	1	33,414,382.28	33,414,382	60.02	.000055
   Residual	8	4,453,817.72	556,727.20
   Total	9	37,868,200.00

   	Coefficients	Standard Error	t Stat	P-value
   Intercept		516.99	1.93	0.09
   X Variable 1			7.75	0.000055

   2. McCourt estimates that November will now have 350 moves. Using this new value, calculate the following expected total, fixed and variable costs for the month of November (round all costs to the nearest dollar):

   Expected fixed cost	$
   Expected variable cost
   Total cost	$

   3. What is the percentage of material handing cost explained by moves:

   percent

   4. Using a P-value of 0.10 for significance, the following is true:

   5. Using a P-value of 0.05 for significance, the following is true:

   Multiple Regression

   In the McCourt Company example, percent of the variability in materials handling cost was explained by changes in the number of moves. As a result, McCourt may want to search for additional explanatory variables. For example, the weight of the items moved might be usefulparticularly if forklifts and other heavy machinery are needed for moving parts and products from one location to another.

   In the case of two explanatory variables, the linear equation is expanded to include the additional variable:

   Y = F + V₁X₁ + V₂X₂

   where

   X₁ = Number of moves

   X₂ = Number of pounds moved

   With three variables (Y, X₁, X₂), a minimum of three points is needed to compute the parameters F, V₁, and V₂. Seeing the points becomes difficult because they must be plotted in three dimensions. Using the scatterplot method or the high-low method is not practical.

   However, the extension of the method of least squares is straightforward. It is relatively simple to develop a set of equations that provides values for F, V₁, and V₂ that yields the best-fitting equation. Whenever least squares is used to fit an equation involving two or more independent variables, the method is called multiple regression. The computations required for multiple regressions are far more complex than in simple (one independent variable) regression and any practical application of multiple regression requires use of regression software. The reliability measures are basically the same for multiple regressions as they were for a regression involving a single variable.

   Applying the Concepts

   The staff of controller of McCourt Company added the variable "pounds moved" to the ten-month data set:

   Month	Materials Handling Cost	Number of Moves	Pounds Moved
   January	$5,600	475	12,000
   February	3,090	125	15,000
   March	2,780	175	7,800
   April	8,000	600	29,000
   May	1,990	200	600
   June	5,300	300	23,000
   July	4,300	250	17,000
   August	6,300	400	25,000
   September	2,000	100	6,000
   October	6,240	425	22,400

   Required:

   1. Using the data on material handling, use regression software such as Microsoft Excel to complete the missing data in the table below (round regression parameters to the nearest cent and other answers to three decimal places):

   McCourt Company SUMMARY OUTPUT
   Regression Statistics
   Multiple R	0.999
   R Square
   Adjusted R Square
   Standard Error	119.600
   Observations	10

   ANOVA
   	df	SS	MS	F	Significance F
   Regression	2	37,768,070.16	18,884,035	1,320.168	9.50629E-10
   Residual	7	100,129.84	14,304.26
   Total	9	37,868,200.00

   	Coefficients	Standard Error	t Stat	P-value
   Intercept		86.068	6.863	0.000239
   X Variable 1		0.340	0.340	9.81E-08
   X Variable 2		0.006	17.446	5E-07

   2. Using the cost equation from Requirement 1, calculate the total expected material handling cost for November if 350 moves are expected and 17,000 pounds of material will be moved (round final answer to the nearest dollar).

   Expected material handling cost: $

   3. For the coming year, McCourt expects 3,940 moves involving a total of 204,000 pounds. Calculate the following:

   Total expected fixed costs (round to the nearest dollar): $ Total material handling cost (add each component in the equation and then round to the nearest dollar): $

   Cumulative Learning Curve

   The learning curveshows how the labor hours worked per unit as the number of units produced increases. The basis of the learning curve is almost intuitiveas we perform an action over and over, we improve, and each additional performance takes less time than the preceding ones. The states that the cumulative average time per unit decreases by a constant percentage, or learning rate, each time the cumulative quantity of units produced doubles. The percentage based learning rate gives the percentage of time needed to make the next unit, based on the time it took to make the previous unit. The learning rate is determined through experience and must have a range between 50 and 100 percent. A 50 percent learning rate would eventually result in no labor time per unitan absurd result. A 100 percent learning rate implies no learning (since the amount of decrease is zero). An 80 percent learning curve is often used to illustrate this model. This is possibly due to the original learning curve work with the aircraft industry found an 80 percent learning curve.

   The cumulative average time and cumulative total time are easily obtained when the cumulative units produced are doubles of the prior cumulative number. For other values of cumulative units, the cumulative average time per unit can be obtained by using by knowing that the cumulative average-time learning model takes a logarithmic relationship:

   Y = pX^q

   where

   Y = Cumulative average time per unit

   X = Cumulative number of units produced

   p = Time in labor hours required to produce the first unit

   q = Rate of learning

   Therefore:

   q = ln (percent learning)/ln 2

   For an 80 percent learning curve:

   q = 0.2231/0.6931 = 0.3219

   So, when X = 3, p = 100, and q = 0.3219, Y = 100 3 ^0.3219 = labor hours (rounded to the nearest tenth). Thus, it takes 100 hours to produce the first unit, an average of hours to produce the first two units ( total hours) and an average of hours per unit to produce the first three units (a total of hours for producing three units). The third unit, therefore, took hours to produce (210.6 160).

   Applying the Concepts

   Medcom Company installs computerized patient record systems in hospitals and medical centers. Medcom has noticed that each general type of system is subject to an 80 percent learning curve. The installation takes a team of professionals to setup and completely test the system.

   Required:

   1. Assume that the first installation takes 800 hours, and the team of professionals is paid an average of $50 per hour.Complete the following table for Medcom Company (round answers to one decimal place):

   Cumulative Number of Systems	Cumulative Average Time per System (in Hours)	Cumulative Total Labor Hours
   1
   2
   4
   8
   16
   32

   2. Using the data table above, calculate the total labor cost of Medcom installing four systems:

   Total Labor cost (round to the nearest dollar): $

   3. Using the data table above, calculate the following for Medcom to produce a third system:

   a. Cumulative average time to produce the third system (round to the nearest tenth): hours

   b. Total time required to produce three systems (round to the nearest tenth): hours

   c. Time to produce the third system (round to the nearest tenth): hours