Solving Tri-criteria: Total Completion Time, Total Earliness, and Maximum Tardiness Using Exact and Heuristic Methods on Single-Machine Scheduling Problems

ABSTRACT


INTRODUCTION
Since 1954, scheduling issues have received a great deal of attention in the literature.Assigning machines to jobs in order to finish them all within specified constraints is the general definition of scheduling [1,2].Efficient scheduling is essential to prevent excessive or underutilization of resources [3].The scheduling problem is a collection of  jobs performed by one machine.Given job ,  ∈  , where  = {1, … , } has an integer-processed time   , due date   .Given schedule  = ((1), (2), . . ., ()) ,  1 =  1 and   = ∑     =1 (where  = 2, 3, … ,  ) are then used to determine the completion time for each job .  =   −    expresses the lateness time of the job  .  = {0, −  } =  {   −   , 0} is used to defined the earliness of job , tardiness in job  is defined as   = {0,   },   =   −   is used to determine the slack time of job .Thus, there is a total completion time ∑ ∈   , total earliness time ∑ ∈   and maximal tardiness   =  ∈ {  } .Smith [4] concerned the total completion time, 1// ∑  problem is minimized using SPT (short processing time) rule and is optimum in 1956.The maximal earliness regarding the 1//∑  problem has been minimized using MST (minimum slack time) rule [5].According to a study by Jackson [6], the Earliest Due Date (EDD) rule was used to minimize the maximum tardiness with respect to the 1//  ; the problem 1//∑  is NP-hard.Any problem with cost functions as subproblems is NP-hard.The challenge is to determine the ideal processing sequence for these jobs on each machine in order to minimize the given objective function.
Researchers focused on only one objective function [7].In practical cases, the decision maker only needs to select one objective function.Nowadays, more studies are conducted on multi-objective planning problems.An overview of multiple and binary scheduling problems was published by Hoogeveen [8].Hierarchical and simultaneous minification are the two main structures used to solve competing criteria [9].The primary criterion is the first and the secondary criterion is the second.In this scenario, one decreases the primary criterion and selects a table with the minimum value of the secondary criterion.In the second method, a Pareto set is formed and the decision maker is the one with the optimal composite objective function [10].Hoogveen [8] presented an algorithm that finds all effective tables for 1//(∑  ,   ) problem.Abdul-Razaq and Ali [11] studied problem 1//(∑  , ∑  ,   ) and found a sub-problem, also solved this problem by branch and bound method.Abdul-Razaq and Motair [9] presented a multiobjective function 1// ∑  + ∑  +   +   and used branch and bound to minimize this problem.Jawad et al. [12] provided the BAB to solve multi-criteria objective function 1// (∑  , ∑  ) problem in the SMSP.Ahmed and Ali [13] suggested BAB and heuristic approaches to minimize the ∑  +   +   for single machine scheduling problem.Al-Tameemi [14] used BAB to solved the problem 1// ∑  + ∑  +   .Arik [15] offered earliness /tardiness with shared due dates and gray processing times.Hameed and Chachan [16] multi-objective minimization the ∑(  +   +   +   ) was proposed for single machine scheduling problems, two local search algorithms (GA and PSO) were also used.Also, Chachan and Hameed [17] used BAB method and local search algorithms to minimize ∑(  +   +   +   ) .In addition, Chachan and Jaafer [18] presented a Branch and Bound algorithm to minimize the ∑(  +   +   +   +   ) with unequal release dates for scheduling (  ) jobs on a single machine.Large and complex problems in the research community are often solved using contemporary heuristic optimization techniques [19].Hassan et al. [20] used a heuristic algorithm to minimize the (  +   + ∑  )in a SMSP.Neamah and Kalaf [21] proved that SPT and EDD rules give efficient (optimal) solutions for two problems 1//(∑  , ∑  ,   ), and 1//∑  + ∑  +   , also, proven special cases, resulting in the most an efficient and optimal solution to these problems.
Within the paper, proposed two new heuristic methods to solve and find efficient solutions three criteria ∑  , ∑  ,   for scheduling problems.We started by organizing it as a tricriteria mathematical model and proposed a sub-problem with three objectives from the original problem.
Below is an outline of the remaining portion of this paper: Section 2 describes the mathematical formulations of tricriteria and analysis of the sub-problem for the proposed problem.Section 3 presents the exact, approximate methods and algorithms for solving the two problems given in Section 2. Section 4 validates the proposed model and demonstrates the effectiveness of the proposed strategy through computational study and results.Moreover, Section 4 presents the results and accompanying discussion.Finally, conclusion and lists of future works are provided in Section 5.

MATHEMATICAL MODEL
The mathematical formulation of the single machine scheduling problem for tri-criteria and tri-objective functions is presented in this section.Firstly, some of the notations included that are utilized in the formulation of tri-criteria and tri-objective functions of the single machine scheduling problem: ACT/S: Average of CPU-Time per second.ANEFS: Average number of efficient solutions.BAB(WDR): BAB method with dominance rules (DRs).BAB(WODR): BAB method without DRs.CT/S: CPU-Time per second.EDD: Jobs are arranged according to their due dates in nondescending order   (where  1 ≤  2 ≤ ⋯ ≤   ); this rule is utilized for minimizing   for problem 1//  [6,7,22].
: OF of (    ) -problem, and   is objective function of ()-problem.
Feasible schedule: Any schedule  ∈  (S is the collection of all schedules) can be considered feasible if it meets the problem's constraints.
MST: Jobs are arranged according to their slack time   =   −   in a non-decreasing order (where  1 ≤  2 ≤ ⋯ ≤   ).For minimizing   with the use of this rule [8].
MOF: Multi-objective function; N: Number MCF: Multi-criteria function.NEFS: Number of efficient solutions.  : N. of jobs, where  is the number of problems tested.OF: objective function regarding MSP could be either maximized or minimized under all possible constraints.
Optimal (OP): The  * schedule is considered optimal in the case when there isn't other schedule  that satisfies   () ≤   ( * ), where  from 1 to (: N. of criteria), assuming a strict inequality for a minimum of one of the conditions that have been mentioned earlier.If not, then  can be considered as dominant over σ * .
SPT: The jobs are being processed in a non-descending order   (i.e.  1 ≤  2 ≤ ⋯ ≤   ), it is commonly known that this rule minimizes ∑  for the 1//∑  problem.

The mathematical model for the 1// (∑𝑪 𝒋 , ∑𝑬 𝒋 , 𝑻 𝒎𝒂𝒙 ) problem
The problem aims to discover an efficient solution that yields the minimal value of the tri-criteria.Total completion time ∑  , total earliness time ∑  , and the maximum tardiness   ; this problem is denoted by:   = (∑  , ∑  ,   ) subject to   ≥   (), This problem is referred to as the (    ) -problem.For (    ) -problem, sub-problem can be concluded that 1// (∑  + ∑  +   ) problem that referred to the ()problem, and it can be defined as follows:

METHODOLOGY
In this section, two exact method (BAB and CEM) and two HMs were introduced for solving the (    )-problem and ()-problem.For the exact approaches, the BAB is utilized as the main approach for solving the problems.Moreover, BAB without DR and BAB with DR were performed.Also, two HMs were proposed and were adopted to find efficient solutions to this problem in a reasonable time.

Exact method
We have presented two exact methods in this subsection (CEM and BAB).The CEM was used as a simple approach that generates all of the feasible tables for choosing the optimal solution.While, the BAB method is the most popular scheduling solution approach.BAB is an illustration of the implicit enumeration method that could identify the optimal solution by methodically reviewing subsets of potential solutions.A search tree with nodes corresponding to these subsets has been utilized for describing BAB.

BAB method to solve the (𝑆 𝐶𝐸 𝑀 𝑇 )-problem
In this subsection, two BAB techniques will be used to solve this problem.
First technique is BAB without DRs (BAB(WODRs)).This method can be summarized as follows: the LB for the non-sequenced section of each node will be based on the SPT rule, and the UB utilized will be based on the MST rule.
The following steps for BAB(WODRs) can be seen below:
Step 4: For every node in the BAB approach's search tree and each partial sequence  of jobs, compute LB()= The objective function's cost of sequencing jobs in  + the cost of un-sequenced jobs arranged according to the SPT rule (where  = SPT).
Step 5: A branch of each node with LB does not dominate the UB.
Step 6: Obtaining a set of solutions at the final level of the search tree, if () the result is indicated,  is added to the set  unless they are dominated by efficient solutions that have been obtained previously in , this process is called filtering .
Second technique is BAB with DRs (BAB(WDRs)).This method could be summarized as follows: The UB and LB of each node for the un-sequenced portion will be based on the SPT rule.To decrease the number of open nodes, which saves time and increases the number of solved problems, this BAB depends on DR, since the size of search tree (number of the nodes) grows as the number of () increases in the BAB approach, particularly in the branching scheme.Thus, it is necessary to decrease this size by removing irrelevant solutions or choosing intriguing ones.The goal of dominance rules is for reducing the available research on scheduling problems.Consequently, as a process for reducing search area and shorten search period.Several Dominance Rules can be used to reduce the current sequence.DRs typically indicate some (all) sections of the path in order to acquire a good value for the objective function, and they can be valuable in determining if a node in the BAB method can be discarded before its lower bound (LB) is determined.DRs are clearly useful when a node can be ignored despite having a less-thanoptimal LB.The DRs are also useful in the BAB approach for eliminating nodes that are dominated by others.These enhancements result in a significant reduction in the number of nodes required to achieve the efficient (optimal) solution.By applying the following theorem: Theorem (1) [23]: If   ≤   and   ≤   , then there's an optimal schedule for (SP) -problem where the job  is processed before the job .
Step 4: For every node in the BAB approach's search tree and each partial sequence  of jobs, compute LB()= The objective function's cost of sequencing jobs in  + the cost of un-sequenced jobs arranged according to the  = SPT rule.
Step 5: Branch from every node within LB ≤ UB and check  → .
Step 6: Obtaining a set of solutions at the final level of the search tree; if F(σ) the result is indicated, σ are added to the set S unless they are dominated by efficient solutions that have been obtained previously in , this process is called S filtering .

HMs for (𝑺 𝑪𝑬 𝑴 𝑻 )-problem and (𝑺𝑷)-problem
Heuristic methods speed up the process of reaching a satisfactory solution.Many researchers have used heuristic algorithms to solve NP-hard problems [24].In this subsection, two heuristic algorithms were proposed (SM-(    ) and DR-(    ) for solving the (    )-problem and the (SP)problem:

SM-(𝑆 𝐶𝐸 𝑀 𝑇 ) method
SM-(    ) method is proposed for solving (    ) -problem and ()-Problem in this subsection [25].Firstly, the objective function using the SPT rule is calculated.Next, arrange the third job in the second position, with the other jobs arranged in accordance with the SPT rule and compute OF, etc., up to  sequences are obtained, then repeat the same procedures when using the MST rule, as described below:

End;
Step 6: To find a set of efficient solutions for (    ) problem, filter set .
Step 7: Output:  represents a set of efficient solutions.

DR-(𝑆 𝐶𝐸 𝑀 𝑇 ) heuristic method
DR-(    ) depends on DRs is proposed for solving (    ) − problem and (SP)-problem.To summarize DR-(    ) method, find a sequence sort with a minimum of   and , corresponding to the DRs, and compute the objective function.DR-(    ) algorithm is summarized in the following steps:
Step 3: Discover the sequence  1 with a non-increasing order of   that does not conflict together with matrix  (DR); if   =   ,where ,  ∈  then order  1 by   , then  =  ∪ { 1 }.
Step 4: Discover the sequence  2 with a non-increasing order of   that does not conflict together with matrix  (DR); if   =   ,where ,  ∈  then order  2 by   , then  =  ∪ { 2 }.
Step 5: Determine the set of the dominant sequence  ′ from .

Results and discussion of the (𝑺 𝑪𝑬 𝑴 𝑻 )-problem
In this subsection, the results of applying the exact methods will be compared with the heuristic methods for the problem (    ).All results from using all presented methods are averages of five examples for each .The simulation result for exact and heuristic methods for solving the 1//(∑   , ∑   ,   ) problem can be analyzed in terms accuracy and CPU-Time.From Table 1 illustrated that CEM gave minimum values for the 1// (∑   , ∑   ,   ) problem compared to the results of the BAB up to  ≤ 11.Also, CEM was taken a long time (CPU-Time) compared to BAB.In additionally, the BAB(WODR) starts to give the minimum values for the (    ) problem compared to the results for BAB(WDR) for  ≤ 7, while BAB(WDR) starts to give the minimum values for the (    ) problem compared to the results for BAB(WODR) for  > 7.
Moreover, from Table 2, BAB(WDR) gave minimum values in terms accuracy and CPU-Time compared with results of BAB(WODR), for  = 12 to 50.Therefore, BAB(WDR) performs better than BAB(WODR), and BAB without DR solved the problem in all cases from  = 12 to 19, but failed to solve the problem when  ≥ 19, whereas BAB with DR (BAB(WDR)) solved the problem in all cases from  = 12 to 55, but failed to solve the problem when  > 50.Moreover, from Tables 1 and 2, the results of the BAB with dominance rules confirmed the number of efficient solutions to the problems less than the number of efficient solutions for BAB without dominance rules and CEM.In addition, from Table 3, the CEM gave best results compared to the heuristic methods (SM-(    ) and DR-(    ).Furthermore, CEM takes a long time in the CPU-Time, whereas the HM SM-(    ) gives better results than DR-(    ) for  ≤ 11.In Table 4, BAB(WDR) gave best showed results when compared with BAB (WODR), SM-(    ) , and DR-(    ) which also showed that BAB without DR solved all cases of the problem from  = 4 to  = 19, and BAB with DR failed to solve all problems for  > 50.In general, the results of BAB method are better when compared to Heuristic Methods up to  = 50.Also, heuristic method SM-(    ) gives better results than DR − (    ) for  ≤ 50 .While DR − (    ) gives better results than SM-(    ) for 50 <  ≤ 5000 , for problem (    ) .From Table 5, the   2 shows the comparison result between BAB and HM for the problem ().Moreover, in Table 8, BAB(WODR) gave best s results when compared with BAB (WDR), SM-() and DR-(), which also showed that BAB with DR solved all problem situations from  = 4 to 17.In addition, all instances of the problem were solved by BAB without DR, from  = 4 to 15, and when  > 15, was unable to solve the problems.However, SM-() and DR-() solved all the problems from  = 4 to  = 4000, but the BAB (WODR) and BAB (WDR) method gave better results.

CONCLUSIONS
In this paper, two new techniques two new Heuristic methods SM-(    ) and DR-(    ) were proposed to solve the tri-criteria problem (1//(∑  , ∑  ,   )), three multi objective (1// (∑  + ∑  +   ) ) machine scheduling problems.In addition, BAB with DRs, BAB without DRs, and CEM as an Exact method were used to compare results  terms of accuracy and computational time.The result showed that, SM-(    ) performs better than DR-(    ) were for all  ≤ 400, while, when  ≥ 400, DR-(    ) gave better results than SM-(    ).Furthermore, for two problems, the result of the BAB with dominance rules for two problems showed a lower number of efficient solutions for all .than the number of efficient solutions of d BAB without dominance rules and CEM.For future work, a new UB and LB can be used for the BAB algorithm to prove its effectiveness in determining the best solution for the MOF.Different machine environments can be used to study more complex problems and/or our proposed problems can be completed with constraints, such as the release date (  ), setup time (  ), and pre-stopping.

Figure 2 .
Figure 2. Results of the comparison between the BAB with and without DR, SM-(), and DR-() for problem () Table 1 illustrated the comparison results of BAB(WODR), BAB(WDR), and CEM for the problem (    ) with  = 4, 5, … , 11.In Table 2, the results of BAB without and with DR for problem (    ), where  = 12: 19,20,30,40,50 were presented.Also, Table 3 showed the comparison results between the proposed heuristics methods (SM-(    ) and DR-(    ) ), with CEM for the problem (    ) with  = 4 to 11.In addition, the results of SM-(    ) and DR-(    ) that were compared to the BAB(WODR), and BAB(WDR) for the problem (    ) have been listed in Table 4, for different values of .Table 5 presented the results of SM-(    ) and DR-(    ) for problem (    ) for different .