production flexibility and capacity investment under demand … · 2013. 4. 8. · verena hagspiel...

Verena Hagspiel Finance Seminar Series, Aarhus University, April 9, 2013 – 1 / 25

Production Flexibility and Capacity Investment under DemandUncertainty

Verena Hagspiel1, Kuno J.M. Huisman2,3, and Peter M. Kort2,4

1Department of Operations, Faculty of Business and Economics, UNIL2Department of Econometrics and Operations Research and CentER, Tilburg University

3ASML Netherlands B.V., Veldhoven, The Netherlands4Department of Economics, University of Antwerp

Motivation

Motivation

• Motivation

• Motivation (2)

• Research Question

Outline

Introduction

Literature

Model

Solution

Results

Conclusion


• Many firms nowadays face high demand volatility:

◦ influences desirability to invest in production capacity

◦ influences choice of capacity level

→֒ Capacity Size

◦ raises the value of being able to adapt production

decision

→֒ Flexibility

Motivation (2)

Motivation

• Motivation

• Motivation (2)


Outline

Introduction

Literature

Model

Solution

Results

Conclusion


Production Flexibility is crucial for Companies:

• Car Industry: Credit crunch recession ⇒ significant drop in demand

⇒ companies had to downscale production ⇒ low utilization rates

Figure 1: Articles, the Guardian, 2009

• LCD industry: During initial stage producing at full capacity.

Later on competition on the supply side led to overcapacity.

Research Question

Motivation

• Motivation

• Motivation (2)


Outline

Introduction

Literature

Model

Solution

Results

Conclusion


Optimal Investment Strategy of Firm

? Optimal investment timing

? Capacity size

? Optimal output rate

→֒ Production Flexibility

⇓

Impact on Investment decision

Outline

Motivation

Outline

• Outline

Introduction

Literature

Model

Solution

Results

Conclusion


• Introduction

• Literature

• Model

• Solution

• Results

◦ Utilization Rate

◦ Capital vs. Labor

◦ Value of Flexibility

• Summary/Conclusion

Model Setup

Motivation

Outline

Introduction

• Model Setup

• Main Aspects

Literature

Model

Solution

Results

Conclusion


• Price setting monopolist

• Demand uncertainty

• Continuous time setting

• Production Flexibility:=

◦ production can fluctuate over time

◦ between zero and capacity level

◦ no adjustment costs

• One time investment → capacity size

• Solution method:

◦ Dynamic programming approach

Main Aspects

Motivation

Outline

Introduction

• Model Setup

• Main Aspects

Literature

Model

Solution

Results

Conclusion


→ Optimize the

• Timing and

• Size of investment

• Output rate

for investment in flexible and inflexible capacity, respectively.

→ Analyze investment in flexible capacity:

◦ Utilization rate

◦ Capital vs. labor intensive industries

→ Compare the two investment strategies (flexible vs. inflexible ) regarding

timing and size.

Literature

Motivation

Outline

Introduction

Literature

• Literature

• Literature (2)

Model

Solution

Results

Conclusion


• Operations Strategy literature:

◦ Van Mieghem and Dada (1999), Anupindi & Jiang (2008) or Chod &

Rudi (2005)

•

irreversible investment

uncertain future rewards

leeway about timing

⇒ Real Options theory

◦ Assumed either flexibility in timing or flexible technology:

• Kulatilaka (1988), Triantis & Hodder (1989), Dixit (1993)

◦ Timing and size: Bar-Illan & Strange (1999)

◦ Different types of flexibility: He & Pindyck (1992), Chronopoulos et

al. (2011)

◦ Closest related: Dangl (1999)

Literature (2)

Motivation

Outline

Introduction

Literature

• Literature

• Literature (2)

Model

Solution

Results

Conclusion


• Production Economics:

◦ Bengtsson (2001): relates RO literature to manufacturing flexibility

Model - Production Quantity

Motivation

Outline

Introduction

Literature

Model• Model - Production

Quantity

• Model - Flexible Capacity

• Solution Method

Solution

Results

Conclusion


• Inverse demand function (linear)

p(qt, t) = θt − γqt

• θt follows the geometric Brownian motion

dθt = αθtdt+ σθtdWt

• Production quantity at time t, qt, price p, γ > 0

• Variable unit production cost c

• In case of unlimited capacity ⇒ optimal production quantity:

q∗t = argmaxq(t) [(p− c)qt] = max

(

0,θt

2γ−

c

2γ

)

Model - Flexible Capacity

Motivation

Outline

Introduction

Literature


Quantity


• Solution Method

Solution

Results

Conclusion


• Capacity K

• Production cannot exceed capacity: 0 ≤ qt ≤ K Therefore,

q′t = min(q∗t ,K)

depending on θt.

• Capacity holding cost chK

• Profit flow at time t equals

π(θt,K) =

−chK for 0 ≤ θt < c I(θt−c)2

4γ − chK for c ≤ θt < 2γK + c II

(θt − γK − c− ch)K for θt ≥ 2γK + c III

• Firm chooses capacity size K at moment of investment.

• Investment cost: I(K) = δKλ, λ < 1 (robustness check: λ > 1)

Solution Method

Motivation

Outline

Introduction

Literature


Quantity


• Solution Method

Solution

Results

Conclusion


• Project value V (θ,K) satisfies

V (θt,K) = Π(θt,K)dt+ E[

V (θt + dθt,K)e−rdt]

with constant discount rate r.

⇒

Vflex(θ,K) =

L1 (K) θβ1 −chK

rfor 0 ≤ θ < c

M1 (K) θβ1 +M2θβ2

+ 14γ

[

θ2

r−2α−σ2 −2cθr−α

+ c2

r

]

−chK

rfor c ≤ θ < 2γK + c

N2 (K) θβ2 + Kr−α

θ −K(Kγ+c+ch)

rfor θ ≥ 2γK + c

β1 (β2) is the positive (negative) root of the quadratic polynomial

1

2σ2β2 + (α−

1

2σ2)β − r = 0.

Solution Method (2)

Motivation

Outline

Introduction

Literature

Model

Solution

• Solution Method (2)

Results

Conclusion


(1) Optimal capacity choice K∗(θ):

maxK(θ) V (θ,K(θ))− cK(θ) for every θ

θ

K I II

c

III

(θ−c)/2γ

I II

III

K*2

K*3

0 100 200 300 400 500Θ

200

400

600

800

1000K*

(2) Optimal investment threshold θ∗

Value of Waiting F (θ) = Value of Investing V (θ)− cK(θ)

◦ waiting region θ < θ∗

◦ stopping region θ > θ∗

Results (Optimal Investment in Region II)

Motivation

Outline

Introduction

Literature

Model

Solution

Results• Results (Optimal

Investment in Region II)

• Results (Optimal

Investment in Region III)

• Results (Capacity

Utilization)

• Robustness (Convex

Investment Cost)

• Capital vs. Labor intensive

• Results (Impact of

Capacity Holding Cost)


Capacity Holding Cost) (2)


Production Flexibility)


Production Flexibility) (2)

• Robustness (Impact of


Conclusion


0 500 1000 1500 20000

500

1000

1500

2000

θ

q*(θ

), K

*(θ)

q*(θ)

K*(θ)

K*(θ*)=747.19

q*(θ*)=378.58

θ*=857.15 θq=1594.37 A

Figure 2: Investment Strategy (Optimal Capacity K∗(θ) and Production

Quantity q∗(θ)). Parameter values: σ = 0.15, α = 0.02, r = 0.1, γ = 1,

c = 100, ch = 100, δ = 1000 and λ = 0.7.

Results (Optimal Investment in Region II)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0 500 1000 1500 20000

500

1000

1500

2000

θ

q*(θ

), K

*(θ)

q*(θ)

K*(θ)

K*(θ*)=747.19

q*(θ*)=378.58

θ*=857.15 θq=1594.37 B

Capacity Restriction

Suspension

Production

q(θ)=min((θ−c)/2γ,K*(θ*))


Quantity q∗(θ)). Parameter values: σ = 0.15, α = 0.02, r = 0.1, γ = 1,

c = 100, ch = 100, δ = 1000 and λ = 0.7.

Results (Optimal Investment in Region III)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0 100 200 300 400 500 6000

50

100

150

200

250

300

K*(θ)

q*(θ)

A θ

q*(θ

), K

*(θ)

K*(θ*)=45.88

q*(θ*)=103.79

θq=191.75 θ*=307.58


Quantity q∗(θ)). Parameter values: σ = 0.05, α = 0.002, r = 0.1,

γ = 1, c = 100, ch = 100, δ = 1000 and λ = 0.7.

Results (Optimal Investment in Region III)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0 100 200 300 400 500 6000

50

100

150

200

250

300

K*(θ)

q*(θ)

B θ

q*(θ

), K

*(θ)

θq=191.75 θ*=307.58

q*(θ*)=103.79

K*(θ*)=45.88Suspension

Capacity Restriction

Production

q(θ)=min((θ−c)/2γ,K*(θ*))


Quantity q∗(θ)). Parameter values: σ = 0.05, α = 0.002, r = 0.1,

γ = 1, c = 100, ch = 100, δ = 1000 and λ = 0.7.

Results (Capacity Utilization)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


Effect of increasing uncertainty on the utilization at moment of investment:

u := q∗(θ∗)K∗(θ∗)

Table 1: Investment strategy with the occupation rate (Parameter values:

α = 0.02, r = 0.1, γ = 1, c = 100, ch = 100, δ = 1000 and λ = 0.7)

σ θ∗ K∗(θ∗) q∗(θ∗) u

0.1 489.78 223.66 194.89 87.14%

0.15 857.15 747.19 378.57 50.67%

0.2 3726.02 16175.4 1813.01 11.21%

Robustness (Convex Investment Cost)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


Table 2: Comparing Investment Strategies with Concave (λ < 1) and Con-

vex (λ > 1) Investment Cost Structure (Parameter values: α = 0.02,

r = 0.1, γ = 1, c = 100, ch = 100 and δ = 1000)

λ σ θ∗ K∗(θ∗) q∗(θ∗) u

0.7 0.1 489.78 223.66 194.89 87.14%

0.7 0.15 857.15 747.19 378.57 50.67%

0.7 0.2 3726.02 16175.4 1813.01 11.21%

1.1 0.1 763.308 287.631 331.654 100%

1.1 0.15 1401.02 866.889 650.508 75.04%

1.1 0.2 7502.41 19461.3 3701.21 19.02%

Capital vs. Labor intensive

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


Flexible

Capacity Utilization Numerical example

Labor intensive

ch = 0, c = 100

low θ∗ = 274.7

(ch < c) K∗(θ∗) = 263.0

q∗(θ∗) = 87.4

ur = 33%

Capital intensive

ch = 100, c = 0

high θ∗ = 279.5

(ch > c) K∗(θ∗) = 118.3

q∗(θ∗) = 139.8

ur = 100%

Figure 4: Capacity utilization

Results (Impact of Capacity Holding Cost)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0 50 100 150200

210

220

230

240

250

260

270

ch

K*(

θ*)

K*(θ*)

0 20 40 60 80 100 120 140 160 180 200250

300

350

400

450

500

550

600

650

700

750

ch

θ*

θ*

Figure 5: Effect of increasing capacity holding cost ch on optimal capacity

size K∗(θ∗) and investment threshold θ∗. (c = 100)

Results (Impact of Capacity Holding Cost) (2)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0 20 40 60 80 100 120 140 160 180 200200

300

400

500

600

700

800

900

ch

θ*

Kfixed

=100

Kfixed

=250

Kfixed

=500

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

400

500

600

700

800

900

Kfixed

θ*

ch=50

ch=100

ch=200

Figure 6: Effect of increasing capacity holding cost ch and capacity size K

on the investment threshold θ∗ when capacity size is a constant parameter.

(c = 100)

Results (Impact of Production Flexibility)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0.025 0.05 0.075 0.1 0.125 0.15 0.175200

400

600

800

1000

1200

1400

1600

uncertainty σ

θ* flexible modelθ* inflexible model

0.025 0.05 0.075 0.1 0.125 0.15 0.1750

500

1000

1500

uncertainty σ

K*(θ*) flexible model

q*(θ*) flexible model

K*(θ*)=q*(θ*) inflexible model

Figure 7: Impact of Flexibility and Uncertainty on the Investment Strategy.

Parameter values: α = 0.02, r = 0.1, γ = 1, c = 100, ch = 100, δ =

1000 and λ = 0.7.

Results (Impact of Production Flexibility) (2)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


0.025 0.05 0.075 0.1 0.125 0.15 0.175300

400

500

600

700

800

900

1000

uncertainty σ

θ* flexible model

θ* inflexibel model

A0.025 0.05 0.075 0.1 0.125 0.15 0.1750

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

uncertainty σ

K*(θ*) flexible modelq*(θ*) flexible modelK*(θ*) inflexible model

B

Figure 8: Impact of Flexibility and Uncertainty on the Investment Strategy.

Parameter values: α = 0.02, r = 0.1, γ = 1, c = 200, ch = 0, δ = 1000

and λ = 0.7.

Robustness (Impact of Production Flexibility)

Motivation

Outline

Introduction

Literature

Model

Solution






Utilization)


Investment Cost)












Conclusion


Investment cost structure I(K) = δKλ with

λ < 1⇒concave or

λ > 1⇒convex

0.025 0.05 0.075 0.1 0.125 0.15 0.1750

500

1000

1500

2000

2500

uncertainty σ

θ* flexible model

θ* inflexible model

0.025 0.05 0.075 0.1 0.125 0.15 0.1750

500

1000

1500

2000

2500

uncertainty σ

K*(θ*) flexible model

q*(θ*) flexible model

K*(θ*)=q*(θ*) inflexible model

Figure 9: Impact of Flexibility and Uncertainty on the Investment Strategy

for Convex Investment Cost.

Conclusions

Motivation

Outline

Introduction

Literature

Model

Solution

Results

Conclusion

• Conclusions

• Extensions


• Our paper extends Real Option theory by considering, besides timing

decision, flexibility in production and capacity choice.

• In contrast to earlier literature we show that two investment cases need

to be taken into account.

• Main Results:

◦ Utilization is decreasing strikingly in demand uncertainty.

◦ Capital vs. labor intensive industry

◦ Impact of capacity holding cost on investment strategy.

◦ Comparison of investment strategy for flexible and inflexible firm.

• Two contrary effects as to the timing of the investment.

• Flexible firm invests in higher capacity. Capacity difference

(flexible vs. inflexible firm) increases in uncertainty.

Extensions

Motivation

Outline

Introduction

Literature

Model

Solution

Results

Conclusion

• Conclusions

• Extensions


• Competitive setting

• Different sources of uncertainty

• Case Study

• ...

production flexibility and capacity investment under demand … · 2013. 4. 8. · verena hagspiel...

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