Branch and Bound Method to Solve Multi Objectives Function

This paper presents a branch and bound algorithm for sequencing a set of n independent jobs on a single machine to minimize sum of the discounted total weighted completion time and maximum lateness, this problems is NPhard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.


1.INTRODUCTION
We consider the scheduling n jobs on a single machine to minimize the bi-criteria problems. Our objective is to find a schedule that minimize sum of discounted total weighted completion time and maximum lateness, with penalty of lateness equal to one the extension of the problem when the penalty of lateness equal or more than one was consider also. The all jobs are available at time zero. This problems is denoted by ∑ ( ) where and ∑ ( ) where (h represent the penalty of lateness).It is minimizing of total discounted weighted completion time which it extends from the minimization of total weighted completion time, Rothkof (1966) [1]. Rothkopf and Smith (1984) [2] considered the total discounted weighted completion time as 1//∑ ( ) . Their study was more general cost function and discounted rate of r , of the problem can be solved as P-type optimally in polynomial time by the Weighted Discounted Shorter Processing Time (WDSPT) rule. The rule that schedules the jobs in non-decreasing order of ratio : [3]. Wang et.al ( 2006) consider the problem ∑ ( ) is NP-hard [4].
Yunqiang Yin et al. (2012) [5] showed that the total weighted discounted completion time is polynomials solvable and optimal by considering the effects of position-dependent learning and time-dependent deterioration simultaneously. The problem total weighted discounted completion time studied by Lin Li et al. (2013) [6] showed the heuristics according to the corresponding problems without learning effect. Guochen Sun andYuewu Li (2013) [7] showed the problem ⁄ ∑ ( ) ⁄ is past-sequence-dependent delivery time and deteriorating jobs. Hongjie Li et al. (2014) [8] studies the problem ⁄ ( ) . / ⁄ considers a scheduling environment in which there are two agents and a set of jobs, each of which belongs to one of the two agents and its actual processing time is defined as a decreasing linear function of its starting time. Each of the two agents competes to process its respective jobs on a single machine and has its own scheduling objective to optimize. The discounted cash flow (DCF) analysis is a method of evaluating a project, company, or assessing using the concepts of the time value of money. All future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows both incoming and outgoing, is the net present value (NPV) which is taken as the value or price of the cash flows. Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient times. Studies of ancient Egyptian and Babylonian mathematics suggest that they used techniques similar to discounting of the future cash flows. Following the stock market crash of 1929, discounted cash flow analysis gained popularity as an evaluation method for stocks. Irving Fisher in his 1930 book The Theory of Interest and John Burr Williams's 1938 text The Theory of Investment Value were the first who formally expressed the DCF method in modern economic terms. The DCF use has increased substantially in institutional ,investment property and business valuation sectors. It is frequently required by client, underwriters, financial advisers and administrators, and portfolio managers. It is used in investment finance, real estate development, corporate financial management and patent valuation.
The discounted cash flow formula is: where CF: cash flow, r: discount rate, n: is the time in years before the future cash flow occurs.
Discounted cash flow analysis is an extension of simple cash flow analysis that takes into account the time value of money and the risks of investing in a project. DCF analysis can be divided into two main categories, the net present value method (NPV) and the internal rate of return method (IRR). Today the DCF model is the most commonly used tool among financial analysts when valuing a firm. It is documented that almost fifty percent of all financial analysts use a DCF method when valuing potential objects to acquire (Hult, 1998  Jackson's earliest due date (EDD) rule1 (i.e., by ordering the jobs according to non-decreasing due dates) Jackson (1955) [9]. Lawler and Moore (1969) [10] showed that the optimal schedule for ( ) in ( ) time. Horn (1974) [11] observed that ( ⁄ ) and ( ⁄ ) were solved by the extended Jackson's rule. Lageweget al. (1976) [12] showed that the problem ⁄ ⁄ and ⁄ can still be solved in polynomial time. Simons (1978) [13] presented a more sophisticated approach to solve the problem ( ⁄ ). Lenstra et al(1977) [14] showed the problem ( ) which is strongly NP-hard. Various branch and bound methods (BAB) exist for solving the problem ( ⁄ ), Baker andSu (1974) (1982) [15] [16]. Sen and Gupta(1983) [17] extended the method to the problem ∑ is NP-hard. The problem ∑ is NP-hard and it's studied by Sen et al. (1988) [18] that developed a branch-and-bound algorithm and derived lower bounds by means of the maximum potential improvement method. Hariri and Potts (1997) [19] proposed a branch and bound (BAB) to solve the ( ) problem.
Some approaches used for the problem of ⁄ ⁄ considerations are given by (Potts and Kovalyov, 2000) [20] and (Allahverdi et al., 2008) [21]. Christos and George (2010) [22] showed that the problem can be solved by simple polynomial-time algorithms. Habibeh and Lai (2010) [23] used genetic algorithm for the problem . RadostawRudek (2011) [24] proved that the problem maximum lateness is NP-hard even if job processing time are described by linear function he proposed BAB algorithm and approximation to verify numerically their efficiency. Yunqiang Yin et al. (2012) [5] showed that the maximum lateness is polynomial solvable and optimal by considering the effects of position-dependent learning and time-dependent deterioration simultaneously. The problems maximum lateness studied by Lin Li et al. (2013) [6] showed the heuristics according to the corresponding problems without learning effect. Suh-Jenq Yang et al. (2013) [25] showed the problem of ( ⁄ ) is consider of past-sequence-dependent delivery times and the effects of deterioration and learning. Morteza and Mehde (2015) [26] proposed a branch and bound (BAB) to solve the problem ⁄ . Spyros T. and Alekos T. (1993) showed that can be found optimal schedule under the condition of unit-length independent of no preemption jobs and identical or uniform machines with respect to the criterion in ( ) which is a significant extension of the well-known maximum lateness [27].

2.Problem formulation
We take into consideration the problem of scheduling n jobs on a single machine to minimize the total cost that can be stated as follows: A set N={1,2,3,….,n} of n independent jobs has to be scheduled on a single machine in order to minimize a given criterion. This study applies the one machine scheduling problem with multiple objective function: The sum of discounted total weighted completion time and maximum lateness which is denoted by ∑ ( ) ⁄ . Under the conditions: Preemption is not allowed, no precedence relation among jobs is assumed and only one job i can be processed at a time. Given a schedule (1,……n), for each job j needs processing time and a positive weighted on the machine and ideally should be completed at its due date Discounted total weighted completion time ∑ ( ) and maximum lateness , can be respectively defined as: Our scheduling problem can be stated mathematically more precisely as follows: Given a schedule (1,2,3,….,n), then for each job can calculate the discounted total weighted completion time ∑ ( ) and maximum lateness . The objective is to find a schedule, ( ( ) ( ) ( )) (belonging to a neighborhood of ( That minimizes the total cost ∑ ( ) ( ) ( ) .
Let S be a set of all schedules,| |= , then we can formulate our problem in mathematical form as: Also the multiple criteria: sum of discounted total weighted completion time and penalty maximum lateness was considered, which it signifies extension of the criteria ∑ ( ) ( ) ( ) under the same condition (E), furthermore is incurred as a penalty for late shipment. That's the problem that the cost of completing job j at a time is denoted:

Decomposition of problem (A)
Because the complexity of problem (A) can be decomposed it into two subproblems which are a simple structure as the follow.
i. ∑ ( ) the discounted total weighted completion time which is solved by WDSPT rule [3] and the formulation mathematical form as: ii. the maximum lateness which is solved by EDD rule [9] and the formulation mathematical form as:

Special Cases
A machine scheduling problem of type NP-hard is not easily solved and it is more difficult when the objective function is multi objective. Using some Mathematical programming methods to find optimal solution for this kind of problem, such as dynamic programming and branch and bound method. Sometimes special cases for this problem can be solved. A special case for scheduling problem means finding an optimal schedule directly without using mathematical programming techniques. A special case, if it exists, depends on satisfying some conditions in order to make the problem easily solvable. These conditions depend on the objective function as well as the jobs [29]. In this section, some special cases for problem (A) and (B) are given.

Case (1):
If the jobs of a schedule ordered according to and satisfy (JIT) job , then gives optimal solution for the problems (A) and (B).

then
. But gives an optimal solution for ∑ ( ). So is optimal solution for the problem (A) and (B). ∎

Proof:
Since is minimized byLawler's algorithm but rule gives minimize to ∑ ( ) and ( ) ( ) Hence algorithm gives an optimal solution for the problem (B). ∎

Case (4) :
If and Then is optimal for the problem (B).

Proof:
Since and then any sequence gives so is optimal for the problem ( is waited time for job j).

Proof:Since
and Then (because ) this means that for each job j, j is just in time (JIT) that is . Then rule gives the optimal solution for the problem (A) and (B). ∎

Case (6):
If and then EDD rule gives an optimal solution for the problem (A) and (B).

Proof:
Since and then any sequence gives optimal for ⁄ ∑ ( ) problem. And EDD is optimal solution for ⁄ ∑ ( ) ⁄ because EDD is optimal for . ∎

Case (7):
If rule gives , then is optimal solution for the problem (B).

Proof:Since
is minimized by rule and ( ) ( ). Hence is optimal for the problem (A). ∎

Case (9):
If and rule satisfies then rule is optimal for the problem (A) and (B).

Proof: Since
∑ be the completion time of the job j and then that is . Hence is optimal for the problem (A) and (B). ∎

Case (10):
If schedule gives for each ( ) then s optimal for the problem (A) and (B).

Case (11):
If ( ) then rule gives an optimal solution for the problem (A).

Proof:
Since then any sequence gives So is optimal for the problem ∑ ( ) .∎

Dominance Rule
Because of branching scheme, the size of the search tree is directly linked to the length of the current sequence (which represents the number of nodes). Hence, a preprocessing step is performed in order to remove as many positions as possible. Reducing the current sequence is done by using several dominance rules. Dominance rules usually specify whether a node can be eliminated before its lower bound is calculated. Clearly, dominance rules are particularly useful when a node can be eliminated which has a lower bound that is less than the optimum solution [29]. Some of dominance rules are valid for minimization of the sum of discounted total weighted completion time and maximum lateness. As in the preprocessing step, similar dominance rules are also used within the branch and bound procedure to cut nodes that is dominated by others. These improvements lead to very large decrease in the number of nodes to obtain the optimal solution.
Below are the three dominance rules that are stated in order to decrease the number of nodes in search tree as well as decreasing the time.

Theorem (5.1):Let
be a partial sequence which it's jobs are schedule for ̅ and T be completion time of the last job in k. If and . Then in optimal schedule for the problem (B).  (4) From (3) and (4) we get in optimal solution for problem (B). ∎

Lemma (5.1):
Let be a partial sequence which it's jobs are schedule for ̅ and T be completion time of the last job in k. If . Then in optimal schedule in the problem(A).

Lemma (5.2):
Let be a partial sequence which it's jobs are schedule for ̅ and T be completion time of the last job in k. If and . Then i j in optimal schedule in the problem (A).

Proof:
Since WDSPT in optimal for the discounted total weighted completion time [3] and EDD in optimal for [9]. ∎

Branch and Bound (BAB) Method [30]
Our BAB method is based on forward sequencing branching rule for which nodes at level k of the search tree are corresponding to initial partial sequence in case if jobs are sequenced in first k positions. The LB at any node is the cost of scheduling jobs (this cost depends on the objective function) and the cost of un sequenced jobs (this cost depends on derived lower bound (LB)). At any level of the BAB method, if a node has LB UB, then this node is dominated. If the branching ends at a complete sequence of jobs then this sequence is evaluated, and if its value is less than the current (UB), this (UB) is reset to take that value. The procedure is then repeated until all nodes have been considered by using back tracking procedure. Backtracking procedure is the movement from the lowest level to the upper level in the BAB method.

6.1Upper bound (UB) Procedure
In this subsection, we propose a heuristic method which is applied once at the root node of search tree in (BAB) to find an upper bound (UB) on the minimization value of problem (A) and (B).

Heuristic (UB)
The Simulated Annealing is suggested to obtain a sequences to be an upper bound (UB) for the problem (A) and (B).
is called cooling parameter in annealing terminology usually , where , -.
Step ( Step (2): Generate a perturbed sequence with one of the neighborhoods operators and set .

Else
Let and .
Step (8): Stop best value of stored in best.

The Lower Bound (LB)
In this subsection, two lower bound and are derived for the problems (A) and (B) respectively.

7.1The Lower Bound For Problem (A)
The lower bound for the problem (A) is based on decomposing (A) of two sub problems ( ) and ( ) as was shown in section (3). Then was calculated to be the lower bound for( ) by ( ) rule (sequencing the jobs in nondecreasing order of Weighted Discounted Shortest Processing Times) [3], and was calculated to be the lower bound for( ) by ( ) rule (sequencing the jobs in non-decreasing order of due date) [9] and then applying Theorem (3.1) to get the first a lower bound for problem (A).

Algorithm
Step (1): Initialize order the un scheduling jobs by using WDSPT rule.
Step (2): Calculate the value of cost function where .∑ ( ) Step (3): Re-order the jobs by using EDD rule.
Step (4): Calculate the value of cost function where .

Algorithm
Step (1) : Initialize order the un scheduling jobs by using WDSPT rule.
Step (4) : Calculate the value of cost function where Step (

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

8.1Test Problems
There exists in the literature a classical way to randomly generate test problems of scheduling problems [33].


The processing time is uniformly distributed in the interval [1,10].


The due date is uniformly distributed in the interval [P(1-TF-RDD/2),P(1-TF+RDD/2)]; where ∑ depending on the relative range of due date (RDD) and on the average tardiness factor (TF).


The an integer weights were generated from uniform distribution [1,10].


The an integer penalty were generated from uniform distribution [1,10].
For both parameters, the values 0.2, 0.4, 0.6, 0.8 and 1.0 are considered. For each selected value of n (where n is the number of jobs), ten problems were generated.

Computational Experience with the Lower and Upper Bound of BAB Algorithm
The BAB algorithm was tested by coding it in MATLAB 7.10.0 (R2010a) and implemented on Intel (R) Core (TM) i7-4500UCPU @ 1. , and the number of unsolved problems (Status).The stopping condition for the BAB algorithm was determined and we consider that the problem is unsolved (state is 1) that the BAB algorithm is stopped after a fixed period of time that is here after 1800 seconds (i.e. after 30 minutes). We observed from tables (1.2) and (1.3), the heuristics of upper bound is good algorithm. It gives the value for objective function equal to optimal or near optimal value for small value of n.