Exact Solutions for Minimizing cost Function with Five Criteria and Release Dates on Single Machine

In this paper, we present a Branch and Bound (B&B) algorithm of scheduling (n) jobs on a single machine to minimize the sum total completion time, total tardiness, total earliness, number of tardy jobs and total late work with unequal release dates. We proposed six heuristic methods for account upper bound. Also, to obtain lower bound (LB) to this problem we modified a (LB) select from literature, with (Moore algorithm and Lawler's algorithm). And some dominance rules were suggested. Also, two special cases were derived. Computational experience showed the proposed (B&B) algorithm was effective in solving problems with up to (16) jobs, also the upper bounds and the lower bound were effective in restricting the search.


Introduction
A scheduling problem is defined as a problem of assigning a set of tasks or jobs to a set of resources or machines in a specific time. Common performance measures criteria are usually in the form ∑fj and fmax =max{fj} Most of the studies that were introduced in the early years of the discovery of the theory of scheduling focused on a single performance measuring criterion. However, in practice, a manager may need to find an acceptable schedule which can simultaneously meet the requirements for several criteria [1].
In this paper, we examine the problem of scheduling with five criteria (referred to above) for measuring performance with the release times, and to the best of our knowledge, this problem was not studied before.
Here, we found it necessary to present some research from the literature which included machine scheduling problems with one or more of the performance criteria contained in our problem. studied this 1/rj/∑WjVj problem, using unequal release dates, to solve this problem some special cases were proofed and using a branch and bound algorithm with up to 30 jobs. Also five local search methods to solve this problem were applied and performance is evaluated to it with up to 60000 jobs, and who showed this problem its NP-hard.
The unit Penalty: For the 1//∑Uj , Moore [20]. Study was among the earliest to consider scheduling to minimize this problem by an algorithm known as Moor algorithm "sometimes known as Hudgson's Algorithm" that solved the problem optimally. With release dates, the problem 1/rj/∑Uj is strongly NP-hard by Lenstra et al [2]. Dauzereperes [21]. Studied this1/rj/∑Uj problem, and determined a lower bound depending on the relaxation of a "Mixed-Integer-Linear-Programming" formulation, presented a heuristic method to solve the problem. A large sample of problems has been tested with up to 50 jobs. The calculations showed the efficiency of the proposed approach by comparison with the lower bound. Philippe et al [22]. To solved this 1/rj/∑Uj problem, who suggesting (B&B) algorithm, with lower bounds based on a "Lagrangian relaxation". Also, they used dominance rules to reduce the search space, suggested techniques are showed solve to optimality cases with up to 200 jobs. Cyril and Samia [23]. In this study showed how good -quality lower and upper bounds that can be calculated for the problem1/rj/∑Uj, using an original mathematical integer programming formulation. Numerical experiences showed the assessing of the proposed approach it up to 160 jobs. Al Zuwaini and mohanned [24,25]. Studied the problem 1/rj/∑(Fj + Uj), and presented a (B&B) algorithm, and application of some dominance rules to solve this problem, finding lower bound by using (SPT)-rule and Moor's algorithm. Computational experience with instances having up to 40 jobs showed that the lower bound was effective in restricting the search. Scheduling problems of multiple performance measures (three or more) with release date, to the moments of writing this paper. We did not find any study submitted to discuss this subject.
From the above, we can say that scheduling problem often increases complexity by increasing the number of performance measures with release date. Also, we can say that our problem (P) is the first study to address scheduling problem with five criteria and with release date.
In this paper, we describe the problem of scheduling of (n) jobs on one-machine with multiple performance measures and release date, with a view to minimizing the sum total completion time, total tardiness, total earliness, number of tardy jobs and total late work, this problem is denoted by 1/ / ∑ …(P), from used form 3-field α ∕ β ∕ by Graham et al [26]. This paper is organized as follows: In next section begins with some notation and basic concepts of one-machine scheduling. Formulation of the problem and decomposition into three sub-problem are given in section 3. Some algorithms are given in section 4. Also, three special cases were presented in section 5. in section 6, the (B&B) algorithm was discussed, an account the upper bound and the derivation of a lower bound, and some dominance rules in section 7. Computational results are presented in section 8.

Notation
The following notations are used in this paper: j : Job index.

The Mathematical Formulation
The problem (P) considered in this paper is to schedule a set N ̅ of n jobs , N ̅ ={1,…,n} on an one-machine. Each job j , j ϵ N ̅ has integer processed time pj , a release date rj , and due date dj. Given a schedule σ =(1,…,n) , then for each job j we calculate the completion time by C1=r1 + p1 , Cj= max {rj , Cj} + pj for j=2,…,n. The tardiness of job j is defined by Tj =max{Cj -dj, 0}, and earliness by Ej =max{dj -Cj , 0}. The unit penalty of job j is defined by Uj = 1 , if Cj> dj; o.w, Uj=0.The late work of job j given by Vj =min{Tj ,pj}. Let be a set of all feasible solutions , and σ is a schedule in . The mathematical form of our problem (P) can be written as : The objective is to find a processing order σ =(σ(1),…,σ(n)) for the problem (P) to minimize the sum of the total completion times, the total tardiness, the total earliness, the number of tardy, and the total late work.
Then go to step (2).
Otherwise if (t ˃ dk ) then find a job j ∈ , with pj is as large as possible and set E=E-, L=L∪ , and t= t-pj , go to step (2).
Step 4: E is the step of early jobs , L is the step of late jobs.
This algorithm to solve the (1//Uj ) problem.

Special Cases (SC)
A special cases (SC) for scheduling problem means getting an optimal schedule (optimal solution) directly without utilize (B&B) method or (DP) technique [28].
In this section we present Two (SC) of our problem (P), which are as follows: Case 1. If Cj= dj ∀ j in a schedule S and the preemptive is allowed then S given an optimal schedule for the problem 1/ rj, pmtn / ∑ .

Branch and Bound (B&B) Algorithm
In this section, we apply (B&B) to get an exact solution for our problem (P). The (B&B) method is strategy to explore the solution space based on the implicit enumeration of the solution. This method is based on the idea of calculate all feasible solutions by a special technique, which among them research tree technique, where helps to present the procedures of this method more clearly. Here, at the root node of the search tree, we suggest six heuristic methods to provide an upper bound (UB) on the cost of the optimal schedule. Also, we derive a especially formula to ensure an lower bound (LB), as following:

Six-Upper Bound (6-UB)
In this section, six heuristic techniques are used for arranging the jobs and valuation the cost problem (P). Among these six-heuristics, we select the lowest value to be the an upper bound, (i.e. UB= min {UB-1,..., UB-6}). This UB is then used in a root node of the search tree in (BAB) method.
Step ( Both of the (UB), (ILB) represent two values a root node in a search tree.
The (B.A.B) method include of essential procedures the following [24]:  Branching is the procedure of dividing mother (original) problem into two or more sub-problems. In the search tree, the sub problems are expressed by nodes. The branching rule specified by use forward branching means the jobs are sequenced one by one from the beginning. Or backward branching i.e. (the jobs are sequenced one by one from the end).  Bounding is procedure the of computing a lower bound on the optimal solution of a sub problems (nodes).  Search strategy is a procedure reflect the method of choosing a node in the search tree to branching from it, usually the branch be from a node with the smallest (LB) in the search tree , with commitment to the following [30].
o If (LB) ≥ (UB), then this sub problem cannot yield a better solution for problem. Thus, we need not continue to branch from the corresponding node in the branching tree.
o If (LB) < (UB), then (UB) is reset to take (LB) value, ( i.e. replace (UB) by (LB) ). This procedure is repeated until all nodes (sub-sets) have been test. The at first noteworthy feature of (B&B) is the use of "Dominance-rules" (DR) that try to exclude nodes prior to calculating (LB) to it [2]. These (DR) are computationally useful as they reduce storage requirements on the computer as well as reducing computation time .

Dominance-Rules (DR)
Dominance rules (DR) usually specify whether a node can be discarded in search tree before its lower bound (LB) is calculated, so it helps reduce search space. Clearly (DR) are particularly useful when a node can be eliminated which has (LB) that is less than the optimal solution. To introduce the (DR) for our problem (P) consider schedules S = (σ, i, j, σ' ) and S' = (σ, j, i, σ') where σ, σ' are two a partial schedule of the remaining n-2 jobs. Let t=∑ ∈ , be the completion time of σ, with rj = r, di≤ dj, and pi≤ pj. Define ( Ci(t) + Ti(t) + Ei(t) + Ui(t) + Vi(t) an the sum of total completion time, tardiness, earliness, number of tardy jobs and late work of job i ) if scheduled at time t and let Fij= (Cij(t) + Tij(t) + Eij(t) + Uij(t) + Vij(t) ) be the sum of total completion time, tardiness, earliness, number of tardy jobs and late work of job i and j, if i precedes j and their processing starts at time t. The following interchange function Δij(t) is used to specify the new dominance properties which gives the cost of interchanging adjacent jobs i and j whose processing start at time t.

Computational Experience
An intensive work of numerical experimentations has been performed subsection (8.1) shows how instances (test problems) can be randomly generated.

Test problems:
We created (10) problems randomly, for each problem, nϵ {5, …, 16} jobs, and for each job j has the following data:  The processing time pj is generated from the discrete uniform distribution [1,10].  Integer due date dj is generated from the uniform distribution [ where Sp= ∑ , (Tf) is the "tardiness factor", and (Rrdd) is the "relative range of the due dates". For the two parameters (Tf) and (Rrdd), the values (0.2, 0.4, 0.6, 0.8, 1.0) are considered.  Integer release date rj is generated for each j from the uniform distribution [1,5].

Computational Experience with the (UB) and (LB) of (B&B) Algorithm
The (B&B) algorithm was tested by coding in (Mat lab 2018) and running on a personal computer Dell Core i7 with Ram 8 GB. Tables 1, 2. Shows the results to problem (P) obtained by (B&B) algorithm, when n ϵ{5,6,…,10} and nϵ{11,12,…,16} respectively. The first column (n) indicate to the number of jobs, the second column (EX) indicate to the number of examples for each instance n, the third column (CEM) complete enumeration method only in Tables 1. The fourth column (optimal) indicate to the optimal solution obtained by(B&B) method, the fifth and sixth columns indicate to upper bound (UB) and initial lower bound(ILB) respectively, and the other columns (NON) are indicate to number of nodes, time (CEM), and time(B&B) , finally, column (status) indicate to the problem solved (0) or not (1). The symbols (*) indicate to the (UB) given an optimal value and (**) indicate to the (ILB) given an optimal value. The (B&B) algorithm was stopped when the sum of (status column ≥3). A condition for stopping the (B&B) algorithm was determined and considering that the problem is unsolved (state is 1). Here, the (B&B) algorithm is stopped, after (1800) second. From Tables 1, 2. We are noticed that the six heuristic of upper bound given good results, it gives the value for objective function equal to optimal or near optimal value.
We also have two other Tables 3, 4. Are the summary of the two previous Tables 1, 2. That show the average computational time of (CEM) and (B&B), the average of nodes, and the unsolved problems. Table 1. The performance of CEM, optimal of (B&B), (UB), (ILB), number of node and (CPU) in seconds of (CEM) and (B&B), for n = (5,7,10 In the Tables 1, 2 Table 1. T.BAB: The time (in seconds) which is required for (B&B).

Conclusion and Future Work
In this paper, we been developed exact solutions for the problem of scheduling (n) jobs on one-machine to minimize the sum total completion time, total tardiness, total earliness, number of tardy jobs and total late work with unequal release dates. A branch