A Mathematical Model for Interplanetary Logistics

Logistics Spectrum, Jan-Mar 2007 by Taylor, Christine, Song, Miao, Klabjan, Diego, de Weck, Olivier, Simchi-Levi, David

(ProQuest-CSA LLC: ... denotes formulae omitted.)

Introduction

This article demonstrates a methodology for designing and evaluating the operational planning for interplanetary exploration missions. A primary question for space exploration mission design is how to best design the logistics required to sustain the exploration initiative. Using terrestrial logistics modeling tools that have been extended to encompass the dynamics and requirements of space transportation, an architectural decision method has been created. The model presented in this article is capable of analyzing a variety of mission scenarios over an extended period of time with the goal of defining interesting mission architectures that enable space logistics. This model can be utilized to evaluate different logistics trades, such as a possible establishment of a push-pull boundary, which can aid in commodity pre-positioning. The model is demonstrated on an Apollo-style mission to both provide an example and validate the methodology.

The development of an interplanetary supply chain requires the unification of two traditionally separate communities: aerospace engineering and operations research. In order to create an effective means of communication between both communities, a distinct terminology has been developed and is detailed extensively in Section I. Specifically, the definition of the commodities or supplies, and the elements or physical containment and propulsion units used to transport the commodities are detailed. Furthermore, the network definition is presented as well as the definition and description of the time expanded network, which is the terrestrial modeling technique employed for the space logistics model. Section II describes the components of the interplanetary logistics problems. Section III presents the problem formulation and constraints. In Section IV a description of the optimization methodology developed to solve this problem is discussed. In Section V the problem formulation and solution methodology is applied for the example of an Apollo-style mission to both explain the implementation and validate the methodology presented. Section VI reviews the contributions of this article and describes continuing work in this area.

I. Problem Definition

The goal of the interplanetary logistics problem is to determine feasible mission architectures to satisfy the demand generated by the needs of exploration. The key concept of the interplanetary logistics problem is that the demand of crew, consumables, equipment and other exploration requirements at in-space locations drives the mission requirements. Therefore, the first required input for the interplanetary logistics problem is the definition of these supplies. For example, if the exploration mission is a sortie style mission to investigate a particular location, the demand might consist of a few crew members at a specific location and the supplies necessary to both support the crew and enable the exploration activities.

Given the demand of the mission, it is necessary to determine how and when the supplies on Earth will be transported to the in-space locations. As missions become more complex and evolve over a period of time, a solution may become less obvious. Since the goal is to minimize the cost of any mission, it is desirable to optimize the timing and method of transport of the supplies to in-space locations. Therefore, it is necessary to define all pathways and structures used for transport, and allow the optimizer to analyze the different architectures to select the best one.

Given this information, the interplanetary logistics problem can determine low cost mission architectures that satisfy the exploration demand. The solution generated will detail the scheduling and assignment of supplies to vehicles for in-space transport and launch scheduling requirements. More importantly, however, the output of this problem can be used to determine a push-pull boundary for the supplies, the potential of a specific location, either on a surface or in-space for storing supplies, benefits of in-situ resource utilization over multiple missions, or even the sensitivity of mission architectures to changes in vehicle parameters.

The first step in developing a model for interplanetary logistics is defining a concrete nomenclature that describes the components of the problem. The problem fundamentally consists of three components: the commodities or supplies that must be shipped to satisfy a mission demand, the elements or physical structures used to both hold and move the commodities, and the network or pathways the elements and commodities travel on. The following sub-sections define the parameters that describe each of these components.

Commodities

The goal of the space logistics project is to determine how to meet the demand for the exploration missions. As such, we are investigating how to optimally ship multiple types of commodities. For the purpose of the logistics problem, a commodity will be defined as a high-level aggregate of a type of supply, such as crew provisions. Thus, we will define a set of k = 1,..., K commodities, each with the following parameters: