Nagurney Humanitarian Logistics Lecture 9

7/27/2019 Nagurney Humanitarian Logistics Lecture 9

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Lecture 9: Critical Needs Supply Chains

Under Disruptions

Professor Anna Nagurney

John F. Smith Memorial Professor

Director Virtual Center for SupernetworksIsenberg School of Management

University of MassachusettsAmherst, Massachusetts 01003

SCH-MGMT 597LG

Humanitarian Logistics and HealthcareSpring 2012

cAnna Nagurney 2012

Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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Critical Needs Supply Chains Under Disruptions

This lecture is based on the paper by Professors QiangPatrick Qiang and Anna Nagurney entitled, A Bi-Criteria

Indicator to Assess Supply Chain Network Performance forCritical Needs Under Capacity and Demand Disruptions, thatappears in the Special Issue on Network Vulnerability inLarge-Scale Transport Networks, Transportation Research PartA (2012), 46 (5), pp 801-812, where references may be found.

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Outline

Background and Research Motivation

The Supply Chain Network Model for Critical NeedsUnder Disruptions

Performance Measurement of Supply Chain Networks forCritical Needs

Numerical Examples

Conclusions



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Background and Research Motivation

From January to October 2005 alone, an estimated97,490 people were killed in disasters globally; 88,117 ofthem lost their lives because of natural disasters (Braine2006).

Some of the deadliest examples of disasters that havebeen witnessed in the past few years:

September 11 attacks in 2001; The tsunami in South Asia in 2004;

Hurricane Katrina in 2005; Cyclone Nargis in 2008; and The earthquakes in Sichuan, China in 2008. The earthquakes in Japan in 2011.

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Disruptions to Critical Needs Supply Chain

Capacities and Uncertainties in Demands

The winter storm in China in 2008 destroyed crop supplies causing sharp food price inflation.

Overestimation of the demand for certain products resulted ina surplus of supplies with around $81 million of MREs being

destroyed by FEMA.

Of the approximately 1 million individuals evacuated afterKatrina, about 100,000 suffered from diabetes, which requiresdaily medical supplies and caught the logistics chain

completely off-guard.

Thousands of lives could have been saved in the tsunami andother recent disasters if simple, cost-effective measures suchas evacuation training and storage of food and medical

supplies had been put into place.Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare


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Goals of Critical Needs Supply Chains

The goals for humanitarian relief chains, for example,include cost reduction, capital reduction, and serviceimprovement (cf. Beamon and Balcik (2008) and Altayand Green (2006)).

A successful humanitarian operation mitigates theurgent needs of a population with a sustainable reductionof their vulnerability in the shortest amount of time and

with the least amount resources. (Tomasini and VanWassenhove (2004)).

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Critical Needs Supply Chains

The model considers the following important factors: The supply chain capacities may be affected by

disruptions;

The demands may be affected by disruptions;

Disruption scenarios are categorized into two types; and

The organizations (NGOs, government, etc.) responsiblefor ensuring that the demand for the essential product be

met are considering the possible supply chain activities,associated with the product, which are represented by anetwork.



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The Supply Chain Network Topology

fOrganization

1

c

s

Manufacturing at the Plants

M1 M2 MnMf f f

c

dd

dd

c

sc

C

3333333333

Transportation

D1,1 D2,1 DnD,1f f f

c c c

D1,2 D2,2 DnD,2

Distribution Center Storage

Transportation

f f f%

ee

ee

A

C

ee

ee

rrrrrrrrj

ee

ee

s

qf f f fR1 R2 R3 RnR

Demand PointsProfessor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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The Supply Chain Network Model for Critical

Needs Under Disruptions: Case I: Demands Can

Be Satisfied Under DisruptionsWe are referring, in this case, to the disruption scenario set 1.

v1ik

pPwk

x1ip , k= 1, . . . , nR,

1i

1; i = 1, . . . , 1, (1)

where v1ik is the demand at demand point k under disruption

scenario 1i ; k= 1, . . . , nR and i = 1, . . . , 1.

Let f

1i

a denote the flow of the product on link a under disruptionscenario 1i . Hence, we must have the following conservation offlow equations satisfied:

f1i

a = pPx1i

p ap, a L, 1i

1; i = 1, . . . , 1. (2)

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Case I: Demands Can Be Satisfied Under

Disruptions

The total cost on a link, be it a manufacturing/productionlink, a transportation link, or a storage link is assumed to be afunction of the flow of the product on the link We have that

ca = ca(f1ia ), a L,

1i

1; i = 1, . . . , 1. (3)

We further assume that the total cost on each link is convex

and continuously differentiable. We denote the nonnegativecapacity on a link a under disruption scenario 1i by u

1ia ,

a L, 1i 1, with i = 1, . . . , 1.

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Case I: Demands Can Be Satisfied Under

Disruptions

The supply chain network optimization problem for criticalneeds faced by the organization can be expressed as follows:Under the disruption scenario 1i , the organization must solvethe following problem:

Minimize TC1i =aL

ca(f1ia ) (4)

subject to: constraints (1), (2) and

x1ip 0, p P,

1i

1; i = 1, . . . , 1, (5)

f1ia u

1ia , a L. (6)

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Case I: Demands Can Be Satisfied

Denote K1i as the feasible set such that

K1i {(x1i , 1i )|x1i satisfies (1), x1i RnP+ and 1i RnL+ }.

Theorem 1 The optimization problem (4), subject to constraints(1), (2), (5), and (6), is equivalent to the variational inequalityproblem: determine the vector of optimal path flows and the vector

of optimal Lagrange multipliers (x1i , 1i ) K1i , such that:

nRk=1

pPwk

Cp(x

1i )

xp+aL

1i a ap

[x

1ip x

1i p ]+

aL

[u1ia pP

x1i p ap]

[1ia

1i a ] 0, (x

1i , 1i ) K

1i , (7)

where Cp(x

1i )

xpaL

ca(f1i

a )fa

ap for paths

p Pwk; k= 1, . . . , nR.Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare


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The Supply Chain Network Model for Critical

Needs Under Disruptions: Case II: Demands

Cannot Be Satisfied Under Disruptions

The disruption scenario set in this case is 2; that is to say,the optimization problem (4) is not feasible anymore. Amax-flow algorithm can be used to decide how much demandcan be satisfied.

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Performance Measurement of Supply Chain

Networks for Critical Needs

Performance Indicator I: Demands Can Be Satisfied:

For disruption scenario1i , the corresponding network performanceindicator is:

E1i1 (G, c, v

1i ) =TC

1i TC0

TC0, (8)

where TC0 is the minimum total cost obtained as the solution to

the cost minimization problem (4).

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Performance Measurement of Supply Chain

Networks for Critical Needs

Performance Indicator II: Demands Cannot Be Satisfied:

For disruption scenario 2i

, the corresponding network performanceindicator is:

E2i2 (G, c, v

2i ) =TD

2i TSD

2i

TD2i

, (9)

where TSD2i is the total satisfied demand and TD

2i is the total

(actual) demand under disruption scenario 2i .

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Definition: Bi-Criteria Performance Indicator for a

Supply Chain Network for Critical Needs

The performance indicator, E, of a supply chain network forcritical needs under disruption scenario sets 1 and 2 andwith associated probabilities p11 , p12 , . . . , p11

and

p21 , p22 , . . . , p2

2, respectively, is defined as:

E = (1i=1

E1i1 p1i ) + (1 ) (

2i=1

E2i2 p2i ) (10)

where is the weight associated with the network indicatorwhen demands can be satisfied, which has a value between 0and 1. The higher is, the more emphasis is put on the costefficiency.


(


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Modified Projection Method (cf. Korpelevich

(1977) and Nagurney (1993))

Step 0: InitializationSet X

1i

0 K1i . Let T = 1 and set such that 0 < 1

Lwhere

L is the Lipschitz constant for the problem.

Step 1: Computation

Compute X1i

T

by solving the variational inequality subproblem:

< (X1i

T

+ F(X1i

T 1

) X1i

T 1

)T,X1i X

1i

T

>, X1i K

1i .

(11)Step 2: Adaptation

Compute X1i

T

by solving the variational inequality subproblem:

< (X1i

T

+ F(X1i

T 1

) X1i

T 1

)T,X1i X

1i

T

>, X1i K

1i .

(12)


M d fi d P M h d ( f K l h


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Modified Projection Method (cf. Korpelevich

(1977) and Nagurney (1993))

Step 3: Convergence VerificationIf max |X

1iT

l X1iT 1

l | e, for all l, with e> 0, a prespecified

tolerance, then stop; else, set T = T+ 1, and return to Step 1.


N i l E l Th S l Ch i T l


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Numerical Examples The Supply Chain Topologyg1Organization

c

dd

ddd

1

2

3

g g gM1 M2 M3e

ee

ee

s

ee

e

ee

C

4 5 6 7 8 9

g gD1,1 D2,1

c c

10 11

g g

,

D1,2 D2,2

ee

eee

C

ee

eee

s

1213 1415 1617

g g g

R1 R2 R3Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare

T l C F i C i i d S l i f


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Total Cost Functions, Capacities, and Solution for

the Baseline Numerical Example Under No

DisruptionsLink a ca(fa) u

0a f

0a

0a

1 f21 + 2f1 10.00 3.12 0.00

2 .5f22 + f2 10.00 6.88 0.00

3 .5f23 + f3 5.00 5.00 0.93

4 1.5f24 + 2f4 6.00 1.79 0.005 f25 + 3f5 4.00 1.33 0.00

6 f26 + 2f6 4.00 2.88 0.00

7 .5f27 + 2f7 4.00 4.00 0.05

8 .5f28 + 2f8 4.00 4.00 2.70

9 f29 + 5f9 4.00 1.00 0.00

10 .5f210 + 2f10 16.00 8.67 0.00

11 f211 + f11 10.00 6.33 0.00

12 .5f212 + 2f12 2.00 3.76 0.00

13 .5f213 + 5f13 4.00 2.14 0.00

14 f214 4.00 2.76 0.10

15 f215 + 2f15 2.00 1.24 0.00

16 .5f216 + 3f16 4.00 2.86 0.00

17 .5f217 + 2f17 4.00 2.24 0.00


Th A Th Di i S i N i l
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There Are Three Disruption Scenarios Numerical

Example Set I

1 The capacities on the manufacturing links 1 and 2 aredisrupted by 50% and the demands remain unchanged.(Disruption type 1)

2 The capacities on the storage links 10 and 11 are disrupted by

20% and the demands at the demand points 1 and 2 areincreased by 20%. (Disruption type 1)

3 The capacities on links 12 and 15 are decreased by 50% andthe demand at demand point 1 is increased by 100%. The

probabilities associated with these three scenarios are: 0.4,0.3, 0.2, respectively, and the probability of no disruption is0.1.(Disruption type 2)

For 1 and 2, we have: TC11 = 299.02 and TC

12 = 361.41;

therefore, E111 =

TC11TC0

TC0 = 0.0296, E121 =

TC12TC0

TC0 = 0.2444.Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare

Th A Th Di ti S i E l
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There Are Three Disruption Scenarios Example

Set I

For 3, we have that E21 = TD

21TSD21

TD2

1= 0.7000.

We let = 0.2 to reflect the importance of being able to satisfydemands and we compute the bi-criteria supply chain performanceindicator as:

E = 0.2 (0.4 E111 + 0.3 E

121 ) + 0.8 (0.2 E

212 ) = 0.1290.


N i l E l S t II


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Numerical Example Set II

Everything is the same as in Example Set 1 above except that

under the third scenario above, the disruptions have decreasedthe demand by 20% at demand point 1 but have increased thedemand by 20% at demand point 2.


N i l E l S t II


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Numerical Example Set II

It is reasonable to assume that the critical needs demands maymove from one point to another under disruptions and it isimportant for a supply chain to be able to meet the demandsin such scenarios. Indeed, those affected may need to beevacuated to other locations, thereby, altering the associated

demands. Under this scenario, we know that all the demandscan be satisfied and we have that TC

13 = 295.00, which

means that E13 = TC

13TC0

TC0= 0.0157. Hence, given the same

weight as in the First Case, the bi-criteria supply chain

performance indicator is now:

E = 0.2(0.4E111 +0.3E

121 +.2E

131 )+0.8(0.2E

212 ) = 0.0177.


Conclusions
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Conclusions

We developed a supply chain network model for critical needs,

which captures disruptions in capacities associated with thevarious supply chain activities of production, transportation,and storage, as well as those associated with the demands forthe product at the various demand points.

We showed that the governing optimality conditions can beformulated as a variational inequality problem with nicefeatures for numerical solution.

We proposed two distinct supply chain network performanceindicators for critical needs products. We then constructed a

bi-criteria supply chain network performance indicator andused it for the evaluation of distinct supply chain networks.The bi-criteria performance indicator allows for thecomparison of the robustness of different supply chainnetworks under a spectrum of real-world scenarios.

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Nagurney Humanitarian Logistics Lecture 9

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