nagurney humanitarian logistics lecture 9
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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
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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)).
Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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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
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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.
Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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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)
Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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.
Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
N i l E l Th S l Ch i T l
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Numerical Examples The Supply Chain Topologyg1Organization
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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
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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.
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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.
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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.
Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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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