Impacts of the Shape of the Demand Distribution on Procurement Risk Mitigation

April 29, 2011POMS 22nd Annual Conference

Anssi Käki and Ahti SaloSchool of Science, Aalto University, Finland

Motivation

• Growing mismatches between demand and supply expose companies to serious operational risks. (Hendricks & Singhal 2005)

• Strategies for mitigating procurement risks can be evaluated with stochastic models.

• Yet, these models are not used in practice. (Kouvelis et al. 2006, Tang 2006)

• We show that the models’ results of can be very sensitive to assumptions about uncertainties.

• Thus, the uncertainties must be well understood when deploying models.

Motivation

• How many components are needed, if demand planners state that ”our demand forecast is 10 000, with a variation of 5 000 items”?

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.5

1

1.5

2

2.5

3

3.5x 10

-4

Demand

f(x)

Probability mass functions

D1

D2

Contents

• We study the capacity reservation option model of Cachon & Larivere (2003).

• We illustrate how the optimal procurement strategy depends on the shape of demand distribution.

• Our results suggest that inaccurate assumptions about uncertainties may lead to non-optimal behaviour.

Capacity reservation option

• A component is procured from a single supplier.• The demand for the component is uncertain.• In the one-period model, at t=0 the manufacturer can:

– Make firm commitments m – Reserve capacity o.

• The component demand d is realized; the manufacturer then decides how much to execute e, restricted by the reserved capacity o. Thus, the total order is m+e ≤ m+o.

Capacity reservation option

• Expected sales for capacity K

• Manufacturer’s profit for revenue r and prices wm, wo, we

• We set wm = $1.0, wo = $0.2, we = $0.9 and r = $2.0.

• Optimal strategy can be derived via maximization of (m,o)

Example with bimodal demand

Product one: the expected sales is 10 000 and sales from 8 000 to 13 000 cover around 50% of probability mass.

Product two: one monopolistic customer accounts for approximately 2/3 of sales. The remaining sales comes from small customers. Expected sales is, again, 10 000.

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

1

2

3

x 10-4

f(x)

Demand

Probability mass functions

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Demand

F(x

)

Cumulative probability functions

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

1

2

3

x 10-4

Demand

f(x)

Probability mass functions

D1

D2

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Demand

F(x

)

Cumulative probability functions

D1

D2

Example with bimodal demand

• Optimal strategies can be determined with numeric integration.

• Even if the optimal profits are identical, strategies differ significantly.

Demand m* o* m*+o* *

Product one

5 910 7 190 13 100 $8 250

Product two

4 440 10 810 15 250 $8 250

Difference +33% -33% -14% 0%

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DemandF

(x)

Cumulative probability functions

D1

D2

*1m*

2m

*2

*2 om

*1

*1 om

Two products and a common component

• A common component of two products with dependencies:❶ Substitute products, such as comparable devices for the same market area.

Demands are likely to be negatively dependent.

❷ Differentiated products for different market segments. There are no demand dependencies.

❸ Complementary products, such as a one device launched separately for two different market areas. Demands are likely to be positively dependent.

❸❷❶

Example samples from joint distributions of two product demands

Two products and a common component

• The optimal strategies and corresponding profits determined with a stochastic optimization model

• For complementary products, the distribution is wider and has heavier right-tail. The option is utilized more.

• Still, the demand fulfilled is on average 3% less it is optimal to prepare for the ”fat-tail”, but the resulting expected profit is lower.

Product type m* o* m*+o* *

Substitute 15 460 8 130 23 590 $18 180

Differentiated 13 880 10 620 24 500 $17 430

Complementary 12 970 12 460 25 430 $16 940

Practical implications

0 5000 10000 15000 200000%

25%

50%

75%

100%

Fixed orders (m)

Pro

fit v

s. O

ptim

al p

rofit

Profit without option

Product one (unimodal)Product two (bimodal)

• How many components are needed, if demand planners state that the demand forecast is 10 000 5 000 items?

• If there is no flexibility (such as the capacity reservation option), the profit can be significantly less.

0

10000

200000

10000

20000

-25%

0%

25%

50%

75%

100%

Firm commitments

Profit with option, bimodal product demand

Option reservations

Pro

fit v

s. O

ptim

al p

rofit

Practical implications

• If there is a flexible alternative (the option in our case), the profit can still drop remarkably.

For example, setting arbitrarily m=8 000, o=4 000 would yield 23% less profits compared to the optimal.

Conclusions

• When evaluating/implementing procurement approaches, careful analysis of uncertainties is essential.

• We have illustrated how the optimal procurement strategy is dependent on:– Demand distribution width and ”fat-tails”– Demand distribution modality– Demand-dependencies between two products that share a

common component.

• Our future work will discuss how copula-based scenarios can be used to address more complex uncertainties, such as several products and uncertainty in supply.

Thank you!

Cachon, G. P. and Lariviere, M. A. (2001). Contracting to assure supply: How to share demand forecasts in a supply chain. Management Science, 47(5):629-646.

Hendricks, K. B. and Singhal, V. R. (2005). Association between supply chain glitches and operating performance. Management Science, 51(5):695-711.

Kouvelis, P., Chambers, C., and Wang, H. (2006). Supply chain management research and Production and Operations Management: Review, trends, and opportunities. Production and Operations Management, 15(3):449-469.

Käki, A. and Salo, A. (2011). Impacts of the Shape of the Demand Distribution on Procurement Risk Mitigation. Proceedings of 22nd Annual POMS Conference. Available at: http://www.pomsmeetings.org/ConfPapers/020/020-0741.pdf

Tang, C. S. (2006). Review: Perspectives in supply chain risk management. International Journal of Production Economics, 103:451-488.

References