Uncertainty Approaches for Solving Generalized Machine Maintenance Problem

DOI: http://dx.doi.org/10.24018/ejers.2020.5.6.1813 Vol 5 | Issue 6 | June 2020 675  Abstract—In this paper, the generalized machine maintenance problem is formulated as linear programming model. The objective is to maximize the percentage production hours available per maintenance cycle of each machine. Data in many real life engineering and economic problems suffers from inexactness. There are different approaches to deal with uncertain optimization problems. In this paper two different approaches of uncertainty, Fuzzy programming and rough interval programming approaches will be introduced. We deal the concerned problem with uncertain data in coefficients of the constraints for the two approaches. A numerical example is introduced to clarify the two proposed approaches. A comparison study between the obtained results of the two proposed approaches and the results of interval approach for Samir A. and Marwa Sh [1] will be introduced.

are precise numbers.However, in the real world systems, the coefficients contained in the objective and constraints are imprecise.Galbraith [5] defines uncertainty as the difference between the amount of information required to perform a task and the amount of information already possessed.There are many approaches that can be adopted to treat uncertain systems.Two of these approaches are fuzzy approach and rough interval approach.Fuzzy optimization problems were considered first by Bellman and Zadeh [6].Thereafter, Tanaka et al. [7] developed the concept of fuzzy mathematical programming in a general level.In fuzzy programming the constraints and/or objective function are viewed as fuzzy sets and their membership functions also need to be known.The first formulation of fuzzy linear programming is proposed by Zimmermann [8].Ebrahimnejad et al. [9] proposed bounded linear programs with trapezoidal fuzzy numbers.Maleki et al. [10] introduced linear programming with fuzzy variables.
The idea of the other approach, rough interval, was proposed by Pawlak [11] as a new mathematical tool to deal with vague concepts.The point of strength of rough set theory than other methods is that, it requires no additional information, external parameters, models or functions to determine membership function and probability distributions.It only uses the information presented within the given data as presented in Düntsch and Gediga [12].Rough set is adopted in many problems as Krysinski proposed in pharmacology [13], Weiguo et al proposed in decision algorithms [14] and Arabani and Nashaei proposed in civil engineering [15].Osman et al. [16] calssified the rough programming problems into three classes according to the place of the roughness.Youness [17] introduced a rough optimal solution and a rough saddle point.Hamzehee et al [18] introduced a problem with rough interval coefficients for linear programming.
In our concerned study, we treat one problem of factory machine maintenance.Our objective is to maximize the percentage production hours available per maintenance cycle of each machine.We introduce the uncertain optimization problems using rough set analysis besides to fuzzy programming approach.We consider that each of the maximum spare part cost, depreciation cost and the maximum waiting time of each machine to be uncertainty rather than constant.
The rest of this paper is organized as follows.The problem formulation is described in section II.In section III the machine maintenance problem with fuzzy is proposed.In section IV the same problem is proposed using rough interval as uncertainty in constraints.A numerical example is provided in section V to clarify the proposed approaches.In section VI a comparison between the obtained results of the two proposed approaches and the results of interval We exhibit the following machine maintenance problem [1] max 1 subject to 1 The objective function (1) maximizes the percentage production hours available per maintenance cycle of each machine.Constraint (2) is on manpower cost associated with the maintenance of each machine.Constraint (3) is on spare part cost associated with the maintenance of each machine.Constraint (4) is on depreciation cost associated with the maintenance of each machine.Constraint (5) is on the maximum hour available for maintenance in each maintenance cycle.
In what follows we consider the parameters that represent each of the maximum spare part cost ( SC ), The maximum depreciation cost ( DC ) and the maximum waiting time of machine i (Wt i ) are uncertainty parameters.

III. THE MACHINE MAINTENANCE PROBLEM WITH FUZZY PARAMETERS
The optimization model of machine maintenance problem with fuzzy parameters is formulated as follows: max 1 1  13)-( 21) can be written in the following equivalent form:

IV. THE MACHINE MAINTENANCE PROBLEM WITH ROUGH PARAMETERS
The optimization model of machine maintenance problem with rough interval parameters is formulated as follows: max 1 Therefore problem (31)-( 36) can be written as max 1 A. Treatment procedure We will treat the uncertainty represented by rough interval coefficient in the constraints as follows: First we will divide the model for the machine maintenance problem with random rough coefficients in constraints into lower and upper interval problems.

B. Lower interval problem:
Then each of the lower and upper interval problems will be divided into two deterministic problems.So the lower interval problem (43)-(48) will give the following two linear problems P1 and P2: Where, the solutions of problems P1 and P2 give the surly optimal range of our machine maintenance problem with rough interval (37-42).
Similarly the upper interval problem (49)-(54) will divided into the following two linear problems P3 and P4: Where, the solutions of problems P3 and P4 give the possibly optimal range of our machine maintenance problem with rough interval (37-42).

V. NUMERICAL EXAMPLE
According to the data of an example which is reported in [1] consider the instance of machine maintenance problem to maximize the percentage production hours available per maintenance cycle of each machine is given by Table I, II and III.The maximum waiting time of machine 1 1 (0.5,1,1.5) The maximum waiting time of machine 2 2 (1,1.5, 2) Wt  2 0.9

 
The maximum waiting time of machine 3 3 (2, 2.5, 3) The maximum waiting time of machine 4 Wt  4 0.7

 
The maximum waiting time of machine 5

 
The maximum waiting time of machine 7 7 (1, 3, 4) The maximum waiting time of machine 8 8 (1,1.5, 2) The maximum waiting time of machine 9 9 (1, 2, 3) By substitute in the model ( 7) -( 12) and using the optimization approach which is described in section III, the deterministic form for this example is obtained as follows: max 0.998x 0.998x 0.999x 0.998x  The maximum waiting time of machine1 The maximum waiting time of machine2 The maximum waiting time of machine3 The maximum waiting time of machine4 The maximum waiting time of machine5 Wt  The maximum waiting time of machine6 Wt  The maximum waiting time of machine7 The maximum waiting time of machine8 The maximum waiting time of machine9 Wt  By substitute in the models ( 55) -( 60), ( 61) -( 66), ( 67) -( 72) and ( 73) -( 78) by using the optimization approach which is described in section IV, the deterministic problems for this example is obtained as follows: Whose solution is Whose solution is The solutions of P3 and P4 give us the possibly optimal range of the given example.So the interval [9.0114,11.2956] is the surly optimal range, the interval [7.1235, 12.3666] is the possibly optimal range and ([9.0114, 11.2956], [7.1235, 12.3666]) is the rough optimal range.The solutions of P1 and P2 give the completely satisfactory solutions and the solutions of P3 and P4 give the rather satisfactory solutions.

VI. COMPARISON STUDY
We test the efficiency of three uncertainty approaches for generalized machine maintenance problem.The three approaches are rough approach, fuzzy approach and interval approach.For the first two suggested approaches we assumed the parameters that represent each of the maximum spare part cost ( SC ), the maximum depreciation cost ( DC ) and the maximum waiting time of machine i (Wt i ) are uncertainty parameters.While in model of Samir A. and Marwa Sh. [1] the production hours (p i ) and the maximum waiting time of machine i (Wt i ) are uncertainty parameters.
For rough problem we obtained four deterministic problems and so we have four solutions, two of them give the completely satisfactory solutions and the others give the rather satisfactory solutions.Fuzzy approach and interval approach [1] have only one deterministic problem and gives one solution.From the obtained results in Table IV, it is clear that the objective function's values are very near to each other for the three uncertainty approaches.The rough approach is preferred because the four solutions obtained by this approach allow a wide range for decision maker to choose a suitable one.
they are constants.It is should be noted that the set of constraints (18-20) have been replaced by the set of constraints (27-29).

TABLE I :
DETERMINISTIC DATA OF MACHINE MAINTENANCE PROBLEM DC 

TABLE III :
DATA FOR ROUGH INTERVAL APPROACH Table IV clarifies the obtained results for the three approaches.

TABLE IV :
RESULTS OF THE THREE UNCERTAINTY APPROACHES