Supply Contract Allocation

Gyana R. Parija

Bala Ramachandran

IBM T.J. Watson Research Center

INFORMS Miami 2001

A Simple Supply Chain

Suppliers Manufacturer Customers

• Risk management • Uncertain customer demand• Long supply lead time

• Fixed quantity supply contracts• Newsvendor solution: Manufacturer covers the risk of uncertain demand

• Supply contracts with quantity flexibility• Supplier and manufacturer share the risk• Price premium required to cover supplier’s cost• Profit sharing on upside demand potential

Business Issues in Managing Supply Contracts

Buyer can manage a portfolio of supply contracts to hedge againstuncertainty and manage procurement costs

Different kinds of supply sources: - Contracts with different kinds of flexibility (quantity, time …)

- Contracts with different Terms & Conditions- Spot Markets

Trade-offs between flexibility to postpone purchase commitment due to demand variability, supplier quantity discounts etc.

Possibility to mitigate inventory risk by optimizing contract quantities and purchasing excess requirements from spot market

Negotiation of competitive prices with component suppliers and contract manufacturers

Drivers impacting Supply Contracts

Demand Forecast and Volatility

Supplier Price & Quantity Discounts

Spot Market Price Volatility

Inventory Carrying costs

Price Decline costs & Salvage value

Risk Tolerance

Industry Supply Demand Balance

Lead Times & Service levels

Capacity Reservation for Multi-product contracts

Related Research Activities

• Research Issues– Analyzing the costs and benefits associated with supplier flexibility

• Manufacturer/buyer: determining the amount of flexibility needed

• Supplier: determining the price premium to be charged

• Channel coordination

– Developing optimized procurement strategy• utilize updated demand forecasts

• rolling horizon flexibility (e.g., buyer commits to purchase a certain quantity every period)

– Capacity reservation• allocation among multiple suppliers

• utilizing spot market

• Current work– Supplier-Manufacturer flexibility model– Procurement / inventory optimization model with supply flexibility

Strategic Sourcing – Allocation between Suppliers and Marketplaces

Determines quantities for strategic supplier contracts

Mitigates inventory risk by optimizing the contract quantity and purchasing excess requirements from spot market

Minimize  Q1,xi

Subject to: for i = 1,… I

for i = 1,… I

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11

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Solution Methodology

Normality assumptions for demand, spot market price

Analytical expressions derived for expected cost, variance of cost, risk of exceeding budget

Grid Search to identify optimum

Reasonable approach for small number of contracts

1. Analytical Formulation with Grid Search

2. Stochastic Programming with OSL Stochastic Extensions

Discretized probability distributions for demand, spot market price

Linear, Mixed-integer Stochastic Programming Problem

Strategic Sourcing – Allocation between Suppliers and Marketplaces

Eg. One strategic supplier, spot market, Budget constraint

Expected Total Procurement cost

1140.00

1160.00

1180.00

1200.00

1220.00

1240.00

1260.00

1280.00

1300.00

0 500 1000 1500

Contract purchase quantity

E(c

ost

)

Probability of exceeding budget

0.000

0.100

0.200

0.300

0.400

0.500

0 500 1000 1500

Contract Procurement Quantity

Prob

abili

ty

Supply sources – strategic supplier, spot market

Determine contract quantity with strategic supplier such that the risk of the procurement cost exceeding budget is < 5%

COST OPTIMAL SOLUTIONContract Quantity – 900

Expected spot purchase - 140Expected cost - $ 1154

Budget Risk – 27%

RISK OPTIMAL SOLUTIONContract Quantity - 1200

Expected spot purchase - 17Expected cost - $ 1163

Budget Risk – 3%

Contract Portfolio Management

Determines quantities to be procured under different supply contracts,given supplier price schedules

Trade-off between contract flexibility, quantity discounts, and spot market purchases

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for j = 1, …, J

for i = 1, …, I and j = 1, …, J

Contract Portfolio Management

 

 

 

Contract Q<600 600 <= Q < 900 900 <= Q < 1300 Q>= 1300

Long Term Contract

1.2 1.16 1.12 1.07

20% Quantity Flexibility Contract

1.3 1.26 1.21 1.15

 

Supplier Price Schedule

Determine contract quantities with strategic supplier such that the risk of the procurement cost exceeding budget is < 20%

0

160

320

480

640

800

960

1120

1280

0

400

8001000

1200

1400

1600

1800

2000

2200

2400

E(cost)

Fixed Quantity Contract

20% Flexibility Contract

Expected Procurement cost

1000-1200 1200-1400 1400-1600 1600-1800 1800-2000

2000-2200 2200-2400

0

160

320

480

640

800

960

1120

1280

0

42000.10.20.30.40.50.60.70.80.9

1

Prob(cost > Budget)

Fixed Quantity Contract

20% Flexibility Contract

Risk of Exceeding Budget

COST OPTIMAL SOLUTIONFixed Quantity Contract – 920

20% Flexibility Contract - 0Expected cost - $ 1203

Budget Risk – 22 %

RISK OPTIMAL SOLUTIONFixed Quantity Contract – 92020% Flexibility Contract - 160

Expected cost - $ 1253Budget Risk – 14 %

Multi-product Contracts with Business Volume Discounts

Aggregate Capacity Reservation for multiple products

Supplier gives business volume discounts based on overall commitment

Trade-off between business volume discounts, inventory liabilities

Minimize:Qi

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x

Subject to:

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for j = 1, …, J

for j = 1, …, J

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Strategic Sourcing – Determining Contract Reservation Prices

Eg. One strategic supplier, spot market, Budget constraint = 1300Contract Quantity = 900 Risk Tolerance = 25%

Reservation Price = 1.04

Supply risk may be specified by a choice of contract quantity – Q1

Determine contract price for which Q1 is optimal

212*1)()( cQcrc

Expected Total cost

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

1600.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Contract Price

Probability of Exceeding B udget

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 0.5 1 1.5

Contract Price

Optimization Solutions and Library(OSL) Stochastic Extensions• OSL Stochastic Extensions is a set of tools and

functions used to obtain an optimal allocation decision

• To apply here, we linearize the function– Generate a list of representative scenarios

along with their probabilities– Create input SMPS files readable by OSL

Stochastic Extensions• Solve using OSL Stochastic Extensions (C++

interface)• Special structured linear MIP amenable to fast

preprocessing techniques in OSLSE

1 Supplier, 1 price class

• MinQ E[Cost] = E[c.Q + ĉ.(D-Q)+ – v.(Q-D)+ ] Q 0

A nonlinear stochastic program in current state becomes:

• MinQ E[Cost] = E[c.Q + ĉ.P – v.S]

Q + P – S = D

Q, P, S 0where P = (D-Q)+ and S = (Q-D)+

Stochastic Programming Formulation- Single Sourcing

Minimize  Qj,xj

Subject to:

for j = 1,… J

)((1

1rSPcQcE

spj

J

jj

11

J

jj

x

jjjjjxbQxb

1

}1,0{j

x

0,, SPQj

Single sourcing - Allocation between strategic supplier and spot market

Quantity discounts from strategic supplier

Input Data

• Purchase price: $ 1.10/unit• Surplus selling price $ 0.55/unit• 1576 scenarios (demand, spot price)

– Demand ~ normal (1000,200)– Spot price ~ normal (1.5,0.3)

Sample Input

D c Probability of Pair• 1015 1.44 0.001• 1015 1.45 0.000577778• 1016 1.45 0.002111111• 1017 1.45 0.001866667• 1018 1.45 0.001777778• 1019 1.45 0.000644444• 1019 1.46 0.001155556• 1020 1.46 0.002022222• 1021 1.46 0.002• 1022 1.46 0.002266667• 1023 1.46 0.000355556• 1023 1.47 0.0018• 1024 1.47 0.002

OSLSE Driver

• EKKContext *env=ekks_initializeContext();

• EKKStoch *stoch=ekks_newStoch(env,"MyStoch",50000); • int type=ekks_readSMPSData(stoch,"supp.core","supp.time","supp.stoch");• ekks_describeFullModel(stoch,1); • ekks_bendersLSolve(stoch,0);• int numints=ekks_markIntegers(stoch);• EKKModel *model=ekkse_getCurrentModel(stoch);• EKKIntegerPresolve *info=(EKKIntegerPresolve *) malloc(sizeof(EKKIntegerPresolve));• ekk_integerPresolve(model,info,0,0);• ekk_branchAndCut(model,NULL,NULL,info,NULL,5,1);• ekks_printNodeSolution(stoch,1,1,COLUMNS);• ekks_printNodeSolution(stoch,1,2,COLUMNS);• ekks_printObjectiveDistribution(stoch);• ekks_deleteStoch(stoch); • ekks_endContext(env);

Output

• Optimal Quantity: 1087• Expected Cost: $ 1229

j suppliers, k discount ranges

• MinQ E[Cost] = E[j k cjkQjk + ĉ.P–v.S]

subject to k xjk = 1, all j

akminxjk Qjk ak

maxxjk

jkQjk + P – S = D

Qjk, P, S, xjk 0 , xjk is binary

akmin ,ak

max discount range constants

Input Data

• 10 suppliers• 5 discount types

– (800,899), (900,999), …, (1200, 1299)

• 50 price combinations

Output

• Order Quantity = 1271

• Supplier : 2

• Discount Range : 5 ($0.89/unit)

• Surplus of 944 units (scenario 10)

• Optimal (Expected) Cost = $ 910.34

Conclusions

• OSLSE Technology– Provides the right modeling environment for

contract portfolio management problems– Optimization problem resolution in reasonable

times

• Deployment – solution based on this industrial strength solver technology can be easily deployed in any commercially available e-commerce suite

Further Work

• Adding other realistic factors to the model such as

– Budget constraints with allowable Risks

• Knapsack constraint in 0-1 variables in the SP formulation – increase in computational work

– Contract terms and service levels and their effects on the allocation decision

Acknowledgements

• Steve Buckley – IBM Research• Kendra Taylor – Georgia Tech• Markus Ettl – IBM Research• Gelonia Dent - IBM Research

Thank You !

Distribution of Pairs

Distribution of Pairs

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 500 1000 1500 2000

Scenario Number

Pro

bab

ilit

y

Stochastic Programming Formulation – Multiple Sourcing

Minimize  Qij,xij

Subject to: for i = 1,… I

for i = 1,… I & j = 1, … J

)((1 1

rSPcQcEspij

I

iij

J

j

11

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jij

x

ijijijijijxbQxb

1

}1,0{ij

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0,, SPQij

Multiple sourcing - Allocation between suppliers and spot market

Quantity discounts from suppliers