manufacturing cost estimation

Manufacturing cost estimationManufacturing cost estimationManufacturing cost estimationManufacturing cost estimation

Learning objectives of the topic• Define the total cost as the sum of fixed and variable costs• Apply cost behavior estimation methods• Apply cost behavior estimation methods

Dr Muhammad Al‐Salamah, Industrial Engineering, KFUPM

Breakdown of total costBreakdown of total cost

• For various manufacturing decisions, the industrial engineer may require to establish a relationship between g y q pthe manufacturing total cost and manufacturing parameters such as volume of production, workforce size etcsize, etc.

• Variable costs change with the parameter levels and the fixed costs do not change.

• Hence, the total cost can be described asC = F + VX

C = costF = fixed costV = variable cost (unit cost)


X = parameter level

Cost behavior estimation methodsCost behavior estimation methods

• Three methods are available for estimating the relationship between the cost and the parameter:p p1. Engineering estimates2. Account analysis3. Statistical (regression) analysis


Engineering methodEngineering method

• Engineering cost estimation is based on measurement and pricing of the work involved in a task.p g

• This method requires the breakdown of work into small components and calculation of the cost associated with each componenteach component.

• Consider the Acer assembly process in AEC.• The assembly process has three stages:The assembly process has three stages:

– Assembly of the components– Installation of the operating systemp g y– Packaging

• Then, take each stage and break it down further to


smaller components.

• For example, the assembly stage requires simple hand tools, assembly table, worker training, etc., y , g,

• Then, these small components are subsequently estimated.

• Engineering cost estimation suggests cost estimates that are mostly optimistic and lower than what they actually are.


Account analysis methodAccount analysis method

• This method is based on the review of each cost account making up the cost and the identification of each as fixed g por variable.

• Consider the East Cement Co which operates in an 8-ho r orkdahour workday.

• The total overhead cost in 8 hours is

Account Fixed VariableUtilities 210 100 Indirect labor 2,000 1,300 Transport 186 3,200 Administration 100 2,172 Others 356 252


Total 2,852 7,024

• The variable costs surely are influenced by the workday, and the variable cost per work hour is:p

878 SR8024,7

=

• If the East Cement would extend the workday to 9 hours, the total overhead cost will become:

754,10 SR)878(92,852 =+

• Account cost estimation is subjective in its approach, hence it is not reliable.


Statistical (regression) cost estimationStatistical (regression) cost estimation

• Statistical cost estimation provides estimates that are less subjective, hence reliable, and more realistic.j , ,

• Statistical cost estimation uses past data and removes random events and reports only the true cost relations.

• This method is valid only on the relevant range of the activity levels.

• Regression analysis a statistical tool follows theseRegression analysis, a statistical tool, follows these steps:1. Collect past cost data2. Fit a least squares regression line3. Test the significance of the linear relationship.


• To establish the regression line, the independent and the dependent variables have to be defined.p

• The independent variable is the activity level X and the dependent variable is the cost C.

• With least squares regression, we like to find the values of the fixed and unit variable costs that will make the sum of the squares of the deviations of the data from the qline as small as it can be:

ECCedeviation iii ECCe −==deviation


• The estimated cost ECi is defined by calculating the values of F and V:

ECi = F + V Xi

• To find the values of F and V, we seek to find the minimum of

∑∑ ==nn

VXFCeD 22 )(∑∑==

−−==i

iii

i VXFCeD11

)(


• The minimum can be found by setting the partial derivatives to zero and solving for F and V:g

0)(2 =−−−=∂ ∑

n

ii VXFCD

0)(2

)(1

=−−−=∂∂

∑

∑=

ni

ii

XVXFCDF

0)(21

==∂ ∑

=i

iii XVXFC

V


• These two equations lead to the normal equations

∑∑==

=+n

ii

n

ii CXVnF

11

∑∑∑===

=+n

iii

n

ii

n

ii CXXVXF

11

2

1


• The normal equations can be solved by substitution:nn

ii

ii

n

XV

n

CF −=

∑∑== 11

n

i

n

in XC

nn

∑∑xy

ii

iii

SSn

CXV =

⎞⎛

−=

∑∑∑ ==

=2

11

1

xx

n

n

ii

i

SX

X⎟⎠

⎞⎜⎝

⎛

−∑∑=12


ii n

X∑=1

ExampleExample

• Consider the production level vs repairs costs:

Repairs costs(1000 riyals)

Production level (units)

C XCi Xi256 50297 100329 150329 150315 125283 88289 94301 103


Repairs cost (1000 riyals)

Production level (units)

Ci Xi Xi2 Xi Ci256 50 2500 12800256 50 2500 12800297 100 10000 29700329 150 22500 49350315 125 15625 39375283 88 7744 24904289 94 8836 27166301 103 10609 31003

Sum 2070 710 77814 214298

0.755799 714340.86

)710(7

)710)(2070(2142982 ==

−=V

219.6471075.02070

5799.717

)710(77814

=−=

−

F


219.647

75.07

F

• The cost at any level of production is given byEC = 219 64 + 0 75 XEC 219.64 + 0.75 X

• It can be read from the equation that the repairs cost increases by SR 750 for every product manufactured.

• The company plans to produce 90 units (notice 90 is in the relevant range), the repairs cost is estimated at

219 64 + 0 75 (90) = SR 287 140219.64 + 0.75 (90) = SR 287,140


• Being a random variable, the cost has a variance and it is equal toq

2σ2

−

−=

nVSS xyyy

for

2n( )

∑ ∑−=nC

CS iiyy

22

• For the previous cost data, the variance is

7.5627

)4340.86(75.03293.43σ2 =−

−=


• The confidence interval on the value of the unit variable cost is given byg y

n StV

2

2,2/σ

α −±

• Hence, it is said that the value of V is in this interval with a 100(1 α)% confidence

xxS

a 100(1 - α)% confidence.• For the previous cost data, the unit variable cost is

09056.72 57V

i.e. the unit variable cost is between 0.66 and 0.84 with

09.05799.71

56.72.57 =±V


95% confidence.

• A measure of the overall quality of the regression line is the coefficient of determination:

yy

xy

SS

VR =2

• For the previous cost data, the value of R2 is

yy

0.993293.434340.8675.02 ==R

• Hence, it is said that 99% of the variability in the repairs cost is explained by changes in the production level.


• If we can assume that the production volume and the repairs cost are random variables and the joint p jdistribution of them follows a bivariate normal distribution, we assess the linear relationship between the repairs cost and the production volume by the valuethe repairs cost and the production volume by the value of the correlation coefficient ρ.

• When ρ = 0, there is no correlation between the two random variables, and we can say the two random variables are independent.


• To test the correlation between the repairs cost and the production volume, we can test the following hypothesis:p , g yp

H0 : ρ = 0H1 : ρ ≠ 0

• The test statistics for this hypothesis is

02nRT −

=

which has the t distribution with n-2 degrees of freedom.20

1 RT

−


• For the previous data, the value of the test statistics is t0= 22.25.

• For a confidence level α = 0.05, t0.025,5 = 2.571.• Since t0.025,5 < t0, we say H0 is rejected and conclude that

there is a strong linear relationship between the repairs cost and the production volume.


manufacturing cost estimation

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