Elsevier

Applied Mathematical Modelling

Design of a location and transportation optimization model including quality of service using constrained multinomial logit

Abstract

The design of an integrated network with decisions about tactical transportation and strategic locations is complicated and challenging. In addition to the need to consider cost issues, the consumers' preferences will also have a significant impact on the resulting network. We propose an integrated transportation and location optimization model for designing logistic networks. Our model uses a hybrid iterative heuristic based on a genetic algorithm and a constrained multinomial logit, which includes consumers' preferences and quality of service variables. Our results show relevant differences when considering, along with network costs, parameters related to the consumers: travel distance, congestion, waiting time, and service time.

Introduction

Designing networks that define both location and transportation is a problem faced in public and private contexts. There have been, and continue to be, research efforts to model and solve this kind of problem, which involves tactical and strategic decisions. Location and transportation decisions are often analyzed separately, but now the problem is being analyzed while jointly considering both decisions.

Location models deal with strategic decisions. The literature is quite extensive in this regard, in general terms, one can identify at least two groups: 1) those related to defining locations that consider the level of service and limited resources, namely along with the lines of a maximal set covering problem, and 2) those related to defining locations to meet demand and minimize total costs. The inclusion of additional transportation decisions is also common in the literature; one can include the transportation network and consider investment and operational costs but also levels of service. Another relevant aspect is the combination of location decisions along with a transport design that includes consumers' preferences and, in general, the modeling of demand.

The combination of location information with transportation decisions for demand modeling is a mathematically complex problem. We believe the main issue is that consumers are agents that maximize their utility, so the network design must address costs and internal aspects of the business and be concerned with the data needed to analyze consumer behavior. These combined aspects lead to complex optimization models with two levels of decisions, considering both costs for the agents that design the network and consumers' characteristics. The last point is complicated and quite relevant because the final network design must better suit consumers' requirements and constraints.

Our literature review is in the next section, and there we make explicit the research gap we are bridging with this paper. In this section, we briefly introduce the topic and relevance of the problem to clarify the paper's objective and highlight the elements of the current research.

This paper focuses on the modeling and resolution of an integrated location and transportation model that considers costs and the quality of service (QoS) variables. On the side of costs, we include location, servers, and transportation costs; the quality of service variables are system capacity, facility congestion, waiting time, and service time.

In our proposal, we integrate a location model and a transportation model, with demand represented as a special kind of logit model the constrained multinomial logit (CMNL [1]), in which a queueing system defines the configuration of the transportation network. The use of the CMNL gives us a more flexible way to represent soft constraints for consumers' parameters and variables. As far as we know, there is no model similar to the one we are proposing. Since the resulting mathematical model cannot be solved using commercial solvers, we propose a combined iterative approach in which we solve the location problem using a genetic algorithm. We solve the transportation model using the CMNL and an optimization problem based on queueing theory.

The framework we propose in this paper includes the model and the heuristic approach to solve it. Besides, we show the applied potential and relevance of this research by using our framework to solve a sample problem related to healthcare coverage in Bogot. Our framework incorporates the following elements:

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We model patients (consumers) as sensitive not only to traveling distances and costs but also to the quality of service variables, such as waiting time, congestion, service time, and system capacity.

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We design networks by applying a strategy that will improve the provided services, shifting the focus from profitability towards a patient-centered system conceived around meeting patient expectations.

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The modeling of types of facilities related to consumers' priorities allows us to design less expensive networks that are better suited to consumers' requirements. For example, in the case of the hospitals we analyze (in Colombia), the most important reason for patients not receiving attention is that they choose to visit a hospital for mild sickness instead of selecting another type of facility that is more appropriate for treating their patients' condition. The result is that patients with mild conditions do not receive appropriate care but exacerbate the congestion at urgent care facilities, affecting the care of more critical patients.

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We also consider other factors that affect the quality of services, such as budget constraints, technical and professional issues related to equipment and personnel, distance, and insurance issues.

Based on our literature review in Section 2, the model we propose is novel and relevant, since it uses this combined approach to include soft constraints with a CMNL, and the combined set of factors we propose related to consumers is also unique. On the methodological side, the heuristic approach is also new, since it combines a genetic algorithm with queueing systems in which the demand is a CMNL. The relevance of the research lies in the kind of problems that the framework can handle and in the fact that the resulting networks show differences from designs that do not consider the variables we include; hence, these variables are not only qualitatively relevant but also lead to different final network designs. The research highlights are summarized below:

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We propose a framework to design networks for transportation and location decisions using various parameters, conditions, and constraints that have not been addressed in the same model.

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We use a CMNL approach to include consumers' requirements in the design; this allows us to make the model more realistic and to add constraints as penalties on the probability obtained from a discrete choice model.

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We propose a genetic algorithm applied to a model that interacts with a transportation model that considers a queueing system with CMNL demand.

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We analyze a numerical case study in the context of a healthcare network. Even though designs of such networks generally consider only costs, healthcare networks should also consider the quality of service variables and consumers' characteristics, such as preferences and constraints.

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Our framework puts strategic location decisions, tactical capacity planning, and operative congestion in a single model that produces a design that balances these factors.

We organize this paper as follows. In Section 2, we present the literature review and describe our contribution. Section 3 details the mathematical modeling and solution strategies for the decision problem, and Section 4 provides numerical results from a healthcare context. Finally, in Section 5, we present conclusions and final remarks.

Section snippets

Literature review

The literature for both location and transportation models is very extense. Since this research proposal is specific for both topics, we focused our analysis in this regard, emphasizing the modeling and methodological aspects. We consider these branches, to be concise and align with the novelty of the paper. Within the context of integrated location and transportation models, a relevant aspect we also develop is discrete choice models alternatives to deal with consumers characteristics and

The decision problem

The problem we are solving helps to design an integrated location and transportation network. Specifically, we have to decide on: opening facilities in a specific geographic location, the type of facility, the availability for types of consumers, the capacity in terms of the number of servers per facility, and the estimation regarding if a client will choose a facility.

Our approach is to take location decisions considering the costs of facilities, its capabilities (type), and capacities, as

Results

In this section, we describe the case study based on the healthcare context at first. Secondly, we analyzed the parameters for the transportation model, aiming to determine the impact of each parameter, i.e., costs, travel and queue waiting time, and travel times, isolated and in conjunction with the other parameters. For such analysis, we analyzed eight combinations in the same number of scenarios, and we describe comparisons with a parameter-absent and a base scenario. Thirdly, we report

Conclusions

This paper proposes a transportation and location model design, including QoS variables, which affect individual preferences. Even though the CMNL allows the inclusion of both system and individual constraints, as far as the authors know, no other works are dealing with the integration of transportation and location problems using CMNL and QoS variables.

Due to the importance of understanding the choice behavior of clients, it has been proven by several authors that CMNL is a suitable

Acknowledgements

The corresponding author was supported by CONICYT, FONDECYT project 11160320.

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