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Online resource minimization

Many real-world problems can be modeled as online optimization problems. In these problems, information arrives over time, while decisions must be made at time points based only on the information then available. Surveys of online algorithms are given in Sgall (1997) and Coffman Jr., Garey and Johnson (1996), while Albers (1997) and Ascheuer et al. (1998) describe practical situations where online algorithms can be used. We investigate a new basic online problem in which work with different deadlines arrives over time, a level of resources must be chosen at each decision point so as to meet all deadlines, and the objective is to minimize the maximum resource usage. The problem is motivated by cases in vehicle fleet planning, warehouse allocation, workspace procurement, and electricity consumption, where the controllable portion of total costs varies with the maximum amount of resource procured. In Section 2, we define our online resource minimization problem, and introduce the notation that will be used throughout. Online algorithms are typically evaluated by means of their competitive ratio, which is the worst-case ratio of the performance of the online algorithm to the performance of an optimal algorithm with perfect information. We end section 2 by deriving a closed-form expression for the optimal value with perfect information. In Section 3 we introduce the α-policy, a simple parameterized policy with parameter α and worst-case ratio α, provided it is feasible. The main result of this section is that, with appropriate parameter choice, the α-policy has as good a worst-case ratio as any other policy. This result applies to any online minimax problem satisfying certain natural conditions. For our problem, we also show that an optimal parameter value α∗ exists. Hence α∗ also equals the optimal competitive ratio. We also show that to find α∗ it is sufficient to study the more restricted version of the problem in which all deadlines coincide with the planning horizon. We introduce two other classes of policies, called the φ-policies and the ψ-policies, in Section 4. Analysis of these policies provides an upper bound on the optimal competitive ratio of α∗ ≤ 3.45. The φ-policy is shown to achieve competitive ratio at best 4. We tackle lower bounds on α∗ in section 5, by finding a closed-form integral expression, and a coupled differential equation system for continuous approximations of the problem. Analytic and numeric solutions of these continuous models lead to a 10,000 period instance that proves α∗ > 2.51. We also conduct a computational study of the α-policy, the φ-policy, and the ψ-policy, and find that the φ-policy usually performs best, and the α-policy typically performs worst. Thus the average-case performance rankings are the reverse of the worst-case performance rankings. These results are described in Section 6.