Optimal Resource Allocation Using Knapsack Problems: a Case Study of Television Advertisements at Ghana Television (GTV)
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Date
2009-08-11
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Abstract
The Knapsack Problems are among the simplest integer programs which are NP-hard. Problems in this class are typically concerned with selecting from a set of given items, each with a specified weight and value, a subset of items whose weight sum does not exceed a prescribed capacity and whose value is maximum. The specific problem that arises depends on the number of knapsacks (single or multiple) to be filled and on the number of available items of each type (bounded or unbounded). Because of their wide range of applicability, knapsack problems have known a large number of variations such as: single and multiple-constrained knapsacks, knapsacks with disjunctive constraints, multidimensional knapsacks, multiple choice knapsacks, single and multiple objective knapsacks, integer, linear, non-linear knapsacks, deterministic and stochastic knapsacks, knapsacks with convex / concave objective functions, etc.
The classical 0-1 Knapsack Problem arises when there is one knapsack and one item of each type. Knapsack Problems have been intensively studied over the past forty (40) years because of their direct application to problems arising in industry (for example, cargo loading, cutting stock, and budgeting) and also for their contribution to the solution methods for integer programming problems. Several exact algorithms based on branch and bound, dynamic programming and heuristics have been proposed to solve the Knapsack Problems. This thesis considers the application of classical 0-1 knapsack problem with a single constraint to selection of television advertisements at critical periods such as Prime time News, news adjacencies, Break in News and peak times. The Television (TV) stations have to schedule programmes interspersed with adverts or commercials which are the main sources of income of broadcasting stations. The goal in scheduling commercials is to achieve wider audience satisfaction and making maximum income from the commercials or adverts. Our approach is flexible and can incorporate the use of the knapsack for Profit maximization in the TV adverts selection problem.
Our work focuses on using a simple heuristic scheme (Simple flip) for the solution of knapsack problems. We show that the results from the heuristic method compares favourably with the well known meta-heuristic methods such as Genetic Algorithm and Simulated Annealing. The thesis shows how television advertisement at critical segments such as prime time news (19:00 GMT) and news adjacencies (five minutes before and after news time) can be prioritized to rake in the maximum returns to support operations of a national Television station (GTV). The computer solution developed could be used for any problem that can be modeled as single 0-1 knapsack problem.
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A Thesis submitted to the Department of Mathematics,
Kwame Nkrumah University of Science and Technology
in partial fulfillment of the requirements for the degree of Master of Science.