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The Economics of Production
Make or Buy Decisions
Capacity Expansion
Learning-Curves
Break-even Analysis
Production Functions
Make or Buy?
Let c1 = unit purchase price c2 = unit production cost (c2 < c1) K = fixed cost of production x = number of units required
Produce if K + c2 x c1x
or x K / (c1 - c2 )
Make or Buy Example
• It costs the Maker Bi Company $20 a unit to purchase a critical part used in the manufacture of their primary product line – a thing-um-a-jig
• It is estimated that the part could be produced internally at a unit cost of $16 after incurring a fixed cost of $20,000 for the necessary equipment.
• What to do?
a thing-um-a-jig
What to do?
Let c1 = $20 c2 = $16 K = $200,000 x = number of units required
Produce ifx K / (c1 - c2 ) = 20,000 /(20 – 16) = 5,000
Nonlinear Cost Function
Let c1 = unit purchase price c2 = K + axb where K, a, b > 0 x = number of units required
If c1 = 20 and c2 = 20,000 + 100x.7 , then
More on that Nonlinear Cost Functionx Make Buy1900 $39,730 $38,0001925 $39,911 $38,5001950 $40,092 $39,0001975 $40,272 $39,5002000 $40,451 $40,0002025 $40,630 $40,5002035 $40,701 $40,7002050 $40,808 $41,0002075 $40,985 $41,5002100 $41,162 $42,0002125 $41,338 $42,5002150 $41,513 $43,0002175 $41,688 $43,5002200 $41,862 $44,0002225 $42,036 $44,5002250 $42,209 $45,0002275 $42,381 $45,500
Strategic Decisions Capacity Expansion
• Capacity Growth Planning– when to construct new facilities
– where to locate facilities
– how large to size a facility
• Economies of scale– advantage of expanding existing facilities
– share plant, equipment, support personnel
– avoid duplication at separate locations
Capacity Expansioncompeting objectives: maximize market share
maximize capacity utilization
time
number units
demand
capacity leads demand
time
number units
demand
capacity lags demand
We need a modellet D = annual increase in demand x = time interval (in yrs) between capacity increases r = annual discount rate, compounded continuously f(y) = cost of expansion of capacity y
assume y = xD, then
cost = C(x) = f(xD) [1 + e-rx + (e-2rx ) + (e-3rx ) + …]= f(xD) [1 + e-rx + (e-rx )2 + (e-rx )3 + …]= f(xD) / [1 – e-rx]
assume f(y) = kya , then
find the x that minimizes C(x)
( )( )
1
a
rx
k xDC x
e
A Diversion - the Geometric Series
2
0
2
0
11 ... ...
1
11 ... ...
1
n n
n
n nrx rx rx rxrx
n
y y y yy
e e e ee
You see? It does
converge.
Discounting – another diversion
Consider the time value of money$1.00 today is worth more than a $1.00 next yearHow much more is it worth?
If r = annual interest rate, then it is worth (1+r) $1.00After two years, it is worth (1+r)2 $1.00 (compounded)
Compounded quarterly for 1 yr =
Compounded continuously for one year =
After t years =
41 / 4r
lim 1 /n r
nr n e
lim 1 /t
n rt
nr n e
More diversionary discounting
A stream of costs: C1, C2, …, Cn incurred at times t1, t2,…, tn has a present value of:
1
i
nrt
ii
C e
Why can’t you show us an example?
For an infinite planning horizonwhere x is the time between expansions:
1 1irx
rxi
CCe
e
The Example
• Chemical firm expanding at a cost ($M) of
– where y is in tons per year.
• Demand is growing at the rate of D = 5,000 tons per year and future costs are discounted at a rate of r = 16 percent
• Find x that minimizes
.62( ) .0107f y y
.62
.16
.0107(5000 )( )
1 x
xC x
e
Capacity Expansion Solution
C(x) - $M
8
9
10
11
12
13
14
15
0 2 4 6 8 10 12
years
.62
.16
.0107(5000 )( )
1 x
xC x
e
5 10.356535.1 10.350425.2 10.34565.3 10.342035.4 10.339645.5 10.338395.6 10.338235.7 10.339115.8 10.340985.9 10.34382
6 10.34757
alternately set C’(x) = 0solve for x.
Learning CurvesBased upon the observation that unit labor hours or costs decrease for each additional unit produced
Units produced
Directlabor hrsper unit
Why does this happen?• Employee learning• reduced set-up times• better routing and scheduling of material (WIP)• improved tool design• more efficient material handling equip. (MHE)• reduced lead-times• improved (simplified) product design• production smoothing• quality assurance• revised plant layout• increased machine utilization
Learning Curve(experience curves)
Y(u) = labor hours to produce the uth unit
assume Y(u) = au-b
a = hours to produce the first unitb = rate at which production hours decline
labor comingto work
Learning Curves
Assume hours to produce unit 2n is a fixedpercentage of the hours to produce unit n
Then for an 80 percent learning curve:
3219.2ln
8.ln
80.2)2(
)(
)2(
bor
an
na
nY
nY bb
b ln(Pct/100)
ln 2b
Observe the simple
formula
Learning Curvesleast-squares analysis
Unit Direct LaborNumber - x Hours -Y(x)
20 35.840 30.160 27.380 25.7100 24.1
Fit Y(x) = ax-b
using Excel
Y(x) = 74x-.243
2-.243 = .845or a 84.5% learning curve
Learning CurvesCumulative Cost
x
xTxV
aiiYxT
aiiYx
i
bx
i
b
)()(
)()(
)(
11
hours to produce ith unit
cumulative direct laborhrs to produce x units
average unit hours toproduce x units
Learning CurvesApproximate Cumulative Cost
b
xa
xb
xaxV
b
xadiiadiiYxT
bb
bxb
x
1)1()(
1)()(
1
1
00
ExampleY(x) = 74x-.243
2-.243 = .845 or a 84.5% learning curve
15.420030,8333/20V(2000)
hr.833,30)2000(75.97)2000(
75.97757.
74
1243.
7474)(
757.
757.757.1243.
0
243.
T
xxx
diixTx
X T(x) V(x)100 3192.396 31.92396200 5395.062 26.97531300 7333.268 24.44423400 9117.509 22.79377500 10795.36 21.59072600 12393.02 20.65504700 13926.95 19.89564800 15408.34 19.26043900 16845.28 18.716981000 18243.86 18.24386
Break-Even AnalysisLet x = number of units produced and sold x = S-1(unit selling price) S(x) = unit selling price F = fixed cost g(x) = variable cost to produce x units
then break-even point occurs when revenue = cost; or
S(x) x = F + g(x)
and profit = revenue – cost orP(x) = S(x)x – [F + g(x)]
Sam Evenon a break
Break-Even Analysis
x
$
F
Breakeven pt
loss
profit
Maxprofit
Revenue curve
Cost curve
loss
Diminishingreturns
Break-Even AnalysisDemand Curve
x
S(x)
S(x) = d + e x + f x2 (quadratic)
d, e, and f are constants to be determined
Break-Even AnalysisDemand Curve
x
S(x)
S(x) = d + e x + f x2
d, e, and f are constants to be determined
Approximate as linearS(x) = d + e x
Break-Even AnalysisUnit Cost
Let M = direct material unit cost ($/unit)L = direct labor rate ($/hour)B = factory burden rateY(x) = direct labor hours to produce unit xC(x) = cost to produce unit x
C(x) = M + L Y(x) + L B Y(x)
= M + (1+B) L Y(x)
= M + (1+B) L a x-b
Learningcurve effect
The Factory Burden Diversion
Manufacturing Costs
Factory burdenDirect costs
Direct laborDirect
material
Indirect material
Indirect labor
Indirect expense
-Supervision-Engineering-Maintenance
-Heating-Lighting-Depreciation-Rent & Taxes
-Office &janitorialsupplies -Paint
Factory Burden - exampleCategory annual costIndirect material $ 6,120Indirect labor 42,800Indirect expenses 22,900
total $71,820
Product annual production labor hours rate wages A 100,000 1000 $9/hr $9,000 B 140,000 1400 $7/hr 9,800 C 80,000 1600 $7/hr 11,200
total 4000 $30,000
burden rate = 71,820 /30,000 = 2.394 per direct labor $
Manufacturing Costs
General Overhead Costs
Profit
Selling Price S(x)
Administrative Costs
MarketingCosts
DevelopmentCosts
Demands
Cumulative Cost
g(x) = M x + L (1+B) T(x)
= M x + L (1+B) [a x1-b / (1-b)]
total cost = F + M x + L (1+B) a x1-b / (1-b)where F is a fixed cost to produce product x
Unit cost:C(x) = M + (1+B) L a x-b
1)(
1
b
xaxT
bLearning curve
Break-Even Analysis -Profit
Profit = P(x) = S(x) x - [F + g(x)]
letting S(x) = d + ex, e < 0
P(x) = (d + e x) x - F - M x - L (1+B) a x1-b /(1-b)
= d x + e x2 - F - M x - g x1-b
where g = L (1+B) a /(1-b)
More Break-Even Analysis
P(x) = (d - M) x + e x2 - g x1-b - F
break-even: set P(x) = 0 and solve for x
maximize profit: set and solve for xdP(x)
= 0 dx
2
1
2
( ) 2 (1 ) 0
( ) 2 (1 )
b
b
dP xd M ex b gx
dx
d P xe b b gx
dx
for e < 0, a max point can exist
Break-Even Analysis - example
P(x) = d x + e x2 - F - M x - g x-b+1
where g = (1+B) L a /(1-b)
Data:d = 100e = - .01F = $100,000M = $4B = .5L = $20 / hra = 10b = .60
P(x) = 100 x - .01 x2 – 100,000
- 4 x – (1+.5) (20) (10) x.4 / .4
= 96x -.01x2 –750 x.4 –100,000
2-.6 = 66%
The Math
2 0.4
0.6
1.6
1/1.6
( ) 96x .01x – 750 x –100,000 0
1382
'( ) 96 .02 300 0
* 4706
''( ) .02 180 0
.02296.07
180
P x
x
P x x x
x
P x x
x
The Graph
P(x)
-$150,000
-$100,000
-$50,000
$0
$50,000
$100,000
$150,000
0 1000 2000 3000 4000 5000 6000 7000
x = 1382
x = 4706
Production Functions
A production function expresses the relationship between an organization's inputs and its outputs. It indicates, in mathematical or graphical form, what outputs can be obtained from various amounts and combinations of factor inputs.
In its most general mathematical form, a production function is expressed as:Q = f(X1,X2,X3,...,Xn) where: Q = quantity of output and
X1,X2,X3,...,Xn = factor inputs (such as capital, labor, raw
materials, land, technology, or management)
Production Functions
There are several ways of specifying this function. One is as an additive production function:
Q = a + bX1 + cX2 + dX3,...
where a,b,c, and d are parameters that are determined empirically.
Another is as a Cobb-Douglas production function
Q = f(L,K,M) = A * (Lalpha) * (Kbeta) * (Mgamma)
where L = labor, K = capital, M = materials and supplies, and Q = units of product.
Cobb-Douglas Production Function
Q = f(L,K,M) = A * (Lalpha) * (Kbeta) * (Mgamma)
Properties of the Cobb-Douglas production function:Decreasing returns to scale: alpha + beta + gamma < 1Increasing returns to scale: alpha + beta + gamma > 1
Let CL, CK, and CM = the unit cost of labor, capital, and material, then
C(L,K,M) = CLL + CKK + CM M
is the total cost function
A Little Production ProblemAn interesting problem: Given a monthly budget of $B, how should the
money be spent to obtain a specified output Q?
Find L, K, and M where L = dollars spent on labor, K = dollars spent on facilities and equipment, and M = dollars spent on material
I know I can work this
one.
.3 .2 .4( , , ) 100
Subject to: $1,000,000
3 ; 2( )
Q f L K M L K M
L K M
M L K L M
The Inevitable Example
labor capital materialA alpha beta gamma
100 0.3 0.2 0.4 L K M Q $8,333 $66,667 $25,000 794,700 RHSbudget 1 1 1 100,000 100,000 -3 1 0 0 -2 1 -2 0 0
Stop the madness.Optimize your production system!
profits
homework: turn-in breakeven problemtext: Chapter 1- 29, 30, 31, 32, 34, 35
36, 37 ,38, 43, 44