A Single-Machine Scheduling Problem with Uncertainty in Processing Times and Outsourcing Costs

We consider a single-machine scheduling problem with an outsourcing option in an environment where the processing time and outsourcing cost are uncertain.Theperformancemeasure is the total cost of processing some jobs in-house and outsourcing the rest. The cost of processing in-house jobs is measured as the total weighted completion time, which can be considered the operating cost. The uncertainty is described through either an interval or a discrete scenario.The objective is to minimize the maximum deviation from the optimal cost of each scenario. Since the deterministic version is known to beNP-hard, we focus on two special cases, one in which all jobs have identical weights and the other in which all jobs have identical processing times. We analyze the computational complexity of each case and present the conditions that make them polynomially solvable.


Introduction
We consider a single-machine scheduling problem with an outsourcing option such that the processing time and outsourcing cost are not known in advance.The uncertainty of two parameters is motivated from unpredictable events, such as the introduction of new machines and disruptions in the finished product's delivery.
In this paper, their uncertainty is described as two types of the scenario set, that is, an interval scenario and a discrete scenario.Let  be the set of scenarios.Let    and    denote the processing time and the outsourcing cost of job  under scenario  ∈ , respectively.In the interval scenario, the processing time and outsourcing cost of job  are given as any value within the intervals [   ,  The scheduling problems under the scenario-based uncertainty above have been studied since Daniels and Kouvelis [1] and Kouvelis and Yu [2] (see Aissi et al. [3] for the comprehensive survey).Furthermore, the deterministic version of the single-machine scheduling problem with an outsourcing option has been extensively studied since Vickson [4] (see Shabtay et al. [5] for comprehensive survey).To our best knowledge, the scheduling problem with an outsourcing option under uncertainty for some parameters has only been studied by Choi and Chung [6].The authors considered the case in which the processing time is uncertain while the outsourcing cost is known.They analyzed the computational complexity for various cases, depending on the following: (i) Whether the performance measure of in-house jobs is the makespan or the total completion time.(ii) Whether the uncertainty is described as the interval scenario or the discrete scenario.
They showed that, for the problem with the makespan as the performance measure of in-house jobs, an interval scenario case is polynomially solvable, while a discrete scenario case is NP-hard.Furthermore, they developed the 2approximation algorithm based on the linear programming and the fully polynomial-time approximation scheme for the discrete scenario case.For the problem with the total completion time as the performance measure of in-house jobs, referred to as Problem TCO, it is known that both scenario cases are NP-hard.Thus, they focused on the case with a special structure of processing times,    =   +   , and proved the polynomiality for both scenario cases.Our problem can be considered a general version of Problem TCO, in that the performance of in-house jobs is the total weighted completion time and the outsourcing cost is uncertain.
The formal definition of our problem is as follows: Consider the set  = {1, 2, . . ., } of  independent jobs that can be either processed on a single in-house machine or outsourced to a subcontractor.Let   be the weight of job .Let (, ) be the schedule such that (i)  and \ are the sets of in-house and outsourced jobs, respectively, (ii)  is a sequence of jobs in .
In this paper, the performance measure for in-house jobs is expressed as the total weighted completion time.Let    () be the completion time of an in-house job  in  under scenario .Our problem is to find an optimal schedule ( * ,  * ) to minimize where is the optimal schedule to minimize   (, ) under scenario .
Let our problem be referred to as Problem P. For both scenarios, Problem P is NP-hard because the problem of finding ( *  ,  *  ) is known to be NP-hard [7,8].For the interval scenario case, however, the optimal schedule under a midpoint scenario  is known to have an approximation factor of two in [9][10][11], where    = (   +    )/2 and    = (   +    )/2 for each  ∈ .Thus, we focus on two cases below: (i) The weight of each job is identical; that is,   = 1 for each  ∈ .
(ii) The processing time of each job is identical under each scenario; that is,    =   for each  ∈  and each  ∈ .
Let cases (i) and (ii) be referred to as Problem P1 and Problem P2, respectively.Note that, for two cases, ( *  ,  *  ) can be obtained in ( 2 ) by the algorithm in [7,8].
The remainder of the paper is organized as follows.Sections 2 and 3 analyze the computational complexity and the polynomially solvable case for Problems P1 and P2.Finally, Section 4 draws conclusions and suggests future research directions.

Problem P1
Both scenario cases of Problem P1 without an outsourcing option, that is, the min-max regret version of 1 ‖ ∑  , are NP-hard [2,12], which implies that both scenario cases of Problem P1 are at least NP-hard, even if the outsourcing costs are certain.In this section, firstly, we prove the NPhardness of the discrete scenario case with certain processing times.Note that the complexity of the interval scenario case with certain processing times is open.Then, we introduce two polynomially solvable cases.

NP-Hardness.
In this subsection, we prove the NPhardness for the discrete scenario case with certain processing times.In this case, the sequence of in-house jobs is identical for an optimal schedule under each  ∈ .Thus, throughout the remainder of the section, we use  instead of (, ) for notational simplicity.
Theorem 1.The discrete scenario case of Problem P1 is NPhard, even if the processing times are certain; that is,    =   for each  ∈ .
Proof.We reduce the partition problem known to be NPcomplete [13], defined below, to Problem P1: given Given an instance of the partition problem, we can construct an instance of Problem P1 with two scenarios, as follows: set  = 2 and  = {1, 2, . . ., 2}.For  = 1, 2, . . ., , where  > 0 is a sufficiently large value.Clearly, this reduction can be done in polynomial time.Note the following: (i) The optimal schedule of scenario 1 is to outsource all jobs.Thus, (ii) The optimal schedule of scenario 2 is to process all jobs in-house.Thus, Henceforth, we show that a schedule  exists such that if and only if there exists a solution to the partition problem.Suppose that there exists a solution  to the partition problem.We can construct a schedule  = {2 − 1 |  ∈ } ∪ {2 |  ∉ }.Then, in , we have the following: (i) Exactly one of the jobs in {2 − 1, 2},  = 1, 2, . . ., , is processed in-house.Thus, under scenarios 1 and 2, the total completion time of in-house jobs is (ii) The total outsourcing cost is, under scenario 1, and, under scenario 2, it is Then, Suppose a schedule Î exists such that Claim.In Î, exactly one of the jobs in {2 − 1, 2},  = 1, 2, . . ., , is processed in-house.
Proof.Let V be the largest index such that both jobs in {2V − 1, 2V} are processed in-house and let  be the largest index such that both jobs in {2 − 1, 2} are outsourced.If V or  does not exist, then let V = ∞ and  = ∞ for consistency of notation.Consider the following two cases.
In this case, exactly one of the jobs in {2 − 1, 2},  = V + 1, V + 2, . . ., , is processed in-house.Then, Note that the value in the right-hand side is the lower bound of the total completion time of in-house jobs in Î.
This is a contradiction.

Polynomiality.
In this subsection, we introduce the conditions that make Problem P1 polynomially solvable.Consider two cases below.
(ii)    = , and the ordered conditions are satisfied.In the interval scenario case, the ordered conditions can be stated as follows: for  = 1, 2, . . .,  − 1. (17) In the discrete scenario case, for each  ∈ , Note that in-house jobs are processed by increasing order of   in an optimal schedule for both cases due to the special structure of the processing time uncertainty.Thus, we use  and    instead of (, ) and    (), respectively, throughout the remainder of the section.
First, we consider case (i).Using the following lemma, the interval scenario case is reduced to the discrete scenario case.
Proof.Suppose that  * ∉  and | * | = .Then, for each scenario  ∈ , The proof is complete.Proof.By Lemma 3, a schedule  ∈  with the minimum () is optimal.The values   ( *  ) for all  ∈  can be obtained in ( 2 ) by the algorithm of Engels et al. [7] and Ghosh [8].The set  can be obtained in ( 2 ) by the algorithm of Choi and Chung [6].Since   () can be calculated in () for each  ∈  and  ∈ , the value () can be calculated in ().Hence, the values () for all  ∈  can be found in ( 2 ).
Before considering the interval scenario case, we introduce the concept of the worst case scenario  = ( 1 ,  2 , . . .,   ) for , defined as follows: let  be the worst case scenario for  if Furthermore, it is observed that the worst case scenario  for  can be constructed as follows [14]: If   V ≤    , then, for each scenario  ∈ , which implies that () ≤ ( * ).Thus, we assume that   V >    .Let  * and  be the worst case scenarios of  * and , respectively.It is observed from (25) that Then, Consider the following three cases.
(i) Jobs in {V, } are processed in-house in  *  .Then, since Let By cases (i)-(iii), () ≤ ( * ).Thus, by repeatedly applying the argument above, we can construct the optimal schedule satisfying Lemma 5 from  * without an increase of the objective value.The proof is complete.Theorem 6.Both scenario cases of Problem P1 with    =  for each  ∈  and each  ∈  and the ordered conditions can be solved in ( 2 ).
Proof.This holds immediately from Lemma 5.

Problem P2
In this section, we show the discrete scenario case of Problem P2 remains NP-hard even if only one of the processing times and outsourcing costs is uncertain and we introduce a polynomially solvable case.Note that the complexity of the interval scenario case with certain processing times is open.Since the processing time of each job is identical for each scenario, in-house jobs are processed in nonincreasing order of   in an optimal schedule.Thus, for notational simplicity, we use  and    instead of (, ) and    (), respectively, throughout this section.

NP-Hardness.
In this subsection, we prove NP-hardness for the discrete scenario case of Problem P2.
Theorem 7. The discrete scenario case of Problem P2 is NPhard, even if the outsourcing cost is certain; that is,    =   for each  ∈ .
(ii) The optimal schedule of scenario 2 is to process each job in {1, 2, . . ., } in-house and to outsource all the other jobs.Thus, We now show that a schedule  exists such that if and only if a solution to the even-odd partition problem exists.
Proof.Let V be the smallest index such that both jobs in {2V − 1, 2V} are not in Q, and let  be the smallest index such that both jobs in {2−1, 2} are in Q.If V or  does not exist, then let V = ∞ and  = ∞ for notational consistency.Consider the following two cases.
By the above claim, we have (42)

Theorem 4 .
Both scenario cases of Problem P1 with    =   +   and    =   +   can be solved in ( 2 ).