Instituto Tecnológico y de Estudios Superiores de Monterrey
Campus Monterrey
School of Engineering and Sciences
A stochastic mixed-integer preparedness model for disaster resource prepositioning
A thesis presented by
Adrian Fernando Rivera Morales Submitted to the
School of Engineering and Sciences
in partial fulfillment of the requirements for the degree of Master of Science
In
Engineering Science
Monterrey Nuevo León, May 15th, 2020
Dedication This thesis is dedicated to:
• My parents for always supporting every decision I make and guiding me to become a better person.
• My brothers for giving me the example and encouraging me to fulfill my goals, both academic and in sports.
• All my friends, both from track and field and from classes. Thanks to them I enjoyed every minute living in Monterrey.
• To God for giving me the opportunity of living this moment.
Acknowledgements
I would like to express my deepest gratitude to all those who have been side by side with me along this journey:
• My advisor, Dr. Neale R. Smith, who helped me to give every step in the right direction and being the best advisor. I honestly recommend any student to work with him, he is a great advisor and knows exactly what must be done.
• Dr. Angel Ruiz and Dr. David Güemes who helped me learn from my mistakes and made possible and excellent work.
• My program director, Dr. Oliver Probst, who accepted me in his program and offered me the academic scholarship.
• Esteban Ogazon, who was basically my mentor my first year of the master’s degree and become a good friend. Thanks to him it was easy to move forward and understand what I had to do.
• The sport’s department of Tecnológico de Monterrey, especially to my coach Francisco Olivares who offered me the scholarship of my career and then of my master’s degree. It would have not been possible to study at this institute without his support.
• Finally, to Tecnológico de Monterrey for the support on tuition and CONACyT with the support for living, without this it would have been very difficult to complete my master’s degree.
A stochastic mixed-integer preparedness model for disaster resource prepositioning
by
Adrian Fernando Rivera Morales Abstract
A key strategic issue in pre-disaster planning for humanitarian logistics is the pre- establishment of adequate capacity and resources that enable efficient relief operations.
This work presents a scenario-based stochastic mixed-integer optimization model to guide the allocation of budget to acquire and position relief assets given a variable demand in each disaster zone, decisions that correspond to the preparedness phase, well in advance before a disaster strikes. The optimization focuses on minimizing the maximum amount of missing resources among the demand locations by transporting resources from storage centers to the affected municipalities. The model was applied to a real case disaster, were the solution presented by the model turned out to be a better option than how they actually solved it. Further comparisons and analyses are presented.
List of Figures
• Figure 2.1 Banks are distributed across 26 of the 32 states in Mexico………18
• Figure 4.1 Real case node network………37
• Figure 4.2 Node network of the experiment………...38
• Figure 4.3 Hurricane Odile’s probability of land impact………38
• Figure 4.4 Scenario 1………39
• Figure 4.5 Scenario 2………40
• Figure 4.6 Scenario 3………40
• Figure 4.7 Scenario 4………41
• Figure 4.8 Scenario 5………42
• Figure 4.9 Stochastic Solution……….42
• Figure 6.1 Pareto chart of the standardized effects for missing resources…………..62
• Figure 6.2 Pareto chart of the standardized effects for total cost………65
List of Tables
• Table 2.1. Number of research articles per journal……….22
• Table 2.2 Comparison of articles………23
• Table 4.1 Basic municipality information………36
• Table 5.1 Resources are not affected by the disaster………..46
• Table 5.2 Resources are lost if the disaster strikes the storage centers………47
• Table 5.3 Real case analysis………...50
• Table 5.4 Average of 5 scenarios………52
• Table 5.5 Average of 5 scenarios plus real case………..53
• Table 6.1 Experimental factors………56
• Table 6.2 Results of the experiments……….57
• Table 6.3 Results for missing resources………....60
• Table 6.4 ANOVA for missing resources………61
• Table 6.5 Results of the total cost………..63
• Table 6.6 ANOVA for total cost………...64
• Table A. Abbreviations and acronyms………68
• Table B. Variables and symbols………..69
Contents
Abstract……….7
List of Figures……….8
List of Tables………...9
Chapter 1 – Introduction……….11
Chapter 2 - Theoretical Framework……….16
Chapter 3 - Model Description………..25
Chapter 4 - Case Study………...31
Chapter 5 - Additional Analyses………….……….40
Chapter 6 - Design of Experiments……….49
Chapter 7 - Conclusion and Recommendations for Future Research………...60
Appendix A……….62
Appendix B……….63
Appendix C……….64
Bibliography………..78
Chapter 1
Introduction
1.1 Motivation
Natural disasters like earthquakes or hurricanes often cause a high degree of damage.
In the last 10 years, the frequency of occurrence of natural disasters and the devastating impact they cause have shown almost exponential growth [1]. This fact can be attributed mostly to poor urbanization strategies and the worsening effects of climate change [2]. In 2017, 335 natural disasters affected over 95.6 million people, killing 9,697 and costing a total of US $335 billion [3]. It is expected that these alarming numbers increase in future years, due to the increment of natural disasters.
The problems caused by natural disasters are countless. People affected by disasters rely on life-saving assistance from humanitarian organizations such as food, shelter, water, etc. The demand for resources and requirements of the victims is an unknown fact, which depends on the location, timing, type and intensity of each disaster [4]. However, a common objective of every relief operation is to deliver aid on time. In other words, similar to traditional supply chains, humanitarian supply chains must be designed to provide ‘‘the right supplies in the right quantities to the right locations at the right time” [5].
Humanitarian logistics is formally defined by Thomas and Mizushima [6] as “the process of planning, implementing and controlling the efficient, cost-effective flow and storage of goods and materials, as well as related information, from the point of origin to the point of consumption for the purpose of meeting the end beneficiary’s requirements”. In humanitarian terms, the end beneficiaries are the people affected by disasters.
Effective logistics management is essential to achieve these objectives. Humanitarian logistics involves procurement, warehousing, inventory management, transportation, and distribution functions, and is the most expensive part of disaster relief operations [7]. Due to the unique characteristics of humanitarian settings, the policies and models developed
inventories [8]. The most important features that make the difference between commercial and humanitarian supply chains, according to Balcik [9], are listed below:
• Objectives. Efficiency is usually the most important objective in commercial inventory management, while satisfying beneficiary needs is always the utmost priority for humanitarian organizations.
• Demand. There is a high level of uncertainty associated with timing, location, type and amount of demand in most humanitarian settings
• Infrastructure. The network infrastructure is generally reliable in commercial settings, whereas post-disaster network may be damaged and involve uncertainties.
• Financial resources. Humanitarian organizations collect most of the donations after disaster occurrences. Moreover, some donations may be earmarked for certain operations and needs. Commercial organizations usually have fewer constraints in allocating budgets for inventory-related expenses.
Due to these particular characteristics, humanitarian operations and crisis/disaster management (HOCM) have increasingly attracted the attention of researchers and practitioners. As a result, there has been an escalating interest in how to improve HOCM [10]. One important aspect, which helps determine the total number of fatalities after a disaster, is the performance of the first-response teams. The quality of the relief efforts can be improved by an effective use of the available technical resources. Because the time, quantity, and quality of the resources are limiting factors, emergency managers need to find an optimal schedule for assigning resources in space and time to the affected areas.
1.2 Problem Statement and Context
This research proposes a scenario-based stochastic mixed-integer optimization model to guide the allocation of a limited budget to acquire and position relief assets, which refer to the donated resources for helping the disaster’s victims, given a stochastic variable demand in each disaster zone to minimize the maximum amount of missing resources among the demand locations.
The costs considered are the cost of transportation, vehicle acquisition, inventory management, and the assignment of warehouses to be used to respond to the disaster.
The model was applied to Hurricane Odile, which is one of the most intense landfalling tropical cyclones in the state of Baja California Sur, Mexico in 2014. This preparedness model deals with the strategic planning and resource allocation for humanitarian aid once it is known a disaster will strike and have an idea of the affected areas
1.3 Research Question
The main research questions are:
• How many assets need to be in place in anticipation of a disaster?
• Where should they be located? We refer to this task as “prepositioning” of assets such as warehouses, medical facilities, available transportation, and temporary shelter space.
The term prepositioning is also commonly employed to refer to the storage of supplies near a potential area in anticipation of an imminent disaster, but we use it in its long-term, strategic sense.
1.4 Solution overview
The research conducted focuses in developing and perfecting a scenario-based stochastic mixed-integer optimization model with the objective of minimizing the maximum amount of missing resources among the demand locations by transporting resources from storage centers to the affected municipalities. In order to achieve this objective in a correct way, the following activities must be done:
• Formulate the humanitarian operation of the crisis management problem (HOCM).
• Determine the performance of the solution compared with a real case and quality metrics.
• Suggest a stochastic solution considering distinct scenarios trying to anticipate where and how the disaster could strike.
• Get enough information on a real case to apply the stochastic model.
• Compare the results obtained with the model and how the real case was solved.
• Formulate a mathematical model to represent different disasters.
• Define characteristics that instances must have to represent the disaster scenarios accurately.
• Program an instance generator.
• Generate several instances to evaluate the solutions.
• Solve the instances with the stochastic model.
• Analyze the solutions.
1.5 Main Contributions
This research involves both practical and theoretical work. Bancos de Alimentos de México (BAMX) could benefit people from suffering by applying the final output of this study. Within the theoretical framework, there is a good contribution regarding humanitarian aid operations.
1.6 Dissertation Organization
The structure of this dissertation includes seven chapters and three appendixes. Chapter 1 presents the introduction of this research. First the motivation, which offers a straightforward background of humanitarian logistics and why it is an important topic to focus on. Then the problem statement is detailed and explained briefly. In addition to these, this chapter presents some research questions, the solution overview, and the main contributions.
Chapter 2 contains the literature review. In this chapter, some published articles are compared and used to evaluate the best features, decisions, and objective functions that a HOCM model must consider.
Chapter 3 is the model description. In this chapter, the details of the stochastic model are explained along with its sets, variables, decision variables and intermediate variables.
The objective function is also described with its respective constraints.
Chapter 4 presents a case study. In this chapter, thanks to BAMX director, we could use data from a real case in Baja California Sur back in 2014, which is used to compare our stochastic model versus the real data.
Chapter 5 present some additional approaches. In this chapter, the real data and our stochastic model is compared to other analysis commonly used in the optimization field.
Chapter 6 presents the design of experiments. In this chapter, a fractional factorial design 2k-p is applied to the data in order to determine which factors affect more the missing resources and the total cost.
Finally, chapter 7 presents conclusions and recommendations for future research.
Appendix A is for abbreviations and acronyms, appendix B for variables and symbols, and appendix C for results and an instance example.
Chapter 2
Theoretical Framework
The purpose of this chapter is to document the state of art regarding the humanitarian operation and crisis management (HOCM) problem, the distinct features and characteristics that HOCM models include, and the solution methods with these models.
Additionally, a brief description of the operations that Bancos de Alimentos de Mexico (BAMX) performs.
By analyzing the characteristics of several HOCM models, it was possible to build a stochastic model that had the best of each model. Many algorithms are presented and a detailed explanation of which features where more relevant for constructing our model.
2.1 BAMX disaster planning problem
BAMX is a Mexican non-profit civil organization, co-founder and member of the Global FoodBanking Network (GFN). With more than 50 food Banks distributed across the country, BAMX is the only food bank network in México and the second largest in the world.
BAMX is focused on rescuing food that is at risk of being wasted by industries, fields, supply centers, supermarkets, restaurants and hotels. Around 60% of the distributed food is fruits and vegetables. The other 40% is mainly composed of grains, groceries and proteins. More than 25,000 persons work in the network, of which 90% are volunteers.
This benefits more than 1’137,000 Mexicans in food poverty.
In addition to rescuing and providing food assistance, BAMX executes various development projects in the communities, focused on:
• Hunger awareness-raising events for the population.
• Developing a model of food education that aims to promote good nutrition.
• Channeling the talent, aspirations and competences of the volunteers.
• Promoting the donation culture among collaborators of the companies.
Figure 2.1 illustrates all the foodbanks that are in Mexico, pointing out each of them with an apple (a red dot), where the states are represented by distinct colors, which represents the geographic region in the country.
Figure 2.1 Banks are distributed across 26 of the 32 states in Mexico
2.1.1 Individual banks operations
Banks inside BAMX network operate independently from each other in a local framework.
These operations include the planning and management of the food donations, its storage in the bank warehouse, and the delivery to the beneficiaries. Each bank has a different number and types of cargo vehicles, in addition to a workforce composed primarily of volunteers.
The network has an estimate of the number of people in food poverty in each region, from which a fixed demand is calculated and assigned to the various banks.
Donations come mainly from frequent donors that deliver the resources directly to the bank’s facilities. The management of each of these donors and their product is assigned to a specific bank by considering the distance, response time, product demand within the zone, and logistic capabilities. The network has agreements with different transportation companies that include the donation of the service, when there is available capacity in the route. Furthermore, community leaders may organize the population to do the collection of resources at the bank's facilities.
Finally, each bank has the responsibility to deliver reports and acknowledgments to the donors of the resources they have received.
2.1.2 BAMX logistic management operations
Besides the local procedures of the banks, there are situations that go beyond their scope of infrastructural and/or organizational capabilities. BAMX central facilities are in charge of coordinating the network logistic decisions in the following situations:
• A large donation comes from a company that is collaborating for the first time, so it has not been assigned to a specific bank.
• The donation is too large to be received at the assigned bank’s facilities.
• The assigned bank does not have the proper transportation or storage qualities to manage the food supplied.
• The resources supplied are not compatible with the needs or customs of the population of the assigned bank’s area.
• The demand of a basic resource cannot be fulfilled by the assigned bank.
• A disaster causes the demand in the region to increase rapidly beyond the local bank´s capacity.
2.1.3 The disaster planning problem description
When a disaster or catastrophe arises, BAMX joins the efforts of the Mexican society to support the relief distribution activities in the affected communities. This often requires urgent changes in the operational planning already established by each individual bank.
Consequentially, many of the situations listed in the previous section take place. The increasing demand of the affected communities can not be satisfied by local banks and there is an unpredictable increase in donations in each region, depending on the nature of the disaster and the perception of its impact by the society. This generates an atmosphere of confusion and chaos for the staff of the banks, which do not have a well- prepared plan for contingencies.
BAMX central logistic department focuses its efforts on organizing the new assignments of the donations driven by the disaster scenario, taking into consideration the need to satisfy the demand of food of the communities they already serve. The following considerations restrict these decisions.
• Every supply must be processed in a bank before being delivered to a community.
This includes documentation, validation of the food quality, and the assembly of basic food packages.
• The affected community may be located in a region that is not regularly attended by the network.
• Food can be stored for a very limited amount of time. Most of the rescued food supplies will not be consumable a few days after they are received.
• Banks have a limited storage capacity and may not have the infrastructure to manage certain types of food (e.g., lack of refrigeration).
• Cargo vehicles can not be relocated to other banks and may be already assigned to a route.
• Only land vehicles are available.
• Donations must be collected as soon as possible, because companies offering
• The time span of the operation may vary from a few days to a few weeks.
• The number of people working in each bank (including volunteers) is assumed to be fixed.
• Roads may not be accessible as a consequence of the disaster.
• Banks may be inoperative as a consequence of a catastrophe.
However, BAMX also has favorable aspects to support their decision making.
• Some disaster scenarios can be known or predicted shortly before they occur.
• Demand levels are already known.
• Donors may offer to deliver the food to the bank’s facilities or to a closer location.
• The donation of transportation services tends to rise.
• Extra transportation capacity can be outsourced at a reduced cost.
• Community leaders and military personnel aid in the delivery process once the supplies reach the affected location.
• The affected community may provide some vehicles to pick up the food at the bank´s facilities.
In order to build a model that can be used in a practical way, some of these considerations are included.
2.2 Analysis of HOCM models
Since the study of this topic is relatively new, the models vary a lot from one another.
Because there are many variables to consider in a model regarding HOCM, the objectives, features, parameters, and decisions diverge in each model.
Two databases were considered to select the articles to be reviewed: Scopus, which according to Elsevier is the largest abstract and citation database of peer-reviewed literature, and ScienceDirect, which is Elsevier’s leading information solution for researchers.
In both databases, the first filter was searching for the words “humanitarian logistics resource allocation” and selecting the publications of the last 3 years (2016, 2017, and 2018). Since these databases search for many types of publications, the next filter was to select just “research articles”, to exclude other review articles, book chapters, conference papers, etc.
After these filters, there were 129 articles in ScienceDirect and 253 articles in Scopus, so the next filter was to select the most important journals on this topic, for which the criteria used for the selection was based on the number of articles written per journal. In table 2.1 you can see the name of the journal and the number of articles in each database.
Table 2.1. Number of research articles per journal
Journal Name
Number of Articles in Scopus
Number of Articles in ScienceDirect International Journal of Disaster Risk Reduction 19 18
European Journal of Operational Research 16 18
Transportation Research: Logistics and Transportation 13 10
Annals of Operations Research 8 -
Production and Operations Management 8 5
International Journal of Production 5 5
Transportation Research Procedia - 4
It was an interesting fact that for both databases the first three journals with more publications were:
• International Journal of Disaster Risk Reduction
• European Journal of Operational Research
• Transportation Research Part E Logistics and Transportation
Therefore, the analysis was done with articles taken from these three journals in both databases. The final filter was to select the articles that included an HOCM model. A total of 24 articles in Scopus and 22 articles in ScienceDirect (some of them repeated) from
After analyzing all these articles, the ones that were more similar and more appropriate to represent the humanitarian operations in Mexico were selected. A total of 12 research articles concerning this topic were deeply analyzed, comparing distinct features, decisions and objective functions for each article’s model. Table 2.2, includes a list of the articles with their corresponding characteristics, where a “1” means that the article includes that characteristic in its model and “-“ means it does not.
Table 2.2 Comparison of articles
The table was done focusing on the main characteristics presented by each article. When two or more articles included the same feature, decision variable or objective function, this new characteristic was added to the table.
Since some characteristics might not be clear enough with just one word, here is the definition for each named characteristic:
Features
• Lineal: the model is linear, which means that it does not include mathematical operations between 2 or more variables in the same restriction.
• Multiperiod: the model is applied for distinct periods in the disaster (prevention, mitigation, preparedness, response, recovery and re-habilitation).
• Multiproduct: considers distinct types of commodities.
• Multimodal: considers distinct types of vehicles.
• Travel time: considers the travel time from department place to the destination.
• Scenarios: the model is applied to distinct scenarios.
• Connectivity disruption: considers that the roads can be unavailable due to the impact of the disaster.
• Inventory disruption: considers that existing inventory can be lost during the disaster.
• Disaster type: for which disasters can the model be applied.
Decisions
• Facility location: the model decides the optimal node (place) to open a facility.
• Transportation: the model decides from where to where transport the commodities (node to node)
• Inventory level: the model decides how much inventory should be assigned to each facility
• Vehicle location: the model decides the number of vehicles assigned to each node.
Objective Function
• Quantity: represents the number of objective functions for each model
• Min Unmet demand: minimize the missing demand in each node
• Max Server demand: maximize the given demand in each node
• Min Response time: minimize de response time for satisfying the demand
• Min Cost: minimize the total cost
• Min Suffering: minimize the suffering of people (humanitarian logistics refers to
“suffering” relating to the unmet demand on average). For example, if the total demand is 100% and it can only provide 80%, it will satisfy 80% of the demand for each node to reduce suffering on average, instead of satisfying 100% of the demand for 80% of the nodes.
• Other: for any other objective function which was not relevant or included in another article.
The last row of Table 1 shows the number of articles that include each feature, decision variables and objective functions, where the highest values can be considered the most relevant features to determine the decision variables according to its objective function.
Until recently, the major thrust of emergency planners was on operational details.
However, this cannot be accomplished without commitments, the most important of which is the pre-establishment of adequate capacity and resources that enable an efficient response.
The most important feature or at least the one that most of them included was the travel time. This makes sense since every time a disaster occurs, help comes from other cities (referred as nodes in the models) so the travel time is a critical factor to satisfy the demand on time. Another important feature is the scenario, which I personally agree with because the impact of each disaster is unpredictable, so it is necessary to give an optimal solution for distinct types of scenarios.
The features with a smaller number of articles are the connectivity and inventory disruption. This is an area of opportunity to make a unique model. These two characteristics are important enough to include them in a HOCM model, because it is not possible to assume that every road will be available between the nodes or that the inventory will be intact after a disaster takes place.
The decisions section is the most important of all the characteristics, since these are the problems that your HOCM model will solve in an optimal way. Knowing this, the four characteristics of the decisions should be included in an HOCM model. Definitely this will make a more complex model which will be harder to exemplify, but it will be worth it.
In the objective function section, it was surprising that most of the articles focused their objective on minimizing the cost. Of course, this is an important objective, but when there is a life-saving situation, spending less money should not be a concern. For this reason, I think that the best objective function to include in our model is to minimize the unmet
demand, that at the same time it will reduce suffering. Certainly, no humanitarian operation has an unlimited budget, so it should be included but as a constraint.
Given this explanation, Chapter 3 presents the stochastic model of this research including these analyses and observations.
Chapter 3
Model Description
This section presents a scenario-based stochastic mixed-integer optimization model to minimize the maximum amount of missing resources among the demand locations that can be transported between zones. The model utilizes discrete decision variables. Before proceeding, a set of assumptions underpinning the model are established in order to limit its scope and to obtain a manageable set of variables. It is important to mention that our model focuses in minimizing the amount of missing resources, but it can minimize the total cost by adding the value obtained of unmet demand as a parameter and changing the objective function to minimizing the budget. The main assumptions are: (1) there is one decision-maker authority with control of the budget, preparedness plan, and response operations, (2) an area is considered as a territorial division, like municipalities or towns, in which a demand for resources can occur and is capable of storing and transporting commodities, (3) while the decision variables were selected to represent the preparedness phase, the model also includes some elements from the response phase.
It is assumed that these decisions, such as route selection, will be re-optimized once it is required, (4) the model is presented as a single objective problem but is designed to permit adding the maximum response time as a secondary minimization objective, and (5) transportation routes will be selected to satisfy cost and time constraints. These selections will be respected, if possible, in the decision making of the response phase.
The model corresponds to the preparedness phase, which means just before the disaster strikes and it can be possible to anticipate approximately where the disaster will strike.
The model should be implemented, in order to obtain better results, two days before the disaster strikes, and follow up through the subsequent days for any changes that have to be done.
Due to the brief time for preparation, the inventory will be referred to as a capacity level.
The donations will not be stored long enough to call it inventory, but it will be necessary to prepare the storage centers with enough capacity to handle certain amount of resources, which will have an impact in the total cost.
For the sets: nodes with storage capacity (L) represent the storage centers that can be used. The demand nodes (M) represent the municipalities that are affected by the disaster. The distribution nodes (N) represent the intermediate municipalities that are used to send resources from L nodes to M nodes. The disaster scenarios (δ) are used to represent different outcomes of the disaster’s impact. The available arcs in the node network will vary from scenario to scenario according to the disaster’s impact zone, which represent the disrupted roads.
For the parameters: transportation cost per weight (tons) sent from node i to node j (Cij) represents the cost that is generated from gasoline, road fees, etc. The cost of storing a unit in node i (Ei) represents the cost of capacity. The storage capacity (tons) in node i (Fi) represents two different things according to the scenario. The first scenario would be one in which the storage capacity is less than the total donations given to that node. In this case, the total units that can be assigned to that node would be restricted by the storage capacity of such node. The second scenario would be one in which the storage capacity is greater than the total donations given to that node. In this case, the total units that can be assigned to that node would be restricted by the total donations to that node.
The time required to travel from node i to node j (Tij) represents the distance between nodes. The demand in node i in scenario s (𝑫𝒊𝒔) represents the total amount of help that is required in each node according to the intensity of the disaster. Vehicle capacity (VCA) works as a constraint to determine how many units can be transported by each vehicle.
The weight of each unit of food aid (W) determines the total amount of units that can fit in each vehicle. The cost of vehicle (VCO) represents the cost of acquiring a vehicle that can then be used to transport units from nodes to nodes. The cost of enabling node i as a warehouse (BC) represents the cost of rent, storage, etc. of a place in each node. The maximum delivery time (TM) works as a restriction of the time it takes from a unit to get
from the storage center to the demand node. The budget (P) works as a constraint that considers the cost of transportation, vehicle acquisition, capacity management and warehouses establishment. Constants (Mt) and (Mf) are relatively large constants compared to (Tij) and (Fij) for modeling purposes only. The probability that scenario s occurs (𝜽𝒔) is used in the objective function as a variable to weigh the maximum proportion of commodity shortage among all areas in each scenario and minimize it.
For the decision variables: the units of food aid assigned to node i (qi) according to its capacity or total donations, the number of vehicles assigned to node i (vi) in order to be able to transport all the food aid to the demand nodes, and if node i is used as a storage center (bi) equals 1, otherwise this equals zero.
For the intermediate variables: (𝒓𝒊𝒋𝒎𝒔 ) represent the units sent from node 𝒊 to node 𝒋 that were stored in node 𝒎 in scenario s, (𝒚𝒔) represents the maximum proportion of commodity shortage among all areas in scenario s, (𝒙𝒔) represents the maximum transportation time needed among all routes in scenario s, (𝒂𝒊𝒋𝒎𝒔 ) is a binary variable that when arc (𝒊, 𝒋) is used to send units that were stored in node 𝒎 in scenario s, equals one;
otherwise it equals zero, and (𝒘𝒋𝒎𝒔 ) represents the latest time that a unit that was stored in node 𝒎, arrives at node 𝒋 in scenario s.
Notation
Sets
𝑳 nodes with capacity 𝑴 demand nodes 𝑵 distribution nodes δ disaster scenarios
Parameters
𝑪𝒊𝒋 transportation cost per weight (𝒕𝒐𝒏𝒔) sent from node 𝒊 to node 𝒋 𝑬𝒊 cost of assigning a unit in node 𝒊
𝑭𝒊 capacity level (𝒕𝒐𝒏𝒔) in node 𝒊
𝑻𝒊𝒋 time required to travel from node 𝒊 to node 𝒋 𝑫𝒊𝒔 demand in node 𝒊 in scenario 𝒔
𝑽𝑪𝑨 vehicle capacity (𝒕𝒐𝒏𝒔) 𝑾 weight of each unit (𝒕𝒐𝒏𝒔) 𝑽𝑪𝑶 cost of vehicle
𝑩𝑪𝒊 cost of enabling node 𝒊 as a warehouse
𝑻𝑴 maximum arrival time allowed after the disaster occurs 𝑷 budget
𝑴𝒕 relatively large constant compared to 𝑻𝒊𝒋 values 𝑴𝒇 relatively large constant compared to 𝑭𝒊𝒋 values 𝜽𝒔 probability that scenario 𝒔 occurs
Decision variables
𝒒𝒊 units assigned to node 𝒊
𝒗𝒊 number of vehicles in each node
𝒃𝒊 if node 𝒊 is used as a warehouse, this equals one; otherwise this equals zero
Intermediate variables
𝒓𝒊𝒋𝒎𝒔 units sent from node 𝒊 to node 𝒋 that were stored in node 𝒎, in scenario s 𝒚𝒔 maximum proportion of commodity shortage among all areas in scenario s 𝒙𝒔 maximum transportation time needed among all routes in scenario s
𝒂𝒊𝒋𝒎𝒔 if the arc (𝒊, 𝒋) is used to send units that were stored in node 𝒎 in scenario s, this equals one; otherwise this equals zero
𝒘𝒋𝒎𝒔 latest time that a unit that was stored in node 𝒎, arrives at node 𝒋 in scenario s
The model is structured as follows:
𝐦𝐢𝐧𝐢𝐦𝐢𝐳𝐞 ∑ 𝜽𝒔𝒚𝒔
𝒔∈𝜹
(1)
𝑺𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐:
∑ 𝒓𝒊𝒋𝒊𝒔
𝒋∈𝑵
− 𝒒𝒊 ≤ 𝟎 𝒊 ∈ 𝑳, 𝒔 ∈ 𝜹 (2)
∑ 𝒒𝒊
𝒊∈𝑵
− 𝒃𝒊𝑭𝒊≤ 𝟎 𝒊 ∈ 𝑳 (3)
∑ ∑ 𝒓𝒊𝒋𝒎𝒔
𝒎∈𝑵 𝒋∈𝑵
− ∑ ∑ 𝒓𝒋𝒊𝒎𝒔 − 𝒒𝒊
𝒎∈𝑵 𝒋∈𝑵
≤ 𝟎 𝒊 ∈ 𝑵, 𝒔 ∈ 𝜹 (4)
𝒘𝒋𝒋𝒔 = 𝟎 𝒋 ∈ 𝑵, 𝒔 ∈ 𝜹 (5)
𝒘𝒋𝒎𝒔 − 𝑻𝒊𝒋𝒂𝒊𝒋𝒎𝒔 − 𝒘𝒊𝒎𝒔 ≥ 𝟎 𝒊, 𝒋, 𝒎 ∈ 𝑵(𝒋 ≠ 𝒎), 𝒔 ∈ 𝜹 (6)
𝒙𝒔− 𝒘𝒋𝒎𝒔 ≥ 𝟎 𝒋, 𝒎 ∈ 𝑵, 𝒔 ∈ 𝜹 (7)
𝒓𝒊𝒋𝒎𝒔 − 𝑴𝒇𝒂𝒊𝒋𝒎𝒔 ≤ 𝟎 𝒊, 𝒋, 𝒎 ∈ 𝑵, 𝒔 ∈ 𝜹 (8)
𝒚𝒔 ≥ 𝑫𝒊𝒔+ ∑𝒋∈𝑵∑𝒎∈𝑵𝒓𝒊𝒋𝒎𝒔 − 𝒒𝒊− ∑𝒋∈𝑵∑𝒎∈𝑵𝒓𝒋𝒊𝒎𝒔
𝑫𝒊𝒔 𝒊 ∈ 𝑴, 𝒔 ∈ 𝜹 (9)
𝒒𝒊𝑾 − 𝒗𝒊𝑽𝑪𝑨 ≤ 𝟎 𝒊 ∈ 𝑳 (10)
∑ ∑ ∑ 𝑪𝒊𝒋𝒓𝒊𝒋𝒎𝒔 𝑾
𝒎∈𝑵 𝒋∈𝑵 𝒊∈𝑵
+ ∑ 𝒒𝒊𝑬𝒊
𝒊∈𝑵
+ ∑ 𝒗𝒊𝑽𝑪𝑶
𝒊∈𝑵
+ ∑ 𝒃𝒊𝑩𝑪𝒊
𝒊∈𝑵
≤ 𝑷 𝒊 ∈ 𝑵, 𝒔 ∈ 𝜹
(11)
𝒙𝒔 ≤ 𝑻𝑴 𝒔 ∈ 𝜹 (12)
𝒂𝒊𝒋𝒎𝒔 , 𝒃𝒊 ∈ {𝟎, 𝟏} (13)
𝒓𝒊𝒋𝒎𝒔 , 𝒒𝒊, 𝒗𝒊∈ 𝒁 ≥ 𝟎 (14)
𝒙𝒔, 𝒚𝒔, 𝒘𝒋𝒎𝒔 ≥ 𝟎 (15)
The objective function (1) is used to minimize the maximum proportion of missing resources among all areas. Constrains (2) ensure that the exit of commodities tagged as originated from a certain area are not greater than the capacity level assigned to that area. Constrains (3) limit the storage capacity. Constrains (4) ensure that the exit flow of a node is not greater than the capacity level plus the entering flow. Constrains (5) are used to set the transportation time to zero when a commodity exits its original warehouse.
Constrains (6) determine the transportation time needed for a commodity to reach an area when it comes from a different node. When a unit from a certain origin does not arrive to a given node, the traveled time remains the same as the last time that commodity was transported. Additionally, these constraints set the arrival time to the latest one, when the units arrive from different areas. In constrains (7), the maximum arrival time among all routes is assigned to 𝒙. This is made in order to add 𝒙 to an objective function in the future. For the time being, it is limited to a value 𝑻 in constraint (12). Constrains (8) determine a unit transported on a certain arc. Constrains (9) determine the maximum proportion of missing units among all areas by the difference between the demand and the final quantity the units, after all transportation is completed. Constrains (10) establish the number of vehicles required in each location given the transportation needs.
Constraint (11) calculates the cost of transportation, vehicle acquisition, capacity
management and warehouses establishment, and restricts it to a budget P. Finally, the declaration of binary and integer variables is presented (12-15).
Chapter 4
Case Study
To validate the model, Hurricane Odile in the state of Baja California Sur, Mexico, was used as a case study. Hurricane Odile is tied for the most intense landfalling tropical cyclone in the Baja California Peninsula during 2014 [23]. Thanks to BAMX (by its acronym in Spanish “Banco de Alimentos de Mexico”), we could obtain enough information from their logistics planning manager to model the scenario. A map of the state of Baja California Sur and border states was used to generate the node network.
The proposed model was then used to determine the nodes that should be used as storage centers, the quantity of units of food aid to be stored in each storage center and the number of vehicles to be assigned to each node in order to transport the food aid to the demand nodes.
Table 4.1 shows the corresponding municipality for each of the node’s number in the network. The state of each of the municipalities is specified as well as the total population, which is used as a parameter to determine the demand in such node when a disaster strikes it.
Figure 4.1 illustrates the node network for the real case scenario. The storage centers used were Los Mochis, Culiacan, Mazatlan, Puerto Vallarta, and Guadalajara, represented by green nodes. The damaged municipalities where El Caribe, Todos los Santos, Vistahermosa, and El Vado, represented by red nodes. Finally, intermediate municipalities near the disaster zone are represented by gray nodes.
Table 4.1 Basic municipality information
Number Municipality State Population
1 Topolobampo Sinaloa 6,361
2 Mazatlan Sinaloa 502,547
3 Vistahermosa Baja California Sur 11,600
4 Los Barriles Baja California Sur 1,056
5 Tijuana Baja California 1,301,000
6 Torreon Coahuila 608,836
7 Hermosillo Sonora 812,229
8 La Paz Baja California Sur 244,219
9 El Vado Baja California Sur 11,600
10 Culiacan Sinaloa 675,773
11 Cabo San Lucas Baja California Sur 81,111
12 Puerto Vallarta Jalisco 203,342
13 Zacatecas Zacatecas 1,579,000
14 El Caribe Baja California Sur 49,600
15 Chihuahua Chihuahua 809,232
16 San Jose del Cabo Baja California Sur 93,069
17 Los Mochis Sinaloa 231,977
18 Ballenas Baja California Sur 11,600
19 Todos los Santos Baja California Sur 6,485
20 Guadalajara Jalisco 1,495,000
21 Cabo Pulmo Baja California Sur 5,800
22 San Juan Baja California Sur 5,300
Figure 4.1 Real case node network
For the real scenario, nodes (2) Mazatlan, (10) Culiacan, (12) Puerto Vallarta, (17) Los Mochis, and (20) Guadalajara were used as the default storage centers and after more than 70 hours of coordination and relocation, the affected families received the aid donated by many Mexicans [24].
Now, for the implementation of our model, we considered all the possible storage centers near the disaster zone as options to be used for this case. Now, nodes (5) Tijuana, (6) Torreon, (7) Hermosillo, (13) Zacatecas, and (15) Chihuahua can also be used as storage centers. This let us with the following network:
Figure 4.2 exemplifies the node network with the possible storage centers to be used indicated by yellow nodes and the intermediate municipalities by gray nodes.
Figure 4.2 Node network of the experiment
A 5-scenario instance was solved to obtain a solution. For each scenario, the damaged nodes and the disrupted roads were modified according to the probabilities of hurricane Odile’s impact zone illustrated in Figure 4.3, trying to anticipate where and how the hurricane could strike.
Figure 4.3 Hurricane Odile’s probability of land impact
For scenario 1, it was assumed that Hurricane Odile affected all the peninsula of BCS, affecting a total of 10 nodes. For the following node networks, damaged municipalities are represented by red nodes, possible storage center by yellow nodes, and intermediate municipalities by gray nodes. Figure 4.4 illustrates the node network for scenario 1.
Figure 4.4 Scenario 1
For scenario 2, the impact zone was reduced to the tip of the peninsula of BCS, affecting only the 6 municipalities that are closer to the ocean. Figure 4.5 shows the node network for scenario 2.
Figure 4.5 Scenario 2
For scenario 3, it was considered that Hurricane Odile followed its path through the Gulf of Baja California (between the states) and affected the first two municipalities, which are Mazatlán (2) and Puerto Vallarta (12). Figure 4.6 illustrates the node network for scenario 3.
Figure 4.6 Scenario 3
For scenario 4, it was considered as well that Hurricane Odile followed its path through the Gulf of Baja California, but now affected the inner states of the state of BCS, affecting a total of 4 municipalities. Figure 4.7 shows the node network for scenario 4.
Figure 4.7 Scenario 4
Finally, for scenario 5 it was assumed that Hurricane Odile continues all the way through the Gulf of Baja California and affected the municipalities where the gulf ends. Figure 4.8 illustrates the node network for scenario 5.
Figure 4.8 Scenario 5
After solving our mixed-integer model considering these 5 scenarios, we obtained the following results shown in Figure 4.9, where the proposed storage centers are represented by blue nodes and the intermediate municipalities by gray nodes.
For this instance, node (7) Hermosillo is a better option than node (20) Guadalajara, which was used in the real case. After implementing the proposed solution with the real data, we obtained that the total amount of missing resources was equal to zero with a total cost of $392,328. In the real case, the missing resources were also zero, but the total cost was
$408,750, meaning our model helped to save $16,422.
Chapter 5
Additional Analyses
The results of two analyses under two different assumptions are presented. For Table 5.1, we assume that the resources in the storage centers will always be available, even though the disaster strikes in those nodes. For Table 5.2, if the selected storage centers are affected by the disaster, the resources in that nodes will be considered lost.
In the first row of each table, the scenario “Real case” shows how the actual disaster was solved. This is useful information to compare with our model’s solution and validate the value of our model.
In rows 2-7, scenarios “Optimal solution” represents the best solution for each one of the 5 scenarios independently and the real case. These solutions are known as “Solution under perfect information”, which are the best way to solve each scenario. For scenarios 1, 2, and 4, the optimal solution is the same since the disaster does not affect any nodes used as storage centers. For scenarios 3 and 5, the optimal solution is better in case 1 since nodes affected by the disaster are used as storage centers, saving transportation and vehicle cost.
In rows 8-13, scenarios “Stochastic solution” represents a stochastic solution considering the 5 scenarios with their respective probability of occurrence, and then evaluating each scenario individually and the real case with this solution. This solution is the best we can get with our model when we can not rely on perfect information and must consider several scenarios. In these scenarios, the costs in both tables are the same since the solution does not change, but for scenario 3 and 5 in case 2, a lot of resources are lost, letting to a high number of missing resources.
In rows 14-19, scenarios “Expected Value” are used as an alternative solution which is commonly used. The first step is to take the results for “Optimal solution” and average them to obtain a solution that considers all the scenarios. Then, this solution is evaluated in each one of the scenarios individually and the real case to obtain the results presented in the table. This analysis is known as the Expected Value Solution. For these scenarios, the same case as in the stochastics solutions occurs for scenarios 3 and 5.
Finally, in rows 20-25, scenarios “Expected Demand” are also used as an alternative solution. In this case, it is necessary to create an additional scenario that considers the average demand in each one of the scenarios. Then, we obtain the optimal solution for this scenario, and evaluate each scenario individually and the real case to obtain the corresponding values. This analysis is known as the Expected Demand Solution. Only scenarios 3 and 5 differ in each table, due to the same explanation as for stochastic solutions.
For all these analyses, in order to obtain the minimum total cost, it was necessary to solve the model and then include the values for unmet demand as a parameter, which will work as a restriction in constrain (9), and finally changing the objective function to minimizing the total cost. This way, it is possible to assure that both, minimum unmet demand and total cost, are considered for the analyses.
Table 5.1 Resources are not affected by the disaster
Scenario Budget Stored
Units
Storage Centers
Missing Resources
Real case $408,750 200 5 0
Optimal solution real case $387,907 200 5 0
Optimal solution scenario 1 $398,995 218 5 0.3207 Optimal solution scenario 2 $398,753 218 5 0.0164 Optimal solution scenario 3 $104,681 215 5 0 Optimal solution scenario 4 $222,885 136 4 0 Optimal solution scenario 5 $128,891 213 5 0 Stochastic solution for real case $392,328 218 5 0 Stochastic solution for scenario 1 $398,795 218 5 0.3207 Stochastic solution for scenario 2 $398,641 218 5 0.0164 Stochastic solution for scenario 3 $147,580 218 5 0 Stochastic solution for scenario 4 $224,841 218 5 0 Stochastic solution for scenario 5 $187,166 218 5 0
Expected Value for real case $452,917 197 8 0.0259 Expected Value for scenario 1 $434,767 197 8 0.3774 Expected Value for scenario 2 $457,092 197 8 0.1639 Expected Value for scenario 3 $171,962 197 8 0.0840 Expected Value for scenario 4 $262,934 197 8 0 Expected Value for scenario 5 $268,254 197 8 0.0833 Expected Demand for real case $409,193 216 6 0.0379 Expected Demand for scenario 1 $410,459 216 6 0.3279 Expected Demand for scenario 2 $418,669 216 6 0.1111 Expected Demand for scenario 3 $178,317 216 6 0 Expected Demand for scenario 4 $223,805 216 6 0 Expected Demand for scenario 5 $180,583 216 6 0
Table 5.2 Resources are lost if the disaster strikes the storage centers
Scenario Budget Stored
Units
Lost Units
Storage Centers
Missing Resources
Real case $408,750 200 0 5 0
Optimal solution real case $387,907 200 0 5 0
Optimal solution scenario 1 $398,995 218 0 5 0.3207 Optimal solution scenario 2 $398,753 218 0 5 0.0164
Optimal solution scenario 3 $181,674 215 0 5 0
Optimal solution scenario 4 $222,885 136 0 4 0
Optimal solution scenario 5 $330,839 213 0 5 0
Stochastic solution for real case $392,328 218 0 5 0 Stochastic solution for scenario 1 $398,795 218 0 5 0.3207 Stochastic solution for scenario 2 $398,641 218 0 5 0.0164 Stochastic solution for scenario 3 $147,580 218 89 5 0.4033 Stochastic solution for scenario 4 $224,841 218 0 5 0 Stochastic solution for scenario 5 $187,166 218 87 5 0.3944
Expected Value for real case $452,917 197 0 8 0.0259 Expected Value for scenario 1 $434,767 197 0 8 0.3774 Expected Value for scenario 2 $457,092 197 0 8 0.1639 Expected Value for scenario 3 $171,962 197 51 8 0.3229
Expected Value for scenario 4 $262,934 197 0 8 0
Expected Value for scenario 5 $268,254 197 51 8 0.3239 Expected Demand for real case $409,193 216 0 6 0.0379 Expected Demand for scenario 1 $410,459 216 0 6 0.3279 Expected Demand for scenario 2 $418,669 216 0 6 0.1111 Expected Demand for scenario 3 $178,317 216 73 6 0.3361
Expected Demand for scenario 4 $223,805 216 0 6 0
Expected Demand for scenario 5 $180,583 216 101 6 0.4667
5.1 Solution under perfect information
When working with perfect information, it is assumed that the director knows beforehand exactly where the disaster will strike and what nodes will be affected. If this is the case, the director would then take an optimal solution for this specific scenario. This would take a total cost of $387,907 allocating 200 resources in 5 different storage centers, letting to a total of zero missing resources, which is the optimal solution for the real case. This is the cost and suffering incurred under perfect information when the nodes affected by the future disasters are perfectly known.
In the traditional analysis, when you can have a close guess of what will really happen considering scenarios, the average of the results of solving each scenario individually under perfect information is used. This analysis was done considering the average of the 5 possible scenarios, and then adding the real case, making it 6 scenarios.
For the 5 scenarios in table 5.1, we obtain an average cost of $250,841 with 0.0674 missing resources. Adding the results for the real case, the average cost is then $273,685 with 0.0562 missing resources.
For the 5 scenarios in table 5.2, we obtain an average cost of $306,629 with 0.0674 missing resources. Adding the results for the real case, the average cost is then $320,176 with 0.0562 missing resources.
5.2 Expected Value Solution
Another common approach to determine the resources to allocate in the respective storage centers is to allocate the average resources in each storage center for each scenario, when they are solved individually. To obtain this value, we get the average resources that were allocated in each scenario when solved independently and evaluate the performance of this information as the solution for the real case. A total of 8 nodes
this solution in the real case, we obtain a total cost of $452,917 with a total of 0.0259 missing resources. This is known as the expected value solution.
The previous analysis could be done since we have knowledge of the real scenario. In the traditional method, the expected value solution is used to solve each scenario individually, and then get an average of these results. The average was calculated considering the 5 scenarios first, and then adding the real case.
For the 5 scenarios in table 5.1, we obtain an average cost of $319,002 with 0.1417 missing resources. Adding the results for the real case, the average cost is then $341,321 and 0.1224 missing resources.
For the 5 scenarios in table 5.2, we obtain an average cost of $319,002 with 0.2376 missing resources. Adding the results for the real case, the average cost is then $341,321 with 0.2023 missing resources.
5.3 Solution under expected demand
One more approach is to solve the problem with the average of the demand of the 5 scenarios. In this case, an additional scenario was created considering the average demand in each node in each of the 5 scenarios, and then the optimal solution was found using the model. The solution includes a total of 6 nodes, allocating 216 resources and using 26 vehicles. Evaluating this solution for the real case, we obtain a total cost of
$409,193 with 0.0379 missing resources.
As in the case of the expected value solution, this analysis could be done since we have knowledge of the real scenario. In the traditional analysis, the solution obtained with the expected demand is evaluated in each scenario individually, and then the average of the solutions is calculated. As well, the average was calculated considering the 5 scenarios first, and then adding the real case.
For the 5 scenarios in table 5.1, we obtain an average cost of $282,367 with 0.0878 missing resources. Adding the real case, the average cost is $303,504 with 0.0795 missing resources. These results will be used in section 4.4.
For the 5 scenarios in table 5.2, we obtain an average cost of $282,367 with 0.2484 missing resources. Adding the real case, the average cost is $303,504 with 0.2133 missing resources.
5.4 Comparisons
In order to quantify the value of our stochastic model, the solutions obtained with the three different approaches are compared with our stochastic solution. The result of the real case is added for an additional comparison. The first comparisons will be considering the real case. Table 5.3 shows the result obtained with our scenario-based stochastic mixed- integer optimization model and the results of the previous analyses applied to the real case, including the real case solution:
Table 5.3 Real case analysis
Analysis Budget Missing Resources
Real Case $408,750 0
Stochastic solution $392,328 0
Perfect Information $387,907 0
Expected Value $452,917 0.0259
Expected Demand $409,193 0.0379
For any circumstance, the approach under perfect information will always be the best possible solution, since you can know exactly what will happen. Comparing this result with the stochastic solution, we get a difference of $4,421 and zero missing resources.
This confirms that our stochastic model is an excellent solution considering that it doesn’t knows exactly what will happen.
The expected value is not ideal for this case, since it suggests using too many storage centers. Comparing it with our stochastic model, we subtract the expected value, which is a total cost of $452,917 and 0.0259 suffering, minus the stochastic solution, $392,328 and 0 suffering, and equals $60,589 and 0.0259 suffering. This is the benefit of solving the stochastic model over the expected value approach, which confirms that this approach is not good since it has a significant difference.
The expected demand is a better analysis than the expected value, having 0.012 more missing resources but using $43,724 less budget. Comparing it with our stochastic model, we get the difference of the expected demand, which is $409,193 with 0.0379 missing resources, minus the stochastic solution, which is $392,328 with 0 missing resources and equals $16,865 and 0.0379 suffering. This is the benefit of solving the stochastic model over the expected demand solution. As we can see, the difference is much lower than for the expected value analysis, which makes this a better method for this specific case.
As how the real case was managed, our stochastic model would have presented a benefit of $16,422 maintaining the zero missing resources. If we could have known exactly the damage caused by the disaster, the savings would have been $20,843. The expected value and the expected demand analyses are not recommended for this case, since the solution presented for both approaches are not better than the solution for the real case.
So far, the evaluations have been only considering the real case. This is possible since we have knowledge of the real scenario and have data to do the comparisons. In the traditional analysis, the way to do these comparisons is by getting an average of possible outcomes, which in our case are represented by scenarios.
In table 5.1 and table 5.2, the results of each of the analyses previously mentioned are shown for each one of the scenarios. In order to follow the traditional method, the average of the 5 scenarios for each analysis was obtained, which are shown in the following table considering each case:
Table 5.4 Average of 5 scenarios Case 1
(resources are not lost)
Case 2
(resources are lost) Analysis Budget Missing
Resources Budget Missing Resources Perfect Information $250,841 0.0562 $306,629 0.0562 Stochastic Solution $271,405 0.0562 $271,405 0.1891 Expected Value $319,002 0.1224 $319,002 0.2023 Expected Demand $282,367 0.0795 $282,367 0.2133
In case 1, the solution under perfect information is the best in both less budget and less missing resources. The stochastic solution obtained with our model is a very good approach, since the missing resources are the same that under perfect information, and there is only a $20,564 difference in budget, which is not bad considering the information in each approach.
In case 2, the budget used under perfect information is greater than the one used for the stochastic solution and even than for expected demand, but the amount of missing resources is much lower, which is the primary concern in our model. In this case, comparing perfect information and our stochastic solution, the average budget is $35,224 more under perfect information but 0.1329 less missing resources, which is a reasonable difference.
For the expected value approach, there is a significant difference comparing it to the stochastic solution. The reason is that in this type of problem, it is not ideal to use too many storage centers due to transportation logistics, and for the expected value there are used 8 different storage centers, while for the stochastic solution are used 5. For both cases, the difference in the budget is the same since the solution does not change, only the missing resources. The difference for the average cost is $47,597, but in case 1 the missing resources are 0.0662 more under expected value and for case 2 are 0.0132
more. The difference in both budget and missing resources let us to conclude that this approach is not useful for this kind of problem.
With the expected demand approach, we obtained better results than fwith the expected value approach. For both cases, the difference in the budget is the same, only the missing resources change. The average cost is $10,962 less for our stochastic solution than for the expected demand. In case 1, the missing resources are 0.0233 less in the stochastic solution and in case 2 are 0.0242 less. These results are close enough to consider this approach as an alternative solution for future disasters. For this specific case, the expected demand solution is a much better approach than the expected value solution.
To continue the traditional method of analysis, the average of the 5 scenarios plus the real case for each analysis was obtained, which are shown in the following table considering each case:
Table 5.5 Average of 5 scenarios plus real case Case 1
(resources are not lost)
Case 2
(resources are lost) Analysis Budget Missing
Resources Budget Missing Resources Perfect Information $273,685 0.0482 $320,176 0.0482 Stochastic Solution $291,559 0.0482 $291,559 0.1621 Expected Value $341,321 0.1086 $341,321 0.1771 Expected Demand $303,504 0.0735 $303,504 0.1882
The conclusions are the same as in the average of the 5 scenarios, with relatively small changes in the average values.
Chapter 6
Design of Experiments
A design of experiments was performed in order to determine which factors have the greatest effect on the solution of the problem with our model. More specifically, a 2k factorial experiment was designed considering the following factors:
1) Budget
2) Magnitude of the disaster 3) Storage center capacity 4) Donations
5) Connectivity disruptions 6) Vehicle’s type
Each combination of factor levels will be referred to as a treatment. Five scenarios were generated for each combination of factor levels. For the scenarios of each treatment, the factors to vary will be the impact zone, the magnitude of the disaster, the total donations and the disrupted roads. A total of 3 replicates, each with 5 scenarios, was solved for each of the treatments.
The corresponding low and high levels were selected considering a central point, which in this case the central points were the actual values for each factor in the real case solution.
The respective low and high levels relative to the central points were selected under the following criteria:
• Budget: for the budget needed to fulfill demand, a -10% for the low level and a +10% for the high level was applied.
