A Mathematical Model for Interplanetary Logistics
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.
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:
* Denote the demand of each commodity as d^sup k^.
* Denote the origin of each commodity as so^sup k^.
* Define the destination of each commodity as sd^sup k^.
* Define the availability interval of each commodity as to^sup k^ = [sto^sup k^, eto^sup k^], where sto^sup k^ is the starting time of the interval and eto^sup k^ is the ending time of the interval.
* Define the delivery interval of each commodity as td^sup k^ = [std^sup k^, etd^sup k^], where std^sup k^ is the starting time of the interval and etd^sup k^ is the ending time of the interval.
* Define the unit mass of each commodity as m^sup k^ when it arrives at the destination.
* Define the unit volume of each commodity as v^sup k^ when it arrives at the destination.
* Define the number of specified waiting sequences as nw^sup k^.
By defining a waiting sequence as part of the commodity input, a number of wait arcs along the path can be specified, which allows onroute destinations to be designated. For each waiting arc sequence I where 0
* Define the static node of the wait sequence as sw^sub l^^sup k^.
* Define the required waiting time period as pw^sub l^^sup k^.
* Define the wait interval for each wait sequence as = [tw^sub l^^sup k^, etw^sub l^^sup k^], where stw^sub l^^sup k^ is the starting time of wait interval l of commodity k, etw^sub l^^sup k^ is the ending time of wait interval l of commodity k, and etw^sub l^^sup k^ - stw^sub l^^sup k^ ? pw^sup l^^sup k^.
It is important to note that in this model a crew member is treated as a commodity. In practice crewed missions are treated differently during mission planning: however, for the purposes of the architectural design tool created by this model, crew can be considered a commodity with highly restrictive parameter values. By narrowing the availability and delivery windows for a crew commodity, the feasible shipment pathways are limited and reasonable architectures for crewed flights can be obtained.
Elements
In order to ship the commodities from the origin to the destination locations, we require ‘containers’ to both hold the commodities and provide propulsion to move the mass through space. These components can be abstracted to a single definition of an element. Elements are physical, indivisible functional units that transport the commodities from origin to destination. An element is classified by the amount of commodity capacity and propulsive capability it possesses. Elements can be divided into two classes: non-propulsive elements M^sub N^ and propulsive elements Mp. The element parameters are (Figure 1) as follows:
* The maximum fuel mass of a propulsive element m, m euro M^sub p^ is denoted by mf^sup m^.
* The specific impulse of the fuel in element m is denoted by I^sub sp^^sup m^.
* The structural mass of element m is denoted by ms^sup m^.
* The mass capacity of element m is denoted by CM^sup m^.
* The volume capacity of element m is denoted by CV^sup m^.
* The cost of element m is denoted by Cost^sup m^.
Networks
In order to transfer the commodities and elements from the origin node to the destination node, the trajectories must be defined. The purpose of the interplanetary logistics model developed in this article is to analyze the multiple choices available for routing all of the commodities and elements to determine the best logistics architecture. To model the different available trajectories, a network model of space is created to represent the possibilities available for transferring commodities to their respective destination. The following sections detail the development of the space network utilized to form the model presented in this article.