UNHCR Project Report

UNHCR AID DISTRIBUTION

ABSAR AHMAD 2012020

FAIZAN ARIF 2012100

HAMZA BILAL 2012124

OMER ZAMAN 2012296

UNHCR AID DISTRIBUTION | 2

Executive Summary Supply Chain of Aid Distribution organizations are one of the most complex supply chains all over the

world. In this report, we will critically analyze the aid distribution of UNHCR based on various aid

distribution supply chain models. Three models are chosen to analyze the supply chain of UNHCR i.e

Humanitarian Relationship Model, Last Mile Aid Distribution Model and Transshipment Model. Each

model is explained and then compared with UNHCR to compare between ideal model and real time

applied model. The later ones mathematical models and are explained using hypothetical examples. In

the end, conclusion is drawn and some future recommendations to better the aid distribution of UNHCR

are discussed.


TABLE OF CONTENTS

Executive Summary 2

INTRODUCTION 4

THE HUMANITARIAN RELATIONSHIP MODEL 4

Relationship with Government 5

Relationship with Donor 5

Relationship with Military 5

Relationship with NGOs 5

Relationship with Logistic / Other Companies 5

Relationship with Aid Agencies 6

Last Mile Aid Distribution Model 6

Hypothetical Example 7

Model Formulation 7

Application in UNHCR 9

Transshipment Model 9

Objective Function 9

Constraints 10

Analysis 11

Recommendations 11

References 14

WORDCOUNT 15

Appendix 16


INTRODUCTION The Office of the United Nations High Commissioner for Refugees was established on December 14,

1950 by the United Nations General Assembly. The agency is mandated to lead and co-ordinate

international action to protect refugees and resolve refugee problems worldwide. Its primary purpose is

to safeguard the rights and well-being of refugees. It strives to ensure that everyone can exercise the

right to seek asylum and find safe refuge in another State, with the option to return home voluntarily,

integrate locally or to resettle in a third country. It also has a mandate to help stateless people

UNHCR’s priority in Pakistan is to achieve lasting solutions for one of the largest and most protracted

refugee situations in the world. Pakistan continues to host approximately 1.5 million refugees. Most are

from Afghanistan and live in refugee villages and urban areas. Since March 2002, UNHCR has

facilitated the return of approximately 3.9 million registered Afghans from Pakistan.

THE HUMANITARIAN RELATIONSHIP MODEL Humanitarian Relationship model engages very different players, who have a very high degree of

heterogeneity in terms of their interests and expertise. The key players are governments, military, aid

agencies, NGOs, Donors and Logistic Companies. The Government has the power to authorize

operations in case of an emergency. The military provides security in areas of emergency for the

organizations. The military and national aid agencies can start acting on their own without the call by the

government. Within the company category, logistics service providers are excellent contributors at each

stage of a disaster-relief operation

through their logistics and supply

chain management core

capabilities. Since each player

within its own specific role can

provide in-kind donations, in the

humanitarian relationship model

the term ‘‘donor’’ refers to those

who exclusively give financial

means to fund aid operations.

Comparing the structure of

UNHCR’s aid and supply with

the model, we find out that

UNHCR lies in the heart of the

model. In the model where everyone is directly connected to everyone, UNHCR lies in the center of it.

Making this modification, we implement this model on UNHCR.


Relationship with Government

UNHCR’s main governmental counterparts for ref

ugees in Pakistan are the Ministry of States and Fr

ontier Regions (SAFRON), the Chief Commission

erate for Afghan Refugees (CCAR) and the Com

missioneratesfor Afghan Refugees (CARs) in the

Provinces.In addition, UNHCR works with the Na

tional Database and Registration Authority (NAD

RA), the Ministry of Foreign Affairs, Ministry of I

nterior, the Economic Affairs Division (EAD) and

the Ministry of Interior.

Since 2002, UNHCR has facilitated the return of 3.8M refugees.

Relationship with Donor

Relationship with Military

UNHCR's humanitarian activities may be linked to the military in

two ways. First, where law and order are lacking and humanitarian

activities are carried out in an insecure environment, peacekeepers

or other international armed forces may be mandated by the Security

Council to ensure the secure delivery of assistance to the victims of

the conflict in question. Second, military resources may be used to

augment the capacity of UNHCR to implement the High

Commissioner's mandate.

Relationship with NGOs

Relationship with Logistic / Other Companies

The objective of UNHCR’s procurement policy is to provide

the beneficiaries with appropriate quality products or services

at the specified time and place and at the lowest total cost.

National NGOs

Tamer-e-Khalq Foundation

Taraqee Foundation – Pakistan

Frontier Primary Health Care –

Pakistan

Union Aid for Afghan Refugees –

Pakistan

Water, Environment and

Sanitation Society – Pakistan

Women Empowerment Organization

UNHCR is working with the National and International NGO

partners to distribute its items to the refugees and disaster

affected areas.

In 2014 it was working with 18 national and 08 international

NGO partners

International NGOs

Church World Service – USA Council for Community

Development Courage Development

Foundation Dost Welfare Foundation Drugs and Narcotics Educational

Services for Humanity

Contributions to UNHCR for the budget year 2015 (as at 18

September 2015) were $ 2,761,507,854 from over 120 sources

which include both countries and the private donors.

Top Goods Procured in 2014 based on

US Dollar

UNHCR Presence


UNHCR hires consultants and specialized companies and their staff for projects.

UNHCR does not purchase from companies engaged directly or indirectly engaged in sale or

manufacture of anti-personnel mines or any similar operations. It also does not engage with companies

who do not follow the practices consistent with the rights set for th in the Convention on the Rights of

the Child.

UNHCR’s Department of Emergency, Security and Supply (DESS) comprises the Procurement Service

(PS), and the Supply Management and Logistics Service (SMLS). These are responsible for global

supply chain management, operational support, planning and reporting on the use of resources.

Relationship with Aid Agencies

In Badin and Thatta, UNHCR has been

working with the National Rural Support

Programme, a Pakistani aid group, which is

delivering the items and establishing small

tent villages of less than 100 families. The

scarcity of dry land on which to pitch the

tents remains a challenge.

Last Mile Aid Distribution Model

Last Mile Aid Distribution

deals with the final stage of

a humanitarian relief chain.

It is concerned with the

delivery of relief supplies to

the calamity affected people

from a local distribution

hub. Local distribution

center or hub is a small

facility situated near

calamity affected area and

holds some inventory of

relief supplies. When the

disaster occurs, LDC

distributes emergency relief

Procurement in UNHCR


supplies to its corresponding demand locations.

The journal gives us two-phase modeling approach to determine a delivery schedule for each vehicle

and make inventory allocation decisions by considering supply, vehicle capacity, and delivery time

restrictions. Logistical problems in the last mile starts from the limitations related to transportation

resources and emergency supplies, difficulties due to damaged transportation infrastructure, and lack of

coordination among relief actors. Last Mile Distribution Model is a mathematical model which takes

certain input data values (i.e vehicle capacity, vehicle type etc.) and quantitatively analyzes three main

operational decisions. These main decisions are

Amount of supplies to be provided at demand location

Determining the number and type of vehicle to be used

Determining the delivery routes for each vehicle

Last mile distribution model minimizes the transportation cost and unsatisfied or late satisfied customers

keeping in mind time and inventory constraints.

Hypothetical Example

In order to understand this model, let’s take a hypothetical example. First of all, the list of required

variables is listed as follows

Model Formulation

Now, we will consider a

simple one LDC and two node

problem. Let us consider a

disaster has occurred and there

are 2000 affected families at

node 1 and 1000 at node 2.

LDC has two mini trucks with

capacity of 500 units and two

large trucks with capacity of

1000 units. The present

inventory at LDC is 3500 units.

Cost for mini truck from LDC to node 1 is 1000 Rs. and for large truck is 2000 Rs. Similarly, Cost for

mini truck from LDC to node 2 is 1500 Rs. and for large truck is 2500 Rs. It is assumed that it is a one

day problem and 1 family requires 1 unit to satisfy its demands.

Variable Description

K Set of Vehicles

R Set of Routes

N Set of all demand locations

N Set of demand locations visited on route r ∈ R

Crk Cost of route r for vehicle k ∈ K

Qk Capacity of vehicle k ∈ K (volume)

Trk Duration (as a fraction of a day) of route r ∈ R for vehicle k ∈ K

dn Demand at n location

D Total Demand

I Inventory at LDC


Given data

K {1, 2}

R {1, 2}

N 2

C11 1000 rs

C12 2000 rs

C21 1500 rs

C22 2500 rs

Q1 500 units

Q2 1000 units

d1 2000 units

d2 1000 units

D 3000 units

I 3500 units

Now, first we will check our inventory constraint

I >= D

3500 >= 3000

So, our inventory constraint is satisfied.

Now, we will check demand constraint

∑ 𝑄𝑘 ≥ 𝑑𝑛

Since, d1 is 2000 and d2 is 1000 units so we can only have two conditions in which this constraint will be

satisfied.

1ST CONDITION: Two Large trucks go to node 1 and two mini trucks go to node 2.

2ND CONDITION: Two mini trucks & one large truck go to node 1 and one large truck go to node 2.

Now, we will calculate cost for each condition

minimize (Total Cost) = ∑ 𝐶𝑟𝑘

For 1st Condition

Total Cost = 2000 + 2000+ 1500 + 1500

Total Cost = 7000 Rs.


For 2nd Condition

Total Cost = 2000 + 1000+ 1000 + 2500

Total Cost = 6500 Rs.

Since, total cost should be minimized, so we will select 2nd condition to transfer our supplies to the

affected area.

This is how Last mile Aid Distribution Model give us no. of vehicles and route schedules and optimizes

our cost. This was a very simple example. Real life examples are much more complex and require more

input data to optimize solution.

Application in UNHCR

UNHCR does not apply Last Mile Aid Distribution Model in mathematical form rather they apply it

based on their experience. They have contract with local transportation companies that deliver them

transport whenever the need arises. The local drivers are aware of every route and they know the

transportation cost for each route and they select the route with minimum cost. So, UNHCR does not

imply with Last Mile Distribution in mathematical form rather it relies on the experience of its drivers.

Transshipment Model Transshipment model is a mathematical model which is usually used by business organizations for

minimizing the cost of transportations and making an efficient supply chain. Using a transshipment

model in aid distribution process could make the supply chain very cost effective and efficient. In this

mathematical model basically we define an objective function which defines the vision of aid

distribution.

Objective Function

So the objective function of this model is to meet unsatisfied demand accumulated over time. However,

cost cannot be completely ignored and so needs to be minimized and included in the objective function.

The “Humanitarian Organizations Code of Conduct” requires that aid must be delivered based on need

and not cost (Sphere Project, 2004; IRFC, 1994b).


Minimize

𝑤 ∑ 𝑈𝑘𝑖𝑡

𝑘𝑖𝑡

+ 𝑥 ∑ 𝐹𝑘𝑗𝑣

𝑘𝑗𝑣𝑡

𝑉𝑘𝑗𝑣𝑡 + 𝑦 ∑ 𝐼𝑘𝑖𝑡

𝑘𝑖𝑡

+ 𝑧 ∑ 𝑉𝑘𝑗𝑣𝑡 ………………………………………..(1)

𝑘𝑗𝑣𝑡

1

Variable w,x,y,z are basically weights that are defined by organization. For Example, if we put w=1,

x=0.001, y=0.01, z=1 in eq(1) we get

1 ∑ 𝑈𝑘𝑖𝑡

𝑘𝑖𝑡

+ 0.001 ∑ 𝐹𝑘𝑗𝑣

𝑘𝑗𝑣𝑡

𝑉𝑘𝑗𝑣𝑡 + 0.01 ∑ 𝐼𝑘𝑖𝑡

𝑘𝑖𝑡

+ 1 ∑ 𝑉𝑘𝑗𝑣𝑡 ………………………………………..(1∗)

𝑘𝑗𝑣𝑡

this means that new objective function (1*) places much greater priority on minimizing need before

transportation and inventory costs.

The weight values of 1, 0.001, 0.01 and 1 in expression are somewhat arbitrary, and here simply reflect

the relative magnitude each particular component in the policy of the organization. We wished the items

to be delivered to recipients as quickly as possible, hence the high relative weighting given to Pkit Ukit.

The units of measurement of each component must also be taken into account. At first glance it may

appear that there is a greater weighting given to inventory than transport. However, the magnitude of the

transportation costs means that its weighting has to be scaled down in order for transportation costs to

approximately equal inventory costs. The danger with using weightings is that if there was a small

transportation cost and a large inventory then these weightings may be inappropriate leading to the

inventory component having a greater influence than the transportation costs weighting. The weightings

have been verified so that this problem should not occur.

Constraints

To establish initial conditions, constraint (A) and (B) specifies the current unsatisfied demand and

inventory of each item at each node, i.e., at the end of day 0:

𝑈𝑘𝑖0 = 𝑈𝑘𝑖0 … … . … . . (𝐴)

𝐼𝑘𝑖0 = 𝐼𝑘𝑖0 … … . … . . (𝐵)

Each time the model is re-run during the course of the relief operation, the values of parameters Uki0 and

Iki0 would be updated to reflect the situation at the time of planning.

𝑇𝑗𝑘𝑣𝑖𝑡 = 𝑇𝑗𝑘𝑣𝑖𝑡0 … … . … . . (𝐶)

Constraint (C) defines the amount of each item sent between pairs of nodes before the disaster has

occurred but which are still in transit and so have not yet arrived at the destination node.

1 all the variables & indices are defined and described in nomenclature.


∑ 𝑇𝑗𝑘𝑣𝑖𝑡 − 𝐿𝑗𝑘𝑣

𝑗𝑣

+ 𝐼𝑘𝑖,𝑡−1 + 𝑈𝑘𝑖,𝑡−1 = ∑ 𝑇𝑗𝑘𝑣𝑖𝑡 + 𝐼𝑘𝑖𝑡

𝑗𝑣

− 𝑈𝑘𝑖𝑡 + 𝐷𝑘𝑖𝑡 … … . … . . (𝐷)

Constraint (D) ensures that no more items are sent than are received and/or taken from the local

inventory. Constraint (D) states that, on any given day and at any given node, the items arriving from

previous nodes, together with the items inherited from the previous day’s ending inventory, less the

unmet demand from the previous day, should equal in quantity the items sent to the next nodes, plus the

demand and the amount put into inventory, less any unsatisfied demand.

𝑉𝑘𝑗𝑣𝑡 ≥0.001 ∑ 𝑇𝑘𝑗𝑣𝑖𝑡𝑊𝑖𝑖

𝑉𝐶𝑣… … . … . . (𝐸)

Constraint (E) ensures that there are enough vehicles to transport items between nodes. The coefficient

0.001 converts the item weights in kilograms to metric tons, the units of vehicle capacity.

0.001 ∑ 𝐼𝑘𝑖𝑡𝑊𝑖 ≤ 𝐶𝑘

𝑖

… … . … . . (𝐹)

Constraint (F) ensures that the weight of all inventory items is within the node capacity limits, again

multiplying by 0.001 to convert from kilograms to metric tons.

Thus our objective function (1) should be minimized under constraints (A-F).

Analysis

After applying these models, it is analyzed that UNHCR is following all three models. Since UNHCR

has direct relations with government of Pakistan, Humanitarian Relationship model is being fully

implemented as military and other aid agencies too have relationship with UNHCR. The two

mathematical models described are not implemented in mathematical form rather they are applied based

on the experience of the drivers.

Recommendations As described in the report, UNHCR does not follow any mathematical model. It distributes its supplies

with the transportation companies and transportation cost is minimized by qualitative analysis of drivers.

The efficiency of aid distribution can be increased significantly if UNHCR apply any mathematical

model. Data should be gathered for all the routes and available vehicles. Excel sheets should be

generated to optimize transportation cost in any given condition. This will be very helpful and efficient

for UNHCR if they are able to apply it.


Journal Article 2 Summary

Title & reference “Humanitarian Logistics and Supply Chain Management” A. Cozzolino, Humanitarian

Logistics

Topic Humanitarian Relationship Model

Main Argument Determining the relationship between different organizations working in a very different

aspects in a disaster / refugee affected area

Finding & most

interesting point

The close relationship between governments, organizations, military and different aid

agencies work in close proximity towards providing aid to refugees and disaster affected

areas

Methodology Qualitative: The level of relationship between different players determine an efficient

distribution of necessary aid to the Individuals

Implications This model is implied such that each players works together to achieve a better future of the

people affected

Relation to our

report

Our report focusses on relation of UNHCR with various different players, it works with to

provide better aid and facilities to the Refugees


Title &

reference

“A transshipment model for distribution and inventory relocation under

uncertainty in humanitarian operations” in “Socio-Economic Planning Sciences” Journal

by Beate Rottkemper and others.

Topic A mathematical model for managing a disaster

Main Argument Unsatisfied need should be satisfied first irrespective of cost or any other factor.

Finding & most

interesting point

It is interesting that a mathematical model that is normally used in business could also be

used for aid distribution.

Methodology Qualitative and Quantitative based on secondary data

Implications Applied on a real time scenario

Relation to our

report

We have used this model to minimize the cost for a continuous running aid programe.



Title & reference Last Mile Distribution in Humanitarian Relief

Journal of Intelligent Transportation Systems, 12(2):51–63, 2008

Topic Last Mile Distribution in Humanitarian Relief

Main Argument The main objective of this journal is to minimize the sum of transportation

Finding & most

interesting point

It is interesting to note that transportation system of aid organizations can also be made

highly cost efficient.

Methodology A quantitative two-phase modeling approach to determine a delivery schedule for each

vehicle and make inventory allocation decisions by considering supply, vehicle capacity,

and delivery time restrictions.

Implications It is applied in UNHCR on experience basis not in mathematical form

Relation to our

report

We critically analyzed aid distribution of UNHCR based on Last Mile Distribution Model


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Word Count= 2307


APPENDIX

Humanitarian Relationship Model ................................................................................................ 17

TRANSSHIPMENT MODEL ...................................................................................................... 18

Abstract: .......................................................................................................................................................... 18

Last Mile Distribution in Humanitarian Relief ............................................................................. 21

SUMMARY ....................................................................................................................................................... 21

Mathematical Modeling ................................................................................................................ 22

Sets .................................................................................................................................................................. 22

Routing parameters ......................................................................................................................................... 22

Demand parameters ........................................................................................................................................ 23

Routing decision variables ................................................................................................................................ 23

Delivery decision variables ............................................................................................................................... 23


Humanitarian Relationship Model

Humanitarian relief-operation management engages very different players, who may have a high degree

of heterogeneity in terms of culture, purposes, interests, mandates, capacity, and logistics expertise. Key

players can be categorized as follow: governments, the military, aid agencies, donors, non-governmental

organizations (NGOs), and private sector companies—among which logistics service providers are

preeminent Governments—host governments, neighboring country governments, and other country

governments within the international community—are the activators of humanitarian logistics stream

after a disaster strikes since they

have the power to authorize

operations and mobilize

resources. In fact, without the

host government authorization,

no other player—with the

exception of national aid

agencies and the military—can

operate in the disaster theater.

On many occasions, the military

has been a very important actor

since soldiers are called upon to

provide primary assistance (i.e.,

hospital and camp installation,

telecommunications, and route

repair) thanks to their high

planning and logistic

capabilities.

Aid agencies are actors through

which governments are able to

alleviate the suffering caused by disasters. The largest agencies are global actors, but there are also many

small regional and country-specific aid agencies. Since each player within its own specific role can

provide in-kind donations, in the humanitarian relationship model the term ‘‘donor’’ refers to those who

exclusively give financial means to fund aid operations. Thus, in addition to country-specific funding

provided by governments in recent years, foundations, individual donors, and companies have become

important sources of funds for aid agencies.

Companies are capable of providing technological support and logistics staff and managers. They also

provide specific services that may no longer be available on the ground immediately after a disaster has

occurred, such as electricity supply, engineering solutions, banking support, and postal services.

Initially, companies are moved to participate in humanitarian efforts because they have observed that

enormous losses are inflicted when disasters interrupt the flow of their business; so they invest in re-


establishing their business continuity. Working to alleviate the economic impact of such disruptions

‘‘makes good business sense’’.

Within the company category, logistics service providers are excellent contributors at each stage of a

disaster-relief operation through their logistics and supply chain management core capabilities. Leading

international logistics service providers, such as Agility, DHL, FedEx, Maersk, TNT, and UPS, have

raised their importance in terms of the resources, assets, and knowledge shared with their humanitarian

counterparts.

TRANSSHIPMENT MODEL

Abstract:

The number of disasters and humanitarian crises which trigger humanitarian operations is ever-

expanding. Unforeseen incidents frequently occur in the aftermath of a disaster, when humanitarian

organizations are already in action. These incidents can lead to sudden changes in demand. As fast

delivery of relief items to the affected regions is crucial, the obvious reaction would be to deliver them

from neighboring regions. Yet, this may incur future shortages in those regions as well. Hence, an

integrated relocation and distribution planning approach is required, considering current demand and

possible future developments.

For this situation, a mixed-integer programming model is developed containing two objectives:

minimization of unsatisfied demand and minimization of operational costs. The model is solved by a

rolling horizon solution method. To model uncertainty, demand is split into certain demand which is

known, and uncertain demand which occurs with a specific probability. Periodically increasing penalty

costs are introduced for the unsatisfied certain and uncertain demand. A sensitivity analysis of the

penalty costs for unsatisfied uncertain demand is accomplished to study the trade-off between demand

satisfaction and logistical costs. The results for an example case show that unsatisfied demand can be

significantly reduced, while operational costs increased only slightly.



Nomenclature for Transshipment model

Sr No

Indices Description

1. t The index t represents a day within the planning horizon. Thus t = 1, ..., 10 means a

ten-day horizon. t = 1 means current day. t = 0 means day before today & t = -1 the day before that.

2. i The index i represent items transported.

3. v v denotes the type of vehicle.

4. j,k The indices j and k represents the network nodes in the supply chain such as

suppliers, airports, seaports, terminals, warehouses, and recipients etc.

Sr No

Variables Description

1. Capt available capacity time in each period t.

2. Iki0 current initial inventory at node k of item i (i.e., at the end of day 0)

3. Dkit demand at node k for item i on day t

4. Uki0 current unsatisfied demand at node k for item i

5. V Cv maximum capacity that each vehicle v can carry in metric tonnes

6. VAkvt number of vehicles available at node k of a vehicle type v on day t (integer)

7. Lkjv lead time from node k to node j using a vehicle of type v

8. Tkjvit0 amount of item i already sent from node k to node j, using a vehicle of type v, i at

past times t =−Lkjv,...,0


9. Fkjv fixed transport cost to include driver, fuel, maintenance etc. of using each vehicle

type v between node k and node j

10. Wi weight of each item i

11. Ck capacity limits of each node k

12. Tkjvit amount of item i sent from node k to node j, using vehicle type v, on day t

13. Vkjvt number of vehicles of type v sent from node k to node j on day t (integer)

14. Ikit inventory at node k of item i at the end of day t

15. Ukit unsatisfied demand (backlog) at node k for item i at the end of day t

Last Mile Distribution in Humanitarian Relief SUMMARY

Last mile distribution is the final stage of the relief chain; it refers to delivery of relief supplies from LDCs to the

people in the affected areas. LDC maybe a tent, a prefabricated unit, or an existing facility.

The journal gives us two-phase modeling approach to determine a delivery schedule for each vehicle and make

inventory allocation decisions by considering supply, vehicle capacity, and delivery time restrictions. Logistical

problems in the last mile starts from the limitations related to transportation resources and emergency supplies,

difficulties due to damaged transportation infrastructure, and lack of coordination among relief actors. The main

operational decisions related to last mile distribution are relief supply allocation, vehicle delivery scheduling, and

vehicle routing. The problems that arise during disaster relief operations may differ depending on various factors,

such as the type, impact, and location of the disaster, and local conditions in the affected regions.

As resource allocation and vehicle routing decisions are closely interrelated, they should be jointly considered. In

this respect, the last mile distribution problem is a variant of the inventory routing problem (IRP).

The main objective of this journal is to minimize the sum of transportation costs and penalty costs for unsatisfied

and late-satisfied demand. The last mile distribution problem determines the best resource allocation among

potential aid recipients in disaster affected areas that minimize the cost of logistics operations, while maximizing

the benefits to aid recipients. More specifically, the last mile distribution problem determines

delivery schedules

vehicle routes

the amount of emergency supplies delivered to demand locations during disaster relief

operations.


The required set of items required may vary greatly by situation depending on factors such as the type and the

impact of the disaster, demographics, and social and economic conditions of the area. We can categorize our

demand as Type1- and Type 2-based on their demand characteristics. Type 1 items are critical items for which

the demand occurs once at the beginning of the planning horizon It includes emergency supplies. Type 1

demands within a short period of time, due primarily to supply unavailability and vehicle capacity limitations. Once

Type 1 items arrive to the demand locations, they are immediately distributed to aid recipients. Therefore, we

assume that no Type 1 inventory is held at any demand location. Type 2 items are items that are consumed

regularly and whose demand occurs periodically over the planning horizon. Type 2 items are shipped to a

demand point, the excess amount can be held for consumption in future periods. We assume that any inventory

holding costs to store these items at the demand points is ignored, because it is likely negligible in relation to the

penalty costs associated with unsatisfied demand. The model serves an “equal allocation principle,” which

allocates supplies proportionally among the demand locations based on demand amounts and population

vulnerabilities, and balances the unsatisfied and late-satisfied demand among demand locations over time. The

relief system is likely unable to optimize the vehicle fleet, in terms of number, capacity, and compatibility after an

emergency. Hence, we assume that the vehicle fleet is comprised of a limited number of vehicles with different

characteristics. Each vehicle can be differentiated based on capacity, speed, and compatibility with various arcs in

the network. In practice, smaller trucks may be used to reach remote areas as roads may be poor or nonexistent,

while larger trucks may be used for areas that are closer and move easily reached. Demand parameters in our

model are based on the assessments done by relief agencies in the affected regions after the disaster

occurrence. The planning horizon parameter used in our model will be the worst case estimate. The length of the

planning horizon must be set to a length much longer than the expected relief horizon; the model will determine

when the delivery of relief supplies will be completed.

Mathematical Modeling

Following variables will be used in the model

Sets

T is set of days in the planning horizon; length of planning horizon.

K is set of vehicles.

R set of routes.

N set of all demand locations.

N set of demand locations visited on route r ∈ R.

E set of demand types: E = {1,2}.

Routing parameters

𝐶𝑟𝑘 is cost of route r for vehicle k ∈ K.


𝑞𝑘 capacity of vehicle k ∈ K (volume).

𝑇𝑟𝑘 duration (as a fraction of a day) of route r ∈ R for vehicle k ∈ K (from Phase 1).

Demand parameters

𝑑𝑖1 demand of type 1 at location i ∈ N (volume per planning horizon).

𝑑𝑖2 demand of type 2 at location i ∈ N on day t ∈ N (volume per day).

𝑝𝑖𝑡1 penalty cost factor for unsatisfied type 1 demand at location i ∈ N by day t ∈ T.

𝑝𝑖𝑡2 penalty cost factor for unsatisfied type 2 demand at location i ∈ N on day t ∈ T.

𝑎𝑡𝑒 amount of type e ∈ E relief supplies arriving to the LDC at the beginning of day t ∈ T.

Routing decision variables

𝑋𝑟𝑡𝑘= 1 if route r ∈ R is used by vehicle k ∈ K on day t ∈ T

0 otherwise.

Delivery decision variables

Yirtke amount of demand of type e ∈ E delivered to location i ∈ N on day t ∈ T by vehicle k ∈ K via

route r ∈ R

𝑊𝑖𝑒 penalty cost associated with unsatisfied type e ∈ E demand on day t ∈ T.

𝑆𝑖𝑡1 fraction of unsatisfied type 1 demand at location i ∈ N by day t ∈ T.

𝑆𝑖𝑡2 fraction of unsatisfied type 2 demand at location on day i ∈ N on day t ∈ T.

𝐼𝑖𝑡2 inventory level of type 2 at location i ∈ N at the beginning of day t ∈ T.


The basic function is given as:


The objective function (1a) minimizes the sum of routing costs and penalty costs for the backordered

Type 1 demand and for the lost Type 2 demand. Constraints (1b) determine the maximum penalty cost

for each day and for each item type. Constraints (1c) find the fraction of unsatisfied (backordered) Type

1 demand at a location over time while (1d) find the fraction of unsatisfied (lost) Type 2 demand at a

location on a day. Constraints (1e) guarantee that the entire Type 1 demand is satisfied by the end of the

planning horizon. Constraints (1f) ensure that the total amount of relief items of each type delivered to

all. Locations on a day is less than or equal to the amount of supplies available at the LDC. Constraints

(1g) and (1h) are vehicle capacity constraints and vehicle time constraints, respectively. Constraints (1i)

ensure that the fraction of unsatisfied demand is between zero and one. Constraints (1j) set the beginning

inventory level to zero at each location for Type 2 items. Constraints (1k) and (1l) are non-negativity

constraints, and (1m) define the binary routing variable.

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