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Target assignment problems consider the optimal assignment of weapons or unmanned vehicles (UVs) to number of targets in order to satisfy certain mission.
Stochastic Task-Assignment Models Nalan Gülpınar, Berç Rustem Imperial College London Department of Computing, 180 Queen’s Gate, London, SW7 2AZ

Abstract In this paper, we are concerned with alternative formulations of the vehicle-task assignment problem in the presence of the uncertainty. Uncertainty arises due to sensing errors, poor intelligence or incorrect information in data. Military applications such as intelligence, surveillance, planning and scheduling, etc, involve decision making in dynamic, distributed, and uncertain environments. Keywords : Uncertainty modelling, stochastic programming, scenarios, worst-case robust strategies, integer programming 1. Introduction Military applications such as intelligence, surveillance, planning and scheduling etc., involve decision making in dynamic, distributed, and uncertain environments. In a large system, multiple sensors may provide incomplete, conflicting, or overlapping data. Moreover, some components or sensors may degrade or become completely unavailable due to failures, weather conditions, or battle damages. Uncertainties in dynamic environment induce different kinds of risks. Robust decisions take into account the uncertainties and the corresponding risks. In this report we are concerned with alternative formulations of the vehicle-task assignment problem in the presence of the uncertainty. Uncertainty arises due to sensing errors, poor intelligence or incorrect information in data. Target assignment problems consider the optimal assignment of weapons or unmanned vehicles (UVs) to number of targets in order to satisfy certain mission goals. For example, in the allocation of UVs to real-world strike operations, the goal is to destroy as many targets as possible subject to various constraints. Typical constraints are the number of available vehicles to perform the mission.

In weapon assignment problems, the target scoring probabilities, group effectiveness or operational effectiveness is maximised or the surviving value of targets is minimised. The other typical objective is to minimise the expected cost or risk of mission. Multicriteria optimisation can be used to measure the trade-off between risk and performance and to permit the decision maker to explore optimal alternatives. We assume that any vehicle can be assigned to any target and all vehicles are homogeneous. The heterogeneous vehicles with coupled mission objectives in UV missions will be considered in the future. Especially, reconnaissance and strike vehicles with different mission goals will be involved to reduce the uncertainty of the information about the environment in mission planning. Scenario Tree The problem of planning requires the definition of possible scenarios as events unfold. The scenarios may be based on geographical locations and successions of events and describe the state at the particular time. Both single stage and multi-stage stochastic decision models rely on future events. This is one way of characterising the uncertainty in terms of

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future scenarios. The future is seen in terms of scenarios that are essentially a discrete set of realisation of uncertainties. In the multi-stage case, each scenario evolves into a set of scenarios in the next stage, with associated probabilities. Instead of forecasts over a single time period, the forecast takes the shape of a scenario tree, which divides the investment horizon into discrete time intervals. The end of the scenario tree (i.e. the final stage of the process) consists of all possible outcomes that one can envisage. Then the optimal plan consists of the achievement (as close as possible) of the desired targets at the end of the operation, given the various possible scenarios and probabilities. Figure 1 represents an example of the scenario tree where s and p denote the events realised with different values and the corresponding branching probabilities, respectively. The subindices represent the label of the nodes at the scenario tree.

mission score is the sum of the individual scores accrued by the vehicles. 2.1 Uncertain Target Score Assume that the target score, denoted by s, is a stochastic random variable. Uncertainty is represented by a number of discrete scenarios St at time t. Consider a time horizon H. Let yej denote total number of vehicles assigned to target j for j ∈ T at scenario e ∈ St for t = 1 , ... , H. The branching probability of scenario e is denoted by pe. The goal is to assign vehicles to targets in different stages (time steps) in order to maximise the expected accumulated value. The expected target score at time t is computed as

Time discounting factors αt (0 ≤ αt ≤ 1) are used to scale the target score, and give a functional relationship between target scores and the time to visit these targets. aij is an entry of the V × T adjacency matrix A indicating to which target each vehicle can possibly be assigned. The task-assignment optimisation problem can be formulated as follows:

Figure 1: An example of scenario tree

2. Vehicle-Task Assignment under Uncertainty Let V and T denote sets of UVs and targets (tasks). The decision variables xij for i ∈ V and j ∈ T, are equal to 1 when a vehicle i is assigned to target j and 0 otherwise. Each target has a score associated with it based on the current classification, and the vehicle accrues that score if it is assigned to that target. If a vehicle is not assigned to a target (or task), it receives zero score. The

In M1, the first set of constraints represents total number of vehicles to be assigned to each target j ∈ Τ at e ∈ St. The second set of constraints ensures that total number of vehicles assigned to tasks cannot exceed the number of vehicles available at time t. The third set basically specifies upper bounds on yej and the last set of constraints define the binary integer variables. We might also impose the following constraints to ensure that each target and

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each vehicle are assigned at most once, respectively:

In M1, Vt represents the total number of available vehicles at time t. In a static environment, the number of available vehicles at each time period is the same. However, this can be easily relaxed and considered as an uncertain parameter in the model when vehicles are under the risk of attack or destruction during the mission. In addition, constraints related to time and fuel restrictions, distance to travel or the number of targets to visit for each vehicle can be added to the model. We assume that total number of available vehicles at any time period does not change and the adjacent matrix is the same at any event in the future. Let xije be a binary variable indicating whether a vehicle i is assigned to target j at event e ∈ St, t = 1 , ... , H. Thus, xije takes 1 if vehicle i is assigned to task j and 0, otherwise, at each scenario e ∈ St. Then the model M1 can be extended by integrating the task-vehicle assignment at each node of the scenario.

The optimal solution provides the taskvehicle assignment at each node in order to increase expected task / target score. 2.2 Uncertain Success Probability Deterministic formulation of the vehicletask assignment problem is described in [6]; but a brief summary is as follows. The scoring scheme defining effectiveness is

based on the definition of a target score. Each target is associated with a score based on the current classification. If a vehicle is not assigned to a target it receives a score of 0. Let yj denote number of unmanned vehicles directed to target j. aij is an entry of the V × T adjacency matrix indicating which target each vehicle can be assigned. uj is an upper bound on variable yj and is defined as: Each target j, for j ∈ T, is assigned a task success probability pj, and a weight that measures the importance of the target. The probability that the task will be successfully carried out for that target depends on the number of vehicles yj which have been assigned to the target in the following way: A target score is the product of this probability and its weighting: Group effectiveness is simply the sum of all individual target scores measured as

Then the goal is to maximise the weighted sum of task success probabilities. The integer nonlinear vehicle-task assignment problem can be formulated as follows:

2.3 Stochastic Success Probability In this section we consider a stochastic formulation of the vehicle-task assignment problem with uncertain success probability. Uncertainty arises on the task success probabilities, pj, due to battle situation, weather conditions, and so on.

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Consequently, they may be treated as being uncertain (stochastic) and therefore, dependent on some random parameter pj = pj (ξ). Therefore, each target j, for j ∈ T, is assigned a task success probability pj (ξ), which is a random parameter. In accordance to the described methodology of managing uncertainties and risks, we model the stochastic behaviour of probabilities pj using scenarios. Namely, probabilities pj(ξ) take different values pj(ξs) = pjs where S is a set of scenarios realized and s ∈ S. Such a scenario set may be constructed, for example, by utilising the historical observations of weapons’ efficiency in different environments, or by using simulated data, experts’ opinions etc. At this stage how to generate these scenarios is not our concern since we eventually consider the minimax strategies over the uncertain probabilities.

In computational experiments, we consider scenarios provided by Martin Kaye from BAE Systems. Although five scenarios lead to purely a single-period formulation of the above model, this can be extended to the multi-stage stochastic programming. At each scenario, criteria of deployment (WD) and supportability (WS) and eight missions (Wj) for j = 1, ... ,8 are used. In Figure 2, the single period scenario tree with five realisations of target scores, si, and the corresponding branching probabilities pi (measured as the weights of the scenarios in the spreadsheet) are presented. Table 1 presents the scenario scores.

Figure 2: Single-period scenario tree provided Table 1: Scenario scores

Similarly, the variable yj or adjacent matrix A can be an uncertain parameter in this formulation. 3. Computational Results

We use the following randomly generated adjacency matrix for finding an optimal assignment of ten vehicles to ten tasks.

3.1 Design of Experiments The optimisation models presented in the previous section are implemented in C++ and use the commercial software CPLEX v8.0 solver to optimise the linear and nonlinear programming problems. All computational experiments are carried out on a 3 GHz Pentium 4, running Linux with 1.5GB RAM.

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Table 2: Vehicle allocation to tasks at each scenario

3.2 Results The total number of vehicles assigned to each task is presented in Figure 3. The maximum expected target score, objective function value, is 8.4525.

Future Work

Figure 3: Total number of vehicles allocated to each task

Figure 4 presents the total number of vehicles assigned to each task at each scenario.

Currently we are working on the implementation of other alternative models; namely M3 and M4. Then we will focus on a robust optimal decision making framework based on the worst-case analysis of the task assignment problem. In this framework the aim is to minimise the risk of uncertainty arising due to stochastic nature of mission planning in order to guard against the worst-case scenarios. References [1] J.R. Birge, F. Louveaux, “Introduction to Stochastic Programming” Springer-Verlag New York, (1997). [2] G.G. denBroeger, R.E. Ellison, and L. Emerling, “On optimum Target Assignments”, Operations Research, 7, (1959), 322– 326.

Figure 4: Total number of vehicles assigned to tasks at each scenario

In order to obtain an assignment of each vehicle to at most one task under different scenarios we impose total number of vehicles as one in model M2. Then the objective function value is obtained as 9.18. Table 2 displays only vehicles assigned to tasks at each scenario. Notice that no vehicle is assigned to tasks W2, W4, W7, W8.

[3] N. Gulpinar, B. Rustem, R. Settergren, “Optimization and Simulation Approaches to Scenario Tree Generation”, Journal of Economics Dynamics and Control, Vol. 28, Issue 7, (2004), 1291-1315. [4] P. Kall, “Stochastic Linear Programming”, Springer, Berlin, (1976). [5] P. Kall, S.W. Wallace, “Stochastic Programming”, Wiley, New York, (1994). [6] S. Ruuth, I. Maros, C. Lucas, G. Mitra, G., “Solution of a Nonlinear Robot Assignment Problem”, in Matson, E. (ed.) Essays in honour of Bjorn Nygreen on his 50th birthday, Norwegian University of Science and Technology, 1996, 49–60. [7] A.S. Manne, “A Target Assignment Problem”, Operations Research, 6, (1958), 346–351. [8] A. Prekopa, “Stochastic Programming, Akademiai Kiado”, Budapest, (1995). [9] B. Rustem, M. Howe, “Algorithms for WorstCase Design and Applications to Risk

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Management”, Princeton University Press, London and New Jersey, (2002).

Acknowledgements The work reported in this paper was funded by the Systems Engineering for Autonomous Systems (SEAS) Defence Technology Centre established by the UK Ministry of Defence. We acknowledge the support given by Jo Thoms, Martin Field, Martin Kaye, and Michele Hughes in the specification of the scenarios.

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