99 CHAPTER 8 CONCLUSION This chapter summarizes the results and contributions of this dissertation. The conclusions detailing the overall implications of the methodologies introduced in this dissertation are drawn and recommendations for future work are also presented in this chapter. The growing complexity of manufacturing process and fierce competition in the market, drive enterprises to optimize their operations as much as possible. Scheduling of project activities with minimum cost is one of the concerned fields of project management to avoid the penalties incurred for delaying the project completion time. In the real world, projects are subject to numerous uncertainties at different levels of planning. Fuzzy project scheduling is one of the approaches that deals with uncertainties in the project scheduling problem. Hence, it has become one of the most fundamental and essential bases of research interest of many researchers. The subject of this dissertation entitled A study on sequencing and scheduling problems under fuzzy environment deals with scheduling problems in the manufacturing field. In this dissertation, we have applied
100 various approaches like critical path method, time cost trade off, flow shop scheduling problem, flow shop scheduling with setup time and flow shop scheduling with transportation time under fuzzy environment and demonstrated the effectiveness of the proposed methods by giving numerical examples. We have basically started from a simple approach to more tailored problem-specific ones. The main focus throughout this dissertation is the minimization of the project completion time, minimization of the project cost and hence maximizing the profit. Chapter one provides a general introduction of different components of the dissertation. This chapter explains some important techniques that are applied to handle uncertainty in project scheduling. The first section of this chapter introduces the scheduling problem and describes how uncertainty influences our ability to address the problems in the real world. It explains a way to reasoning with uncertainty in the scheduling domain. This chapter also comprises a literature review to investigate the current state of project scheduling under uncertainty. It explores the important limitations of the current practice of project scheduling under uncertainty. It determines the need, scope and objectives of the new approaches. Chapter two gives the fundamental concepts of fuzzy sets, different types of fuzzy numbers, ranking of fuzzy numbers and arithmetic operations of fuzzy numbers, which are essential for the development of this dissertation. Fuzzy critical path and fuzzy critical path length are very useful for the
101 project managers to take decision in planning and scheduling the complex projects. Chapter three briefly reviews the Fuzzy critical path analysis and describes the current popular techniques to find the critical path under uncertainty. In this chapter without defuzzifying the fuzzy activity durations, we propose a new method to find the critical path in a project network. In the present study imprecise variables are represented by trapezoidal fuzzy numbers. It is also found that the result obtained in this approach coincides with the existing earlier result which reveals that the proposed method is effective in determining the critical path in the fuzzy sense. Reducing the original project duration which is called crashing project networks in many studies which is aimed at meeting a desired deadline with the lowest amount of cost, is one of the most important and useful concepts for project managers. This dissertation aims at the development of an efficient approach with fuzzy activity time and fuzzy activity cost for project time-cost optimization, incorporating the vagueness or fuzziness of the dynamic conditions of the real world. It provides an efficient computational technique for time-cost optimization project scheduling problem incorporating the uncertainty. In Chapter 4, the concept of fuzzy time cost trade off is discussed. In this chapter, we use triangular fuzzy numbers to effectively deal with the ambiguities involved in the activity durations. Applying a new type of fuzzy arithmetic [36] and a fuzzy ranking method [37] we propose a
102 method for finding the optimal duration by crashing the fuzzy activities of the project network without converting the fuzzy activity time to classical numbers. The validity of the proposed method is examined with a numerical example which proves the advantages of the proposed method over existing methods available in the literature. By Scheduling, we assign a particular time for completing a particular job. The main objective of scheduling is to arrive at a position where we will get minimum processing time. In chapter 5 we discuss fuzzy flow shop scheduling problem and we make an attempt to obtain improved solutions to the given problem. The problem examined here is n job, m machine fuzzy flow shop problem under uncertain conditions. This study tries to solve the problem of a flow shop scheduling with the objective of minimizing the makespan. We started this chapter by presenting an overview of the scheduling theory. The terminologies of scheduling and algorithm for fuzzy flow shop scheduling are discussed in this chapter. Here, we formulated fuzzy flow shop scheduling problems where the processing time of each job at each machine was given as a triangular fuzzy number. We have proposed a new approach to minimize the rental cost of the machines where processing time of the machines is uncertain. A numerical example has been provided to explain the effectiveness of the proposed method. The results obtained are compared with the results available in the literature and are found to be
103 more cost effective. The model offers a new methodology for quantifying uncertainty in project scheduling and adds significant capabilities to fuzzy flow shop scheduling. Scheduling activities profoundly depend on the time or costs required to prepare the facility for performing the activities. It is a fact that there is tremendous savings when setup time or cost is explicitly incorporated in scheduling decisions in various real world industrial environments. However, this fact has been ignored in the vast majority of existing scheduling literature. In chapter 6 we emphasize the importance and benefits of explicitly considering fuzzy setup time or fuzzy setup cost in scheduling theory. A review of the latest research on scheduling problems with fuzzy setup time is also provided in this chapter. The objective of this chapter is to develop a new algorithm to minimize the rental cost for two stages specially structured flow shop scheduling problems under a specified rental policy with fuzzy processing time and fuzzy setup time which are represented by triangular fuzzy numbers. In this chapter, we have implemented an algorithm, which is an attempt to provide improvement in the solutions to fuzzy scheduling problems with set up time. We have discussed the results obtained by the proposed algorithm by providing a numerical example. We have also compared the results of our algorithm with the results obtained by Deepak Gupta et al. [27]. It can be concluded that solution derived by the proposed method is more
104 flexible and optimal in nature. Most machine scheduling models assume that jobs are delivered instantaneously from one location to another without transportation time being involved. In chapter 7 we relax this assumption, since there are practical scheduling situations in which certain time is required by jobs for their transportation from one machine to another machine. Chapter 7 deals with a three-machine fuzzy flow shop problem with triangular fuzzy processing time and fuzzy transportation time. In this chapter we propose a method for minimize the rental cost of the machines by minimizing the makespan without converting the fuzzy transportation time and fuzzy processing time to classical numbers. Besides giving an overview of the literature, in this chapter we have introduced the main concepts of fuzzy flow shop scheduling with transportation time. The new algorithm has been justified by an example taking bi-criterion three machines, five jobs flow shop problem in which both processing time and transportation time are represented by triangular fuzzy numbers. This study has demonstrated the feasibility of applying fuzzy set theory to sequencing and scheduling problems. In this dissertation, we have discussed various techniques to solve scheduling problems under uncertain conditions. Here we make an attempt to obtain improved scheduling which is found to be cost effective and which will increase the profitability and productivity of the organization.
105 Most of the existing methods convert the given fuzzy scheduling problem into a crisp problem and find the solution which results in the loss of fuzzy nature of the data. Also they fail to the capture uncertainty properly and produce inaccurate, inconsistent and unreliable results. The methods proposed in this dissertation preserve imprecise nature of the parameters till end. This is the uniqueness of the proposed dissertation. Without converting the fuzzy variables to classical numbers, this dissertation proposes algorithms which are an attempt to provide improvement in the solutions in fuzzy project scheduling. The algorithms proposed here provide better results which are more cost effective as compared with the algorithms available in the literature. This will give a clear vision to the decision maker to adopt a better strategy in taking the right decision at the right time. The conceptual framework for applications of fuzzy theory in project scheduling needs further developments in order to make it fully applicable to very large complex projects. There are several potential extensions to the ideas presented in this dissertation. Further improvements can be made in our algorithms so as to bring out much more enhanced solutions. The critical path analysis and time-cost trade-off problems under fuzzy environment can be extended by using multi objective linear programming, making use of arithmetic operations and ranking method for triangular fuzzy numbers, adopted in this dissertation. The above said problems can also be solved by using some other arithmetic operations, ranking method,
106 which may give more sharper solutions. This dissertation can be extended to m machines n jobs problems as a general case under fuzzy environment. Work can also be extended considering various parameters like job blocking, single and multiple transporting facilities, machine break down, weightage of jobs etc. Parallel machines concept in fuzzy environment, fuzzy job shop, fuzzy open shop problems can also be considered for future research work.