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- Introduction
- Objectives
- The ORMP problem
- The HLBP problem
- Comparison
- Problem definition
- Analysis
- Randomized algorithms
- Conclusion

- The objective is to minimize the maximum level of resource allocated at any time during the planning period, this problem is called Online Resource Minimization Problem (ORMP)
- The objective is to minimize the maximum amount of work assigned to any machine, the problem is called the Hierarchical Line Balancing Problem (HLBP)

- Both ORMP and HLBP are special cases of a more general problem, the Online Min-Max Problem (OMMP)
- The quality of the algorithm is evaluated by the competitive ratio

- A simple parameterized deterministic algorithm, called the -policy, with parameter and competitive ratio , provided it produces a feasible solution
- The -policy is also optimal among all randomized algorithms

- Work with different deadlines arrives over time and has to be performed using a resource. The quantities of work that arrive as well as their deadlines become known only at the times of arrival. At a given set of time points, the decision maker decides how much resource to allocate and what part of the available work to perform at that time

- Work with different requirements arrives over time and has to be assigned to a collection of machines with different capabilities. The machines form a linear hierarchy based on their capabilities
- The amount of work that arrives as well as the required machine capabilities become known only at the time of arrival

- ORMP appears to be a new problem, and there is no existing literature discussing it
- For HLBP, the optimal competitive ratio tends to e when the number of machines goes to infinity. When the machines have same capability, the competitive ratio is 2/1-m for the m identical parallel machines [Graham]

- An instance of the OMMP is a finite sequence
(a(1), ….a(T)) of length T

- The set of all instances with parameter is denoted by For example, the set of all instances of length T is denoted by T
- let t = (a(1), ….a(T)) denote the first t elements of instance ; that is, t denotes the history of instance up to time t<=T
- For any instance of length T, a solution r is a sequence (r(1), …...r(T)) R+Tof T nonnegative real numbers

- ()(t): the decision at time t for instance under
- r(t) denotes the decision ()(t) when is fixed
- For any instance T and any deterministic algorithm the value () = max {r(1), r(2), …, r(t)}
and the optimal value is *() = inf r ( {max {r(1), r(2), …, r(t)}}

The -policy is an online policy with parameter >= 1, defined by: ri () = *(i) for all i

is the worst-case competitive ratio of policy over all instances with time

= Sup {()/ *()}

An instance of the OMMP is a finite sequence

(a(1), ….a(T)) of length T

- The set of all instances with parameter is denoted by For example, the set of all instances of length T is denoted by T
- let t = (a(1), ….a(T)) denote the first t elements of instance ; that is, t denotes the history of instance up to time t
- For any instance of length T, a solution r is a sequence (r(1), …...r(T)) R+Tof T nonnegative real numbers

- A simple parameterized algorithm, called the -policy, with parameter and competitive ratio , provided that it is feasible
- To be feasible a solution must satisfy 3 constraints:
1) the total amount of work performed at time i cannot exceed the amount of work accomplished with ri resources

2) all work must be performed with the respective deadlines

3) the work cannot be performed before it has arrived

- For any algorithm DO if < then the -policy with parameter = achieves the same competitive ratio, =
Proof:

- ()(t): the decision at time t for instance under
- r(t) denotes the decision ()(t) when is fixed
- For any instance T and any deterministic algorithm the value () = max {r(1), r(2), …, r(t)}
and the optimal value is *() = inf r ( max {r(1), r(2), …, r(t)}}

The -policy is an online policy with parameter >= 1, defined by: ri () = *(i) for all i

- For some problems, randomized algorithms can have better competitive ratios than deterministic algorithms (Motwani and Raghavan and Hoogeveen and Vestjens)
- A randomized algorithm can have better competitive ratios for the OMMP than any deterministic algorithm.

- The -policy theory developed in this paper is a powerful tool for finding worst-case optimal algorithms for online min-max problems
- With an appropriate choice of parameters , the -policy has as good a competitive ratio as any other deterministic algorithm
- Under mild conditions an optimal parameter value exists, so the -policy is

- B. Hunsaker y, A. J. Kleywegt, M. W. P. Savelsbergh, andC. A. Tovey, Optimal Online Algorithms for Minimax Resource Scheduling , SIAM J. Discrete Math. Vol. 16, No. 4, 2003, pp. 555-590
- A. J. Kleywegt, V. S. Nori, M. W. P. Savelsbergh, and C. A. Tovey, Online resource minimization, in Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, January 1999, SIAM, Philadelphia, 1999, pp. 576-585.