A unified approach for single machine scheduling with positiondependent workloads and positional penalties
Abstract
In this paper, we provide unified methods for solving scheduling problems with convex resource allocation, concentrating on objective functions that can be expressed as a scalar product of the actual processing times vector and a jobindependent and positiondependent penalties vector. We assume general positiondependent workloads, i.e., the workloads are not restricted to be either monotone functions of the job positions or to specific functions. The first unified method focuses on minimizing a scheduling measure subject to a constraint on the resource consumption, whereas the second addresses the complementary problem of minimizing the consumed resource given an upper bound on the scheduling measure. For each unified approach, we provide a methodical analysis and consequently provide a \(O\left( {n^{3} } \right)\) solution algorithm. Furthermore, to demonstrate the unified schemes, we solve several scheduling measures that involve earliness and tardiness penalties.
Keywords
Single machine scheduling Convex resource allocation Positiondependent workloads Positional penalties1 Introduction
Classical deterministic scheduling theory assumed that the jobs processing times are given constants. Later studies challenged this paradigm and introduced positiondependent processing times or controllable processing times. Studies involving positiondependent processing times are derived from the notion that processing times can be decreased due to a learning effect or increased implying an aging effect or deteriorating jobs. Generally, the optimal job sequence can be achieved by a simple \(O\left( {n\log n} \right)\) matching procedure, i.e., matching \(n\) jobs to \(n\) positions (see [9]). The most inclusive representation of the variable job processing times phenomenon presumes general positiondependent processing times, and accordingly no restrictions on the jobs functional form are imposed. In these scheduling problems, the input includes a matrix of \(n \times n\) jobposition processing times. Consequently, the computational complexity of the solution algorithm increases significantly, as in most cases the solution requires solving a Linear Assignment Problem (LAP), which is known to be solved in \(O\left( {n^{3} } \right)\) time. Widespread research has been carried out addressing this phenomenon. Recent studies on various variable job processing time problems include among others: [2, 4, 5, 6, 17, 19, 20, 36, 38, 39, 40, 41].
Scheduling models with controllable processing times are very extensive and influential. Similar to variable processing times research, studies with controllable processing times adopt the principle that processing times are not fixed. Unlike the jobposition method, in this scheme the processing times are not a function of the position/time, but rather a function of the resources allocated to the job operations. This modernized methodology aims to address various reallife problems where the improvement in the deployment of inputs in the production system is realized by allocating extended resource to job processing. Most studies suppose that the actual job processing time is either a linear or nonlinear function of the amount of resource allocated to the processing of the job. In this study, we focus on the latter and more practical function. Thus, we assume that the input includes positiondependent workloads of the jobs processing operation. In fact, the actual job processing times is a convex decreasing function of the amount of resource allocated to the processing of the job. According to a wideranging survey of Shabtay and Steiner [25], the convex function reflects the fundamental law of diminishing marginal returns, validating that productivity increases at a decreasing rate with the utilized amount of resource. Moreover, [25] provides numerous practical problems involving various objective functions on different machine settings varying from single machine to openshops, demonstrating the importance of this research class in scheduling theory. For instance, [25] points out that convex resource allocation can be found in “many actual government and industrial operations and … very large scale integration (VLSI) circuit design, where the product of the silicon area (resource) and the square of time spent equals a constant value (the workload) for an individual job”. Recent articles with convex resource allocation include [6, 7, 14, 18, 19, 26, 27, 28, 29, 30, 34, 41].
Type 1: Minimizing the total integrated cost of the scheduling measure and consumed resource.
Type 2: Minimizing the scheduling measure subject to an upper bound on the available resource.
Type 3: Minimizing the consumed resource subject to an upper bound on the scheduling measure.
Type 4: Identifying the set of Paretooptimal schedules for the scheduling measure and the consumed resource.
Recently, Oron [24] combined the methods of positiondependent processing times and convex resource allocation by introducing the realistic and challenging model of positiondependent workloads. The author assumed a continuous nonrenewable resource, implying that any amount of available resource can be allocated to any given job. Specifically, he considered decreasing workloads as a result of a positiondependent learning effect. The paper focused on Type 2 problems, i.e., minimizing different scheduling measures subject to a given upper bound on the available resource. The studied scheduling measures were: makespan; total completion time (flowtime); linear combination of the makespan and total flowtime on a single machine; and minimizing total flowtime on \(m\)identical parallel machines. Moreover, the author suggested that minimizing Total Absolute Deviation of Completion times (TADC) and minimizing Total Absolute Deviation of Waiting Times (TADW) can be solved using the same approach.
Leyvand et al. [13] provided a unified mechanism for scheduling with convex resource consumption functions using positional penalties, and focused on Type 1 problems. The authors proved that scheduling problems with positionindependent workloads, where the objective functions are the scalar products of two vectors, i.e., the (actual) processing times vector and the positional weights vector, can be solved in polynomial time by applying a unified method.
In this paper, we further generalize the results of [13] and [24]. We consider scheduling problems involving general positiondependent and positional penalties and make available a unified tool for solving such problems. We prove that the results of [24] can be further extended to solve problems where the workloads are not restricted to monotone functions of the job starting times, functions of the job positions, or to specific linear/exponential functions.
Following [13] and [24], we restrict our study to objective functions that can be expressed as a function of the jobs processing times and the positional weights. Unlike [13], who provided a unified method for the Type 1 problems and [24] who studied Type 2 problems, we provide unified methods both for Types 2 and 3.
In the first part of this study, we provide a unified scheme for minimizing a general scheduling measure that is a scalar product of the actual processing times and the positional weights vectors, subject to a constraint on the available amount of resource that can be consumed. Similar to [24], we prove that these problems can be solved in \(O\left( {n^{3} } \right)\) time. Once we obtain the solution algorithm for the general problem, we apply it with minor modifications to solve three different scheduling problems with earliness and tardiness penalties. Specifically, we consider the following objective functions: the total weighted earliness plus the total weighted tardiness with common unrestrictive duedate; minsum common duewindow; and minsum common duewindow based on flowallowance. In the second part, we introduce a unified scheme for minimizing the consumed resource subject to an upper bound on the scheduling measure and apply the results to solve the same earliness/tardiness problems.
The article is comprised of five major sections and several subsections. In Sect. 2, we provide the notations and formulation of all studied problems. Section 3 is dedicated to minimizing scheduling measures: In Sect. 3.1, we introduce the unified method and in subsequent Sects. 3.2–3.4, we solve total weighted earliness and tardiness with common unrestrictive duedate, minsum common duewindow, and minsum common duewindow based on flowallowance, respectively. In Sect. 4, we concentrate our attention on the unified method for minimizing the total amount consumed. In Sect. 5, we provide conclusions and future research topics.
2 Notations and formulation
We consider an \(n\)job single machine scheduling problem. We assume, as is very prevalent in scheduling theory, that all the jobs are available at time zero, and preemption is forbidden. The actual processing time \(p_{jr}\) of job \(j, j = 1, \ldots ,n\) if assigned to position \(r, r = 1, \ldots ,n\) is defined as \(p_{jr} = \left( {\frac{{w_{jr} }}{{u_{j} }}} \right)^{k}\), where \(w_{jr}\) is a strictly positive value emulating the workload of the processing operation of job \(j\) in position \(r\), \(u_{j}\) is a decision variable reflecting the amount of resource assigned to the processing operation, and \(k\) is a given positive factor.
After we solve this general problem, we apply the devised solution procedure to solve three different scheduling problems involving positional weights. To formulate the additional problems, we need several new notations as follows.
For a given schedule, the completion time of job \(j, j = 1, \ldots ,n\) is denoted by \(C_{j}\). Let, \(d_{j}\) denote the duedate of job \(j, j = 1, \ldots ,n\). Jobs completed before the duedate are considered as early, and jobs completed after the duedate are considered as tardy. The (single) ontime job is not penalized whereas the early and tardy jobs incur penalties. The earliness of the job \(j\) is defined as \(E_{j} = \hbox{max} \left\{ {0, d_{j}  C_{j} } \right\}\) and the tardiness of job \(j\) is defined as \(T_{j} = \hbox{max} \left\{ {0, C_{j}  d_{j} } \right\}, j = 1, \ldots ,n\).
We first apply the unified approach to minimize the total weighted earliness plus the total weighted tardiness with a common unrestrictive duedate (URDD). We denote the common duedate by \(d\), such that \(d_{j} = d, j = 1, \ldots ,n\). We assume that the earliness penalty and the tardiness penalty are strictly positive and not necessarily equal. Let \(\alpha\) denote the earliness unit penalty and \(\beta\) denote the tardiness unit penalty, thus the objective function is, \(Z_{1} = \mathop \sum \nolimits_{j = 1}^{n} \alpha E_{j} + \mathop \sum \nolimits_{j = 1}^{n} \beta T_{j}\).
The last problem we solve, using the unified approach, addresses the duewindow assignment and scheduling problem based on common flowallowance (SLKDW), presented by [23]. Similar to problem PZ1, jobs that are completed within the time interval (duewindow) are not penalized and otherwise are penalized according to their earliness/tardiness. Once more, we seek instantaneously the optimal job sequence and the start time and size of the window. In this method of duewindow problems, there are two common flowallowance constants denoted by \(q_{1}\) and \(q_{2}\), which are decision variables, such that \(q^{2} \ge q^{1}\). Unlike problem PZ1, the duewindows, denoted by \(D_{j}\), are jobdependent such that \(D_{j} = \left[ {d_{1j} , d_{2j} } \right]\), \(j = 1, \ldots ,n\), where \(d_{ij} = p_{j} + q_{i} ,i = 1,2, j = 1, \ldots ,n\).
3 A unified approach for minimizing a scheduling measure
In this section, we establish the theoretical fundamentals of our unified approach to solving scheduling problems involving positional weights. In all problems, we seek to minimize a cost function subject to a constraint that the total amount of consumed resource cannot exceed a given upper bound. In SubSect. 3.1, we analyze the theoretical problem and introduce a general solution procedure based on solving a Linear Assignment Problem (LAP). In the succeeding Sects. 3.2–3.4, we solve practical problems founded on the general procedure with very minor adaptations, while keeping the computational complexity.
3.1 Problem PZ0: \({\mathbf{1}}\left {\sum\nolimits_{{\varvec{j}{\mathbf{ = 1}}}}^{\varvec{n}} {\varvec{u}_{\varvec{j}} {\mathbf{ \le }}\varvec{U}} } \right\varvec{Z}_{{\mathbf{0}}}\)
Next, we utilize the Karush–Kuhn–Tucker (KKT) conditions to achieve an optimal solution (see [10]).
Theorem 3.1.1
Assume that \(Z\left( \varvec{u} \right)\) and \(\theta \left( \varvec{u} \right)\) are differentiable functions satisfying certain regularity conditions. Then \(\varvec{u}^{\varvec{*}} = \left( {u_{1}^{*} ,u_{2}^{*} , \ldots ,u_{n}^{*} } \right)\) can be an optimal solution for the nonlinear programming problem only if there exists a number \(\lambda\) such that all KKT conditions are satisfied.
We emphasize that the objective function and the single constraint are convex functions of the amount of resource allocated to jobs \(j = 1, \ldots ,n\), and the problem is to find the global minimum of a convex function under linear constraint. Thus, from Theorem 3.1.1, NLP 3.1.1 can be solved to optimality by using the KKT conditions.
 i.
\(\varvec{u} = \left( {u_{1} ,u_{2} , \ldots ,u_{n} } \right)\) is the vector of multiple decision variables;
 ii.
\(\lambda\) is the single Lagrange multiplier (the shadow price of the objective function);
 iii.
\(\theta \left( {\varphi ,\varvec{u}} \right) = \mathop \sum \limits_{j = 1}^{n} u_{j}\).
 1.
\(\frac{{\partial Z_{0} \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} + \lambda \frac{{\partial \theta \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} \ge 0\)
 2.
\(u_{j} \left( {\frac{{\partial Z_{0} \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} + \lambda \frac{{\partial \theta \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} } \right) = 0\)
 3.
\(\theta \left( {\varphi ,\varvec{u}} \right)  U \le 0\)
 4.
\(\lambda \left( {\theta \left( {\varphi ,\varvec{u}} \right)  U} \right) = 0\)
 5.
\(u_{j} \ge 0, j = 1, \ldots ,n\)
 6.
\(\lambda \ge 0\)
We start our discussion by assuming that the job sequence, \(\varphi\), has been achieved already and seek the expressions for the optimal amount of resource allocated to each job and the optimal actual processing time. Once these equations are realized, we reinsert them to the cost function to establish the optimal cost.
This result validates the outcome of [24], with the added advantage that the workloads are general positiondependent and are not restricted to any specific function.
The formal unified approach solution procedure consists of several easytofollow steps.
Algorithm 3.1.1
 1.
Determine the positional weights \(\xi_{r} , r = 1, \ldots ,n\).
 2.
Apply LAP 3.1.1 to obtain an optimal sequence and the workloads \(w_{jr} , j,r = 1, \ldots ,n\).
 3.
Use Eq. (8) to determine \(W\).
 4.
Utilize Eq. (9) to calculate the amount of resource allocated to each job \(u_{j} , j = 1, \ldots ,n\).
 5.
Apply Eq. (10) to compute the actual processing times \(p_{jr} , j, r = 1, \ldots ,n\).
 6.
Use Eq. (12) to obtain the optimal cost function value \(Z_{0}^{*}\).
Theorem 3.1.2
The computational complexity of Algorithm 3.1.1 is \(O\left( {n^{3} } \right)\).
Proof
Step 1: Is carried out in \(O\left( n \right)\) time. Step 2: The computational complexity required for solving the assignment problem is known to be \(O\left( {n^{3} } \right)\). Steps 3–6: Each of these steps is done at most in \(O\left( n \right)\) time. Thus, the total complexity of the solution procedure is dominated by Step 2 and is \(O\left( {n^{3} } \right)\).
We note that if the input is a vector of \(n\) positionindependent workloads (as opposed to a matrix of \(n \times n\) jobposition workloads), applying LAP 3.1.1 is unnecessary, and we need to assign jobs to positions according to the HLP rule (see [9]), implying that the computational effort is reduced to \(O\left( {n\log n} \right)\).
In the next subsections, we apply the unified approach to address three fundamental scheduling problems involving duedates. Based on Algorithm 3.1.1, with minor adaptions, we provide \(O\left( {n^{3} } \right)\) time polynomial solution procedures for all studied problems.
3.2 Problem PZ1: \(1\left {\varvec{URDD},\mathop \sum \limits_{{\varvec{j} = 1}}^{\varvec{n}} \varvec{u}_{\varvec{j}} \le \varvec{U}} \right\varvec{Z}_{1} \varvec{ }\)
The first scheduling measure we consider is the total weighted earliness plus the total weighted tardiness, where all jobs share a common unrestrictive duedate. In the context of this study, we define the unrestrictive duedate as sufficiently late so as not to affect the scheduling decisions. Baker and Scudder [3] provide several important properties of an optimal schedule which are still valid for our problem:
Property 3.2.1
There are no idle times between consecutive jobs.
Property 3.2.2
An optimal schedule exists in which \(d = C_{l}\) for some \(l\).
Property 3.2.3
Property 3.2.3 implies that jobs in positions \(1, \ldots ,l  1\) are early and penalized as such, while the job in position \(l\) is ontime and thus not penalized, whereas jobs in positions \(l + 1, \ldots ,n\) are tardy and penalized. Consequently, it is straightforward to define the positional weight of job \(j\) if assigned to position \(r\).
Property 3.2.4
We note that the positional weight of the job assigned to the first position is zero, thus no resource should be allocated to reduce the processing time of this job, leading to the following property:
Property 3.2.5
The total available resource should be allocated to the jobs in positions \(2, \ldots ,n\).
 1.$$u_{j} = 0\;{\text{if}}\;X_{j1} = 1.$$(15)
 2.$$p_{j1} = w_{j1} \;{\text{if}}\;X_{j1} = 1.$$(16)
 3.
Now, we are ready to provide the solution algorithm for problem PZ1:
Algorithm 3.2.1
 1.
 2.
Apply LAP 3.1.1 to obtain an optimal sequence.
 3.
If \(X_{j1} = 1\), set \(u_{j} = 0\) and \(p_{j1} = w_{j1}\).
 4.
Utilize Eq. (17) to calculate \(W\).
 5.
Apply Eq. (18) to compute \(u_{j} , j = 1, \ldots ,n, r = 2, \ldots ,n\).
 6.
Use Eq. (10) to determine \(p_{jr} , j = 1, \ldots ,n, r = 2, \ldots ,n\).
 7.
Use Eq. (12) to obtain \(Z_{1}^{*}\).
Numerical example 3.2.1
The common (unrestrictive) duedate is \(d = 400\), the cost parameters are \(\alpha = 3\) and \(\beta = 4\), the total amount of available resource is \(U = 10\) and \(k = 0.5\).
Solution for Numerical example 3.2.1—optimal sequence, resource allocation, actual processing times, and cost
\(\xi_{r}\)  0.0  3.0  6.0  9.0  12.0  12.0  8.0  4.0. 
\(\varphi^{*}\)  4  3  8  1  2  7  5  6 
\(w_{jr}\)  37.000  17.000  25.000  14.000  1.000  2.000  4.000  5.000 
\(u_{j}^{*}\)  0.000  1.116  2.014  2.175  1.093  1.378  1.325  0.899 
\(p_{jr}^{*}\)  37.000  3.903  3.523  2.537  0.956  1.205  1.738  2.358 
\(Z^{*}\)  104.950 
3.3 Problem PZ2: \(\varvec{ }1\left {\varvec{CONDW},\mathop \sum \limits_{{\varvec{j} = 1}}^{\varvec{n}} \varvec{u}_{\varvec{j}} \le \varvec{U}} \right\varvec{Z}_{2}\)
The classical assignment and scheduling problem with common duewindow on single machine was first introduced by [15], and since then studied in numerous papers as evident by the survey of Janiak et al. [11]. Based on preceding papers, [11] propose that “There is a variety of applications of due window scheduling models in JIT manufacturing, semiconductor manufacturing, chemical processing, PERT/CPM scheduling as well as in IT”. More recent articles include [14, 16, 17, 19, 27, 28, 29, 32, 36, 37, 40, 42].
The following properties are known in scheduling theory for the classic problem presented by [15], i.e., with constant processing times, and are still valid for our problem.
Property 3.3.1
An optimal schedule exists in which \(d_{1} = C_{l}\) for some \(l\), and \(d_{2} = C_{m}\) for some \(m\), such that \(m \ge l\).
Property 3.3.2
Property 3.3.3
Given Properties 3.3.1–3.3.3, applying the unified approach is straightforward.
Algorithm 3.3.1
 1.
 2.
Apply LAP 3.1.1 to obtain an optimal sequence.
 3.
Utilize Eq. (8) to compute \(W\).
 4.
Use Eq. (9) to obtain \(u_{j} , j,r = 1, \ldots ,n\).
 5.
Apply Eq. (10) to determine \(p_{jr} , j, r = 1, \ldots ,n\).
 6.
Utilize Eqs. (22) and (23) to calculate the optimal common duedates, \(d_{1}^{*}\) and \(d_{2}^{*}\).
 7.
Use Eq. (12) to find \(Z_{2}^{*}\).
Numerical example 3.3.1
We solve the \(8 \times 8\) jobposition workloads problem given in Table 1, with the total amount of resource \(U = 10\) and \(k = 0.5\). The cost parameters are \(\alpha = 11,\beta = 20, \gamma = 3\) and \(\delta = 7\).
Solution for Numerical example 3.3.1—optimal sequence, resource allocation, actual processing times, common duedates, and cost
\(\xi_{r}\)  24.0  35.0  46.0  56.0  56.0  56.0  40.0  20.0 
\(\varphi^{*}\)  6  3  8  1  2  7  5  4 
\(w_{jr}\)  10.0  17.0  25.0  14.0  1.0  2.0  4.0  22.0 
\(u_{j}^{*}\)  0.941  1.444  1.970  1.851  0.768  0.968  0.974  1.083 
\(p_{jr}^{*}\)  3.260  3.431  3.562  2.750  1.141  1.438  2.026  4.506 
\(d_{1}^{*}\)  10.254  
\(d_{2}^{*}\)  15.582  
\(Z^{*}\)  831.763 
3.4 Problem PZ3: \(1\left {\varvec{SLKDW},\mathop \sum \limits_{{\varvec{j} = 1}}^{\varvec{n}} \varvec{u}_{\varvec{j}} \le \varvec{U}} \right\varvec{Z}_{3}\)
The duedate assignment and scheduling problem based on flowallowance (slack, SLK) method was introduced by [1] and later was generalized by [23] to slack with duewindow setting (SLKDW). In this subsection, we demonstrate the application of the unified scheme to solve the SLKDW problem. Unlike the CONDW method, in SLKDW the jobdependent duedates can be expressed as a simple linear function of the job processing times. Generally, each of the duedates is defined as the sum of its processing time and a jobindependent constant. Significant papers considering different SLK objective functions include: [4, 12, 14, 22, 30, 31, 33, 35, 37, 40].
As with the discussion in Sect. 3.3 for the CONDW method, we start with several important and applicable results presented for the constant processing time problem. [23] and [33] proved the following properties for the duewindow based on common flowallowance problem.
Property 3.4.1
An optimal schedule exists in which \(q_{1} = C_{l}\) for some \(l\), and \(q_{2} = C_{m}\) for some \(m\), such that \(m \ge l\).
Property 3.4.2
Property 3.4.3
Since the positional weight of the job assigned to the last position is zero, we do not allocate resource to this job, and present the following properties:
Property 3.4.4
The total available resource should be allocated to the jobs in positions \(1, \ldots ,n  1\).
 i.$$u_{j} = 0\; {\text{if}}\;X_{jn} = 1.$$(27)
 ii.$$p_{jn} = w_{jn} \;{\text{if}}\;X_{jn} = 1.$$(28)
 iii.
Algorithm 3.4.1

Step 1: Utilize Eqs. (24)–(26) to calculate \(l,m\) and \(\xi_{r} , r = 1, \ldots ,n\), respectively.

Step 2: Apply LAP 3.1.1 to obtain an optimal sequence.

Step 3: If \(X_{jn} = 1\), then \(u_{jn} = 0\) and \(p_{jn} = w_{jn}\).

Step 4: Use (29) to calculate \(W\).

Step 5: Utilize (30) to obtain \(u_{j} , j = 1, \ldots ,n, r = 1, \ldots ,n  1.\)

Step 6: Apply (10) to compute \(p_{jr} , j = 1, \ldots ,n, r = 1, \ldots ,n  1.\)

Step 7: Use (31) and (32) to calculate the optimal common flowallowance constants, \(q_{1}^{*}\) and \(q_{2}^{*}\).

Step 8: Use (12) to calculate \(Z_{3}^{*}\).
Numerical example 3.4.1
We solve a problem with the same input data as assumed in Numerical example 3.3.1.
Solution for Numerical example 3.4.1—optimal sequence, resource allocation, actual processing times, common flowallowance constants, duedates, and cost
\(\xi_{r}\)  35.0  46.0  56.0  56.0  56.0  40.0  20.0  0.0 
\(\varphi^{*}\)  6  3  7  2  8  5  1  4 
\(w_{jr}\)  10.0  17.0  6.0  14.0  2.0  3.0  12.0  22.0 
\(u_{j}^{*}\)  1.355  1.940  1.563  2.074  1.084  0.992  0.992  0.0 
\(p_{jr}^{*}\)  2.717  2.960  1.959  2.598  1.358  1.739  3.479  22.0 
\(q_{1}^{*}\)  5.676  
\(q_{2}^{*}\)  11.592  
\(d_{1j}\)  8.393  8.636  7.635  8.275  7.035  7.416  9.155  27.676 
\(d_{2j}\)  14.309  14.552  13.551  14.190  12.950  13.331  15.071  33.592 
\(Z^{*}\)  701.654 
 1.
The common duedate problem can be solved by setting the positional weights (see Property 3.3.3) to: \(\xi_{r} = \left\{ {\begin{array}{*{20}c} {\alpha \left( {r  1} \right) + n\gamma , r = 1, \ldots ,l} \\ {\beta \left( {n  r + 1} \right), r = l + 1, \ldots ,n} \\ \end{array} } \right.\).
 2.
The common duedate based on flowallowance problem can be solved by modifying the positional weights (Property 3.4.3) to: \(\xi_{r} = \left\{ {\begin{array}{ll} {\alpha r + n\gamma , r = 1, \ldots ,l} \\ {\beta \left( {n  r} \right), r = l + 1, \ldots ,n} \\ \end{array} } \right.\).
The optimal flowallowance can be found by \(q = \mathop \sum \nolimits_{j = 1}^{l} p_{jr}\).
Since the main difference is in the equations of the positional weights, the computational complexity is dominated by the running time of LAP 3.1.1, and thus remains \(O\left( {n^{3} } \right)\).
Supplementary to the above solved problems, in the “Appendix” we briefly indicate an additional two groups of scheduling problems that can be solved to optimality by using the unified method.
4 A unified approach for minimizing resource consumption cost
Similar to Sect. 3 in this section, we start with devising the principles of the unified approach to solving scheduling problems involving positional weights. Unlike Sect. 3, we now seek to minimize the total amount of consumed resource given an upper bound on the scheduling measure. In Sect. 4.1, we solve the general problem and in later subsections address practical problems.
4.1 Problem PU0:\(1\left {\varvec{Z}_{0} \le \varvec{Q}} \right\varvec{U}\)
Similar to NLP 3.1.1, NLP 4.1.1 can be solved to optimality by applying the KKT conditions.
 1.$$\frac{{\partial U\left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} + \eta \frac{{\partial \phi \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }} \ge 0$$
 2.$$u_{j}^{*} \left( {\frac{\partial U}{{\partial u_{j} }}\left( {\varphi ,\varvec{u}} \right) + \eta \frac{{\partial \phi \left( {\varphi ,\varvec{u}} \right)}}{{\partial u_{j} }}} \right) = 0$$
 3.$$\phi \left( {\varphi ,\varvec{u}} \right)  Q \le 0$$
 4.$$\eta \left( {\phi \left( {\varphi ,\varvec{u}} \right)  Q} \right) = 0$$
 5.$$u_{j} \ge 0, j = 1, \ldots ,n$$
 6.$$\eta \ge 0$$
Similar to Sect. 3, we start with a given sequence and seek the optimality conditions.
Analyzing (42), we claim that, similar to Sect. 3.1, the optimal sequence that minimizes the total amount of consumed resource can be achieved by minimizing the expression \(\mathop \sum \nolimits_{j = 1}^{n} \mathop \sum \nolimits_{r = 1}^{n} \xi_{r}^{{\frac{1}{1 + k}}} w_{jr}^{{\frac{k}{1 + k}}} X_{jr}\).
Consequently, we can apply the Linear Assignment Problem LAP 3.1.1, provided in Sect. 3, to obtain the optimal sequence.
Algorithm 4.1.1
 1.
Find the positional weights.
 2.
Apply LAP 3.1.1 to obtain an optimal sequence.
 3.
Utilize Eq. (39) to determine \(V\).
 4.
Apply Eq. (40) to obtain \(u_{j} , j, r = 1, \ldots ,n\).
 5.
Utilize Eq. (41) to calculate \(p_{jr} , j, r = 1, \ldots ,n\).
 6.
Use Eq. (43) to find the optimal total consumed resource \(U^{*}\).
Theorem 4.1
The computational complexity of Algorithm 4.1 is \(O\left( {n^{3} } \right)\).
In the following subsections, we address the same problems studied in Sects. 3.3–3.4. For the sake of brevity, only the adapted equations and solution algorithms are provided, complemented by numerical examples. Again, as the main difference is manifested in the calculation of the positional weights, our closing discussion of Sect. 3 is still valid and the computational complexity of the solution procedures remains \(O\left( {n^{3} } \right)\).
4.2 Problem PU1: \(1\left {\varvec{URDD},\varvec{ Z}_{1} \le \varvec{Q}} \right\varvec{U}\)
Algorithm 4.2.1
 1.
Use (13) and (14) to calculate the index of the ontime job and the positional weights, respectively.
 2.
Apply LAP 3.1.1 to obtain an optimal sequence.
 3.
If \(X_{j1} = 1\), set \(u_{j} = 0\) and \(p_{j1} = w_{j1}\).
 4.
Utilize Eq. (44) to compute \(V\).
 5.
Utilize Eq. (45) to calculate \(u_{j} , j = 1, \ldots ,n, r = 2, \ldots ,n\).
 6.
Apply Eq. (46) to find \(p_{jr} , j = 1, \ldots ,n, r = 2, \ldots ,n\).
 7.
Use (43) to obtain \(U^{*}\).
Numerical example 4.2.1
Solution for Numerical example 4.2.1—optimal sequence, resource allocation, actual processing times, and total consumed resource
\(\xi_{r}\)  0.0  15.0  30.0  40.0  32.0  24.0  16.0  8.0 
\(\varphi^{*}\)  7  3  6  5  4  2  1  8 
\(w_{jr}\)  40.0  21.0  9.0  4.0  9.0  4.0  17.0  2.0 
\(u_{j}^{*}\)  0.000  0.367  0.440  0.406  0.459  0.289  0.357  0.110 
\(p_{jr}^{*}\)  40.000  7.561  4.524  3.137  4.428  3.719  6.897  4.258 
\(U^{*}\)  2.429 
4.3 Problem PU2: \(1\left {\varvec{CONDW},\varvec{ Z}_{2} \le \varvec{Q}} \right\varvec{U}\)
Algorithm 4.1.1 with very minor changes can be applied to solve problem PU2:
Algorithm 4.3.1
Numerical example 4.3
We solve the \(8 \times 8\) jobposition workloads problem given in Table 2, \(Q = 1250\), \(k = 0.5\) and the cost parameters are: \(\alpha = 10,\beta = 17, \gamma = 7\) and \(\delta = 9\).
Solution for Numerical example 4.3.1—optimal sequence, resource allocation, actual processing times, common duedates, and total consumed resource
\(\xi_{r}\)  56.0  66.0  72.0  72.0  68.0  51.0  34.0  17.0 
\(\varphi^{*}\)  1  4  6  5  3  2  7  8 
\(w_{jr}\)  8.0  7.0  9.0  4.0  18.0  4.0  42.0  2.0 
\(u_{j}^{*}\)  1.028  1.098  1.265  0.965  1.534  0.767  1.282  0.293 
\(p_{jr}^{*}\)  2.789  2.525  2.668  2.036  3.426  2.284  5.725  2.614 
\(d_{1}^{*}\)  5.314  
\(d_{2}^{*}\)  10.018  
\(U^{*}\)  8.231 
4.4 Problem PU3: \(1\left {\varvec{SLKDW}, \varvec{Z}_{3} \le \varvec{Q}} \right\varvec{U}\)
Algorithm 4.4.1
 1.
 2.
Apply LAP 3.1.1 to obtain an optimal sequence.
 3.
If \(X_{jn} = 1\), set \(u_{j} = 0\) and \(p_{jn} = w_{jn}\).
 4.
Utilize Eq. (47) to compute \(V\).
 5.
Apply Eq. (48) to calculate \(u_{j} , j = 1, \ldots ,n, r = 1, \ldots ,n  1\).
 6.
Utilize Eq. (49) to find \(p_{jr} , j = 1, \ldots ,n, r = 1, \ldots ,n  1\).
 7.
 8.
Use (43) to calculate \(U^{*}\).
Numerical example 4.4.1
Solution for Numerical example 4.4.1—optimal sequence, resource allocation, actual processing times, common flowallowance constants, duedates, and total consumed resource
\(\xi_{r}\)  66.0  72.0  72.0  68.0  51.0  34.0  17.0  0.0 
\(\varphi^{*}\)  1  4  6  5  8  2  3  7 
\(w_{jr}\)  8.0  7.0  9.0  4.0  17.0  4.0  26.0  44.0 
\(u_{j}^{*}\)  0.835  0.846  0.920  0.676  0.904  0.426  0.501  0.0 
\(p_{jr}^{*}\)  3.096  2.876  3.128  2.433  4.337  3.065  7.207  44.0 
\(q_{1}^{*}\)  3.096  
\(q_{2}^{*}\)  9.100  
\(d_{1j}\)  6.192  5.972  6.224  5.529  7.433  6.161  10.303  47.096 
\(d_{2j}\)  12.196  11.976  12.228  11.533  13.437  12.165  16.307  53.100 
\(U^{*}\)  5.107 
5 Conclusion
We studied scheduling problems where the objective function is a scalar product of the processing times vector and the positional penalties vector. We considered controllable processing times jobs with convex resource allocation. Specifically, we focused on positiondependent workloads and provided unified methods for two types of problems: minimizing a scheduling measure subject to a constraint on the resource consumption and the complementary problem of minimizing the consumed resource given an upper bound of the scheduling measure. We proved that by applying our solution algorithms each of these cases can be solved to optimality in \(O\left( {n^{3} } \right)\) time. We demonstrated the use of the unified schemes to solve several scheduling measures with earliness and tardiness penalties.
To the best of our knowledge, no study has been carried out on convex resource allocation on proportionate flowshops, thus a challenging future research might be minimizing the makespan or/and flowtime even with positionindependent workloads in this setting. Several scheduling problems with positiondependent processing times have been studied on identical parallel machines. Thus, another interesting direction for future research is the extension of these problems to positiondependent workloads.
Notes
Acknowledgements
This research did not receive any specific Grant from funding agencies in the public, commercial, or notforprofit sectors.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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