Refined conditions for Vshaped optimal sequencing on a single machine to minimize total completion time under combined effects
Abstract
We address singlemachine scheduling problems for which the actual processing times of jobs are subject to various effects, including a positional effect, a cumulative effect and their combination. We review the known results on the problems to minimize the makespan, the sum of the completion times and their combinations and identify the problems for which an optimal sequence cannot be found by simple priority rules such as Shortest Processing Time (SPT) and/or Longest Processing Time (LPT). Typically, these are problems to minimize the sum of the completion times under a deterioration effect, and we verify under which conditions for these problems an optimal permutation is Vshaped (an LPT subsequence followed by an SPT subsequence). We demonstrate that previously used techniques for proving that an optimal sequence is Vshaped are not properly justified. We use the corrected method to describe a wide range of problems with a pure positional effect and a combination of a cumulative effect with a positional effect for which an optimal sequence is Vshaped. On the other hand, we show that even the refined approach has its limitations.
Keywords
Scheduling Single machine Positional effect Cumulative effect Total completion time Vshaped1 Introduction
Since the early 1990s, there has been a considerable interest in enhanced scheduling models in which the processing times of jobs are affected by their locations in the schedule. Mathematically, this is formalized in terms of various timechanging effects. In this paper, we clarify the status of a number of singlemachine problems with various timechanging effects.
We consider scheduling problems with changing times in which the jobs of set \(N=\left\{ 1,2,\ldots ,n\right\} \) are to be processed on a single machine. Each job \(j\in N\) is associated with its “normal” processing time \(p_{j}\). It is convenient to think of normal processing times as the time required under normal processing conditions of the machine, which might change during the processing, thereby affecting the actual processing times.
In the literature on scheduling with changing processing times, traditionally there is a distinction between socalled deterioration effects and learning effects. Informally, under a deterioration effect, the later a job is placed in a schedule, the longer it takes to process it. This phenomenon is often found in manufacturing: If a machine loses its initial processing quality, it increases the actual processing times of some later scheduled jobs. Under a learning effect, the opposite is observed: The later a job is scheduled, the shorter its actual processing time is. To illustrate a learning effect, a machine may be thought of as a human operator who gains experience during the process, which leads to a certain processing time reduction.
Consideration of timechanging effects should not be limited to monotone effects only, such as deterioration and learning. For example, if a human operator processes jobs on certain equipment, then during the process that equipment might be subject to wear and tear, i.e., it might deteriorate with time, however, the operator simultaneously gains additional skills by learning from experience. This gives rise to a combined effect which has a nontrivial influence on the actual processing times.

Positional effects the actual processing time of a job is a function of its normal processing time and the position it takes in a schedule; see a focused survey by Rustogi and Strusevich (2012b) and a discussion in Agnetis et al. (2014);

Start timedependent effects the actual processing time of a job is a function of its normal processing time and its start time in a schedule; see the book Gawiejnowicz (2008) which gives a detailed exposition of scheduling models with this effect;

Cumulative effects the actual processing time of a job depends on its normal processing time and a function of the normal processing times of previously scheduled jobs; see Kuo and Yang (2006a, b), where a similar effect is introduced.
If job j is sequenced in position \(\pi (r) \) of permutation \( \pi =\left( \pi (1) ,\pi (2) ,\ldots ,\pi \left( n\right) \right) \), its completion time is denoted either by \(C_{j}\left( \pi \right) \) or by \(C_{\pi \left( r\right) }\), whichever is more convenient. Let \(\Phi \left( \pi \right) \) denote an objective function to be minimized. Popular objective functions include the maximum completion time \(C_{\max }\left( \pi \right) \), also known as the makespan; the sum of the completion times \(F\left( \pi \right) =\sum _{j\in N}C_{j}\), also known as the total completion time; and a more general function \(\xi C_{\max }+\eta \)\(\sum C_{j}\), with nonnegative coefficients \(\xi \) and \(\eta \).
In (3), f is a continuous differentiable function, common to all jobs. In the case of learning, \(f:\left[ 0,+~\infty \right) \rightarrow \left( 0,1\right] \) is a nonincreasing function, while in the case of deterioration, \(f:\left[ 0,+\,\infty \right) \rightarrow \left[ 1,+\,\infty \right) \) is a nondecreasing function.
We denote the singlemachine problems of minimizing an objective function \( \Phi \) subject to the effects (3) and (4) by \(1\vert p_{j}(r)=p_{j}f(P_{r})\vert \Phi \) and \(1\vert p_{j}(r) =p_{j}f(P_{r}) g(r) \vert \Phi \), respectively.
This paper is organized as follows. In Sect. 2, we consider the problems \(1\vert p_{j}(r)=p_{j}g(r)\vert \Phi \) with a pure positional effect, where \(\Phi \in \{ C_{\max },\sum C_{j},\xi C_{\max }+\eta \sum C_{j}\} \). Reviewing this wellstudied class of problems, we stress that some of the problems \( 1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\) with a positional deterioration effect cannot be solved by a priority rule. Traditionally, in scheduling with variable processing times, if a solution algorithm is not known, “the second best thing” would be to establish some property of an optimal sequence, e.g., the Vshapeness. We describe the refined procedure that may convert a given permutation to a Vshaped permutation without increasing the value of the function. We give conditions when such a procedure leads to an optimal Vshaped sequence. We show that these conditions hold if g is a concave function of r, a polynomial function \( g(r) =r^{a}\), \(a>0\), and an exponential function \(g(r) =\gamma ^{r1}\), \(\gamma >1\). On the other hand, the established conditions need not hold for a convex function g(r) .
In Sect. 3, we review the problems \(1\vert p_{j}(r) =p_{j}f(P_{r}) g(r) \vert \Phi \) with a combined effect. While some of these problems accept an optimal sequencing policy based on either the SPT or LPT rule, problem \(1\vert p_{j}(r) =p_{j}f(P_{r}) \vert \sum C_{j}\) with a pure cumulative deterioration effect given by a concave function f, including a polynomial function \(f(P_{r}) =( 1+P_{r})^{A}\), \( 0<A<1\), is not solvable by a priority rule. Refining this result, we show that for this problem, an optimal permutation is not even Vshaped.
In Sect. 4, we look at the problems in which the cumulative effect is normalized, i.e., function f does not just depend on \(P_{r}\), the sum of normal processing times of the jobs sequenced prior to position r, but on the ratio \(P_{r}/P\), where P is the sum of all processing times. By contrast with problem \(1\vert p_{j}(r) =p_{j}f(P_{r})\vert \sum C_{j}\) with a nonnormalized deterioration effect given by a concave function f, problem \(1\vert p_{j}(r) =p_{j}f(P_{r}/P) \vert \sum C_{j}\) with a normalized cumulative effect is solvable by the SPT rule under certain conditions which, for example, hold for polynomial functions f. For problem \(1\vert p_{j}(r) =p_{j}f(P_{r}/P) g(r)\vert \sum C_{j}\), we establish conditions for an optimal permutation to be Vshaped. Although the conditions hold for a wide range of problems, they do not hold for the model in which both functions f and g are polynomial. The latter problem has been studied by Lu et al. (2015), where relying on a wrong proof technique, the authors claim that the problem admits an optimal Vshaped sequencing policy.
Section 5 contains concluding remarks. In particular, we emphasis that if our refined technique for proving the Vshapeness of an optimal permutation fails for some problems that does not mean that such problems do not admit an optimal Vshaped sequencing policy. It only implies that more advanced methods have to be used for proving or disproving the Vshapeness of an optimal permutation.
2 Pure positional effects: algorithms and Vshapeness
In this section, we consider a range of problems \(1\vert p_{j}(r) =p_{j}g(r) \vert \Phi \) with a pure positional effect (2) to minimize a function \(\Phi \in \{ C_{\max },\sum C_{j},\xi C_{\max }+\eta \sum C_{j}\} .\)
We start with a brief review of the results on problem \(1\vert p_{j}(r) =p_{j}g(r) \vert \Phi \). A systematic exposition of the relevant material is contained in Chapter 7 of the monograph Strusevich and Rustogi (2017). Then, we focus on problem \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\), which under an arbitrary deterioration effect does not admit solution by a priority rule. We derive conditions on the positional factors g(r) , \(1\le r\le n\), which guarantee that an optimal permutation is Vshaped.
2.1 Polynomialtime algorithms: a review
Most of the problems related to \(1\vert p_{j}(r) =p_{j}g(r)\vert \Phi \) can be solved in polynomial time by reducing them to the classical problem of minimizing a linear form over permutations. For completeness, we present the latter problem and an algorithm for its solution below.
Algorithm Match
Input: Two (unsorted) arrays \({\mathbf {a}}=\left( a_{1},a_{2},\ldots ,a_{n}\right) \) and \({\mathbf {b}}=\left( b_{1},b_{2},\ldots ,b_{n}\right) \)

Step 1. If required, renumber the components of array \({\mathbf {b}}\) so that (9) holds.
 Step 2. Output a permutation \(\varphi \) such thatholds.$$\begin{aligned} a_{\varphi (1) }\le a_{\varphi (2) }\le \cdots \le a_{\varphi \left( n\right) } \end{aligned}$$(12)

if the sequence \(W(r) ,1\le r\le n\), of positional weights is not monotone, then an optimal permutation can be found by matching smaller components of the array of the positional weights to larger components of the other array of processing times;

if the sequence W(r) , \(1\le r\le n\), of positional weights is monotone nondecreasing, then an optimal permutation can be found by ordering the jobs in accordance with the LPT priority rule applied to the normal processing times \(p_{j}\);

if the sequence W(r) , \(1\le r\le n\), of positional weights is monotone nonincreasing, then an optimal permutation can be found by ordering the jobs in accordance with the SPT priority rule applied to the normal processing times \(p_{j}\).
Theorem 1
Problem \(1\vert p_{j}(r) =p_{j}g(r)\vert C_{\max }\) under a general positional effect (2) reduces to minimizing a linear form (13) with \(W(r)=g(r),~1\le r\le n\), and \(\Gamma =0\), and is solvable in \( O(n\log n) \) time by Algorithm Match. In the case of a learning effect (7), an optimal permutation is obtained in \(O(n\log n)\) time by renumbering the jobs in the SPT order. In the case of a deterioration effect (8), an optimal permutation is obtained in \(O(n\log n) \) time by renumbering the jobs in the LPT order.
Factors g(r), \(1\le r\le n\), that satisfy (8) define a deterioration effect. Since for any r, \(1\le r\le n1\), we have that \( g(r)\le g(r+1)\) but \(nr+1>n\left( r+1\right) +1\), we cannot guarantee that the positional weights W(r), \(1\le r\le n\), form a monotone sequence. Thus, there is no evidence that a solution to problem \(1\vert p_{j}(r)=p_{j}g(r)\vert \sum C_{j}\) with a deterioration effect can be obtained by a priority rule.
It is straightforward to verify that problem \(1\vert p_{j}(r) =p_{j}g(r) \vert \xi C_{\max }+\eta \sum C_{j}\) reduces to minimizing the linear form (13) with the positional weights \(W(r)=(\xi +(nr+1)\eta ) g(r),~1\le r\le n\). Similarly to the problem of minimizing total completion time, here in the case of a positional learning effect (7), the sequence W(r), \(1\le r\le n\), is nonincreasing, so that the solution can be found by the SPT rule. Otherwise, unless \(\eta =0\), the sequence of positional weights need not be monotone, so that an optimal solution can be found by Algorithm Match, but not by a priority rule.
The following statement summarizes the status of problems \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\) and \( 1\vert p_{j}(r) =p_{j}g_{j}(r)\vert \xi C_{\max }+\eta \sum C_{j}\).
Theorem 2
Under a general positional effect (2), problems \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\) and \(1\vert p_{j}(r)=p_{j}g_{j}(r)\vert \xi C_{\max }+\eta \sum C_{j}\) reduce to minimizing a linear form (13) with \(\Gamma =0\) and with \( W(r)=(nr+1) g(r),~1\le r\le n\), and \(W(r)=(\xi +( nr+1) \eta ) g(r),~1\le r\le n\), respectively. Both problems are solvable in \(O( n\log n) \) time by Algorithm Match. In the case of a learning effect (7), for each of these problems, an optimal permutation is obtained in \(O(n\log n)\) time by renumbering the jobs in the SPT order. In the case of a deterioration effect (8), both problems do not admit a priority rule solution for an arbitrary nondecreasing array g(r) , \(1\le r\le n\), of jobindependent positional factors, unless \(\eta =0\).
Solution algorithms for problems with a jobindependent positional effect
\(\Phi \)  Factorsg(r)  Algorithm 

\(C_{\max }\)  Arbitrary nonmonotone  Match 
\(C_{\max }\)  Arbitrary deterioration (8)  LPT 
\(C_{\max }\)  Arbitrary learning (7)  SPT 
\(\sum C_{j}\)  Arbitrary nonmonotone  Match 
\(\xi C_{\max }+\eta \sum C_{j}\)  Arbitrary nonmonotone  Match 
\(\sum C_{j}\)  Arbitrary learning (7)  SPT 
\(\xi C_{\max }+\eta \sum C_{j}\)  Arbitrary learning (7)  SPT 
\(\sum C_{j}\)  Polynomial deterioration (14), \(A<\log _{2}\left( \frac{n}{n1}\right) \)  LPT 
\(\sum C_{j}\)  Polynomial deterioration (14), \(A>\log _{2}^{1}\left( \frac{n}{n1} \right) \)  SPT 
\(\sum C_{j}\)  Exponential deterioration (15), \(\gamma \ge 2\)  LPT 
Problem \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\) can be solved by a priority rule under additional assumptions regarding positional factors g(r) , \(1\le r\le n\), that define a deterioration effect. In particular, it is proved in Gordon et al. (2008) that problem \(1\left p_{j}(r) =p_{j}g(r) \right \sum C_{j}\) under an exponential positional deterioration effect (15) is solvable by the LPT rule if \(\gamma \ge 2\), while no priority rule solution exists for this problem if \(1<\gamma <2\). For problem \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\) under a polynomial deterioration effect given by (14), the conditions on A that guarantee that the problem can be solved either by the SPT rule or by the LPT rule are given in Chapter 7.2.2 of the monograph Strusevich and Rustogi (2017).
We summarize the results on the solution algorithms for problems \( 1\vert p_{j}(r) =p_{j}g(r)\vert \Phi \) with \(\Phi \in \{ C_{\max },\sum C_{j},\xi C_{\max }+\eta \sum C_{j}\} \) in Table 1.
2.2 Vshapeness
In this section, we study problem \(1\vert p_{j}(r)=p_{j}g(r)\vert \sum C_{j}\), provided that array g(r) , \(1\le r\le n\), is nondecreasing, i.e., satisfies (8) and defines a deterioration effect. We derive conditions on g(r) which guarantee that for problem \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\), there exists a Vshaped optimal permutation.
Recall that a Vshaped permutation consists of an LPT subsequence of jobs followed by an SPT subsequence of the remaining jobs; one of these subsequences may be empty.
Procedure Peak(r)

Step 1. Compute \(G(\pi ) ,\) the joint contribution of the jobs \(\pi (r1) \), \(\pi (r) \) and \(\pi (r+1) \) to the objective function \(F(\pi ).\)

Step 2. Create permutations \(\pi ^{\prime }\) and \(\pi ^{\prime \prime }\) obtained from \(\pi \) by interchanging job \(\pi (r) \) with the adjacent jobs, i.e., with \(\pi (r1) \) and \(\pi (r+1) \), respectively. Compute \(G(\pi ^{\prime }) \) and \( G(\pi ^{\prime \prime })\).
 Step 3. Ifthen either \(\pi ^{\prime }\) or \(\pi ^{\prime \prime }\) is a permutation with a value of the objective function that is at most \(\Phi (\pi ) \) and which does not have a peak in position r.$$\begin{aligned} G(\pi ) \ge \min \{ G(\pi ^{\prime }), G(\pi ^{\prime \prime })\}, \end{aligned}$$(20)
Lemma 1
Proof
Lemma 1 implies that for problem \(1\vert p_{j}(r) =p_{j}g(r)\vert \sum C_{j}\), there exists a Vshaped optimal permutation for fairly general types of deterioration effects.
Theorem 3
Proof
Computations for Example 1
(1, 2, 3)  \(\left( 1,3,2\right) \)  (2, 1, 3)  (2, 3, 1)  (3, 1, 2)  (3, 2, 1)  

\(C_{\pi (1) }\)  1.0  1.0  2.0  2.0  3.0  3.0 
\(C_{\pi (2) }\)  3.8  5.2  3.4  6.2  4.4  5.8 
\(C_{\pi (3) }\)  12.8  11.2  12.2  9.2  10.4  8.8 
\(\sum C_{\pi (j) }\)  17.6  17.4  17.8  17.4  17.8  17.6 
Notice that the statement regarding Case (c) of Theorem 3 is given in Mosheiov (2005). The proof there is based on a peakremoving process, similar to Procedure Peak(r); however, for the three jobs \(\pi ( r1) ,\pi (r) \) and \(\pi ( r+1) \) such that (19) holds, the contributions to the objective function are not defined correctly. Indeed, in Mosheiov (2005), it is assumed that if the jobs \( \pi ( r1) ,\pi (r) \) and \(\pi \left( r+1\right) \) are processed in this order, then their actual processing times contribute to the objective function three times, two times and one time, respectively. However, the contributions of actual processing times should be computed with respect to the positions of the jobs from the rear of the schedule, as done in the corrected expression (21).
We conclude this section by demonstrating that Theorem 3 does not hold for convex nondecreasing functions g(r) .
Example 1
3 Pure and combined cumulative effects: review and Vshapeness
Theorem 4
For problem \(1\vert p_{j}(r) =p_{j}f(P_{r}) g(r) \vert \Phi \) with \(\Phi \in \{ C_{\max },\sum C_{j},\xi C_{\max }+\eta \sum C_{j}\} \) under an effect (4), an optimal permutation can be found in \(O(n\log n) \) time by sorting the jobs in accordance with the SPT rule, provided that f is convex on \([0,+\,\infty ) \) and array g(r) , \(1\le r\le n\), is nonincreasing.
The conditions of Theorem 4 imply that array g(r) , \(1\le r\le n\), defines a positional learning effect. If function f is defined by (26), then it is convex if either \( A<0\) (learning) or \(A>1\) (fast deterioration). We exclude from consideration the case that \(A=0\), since no cumulative effect takes place.
The case of a combined effect (4), provided that function f is concave and the array g(r) , \(1\le r\le n\), is nondecreasing, is not fully symmetric to that presented in Theorem 4, and only the makespan \(C_{\max }\) can be minimized by a priority rule, this time LPT.
Theorem 5
For problem \(1\vert p_{j}(r) =p_{j}f(P_{r}) g(r) \vert C_{\max }\) under an effect (4), an optimal permutation can be found in \(O(n\log n) \) time by sorting the jobs in accordance with the LPT rule, provided that function f is concave on \([0,+\,\infty ) \) and the array g(r) , \(1\le r\le n\), is nondecreasing.
The conditions of Theorem 5 imply that array g(r) , \(1\le r\le n\), defines a positional deterioration effect. If function f is defined by (26), then it is concave if \(0<A\le 1\) (slow deterioration).
The results presented in Strusevich and Rustogi (2017) are summarized in Table 3, including their implication for a polynomial cumulative effect (26). Here, in the second column, we use symbols \(\nearrow \) and \(\searrow \) to indicate whether the sequence g(r) , \(1\le r\le n,\) is nondecreasing or nonincreasing, respectively. Additionally, we write \(g=1\) if \(g(r) =1,\)\(1\le r\le n.\)
Results for problems with a combined cumulative effect (4)
Condition on f  Condition on g  Objective  Rule 

\(f\mathrm {convex}\)  \(g\searrow \)  \(C_{\max }\)  SPT 
\(f\mathrm {convex}\)  \(g\searrow \)  \(\sum C_{j}^{z}\)  SPT 
\(f\mathrm {convex}\)  \(g\searrow \)  \(\xi C_{\max }+\eta \sum C_{j}^{z}\)  SPT 
\(f\mathrm {concave}\)  \(g\nearrow \)  \(C_{\max }\)  LPT 
\(f\mathrm {concave}\)  \(g=1\)  \(\sum C_{j}\)  Open 
\(f=\left( 1+P_{r}\right) ^{A}, A<0\) or \(A\ge 1\)  \(g\searrow \)  \(C_{\max }\)  SPT 
\(f=\left( 1+P_{r}\right) ^{A}, A<0\) or \(A\ge 1\)  \(g\searrow \)  \(\sum C_{j}^{z}\)  SPT 
\(f=\left( 1+P_{r}\right) ^{A}, A<0\) or \(A\ge 1\)  \(g\searrow \)  \(\xi C_{\max }+\eta \sum C_{j}^{z}\)  SPT 
\(f=\left( 1+P_{r}\right) ^{A}, 0<A\le 1\)  \(g\nearrow \)  \(C_{\max }\)  LPT 
\(f=\left( 1+P_{r}\right) ^{A}, 0<A<1\)  \(g=1\)  \(\sum C_{j}\)  Open 
Computations for Example 2
(1, 2, 3)  \(\left( 1,3,2\right) \)  (2, 1, 3)  (2, 3, 1)  (3, 1, 2)  (3, 2, 1)  

\(C_{\pi (1) }\)  1.0000  1.0000  26.0000  26.0000  27.0000  27.0000 
\(C_{\pi (2) }\)  37.7696  164.5791  31.1962  39.1838  32.2915  166.2961 
\(C_{\pi (3) }\)  180.6401  171.9275  174.0667  179.1981  172.3058  173.6446 
\(\sum C_{\pi (j) }\)  219.4097  363.5066  231.2629  219.3818  231.5973  365.9407 
Example 2
4 Pure and combined normalized cumulative effects
4.1 Pure cumulative normalized effect: SPT
Theorem 6
Theorem 6 is used to prove the following lemma.
Lemma 2
Proof
Theorem 7
Proof
Corollary 1
For problem \(1\vert p_{j}(r) =p_{j}f(P_{r}/P)\vert \sum C_{j}\) under a normalized deterioration effect (27) with \(0<A<1\), an optimal permutation can be found in \(O( n\log n) \) time by sorting the jobs in accordance with the SPT rule.
On the other hand, consider a logarithmic function \(f(\frac{x}{P})=\ln \left( e+\frac{10x}{P}\right) \), such that \(f\left( 0\right) =1\). Function f is concave and has a nondecreasing secondorder derivative; however, inequality (30) does not hold for all \(y\in (0,0.14623), \) so that Theorem 7 cannot be applied.
Observe a striking impact of the normalized effect: In the case of a polynomial function f, problem \(1\vert p_{j}(r)=p_{j}f( P_{r}/P)\vert \sum C_{j}\) with a normalized deterioration effect is solvable by the SPT rule, while for problem \( 1\vert p_{j}(r) =p_{j}f(P_{r})\vert \sum C_{j}\) with a nonnormalized effect, Example 2 demonstrates that an optimal permutation does not have to be Vshaped.
4.2 Combined cumulative normalized effects: Vshapeness
This subsection is aimed at resolving the status of problem \(1\vert p_{j}(r) =p_{j}f( P_{r}/P) g(r)\vert \sum C_{j}\) for a wide range of functions that define the combined effect, including those functions that are considered in Lu et al. (2015). Recall that Lu et al. (2015) address problem \(1\vert p_{j}(r) =p_{j}f(P_{r}/P) g(r)\vert \sum C_{j}\) and claim that for \(f( P_{r}/P) \) given by a normalized polynomial function (27) with \(0<A<1\) and \(g(r) =r^{a}\) for \( a>0\), an optimal permutation is Vshaped. However, the proof technique used in Lu et al. (2015) is based on Mosheiov (2005) and is therefore incorrect. That leaves the status of the problem open.
Notice that it follows from Sect. 2.2 that for problem \( 1\vert p_{j}(r) =p_{j}f(P_{r}/P) g(r)\vert \sum C_{j}\) for an optimal permutation to be Vshaped, the function g(r) that defines the positional effect should satisfy the conditions of Lemma 1. Our proof is split into two parts, depending on which of the two conditions of Lemma 1 holds for position r.
Lemma 3
For function \(\psi (t) \) defined by (32), such that \(\lambda >1\), \(\mu \ge 1\) and function \(f(X/P):[0,P] \rightarrow [0,+\,\infty ]\) is a concave normalized function, the inequality \(\psi (t) \ge 0\) holds for all nonnegative t such that \(X+\lambda t\le P\).
Proof
As in Sect. 2.2, assume that a permutation \(\pi =(\pi (1), \pi (2),\ldots ,\pi (n))\) which is optimal for problem \(1\vert p_{j}\)\((r) =p_{j}f\left( P_{r}/P\right) g\left( r\right) \vert \sum C_{j\text { }}\)exhibits a peak in position r, i.e., (19) holds for three consecutive positions \(r1,r\) and \(r+1\), \(2\le r\le n1\).
Lemma 4
For problem \(1\vert p_{j}(r) =p_{j}f(P_{r}/P) g(r) \vert \sum C_{j}\), let a cumulative deterioration effect be defined by a function f that is a concave differentiable nondecreasing normalized function. For a permutation \(\pi =(\pi (1),\ldots ,\pi (r1)\), \(\pi (r), \pi (r+1) ,\ldots ,\pi (n))\) such that for some r, \(2\le r\le n1 \), (19) holds, let \(\pi ^{\prime }=((\pi (1), \ldots ,\pi (r)\), \(\pi (r1), \pi (r+1) ,\ldots ,\pi (n))\) be obtained from \(\pi \) by interchanging jobs \(\pi \left( r1\right) \) and \(\pi (r). \) Then, inequality (22) implies that \(G(\pi ) \ge G(\pi ^{\prime }) \).
Proof
Now, we consider the more intricate case of problem \(1\vert p_{j}(r) =p_{j}f( P_{r}/P) g(r)\vert \sum C_{j} \) assuming function g satisfies condition (23).
The following lemma is an analog of Lemma 2.
Lemma 5
Proof
The proof is similar to the proof of Lemma 2 and is based on multiple applications of the Lagrange mean value theorem, i.e., Theorem 6.
Lemma 6
Proof
We now examine how the derived conditions can be applied to a specific problem.
Consider first problem \(1\vert p_{j}(r)=p_{j}f( P_{r}/P) g(r)\vert \sum C_{j}\), where \(f( P_{r}/P) \) is a normalized polynomial function (27) with \( 0<A<1\) and \(g(r) =r^{a}\) for \(a>0\). As mentioned earlier, Lu et al. (2015) claim that for that problem, an optimal permutation is Vshaped, although no rigorous proof has been given.
Example 3
Notice that this counterexample does not mean that for the described instance, an optimal permutation is not Vshaped. In fact, it can be verified by full enumeration that an optimal solution is delivered by a SPT permutation, e.g., the one that keeps the jobs in the order of their numbering. The example only demonstrates that in general, an optimal permutation cannot be derived by Procedure Peak\(\left( r\right) \) from an arbitrary sequence of jobs, i.e., Procedure Peak(r) may fail for a particular r. We therefore need another technique, different from simple peak removing, to verify whether an optimal permutation is Vshaped or not.
5 Conclusion
In this paper, we refine the proof technique previously employed for proving the existence of an optimal Vshaped sequencing policy for a range of scheduling problems with various timechanging effects such as positional, cumulative and their combination. The refinement is achieved by presenting a corrected formula for a contribution that an individual job makes to the objective function, typically, the sum of the completion times.
For pure positional effects, we give conditions for an optimal Vshaped policy that hold for the popular polynomial and exponential effects, as well as for nonmonotone concave effects. For problems under a combination of a cumulative concave normalized effect and a positional effect, we also derive conditions which hold for a wide range of problems. However, they do not hold for the problem in which both cumulative and positional effects are polynomial, which contradicts the claim made by Lu et al. (2015).
The presented counterexamples show limitations of the discussed proof technique for proving Vshapeness and show the necessity for a more powerful method.
Notes
References
 Agnetis, A., Billaut, J.C., Gawiejnowicz, S., Pacciarelli, D., & Soukhal, A. (2014). Multiagent scheduling., Models and algorithms Berlin: Springer.CrossRefGoogle Scholar
 Gawiejnowicz, S. (2008). Timedependent scheduling. Berlin: Springer.CrossRefGoogle Scholar
 Gordon, V. S., Potts, C. N., Strusevich, V. A., & Whitehead, J. D. (2008). Single machine scheduling models with deterioration and learning: Handling precedence constraints via priority generation. Journal of Scheduling, 11, 357–370.CrossRefGoogle Scholar
 Hardy, G., Littlewood, J. E., & Poylia, G. (1934). Inequalities. London: Cambridge University Press.Google Scholar
 Kuo, W.H., & Yang, D.L. (2006a). Minimizing the makespan in a single machine scheduling problem with a timebased learning effect. Information Processing Letters, 97, 64–67.CrossRefGoogle Scholar
 Kuo, W.H., & Yang, D.L. (2006b). Minimizing the total completion time in a singlemachine scheduling problem with a timedependent learning effect. European Journal of Operational Research, 174, 1184–1190.CrossRefGoogle Scholar
 Lu, Y.Y., Wang, J.J., & Huang, X. (2015). Scheduling jobs with position and sumofprocessingtime based processing times. Applied Mathematical Modelling, 39, 4013–4021.CrossRefGoogle Scholar
 Mosheiov, G. (1991). \(V\)shaped policies for scheduling deteriorating jobs. Operations Research, 39, 979–991.CrossRefGoogle Scholar
 Mosheiov, G. (2001). Scheduling problems with a learning effect. European Journal of Operational Research, 132, 687–693.CrossRefGoogle Scholar
 Mosheiov, G. (2005). A note on scheduling deteriorating jobs. Mathematical and Computer Modelling, 41, 883–886.CrossRefGoogle Scholar
 Rustogi, K., & Strusevich, V. A. (2012a). Single machine scheduling with general positional deterioration and ratemodifying maintenance. Omega, 40, 791–804.CrossRefGoogle Scholar
 Rustogi, K., & Strusevich, V. A. (2012b). Simple matching vs linear assignment in scheduling models with positional effects: A critical review. European Journal of Operational Research, 22, 393–407.CrossRefGoogle Scholar
 Strusevich, V. A., & Rustogi, K. (2017). Scheduling with timeschanging effects and ratemodifying activities. Switzerland: Springer.CrossRefGoogle Scholar
 Wu, C.C., & Lee, W.C. (2008). Singlemachine scheduling problems with a learning effect. Applied Mathematical Modelling, 32, 1191–1197.CrossRefGoogle Scholar
 Yin, Y., Xu, D., Sun, K., & Li, H. (2009). Some scheduling problems with general positiondependent and timedependent learning effects. Information Sciences, 179, 2416–2425.CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.