Single Machine Group Scheduling with Position Dependent Processing Times and Ready Times

We investigate a single machine group scheduling problem with position dependent processing times and ready times. The actual processing time of a job is a function of positive group-dependent job-independent positional factors. The actual setup time of the group is a linear function of the total completion time of the former group. Each job has a release time. The decision should be taken regarding possible sequences of jobs in each group and group sequence to minimize the makespan. We show that jobs in each group are scheduled in nondecreasing order of its release time and the groups are arranged in nondecreasing order of some certain conditions. We also present a polynomial time solution procedure for the special case of the proposed problem.

In the second place, many manufactures have implemented the concept of group technology (GT.Burbidge [15]); it is conventional to schedule continuously all jobs from the same group.Group technology that groups similar products into families helps increase the efficiency of operations and decrease the requirement of facilities (Mitrofanov [16], Janiak and Kovalyov [17], and Webster and Baker [18]).In this paper, we do not assume that all maintenance periods are identical and allow each one of them to leave the processing conditions of the machine in a different state.We deal with a more general concept of a group setup time performed by the processing group.Thus, the effects that change the actual group setup time may become additionally dependent on the actual processing times of jobs before the proposed group.
Rustogi and Strusevich [10] presented real-life examples: "These general positional effects can be found in practice as well.Extending the coursework marking example above, after marking a certain number of scripts, the teacher might get tired or bored, her attention becomes less focused and each new script may even take longer to mark than the one before.We are sure our academic colleagues know this feeling, and they also know the remedy: take a break, have a cup of coffee." However, to the best of our knowledge, only few results concerning scheduling models and problems with position dependent processing times and group technology simultaneously are known.But combining the group technology with start time dependent processing times is more common.For the case that setup time of each group is a fixed constant, Wang et al. [19] considered single machine group scheduling in which the actual processing time of a job is a general linear decreasing function of its starting time; for the makespan minimization problem and total completion time minimization problem they showed that some problems can be solved in polynomial time.Xu et al. [20] considered the single machine scheduling problems with group technology and ready times; the job processing times are described by a function which is proportional to a linear function of time; 2 Mathematical Problems in Engineering the setup times of groups are assumed to be fixed and known; it showed that minimizing the makespan with ready times can be solved in polynomial time and proposed a heuristic algorithm.J.-B.Wang and J.-J.Wang [21] continued the work of Xu et al. [20]; they considered a more general deterioration model than the group setup times and job processing times; the objective is to minimize the makespan.They showed that the problem can be solved in polynomial time when start time dependent processing times and group technology are considered simultaneously.
Motivated by the ideas of Rustogi and Strusevich [14] and J.-B.Wang and J.-J.Wang [21], we consider the single machine scheduling problem with ready times of the jobs under the group technology assumption and position dependent processing times.Our objective is to find the optimal group sequence and the optimal job sequence to minimize the makespan.The rest of the paper is organized as follows.In the next section we describe the formulation of our problem.In Section 3, we consider the solution method for minimizing makespan.In Section 4, a reduced model will be introduced and proposed a polynomial time algorithm.The conclusion is given in the last section.

Problem Description
The independent jobs  1 ,  2 , . . .,   have to be processed on a single machine.They are nonpreemptive and to be grouped into  groups:  [1] ,  [2] , . . .,  [𝑘] .The jobs in the same group are consecutively processed as long as the job has arrived; a setup time is required if the machine switches from one group to another and all setup times are positive.Assume that each group contains a total of   jobs, so that the permutation of jobs in the th group is given by  [] = { [] (1),  [] (2), . . .,  [] ( [] )}, where ∑  =1  [] = .Let   [] () represent the ready (arrival) time of the th job in group   .Depending on the choice of groups and the order in which they are performed, the actual processing time of a job   ( =  [] ()), scheduled in position  (1 ≤  ≤   ) in group   (1 ≤  ≤ ), is given as follows: where  [] () denotes positive group-dependent jobindependent positional factors of the th job in group   ; we call the values  [] () deterioration factors.Furthermore, since the number of jobs,   , in each group and the total duration of each group,   , are known, the actual setup time   is defined as follows: where   (>0) is the deterioration rate of the group   .The deterioration factors  [] () are given in the form of a collection of ordered array of numbers: 1 ≤  [] (1) ≤  [] (2) ≤ ⋅ ⋅ ⋅ ≤  []  () |  = 1, 2, . . ., ;  = 1, 2, . . .,   } represent the makespan of a given schedule.Using the three-field notation schema in scheduling problems (Graham et al. [22]), the makespan minimization problem is denoted as 1 |   ,  []   () =    [] (),   =    −1 , GT |  max , where GT denote group technology.

The Solution Method
In the following section, we will give the solution method so that the single machine minimization scheduling problem with deteriorating jobs and ready times can be solved under certain conditions.Firstly, we consider that all jobs can be processed in one group; that is,  = 1.

Lemma 1.
For the problem 1 |   ,   () =   () |  max , where () meets ( 3), the optimal job sequence can be obtained by sequencing the jobs in nondecreasing order of   .
Proof.Suppose that  = { 1 ,   ,   ,  2 } and   = { 1 ,   ,   ,  2 } are two job sequence, where  1 and  2 denote a partial sequence (note that  1 and  2 may be empty), and the difference between  and   is a pairwise interchange of two adjacent jobs   and   .In addition, the completion time of the last job of  1 in sequence (  ) is denoted by .Then, the completion times of job   and   under  are () = max {  () ,   } +    ( + 1) = max { +    () +    ( + 1) ,   +    () Similarly, the completion times of jobs   and   under   are Suppose that   ≤   , based on ( 6) and ( 8), we have Based on the formula (3)   ≥   , () ≤ ( + 1), and the values of   ,   , and , we divide these values into three cases as follows.
(1)   ≤   ≤ ; that is, jobs   and   have arrived at time ; then (2)   ≤  <   ; that is, job   has arrived before time , and   has not arrived at time ; then (3)  <   ≤   ; that is, jobs   and   have not arrived at time ; then In conclusion, we have   (  ) ≥   ().Repeating this interchange argument, an optimal schedule can be obtained by sequencing the jobs in nondecreasing order of   .
Proof.In the same group, the result of ( 1) can be easily obtained by Lemma 1. Next, we consider the case in item (2).
Let  and   be a pairwise interchange of two adjacent groups   and   , that is,  = [ and the completion time of the group   is Under   , the completion time of the group   is and the completion time of the group   is Suppose that that is Based on ( 19) and ( 21), we have Therefore   [] (  ) (  ) ≥   [] (  ) (); this completes the proof.
Proof.Similar to Theorem 2, using the two exchange methods, under , based on (27), we can obtain the completion time of and the completion time of Under   , the completion times of   and   are   [] ()  [] () + Therefore   [] (  ) (  ) ≥   [] (  ) (); this completes the proof.
Step 3. Groups are scheduled in nondecreasing order of (  ) =   [] (()) − ∑ ()−1 =1   [] ()  [] ().Obviously, the complexity of obtaining the optimal job sequence within a certain group   is (  log   ) and that of obtaining the optimal group sequence is ( log ).It is easy to show that ∑  =1 (  log   ) ≤ ( log ).Hence, the complexity of Algorithm 4 is at most ( log ).In addition, we demonstrate Algorithm 4 by the following example.
Example 5.There are eight jobs ( = 8) divided into three groups ( = 3) to be processed on a single machine.

Solution.
According to Algorithm 4, we solve Example 5 as follows.

Conclusion
In this paper we have considered the scheduling problem with group technology and position dependent processing times, for the case that group setup times are linearly related to the completion time of the former group and the job processing times are the general nondecreasing function of the positional factors.We showed that the makespan minimization problem with ready times can be solved under certain conditions.A reduced model can be considered as special case of our general model.Furhermore, we present an ( log ) time algorithm to solve the proposed problem.