1 Evaluating procurement strategies under uncertain demand and risk of component unavailability Anssi Käki and Ahti Salo Systems Analysis Laboratory Aalto

1

Evaluating procurement strategies under uncertain Evaluating procurement strategies under uncertain

demand and risk of component unavailabilitydemand and risk of component unavailability

Anssi Käki and Ahti SaloSystems Analysis Laboratory

Aalto University School of Science and TechnologyP.O.Box 11100, 00076 Aalto

FINLAND

Evaluating procurement strategies Anssi Käki and Ahti Salo 2

Manufacturer’s problemManufacturer’s problem

What procurement policies are best when there are– Uncertainties in end product demand and supplier capability

– Inter-dependencies between uncertainties.

* E.g. Martínez-de-Albéniz and Simchi-Levi (2003) consider similar options.

ProductsComponentsSuppliers Market

Material flow

Common

Product specific

To minimize costs and hedge supply risks, the manufacturer

can use normal orders or capacity reservation options*.


Research perspectiveResearch perspective Typical risk mitigation strategies include

– Supplier diversification (supply uncertainty)*– Common components (demand risk pooling)**.

ProductsComponentsSuppliers Market

* See Tang (2006) for literature review, Kleindorfer&Wu (2003) and Federgruen&Yang (2008) for models. ** E.g. Groenevelt &Rudi (2000), Van Mieghem (2004).

Material flow

Correlated uncertainty

Common

Product specific

Our approach is novel, for it combines following aspects:– Non-stationary and inter-dependent (correlated) uncertainties– Uncertainty modeling without probability distributions– Risk mitigation with options instead of supplier diversification– Stochastic demand and supply (costs are deterministic).


Research questions and approachResearch questions and approach

Our initial research questions include:

1. When does capacity reservation option reduce the expected and worst case

procurement cost?

2. What is the impact of common component on costs?

3. Does negative correlation between demand and supply capability increase

costs?

* Adopted from Hochreiter and Pflug (2007).

To answer these questions, we propose a framework with following

steps*:

1. Data preprocessing / ”realistic” initial assumptions

2. Multivariate scenario generation and

3. Building and solving of a stochastic cost-minimization model.


Stochastic optimization modelStochastic optimization model

Unit costs include i) fixed order, ii) capacity reservation,

iii) capacity execution, iv) inventory holding and scrap and

v) shortage.

]]))[(min[(min212,102,02,01,0 ,,

cce

crqq

fEfEf

Initial, first and second stage costs


Decision steps: Initial fixed ordersDecision steps: Initial fixed orders


Decision steps: Capacity reservationsDecision steps: Capacity reservations


Decision steps: Capacity executionDecision steps: Capacity execution


CostsCosts


Example of one product, component and perfectly Example of one product, component and perfectly reliable supplierreliable supplier

5.119

1.0)5025.02025.03025.0(

1.0405.09.01005.0

2.0100150]costs Total[

E Initial stage: Order & Reservation

First stage: Execution & holding

Second stage: Scrap


Without option, the optimal policy is q0,1=50, q0,2=100 and

With option Without option

Example cont’dExample cont’d

.5.1195.159

1.0)5025.012025.013025.0405.0(

1150]costs Total[

E

105

9.01005.0

1.0100150]sales Total[

E

150sales Total

OrderHolding & scrap

100

1005.050]useCapacity [

E

150useCapacity

Supplier perspective:

Supplier benefit depends on how expected extra capacity can be used with options


0 1 2 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Stage

Sal

es

Example demand scenarios, 10 x 10 tree

Scenario trees are built withScenario trees are built withmoment matching method*moment matching method*

Demand: » Expected product sales

» Variance and skewness

» Correlation between sales

Supply capability:» Expected capability (0…100%)

» Variance and skewness

» Correlation between suppliers

Correlation between aggregated

demand and supply

1st stage targets:E[D]=500Var[D]=10 000Skew[D]=2

2nd stage targets:E[D] i,2= 5 x Di,1

Var[D] = 5 x Var[D]Skew[D]=2

1 2 86

88

90

92

94

96

98

100

Stage

Sup

ply

capa

bilit

y %

Example supply capability scenarios, 10 x 10 tree

1st stage targets:E[S]=97%Var[S]=10%Skew[S] = -0.5

2nd stage targets:E[S] i,2= Si,1

Var[S] =Var[S]Skew[S]=-0.5

* Hoyland and Wallace (2001).


Heuristic for multivariate scenario generationHeuristic for multivariate scenario generation To maintain other statistical properties (marginal distributions) while varying correlation

(joint distribution), we use a ”scenario enumeration heuristic”.

Enumeration heuristic


0 2000 4000 6000 8000 10000 0

2000

4000

6000

8000

10000

12000Negatively correlated products

0 2000 4000 6000 8000 10000 0

2000

4000

6000

8000

10000

12000Uncorrelated products

0 2000 4000 6000 8000 10000 0

2000

4000

6000

8000

10000

12000Positively correlated products

Demand scenarios of two productsDemand scenarios of two products

Scenario enumeration: demand of product one (y-axis value) remains unchanged

Plotted data contains 2nd stage values of 10 x 10 trees with equal probabilities. Red lines are OLS regression lines; they are statistically significant in positive and

negative case (p<0.01).


Demand vs. supply scenariosDemand vs. supply scenarios

0 1000 2000 3000 4000 5000 6000 7000 8000 86

88

90

92

94

96

98

100

Average of aggregated demand

Su

pp

ly c

ap

ab

ility

ave

rag

e

Uncorrelated

0 1000 2000 3000 4000 5000 6000 7000 86

88

90

92

94

96

98

100

Average of aggregated demand

Su

pp

ly c

ap

ab

ility

ave

rag

e

Negatively correlated

Plotted data contains 2nd stage values of 10 x 10 trees with equal probabilities. Negative-case OLS regression line is statistically significant (p<0.01).


Sample of four multivariate scenario treesSample of four multivariate scenario trees

Some properties are in common for

all scenarios, for example: Scenarios represent different

business environments, for example:

Demand SupplyD1 D2 S1 S2

E 2453 2590 95.6 % 95.7 %

Std 2122 2244 3.3 3.3

Skew 1.24 1.28 -0.27 -0.33

Scenario Correlation

Between demands Demand vs. supply capability

Complementary products- E.g. same products for different sales areas

0.38 -0.40

Substitute products- E.g. similar products for same sales area

-0.36 -0.35

Only demand-supply dependency- E.g., products independent, but market demand drives supply capability

0.02 -0.41

No inter-dependencies- E.g., differentiated products and supply capability not demand-driven

-0.02 0.01


10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 >300000

0.2

0.4

0.6

0.8

1

Cost of scenario

Pro

ba

bili

ty

Probability distribution of costs

59%

10%

9%

4%

18%

Distrubution of expected costs

OrderReservationExecutionHoldingShortage

12000 13000 14000 15000 16000 17000 18000 190000

0.2

0.4

0.6

0.8

1

Cost of scenario

Pro

ba

bili

ty


59%

13%

14%

5%

10%Distrubution of expected costs


Worst case risks grow, if inter-dependenciesWorst case risks grow, if inter-dependenciesoccuroccur

No inter-dependecies Complementary products

E 15933CVaR (5%) 34800

E 16056 +1 %CVaR (5%) 44700 +28 %

>


9000 10000 11000 12000 13000 14000 150000

0.2

0.4

0.6

0.8

1

Cost of scenario

Pro

ba

bili

ty


69%

9%

11%

4%8%



10000 11000 12000 13000 14000 150000

0.2

0.4

0.6

0.8

1

Cost of scenario

Pro

ba

bili

ty


66%6%

9%

4%

15%



Use of common component can aggregate Use of common component can aggregate worst case riskworst case risk

No inter-dependencies Complementary products

E 11941.00CVaR (5%) 28900.00

E 13781.00 +15 %CVaR (5%) 44300.00 +53 %

> >


High demand drives costs more comparedHigh demand drives costs more comparedto low supply capabilityto low supply capability

0 2000 4000 6000 8000 10000 12000 14000 16000 0

10000

20000

30000

40000

50000

60000

Average demand

Cos

ts

Costs vs. average demand


86 88 90 92 94 96 98 100 0

10000

20000

30000

40000

50000

60000

Average supply

Cos

ts

Costs vs. average supply


No inter-dependencies Complementary products

0 5000 10000 15000 0

10000

20000

30000

40000

50000

Average demand

Cos

ts

Costs vs. average demand


86 88 90 92 94 96 98 100 102 0

10000

20000

30000

40000

50000

Average supply

Cos

ts

Costs vs. average supply



Preliminary resultsPreliminary results

Our approach allows systematic analysis of the performance of

procurements policies Initial observations:

– Capacity reservation option seems to reduce costs (minimum reduction 5%,

depending on scenario and setup).

– Use of common components has an impact on expected costs, which is highest

with complementary products > non-correlated > substitute products.

– Maximum costs can be significantly higher in case of complementary products

and a common component.

– There is some evidence that negative correlation between demand and supply

capability would increase especially worst case costs.


Next stepsNext steps

Improve uncertainty modeling:– Detailed assessment of supplier capability

– Analysis and improvement of scenario enumeration heuristic.

Supplement the optimization model with risk constraints*. Investigate model expansion with respect to time stages and other

variables, such as components, products and suppliers. Evaluate new strategies, such as forecast-sharing based procurement.

* E.g. Sodhi (2005) considers ”Demand-at-Risk” and ”Inventory-at-Risk”.


ReferencesReferencesFedergruen, A. and Yang, N. (2008). Selecting a portfolio of suppliers under demand and supply risks. Operations Research,

56(4):916-936.

Groenevelt, H. and Rudi N. (2000). Product design for component commonality and the effect of demand correlation.

Working paper, University of Rochester, Rochester, NY

Hochreiter, R. and Pflug, G. C. (2007). Financial scenario generation for stochastic multi-stage decision processes as facility

location problems. Annals of Operations Research, 152(1):257-272.

Hoyland, K. and Wallace, S. W. (2001). Generating scenario trees for multi-stage decision problems. Management Science,

47(2):295-307.

Kleindorfer, P. R. and Wu, D. J. (2003). Integrating long- and short-term contracting via business-to-business exchanges for

capital intensive industries. Management Science, 49(11):1597-1615.

Martínez-de-Albéniz, V. and Simchi-Levi, D. (2003). A portfolio approach to procurement contracts. MIT Sloan School of

Management Paper 188, Available at http://ebusiness.mit.edu/research/papers/188DSleviPortfolioApproach.pdf.

Sodhi, M. S. (2005). Managing demand risk in tactical supply chain planning for a global consumer electronics company .

Production and Operations Management, 14(1):69-79.

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

103:451–488.

Van Mieghem, J. A. (2004). Commonality strategies: Value drivers and equivalence with flexible capacity and inventory

substitution. Management Science, 50(3):419-424.


Appendix – Computational aspectsAppendix – Computational aspects

Scenario trees by moment matching is hard:– Non-linear, non-convex optimization problem

– With constant probabilities, amount of variables is N1+N1xN2+N1xN2xN3+…,

where Nn = amount of nodes of stage n

– If probabilities are decision variables, problem is even harder

– There are more efficient heuristics available*

Test runs show that the stochastic optimization model is

solvable with e.g. 100 x 100 = 10 000 scenarios (solving time

less than one minute with Lenovo SL500 laptop and CPLEX

12.0).

* See: Hochreiter, R. (2009). Algorithmic aspects of scenario-based multi-stage decision process optimization. In: Rossi, F., Tsoukias, A. (eds.) Algorithmic Decision Theory 2009. LNCS, vol. 5783, pp. 365–376. Springer, Heidelberg.

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1 Evaluating procurement strategies under uncertain demand and risk of component unavailability Anssi Käki and Ahti Salo Systems Analysis Laboratory Aalto