An effective genetic algorithm approach to multiobjective resource allocation problems (MORAPs)

doi:10.1016/j.amc.2003.10.057

Applied Mathematics and Computation

Volume 163, Issue 2, 15 April 2005, Pages 755-768

https://doi.org/10.1016/j.amc.2003.10.057 Get rights and content

Abstract

Dynamic programming (DP) is a mathematical procedure designed primarily to improve the computational efficiency of solving select mathematical programming problems by decomposing them into smaller, and hence computationally simpler, subproblems. In solving Multiple objectives Dynamic Programming Problem (MODP), classical approaches reduce the multiple objectives into a single objective of minimizing a weighted sum of objectives. The determination of these weights indicate the relative importance of the various objective. Also, if the problem scale increases, it become difficult to be dealt with even in the case of single objective because of the rapid expansion of the number of states to be considered. In this paper, we investigated the possibility of using genetic algorithms (GAs) to solve multiobjective resource allocation problems (MORAPs). This procedure eliminates the need of any user defined weight factor for each objective. Also, the proposed approach is developed to deal with the problems with both single or multiple objectives. The simulation results for multiobjective resource allocation problems (MORAP) shows that genetic algorithms (GAs) may hopefully be a new approach for such kinds of problems.

Introduction

Dynamic programming (DP) is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. More so than other optimization techniques, DP provides a general framework for analyzing many problem types. Within this framework a variety of optimization techniques can be employed to solve particular aspects of a more general formulation [10], [13]. Resource allocation problem RAP is the process of allocating resources among the various projects or business units. Resource may be a person, asset, material, or capital which can be used to accomplish a goal. A project may be a set of related tasks which have a specific goal. A goal my be objective or target, usually driven by specific future financial needs. The “best” or optimal solution may mean maximizing profits, minimizing costs, or achieving the best possible quality. An almost infinite variety of problems can be tackled this way, but here are some typical examples:

Finance and investment: Working capital management involves allocating cash to different purposes (accounts receivable, inventory, etc.) across multiple time periods, to maximize interest earnings.

Capital budgeting: involves allocating funds to projects that initially consume cash but later generate cash, to maximize a firm's return on capital.

Portfolio optimization: creating “efficient portfolios” involves allocating funds to stocks or bonds to maximize return for a given level of risk, or to minimize risk for a target rate of return.

Job shop scheduling: involves allocating time for work orders on different types of production equipment, to minimize delivery time or maximize equipment utilization. And many other application that can be formulated as resource allocation problem.

On the other hand, there has recently been a great deal of interest in promising genetic algorithm (GA) and its application to various disciplines including optimization problems [3], [4], [8]. Genetic algorithms are also being applied to a wide range of optimization and learning problems in many domains. Genetic algorithms lend themselves well to optimization problems since they are known to exhibit robustness, require no auxiliary information, and can offer significant advantages in solution methodology and optimization performance. An useful feature of genetic algorithm is to handle multiobjective function optimization [2], [4]. GAs operate on a population of candidate solutions encoded to finite bit string called chromosome. In order to obtain optimality, each chromosome exchanges information by using operators borrowed from natural genetic to produce the better solution. GAs differ from other optimization and search procedures in four ways [8]:

(1)
GAs work with a coding of the parameter set, not the parameters themselves. Therefore GAs can easily handle the integer or discrete variables.
(2)
GAs search from a population of points, not a single point. Therefore GAs can provide a globally optimal solutions.
(3)
GAs use only objective function information, not derivatives or other auxiliary knowledge. Therefore GAs can deal with the non-smooth, non-continuous and non-differentiable functions which are actually existed in a practical optimization problem.
(4)
GAs use probabilistic transition rules, not deterministic rules.

This chapter is organized as follows, the formulation of the multiobjective resource allocation Problem (RAP) are described in Section 2. Section 3 gives out representation MORAP as a network model. In Section 4, we introduce multiobjective resource allocation problem via genetic algorithm. The simulation result are discussed in Section 5. And conclusion follows in Section 6.

Section snippets

Multiobjective resource allocation problem MORAP

The general form of the multiobjective resource allocation problem is as follows [1]:
MORAP: $Max Z_{1} (x_{1},x_{2},…,x_{n})=∑_{k=1}^{n} z_{k}^{1} (x_{k})$ $Max Z_{2} (x_{1},x_{2},…,x_{n})=∑_{k=1}^{n} z_{k}^{2} (x_{k})$ $………$ $Max Z_{P} (x_{1},x_{2},…,x_{n})=∑_{k=1}^{n} z_{k}^{p} (x_{k})$ $subject to ∑_{k=1}^{n} g_{k} (x_{k})⩽S,$ $g_{k} (x_{k})⩾0,$ $x_{k} ⩾0.$ The reason for the label “resource allocation problem” is that finding optimal values x_k (decision variables) is equivalent to allocating the “resources” s among the “activities” stages g_k(x_k) in such a way that objective function Z(x₁,…,x_n) is maximized.

The efficient solution

Representation MORAP as a network model

Resource allocation problem is to find the best way to allocate scarce resources. The resources may be raw materials, machine time or people time, money, or anything else in limited supply. The “best” or optimal solution may mean maximizing profits, minimizing costs, or achieving the best possible quality. There are many applications to resource allocation problem such as:

1.
How many inspectors to allocate to each river (or region) to monitor pollution?
2.
How many patrol cars to allocate to different

Multiobjective resource allocation problem via genetic algorithm

Multiobjective optimization problems give rise to a set of Pareto-optimal solutions, none of which can be said to be better than other in all objectives. In any interesting multiobjective optimization problem, there exists a number of such solutions which are of interest to designers and practitioners. Since no one solution is better than any other solution in the Pareto-optimal set, it is also a goal in a multiobjective optimization to find as many such Pareto-optimal solutions as possible.

Simulation results

In this section, an experimental verification of the proposed algorithm is carried out. We apply the proposed algorithm for multiobjective resource allocation problem (taken from [1]), then we compare the results MORAP obtained by dynamic programming [1], [5] and that obtained using genetic algorithms to verify the convergence of the algorithm for the optimal solution.

Consider the problem (taken from [1]) of allocating 6 workers to a certain set of 4 jobs. Table 1, Table 2 provide the expected

Conclusions

Although resource allocation problem can be approached by dynamic programming, with the increased problem scale, many stages and states must be considered, which will greatly affect the efficiency of the procedure dynamic programming in getting the optimal solution. Moreover, the DP has the difficulty when the problem scale increases, it becomes difficult to be dealt with even in the case of single objective because of the rapid expansion of the number of the states to be considered. So we

References (13)

M.L. Hussein et al.
A fuzzy dynamic approach to the multicriterion resource allocation problem
Fuzzy Sets and Systems
(1995)
T. Murata et al.
Multiobjective genetic algorithm and its application to flowshop Scheduling
Computers and Industrial Engineering
(1996)
M.A. Abo-Sinna, S.M. Lee, M.S.A. Osman, A.M. Sarhan, A dynamic programming approach for solving bicriterion resource...
C. Coello
A comprehensive survey of evolutionary-based multiobjective optimization techniques, knowledge and information systems
An International Journal
(1999)
K. Deb
An introduction to genetic algorithms
Sadhana
(1999)
M. Gen et al.
Genetic Algorithms and Engineering Optimization
(2000)

There are more references available in the full text version of this article.

Cited by (58)

Integer optimization models and algorithms for the multi-period non-shareable resource allocation problem
2024, European Journal of Operational Research
The resource allocation problem (RAP) determines a solution to optimally allocate limited resources to several activities or tasks. In this study, we propose a novel resource allocation problem referred to as multi-period non-shareable resource allocation problem (MNRAP), which is motivated by the characteristics of resources considered in the stem cell culture process for producing stem cell therapeutics. A resource considered in the MNRAP has the following three characteristics: (i) resource consumption required to perform an activity and available resource capacity may change over time; (ii) multiple activities cannot share one resource; and (iii) resource requirements can be satisfied through the combination of different types of resources. The MNRAP selects some of the given activities to maximize the overall profit under limited resources with these characteristics. To address this problem, pattern-based integer programming formulations based on the concept of resource patterns are proposed. These formulations attempt to overcome the limitations of a compact integer programming formulation, the utilization of which is challenging for large-scale problems owing to their complexity. Further, based on a branch-and-price approach to solving pattern-based formulations, effective heuristic algorithms are proposed to provide high-quality solutions for large instances. Moreover, through computational experiments on a wide range of instances, including real-world instances, the superiority of the proposed formulations and heuristic algorithms is demonstrated.
Bi-level programming and multi-objective optimization for the distribution of resources in hierarchical organizations
2024, Applied Mathematical Modelling
The decision regarding the hierarchical distribution of resources is crucial for many organizations that want to allocate such resources efficiently. At the upper level of the hierarchy, a decision-maker may have an objective and a set of feasible solutions partially determined by the lower level. Nevertheless, the upper level can influence but not control the decision-maker at the lower level. Moreover, the objectives of the upper and lower levels are conflicting. Hence, the hierarchical distribution of resources can be formulated as a bi-level programming model. This paper reformulates the hierarchical allocation of resources. The Hooke-Jeeves (HJ) algorithm and the hybrid scatter search–Nelder-Mead (SSNM) method are proposed to solve the newly formulated problem. The problem is also analyzed from a multi-objective perspective to equally prioritize the upper and lower levels while subjecting each to an interdependent set of constraints; a multi-objective scatter search algorithm (MSSA) was developed for that purpose. Tailor-made instances were also designed for the implementation of the developed algorithms. The findings demonstrated that the HJ algorithm provides the best solution according to the upper-level objective function. However, it does not guarantee the best result for the lower level. In comparison, MSSA has a set of best possible solutions for both objectives. The hybrid SSNM method could not match the results provided by the former methods.
A discrete particle swarm optimization method for assignment of supermarket resources to urban residential communities under the situation of epidemic control
2021, Applied Soft Computing
When contagious diseases hit a city, such as MERS, SARS, and COVID-19, the problem arises as how to assign the limited supermarket resources to urban residential communities for government measures. In this study, in order to solve the assignment problem from supermarket resources to urban residential communities under the situation of the epidemic control, the discrete multi-objective particle swarm algorithm can be improved by introducing some new strategies, and the probability matrix can be used to simulate the many-to-many assignment relationship between residential communities and supermarkets. The ultimate purpose of this research is to achieve an optimal way to balance the two conflicting objectives, i.e. minimization of the cross-infection risk and maximization of the service coverage rate. Also, the optimization considers the accessible distance limit and the service capacity constraints of supermarkets for the feasible scheme. For this aim, we redefine the subtraction operator, add operator and multiply operator to generate the Pareto optimal solutions, and introduce a new study strategy based on the idea of differential evolution in the particle swarm algorithm (PSO-DE). In this work, we take the COVID-19 epidemic outbreak in Wuhan city of China as an example in the experiment. The simulation results are compared with the Genetic Algorithm (GA), Simulated Annealing (SA), Ant Colony Algorithm (ACO) and the Particle Swarm Optimization with Roulette Wheel Selection (PSO-R), and these results have been shown that the algorithm PSO-DE proposed in this work has a better optimization performance in both objectives.
Using a heuristic algorithm to design a personalized day tour route in a time-dependent stochastic environment
2018, Tourism Management
Citation Excerpt :
Zheng et al. (2017) applied a method involving a genetic algorithm and a difference evolution algorithm to evolve the solutions. The genetic algorithm is widely applicable to discrete optimization problems (Osman, Abo-Sinna, & Mousa, 2005), and the difference evolution algorithm is especially suitable for dealing with continuous optimization problems (Plagianakos, Tasoulis, & Vrahatis, 2008). Other than this study, discussion of optimizing solutions with both discrete and continuous decision variables has been negligible in the research.
A substantial transformation has occurred in tourist behavior in the postmodern tourism era, where the tourism market is dominated by the demand for tailored experiences. Therefore, the design of personalized tourist routes plays a fundamental role in improving tourists’ travel experiences and the success of tourist attractions. This study proposes a hybrid heuristic algorithm based on random simulation (RS-H²A) to design a personalized day tour route in a time-dependent stochastic environment. To evaluate the performance of this algorithm, we conducted a case study at Jiuzhai Valley in Sichuan, China. The results of paired sample t-tests indicated that the proposed algorithm performed significantly better than existing methods. Furthermore, our proposed approach had the ability to design more realistic and personalized routes for tourists than previous methods. We designed an experiment to further explore how uncertain environments affect tourists with different levels of risk awareness.
A novel interactive preferential evolutionary method for controller tuning in chemical processes
2015, Chinese Journal of Chemical Engineering
Citation Excerpt :
Xu and Yager [5] have proposed to divide a decision cycle into a plurality decision cycle and set the dynamic multiple attribute for more uncertainty attribute decision problems. Osman et al. [6] have introduced optimization algorithms to obtain attribute weights in MADM. Little attention has been paid to decision-makers' subjective preferences, which plays dominant roles in dealing with decision-making issues.
In response to many multi-attribute decision-making (MADM) problems involved in chemical processes such as controller tuning, which suffer human's subjective preferential nature in human–computer interactions, a novel affective computing and preferential evolutionary solution is proposed to adapt human–computer interaction mechanism. Based on the stimulating response mechanism, an improved affective computing model is introduced to quantify decision maker's preference in selections of interactive evolutionary computing. In addition, the mathematical relationship between affective space and decision maker's preferences is constructed. Subsequently, a human–computer interactive preferential evolutionary algorithm for MADM problems is proposed, which deals with attribute weights and optimal solutions based on preferential evolution metrics. To exemplify applications of the proposed methods, some test functions and, emphatically, controller tuning issues associated with a chemical process are investigated, giving satisfactory results.
Comparison of multi-objective optimization techniques applied to off-gas management within an integrated steelwork
2014, Applied Energy
Process optimization is of utmost importance for a correct management within any industrial field and especially within the energy and carbon intensive ones. Increasingly stringent regulations aiming at reducing the environmental impact force the companies to lower their CO₂ emissions while preserving the economical sustainability of their production processes. Therefore the need arises to face these challenges by formulating optimization problems involving multiple objectives, which correspond to different and often conflicting requirements.
In the present work the process gases network of a real integrated steelwork is analyzed and modeled to the aim of formulating an optimization problem considering two conflicting objectives. The multi-objective optimization problem is firstly faced through linear programming (LP) and solved by exploiting the ε constraint method. As an alternative, an innovative evolutionary algorithm is proposed, which presents the advantage of allowing for a more flexible expression of the process model and constraints as well as an easier integration of different modules. The two approaches are discussed and compared. The results shown that surely multi-objective optimization is surely a viable solution, as CO₂ emissions can be reduced by more than 4% and profit increased up to 20% compared to the presented base cases in normal operating conditions. The savings are even higher if off-design plant operation conditions subsist. The LP formulation provides faster generation of analytically optimum solutions, which lead to higher savings. However, in the generation of intermediate trade-offs and complete exploration of the solution search space the evolutionary approach is more flexible and more suitable to this application in the long-term.

View all citing articles on Scopus

View full text

An effective genetic algorithm approach to multiobjective resource allocation problems (MORAPs)

Abstract

Introduction

Section snippets

Multiobjective resource allocation problem MORAP

Representation MORAP as a network model

Multiobjective resource allocation problem via genetic algorithm

Simulation results

Conclusions

Fuzzy Sets and Systems

Computers and Industrial Engineering

A comprehensive survey of evolutionary-based multiobjective optimization techniques, knowledge and information systems

An International Journal

An introduction to genetic algorithms

Sadhana

Genetic Algorithms and Engineering Optimization