^{1}

^{2}

^{1}

^{2}

We consider a single-machine scheduling problem with an outsourcing option in an environment where the processing time and outsourcing cost are uncertain. The performance measure 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 be NP-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.

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

The scheduling problems under the scenario-based uncertainty above have been studied since Daniels and Kouvelis [

Whether the performance measure of in-house jobs is the makespan or the total completion time.

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 2-approximation 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

The formal definition of our problem is as follows: Consider the set

In this paper, the performance measure for in-house jobs is expressed as the total weighted completion time. Let

Let our problem be referred to as

The weight of each job is identical; that is,

The processing time of each job is identical under each scenario; that is,

Let cases (i) and (ii) be referred to as

The remainder of the paper is organized as follows. Sections

Both scenario cases of Problem P1 without an outsourcing option, that is, the min–max regret version of

In this subsection, we prove the NP-hardness 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

The discrete scenario case of Problem P1 is NP-hard, even if the processing times are certain; that is,

We reduce the partition problem known to be NP-complete [

Given an instance of the partition problem, we can construct an instance of Problem P1 with two scenarios, as follows: set

where

The optimal schedule of scenario 1 is to outsource all jobs. Thus,

The optimal schedule of scenario 2 is to process all jobs in-house. Thus,

Henceforth, we show that a schedule

Suppose that there exists a solution

Exactly one of the jobs in

The total outsourcing cost is, under scenario 1,

Then,

Suppose a schedule

In this case, exactly one of the jobs in

In this case, exactly one of the jobs in

Let

In this subsection, we introduce the conditions that make Problem P1 polynomially solvable. Consider two cases below.

Note that in-house jobs are processed by increasing order of

First, we consider case (i). Using the following lemma, the interval scenario case is reduced to the discrete scenario case.

For the interval scenario case of Problem P1 with

Now, we focus on the discrete scenario case. Without loss of generality, we assume that

For Problem P1 with

Suppose that

Both scenario cases of Problem P1 with

By Lemma

Henceforth, we consider case (ii).

If

Suppose that, in an optimal schedule

Let

First, we consider the discrete scenario case. Then, it is observed that

Before considering the interval scenario case, we introduce the concept of the

Jobs in

Exactly one job in

No job in

By cases (i)–(iii),

Both scenario cases of Problem P1 with

This holds immediately from Lemma

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 this subsection, we prove NP-hardness for the discrete scenario case of Problem P2.

The discrete scenario case of Problem P2 is NP-hard, even if the outsourcing cost is certain; that is,

We reduce the even-odd partition problem known to be NP-complete [

Given an instance of the even-odd partition problem, we can construct an instance of Problem P2 with two scenarios, as follows: set

and, for

where

The optimal schedule of scenario 1 is to process all jobs in-house. Thus,

The optimal schedule of scenario 2 is to process each job in

We now show that a schedule

Suppose that there exists a solution

Exactly one of the jobs in

The total weighted completion time is, under scenario 1,

Then,

Suppose that a schedule

In this case, exactly one of the jobs in

In this case, exactly one of the jobs in

By the above claim, we have

The discrete scenario case of Problem P2 is NP-hard, even if the processing time of each job is identical under all scenarios; that is,

Given an instance of the partition problem, we can construct an instance of Problem P2 with two scenarios, as follows: set

where

In this subsection, we prove the polynomiality of the case with

Using the following lemma, the interval scenario case is reduced to the discrete scenario case.

For the interval scenario case of Problem P2 with

Since the proof is almost similar to that of Lemma

Now, we focus on the discrete scenario case. Let

For Problem P2 with

Since the proof is almost similar to that of Lemma

Both scenario cases of Problem P2 with

This holds immediately from Lemmas

We considered the min–max regret version of a single-machine problem with an outsourcing option such that

the processing time or the outsourcing cost of each job is uncertain,

the cost for processing jobs in-house is expressed as the total weighted completion time.

The uncertainty was described as being of two types with a discrete scenario and an interval scenario. Since the deterministic version of the problem is known to be NP-hard, we considered two cases with identical weights or identical processing times for each scenario. For the first case, we proved the NP-hardness of its discrete scenario case, even if the processing time is certain. Furthermore, we presented two special structures for the processing time and outsourcing cost, each of which makes the first case polynomially solvable. For the second case, we proved the NP-hardness of its discrete scenario case, even if one of the outsourcing cost and the processing time is uncertain. Finally, we presented one special structure for the processing time and outsourcing cost that makes the second case polynomially solvable.

For Problems P1 and P2, it is observed that the complexities remain NP-hard, only if one of the processing time and the outsourcing cost is uncertain. However, their complexities move to the polynomially solvable class, if the impact of uncertainty on each job under each scenario is the same, which is described by

For future work, it would be interesting to consider the following:

Developing the algorithm for Problem P whose performance is verified through analyzing the approximability or conducting the numerical experiments.

Analysis of computational complexity for the interval scenario cases of Problems P1 and P2 when the processing time is certain.

Analysis of the min–max version of Problems P1 and P2 to find an optimal schedule

Note that the discrete scenario case of the min–max version of Problem P1 is NP-hard, even if only the outsourcing cost is uncertain. The reduction from the partition problem is as follows: For

where

The authors declare that they have no conflicts of interest.

This study was financially supported by the research fund of Chungnam National University in 2016.