Open AccessArticle

Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost

Yi-Chun Wang

Si-Han Wang

and

Ji-Bo Wang

School of Science, Shenyang Aerospace University, Shenyang 110136, China

Author to whom correspondence should be addressed.

Mathematics 2023, 11(1), 222; https://doi.org/10.3390/math11010222

Submission received: 3 November 2022 / Revised: 14 December 2022 / Accepted: 30 December 2022 / Published: 2 January 2023

(This article belongs to the Special Issue Industrial Mathematics in Management and Engineering)

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Abstract

Under just-in-time production, this paper studies a single machine common due-window (denoted by CONW) assignment scheduling problem with position-dependent weights and resource allocations. A job’s actual processing time can be determined by the resource assigned to the job. A resource allocation model is divided into linear and convex resource allocations. Under the linear and convex resource allocation models, our goal is to find an optimal due-window location, job sequence and resource allocation. We prove that the weighted sum of scheduling cost (including general earliness–tardiness penalties with positional-dependent weights) and resource consumption cost minimization is polynomially solvable. In addition, under the convex resource allocation, we show that scheduling (resp. resource consumption) cost minimization is solvable in polynomial time subject to the resource consumption (resp. scheduling) cost being bounded.

Keywords:

scheduling; assignment problem; resource allocation; positional-dependent weights; earliness–tardiness

MSC:

90B35

1. Introduction

In scheduling models and problems, a due-window assignment has attracted growing attention, particularly in just-in-time (JIT) philosophy (Yin et al. [1]; Janiak et al. [2]; Liu [3]; Liu et al. [4]; Rolim and Nagano [5]; Qian and Zhan [6]). More recently, Wang et al. [7] studied a single machine production scheduling with the due-window assignment. The objective is to minimize the total cost, which is a weighted sum function of earliness, tardiness, due-window starting time, and due-window size, where the weights only depend on their position in a sequence (i.e., position-dependent weights, Wang et al. [8] and Wang et al. [9]). Sun et al. [10] and Qian and Han [11] investigated a proportionate flow shop with the CONW and slack due-window (denoted by SLKW) assignments. The objective is to minimize the generalized weighted sum of numbers of early and late jobs, earliness–tardiness penalties and due-window assignment cost, where the weights are the position-dependent weights. Yue and Zhou [12] studied single machine scheduling with the due-window assignment and stochastic processing times. Wang [13] considered single machine scheduling with past-sequence-dependent setup times. Under common, slack and different due-date assignments, he proved that a general earliness–tardiness cost minimization can be solved in polynomial time. Shabtay et al. [14] explored the single machine scheduling problem with common due-date/due-window assignments. They proved that a non-regular objective cost minimization can be solved in polynomial time under some conditions. They also showed that the problem is NP-hard when the length of the due-window is bounded. Jiang et al. [15] scrutinized the seru scheduling problem with a learning effect. Under multiple due-window assignments, they showed that some cases of the problem are polynomially solvable. Liu et al. [16] considered the single machine due-window assignment scheduling with past-sequence-dependent setup times. Under three due-window assignment models, the objective is to minimize a non-regular cost function (including the weighted sum of earliness–tardiness, number of early and tardy jobs, due-window assignment cost). They proved that this problem is polynomially solvable.

Recently, Wang et al. [17] explored single-machine problems with due-window assignment and positional-dependent weights. Under common and slack due-window assignments, the goal is to minimize the weighted sum of the number of early and late jobs, earliness–tardiness and due-window assignment costs, where the weights are positional-dependent weights. They demonstrated that some versions of common and slack due-window assignments are polynomially solvable. However, in real-world production processes, job processing times are complicated, which are controlled by allocating additional resources (e.g., fuel and gas, et al., see Shabtay and Steiner [18], Liu et al. [19], Lu et al. [20], Wang et al. [21], Zhao [22], Lu et al. [23], and Yan et al. [24]). Hence, in this investigation, we extend the results of Wang et al. [17] by considering resource allocation that includes the one given in Wang et al. [17] as a special case. We summarize the contributions as follows: (1) We investigate single machine scheduling with resource allocation and CONW due-window assignment; (2) Under linear resource allocation, we provide an analysis for the weighted sum of scheduling cost (including generalized earliness–tardiness penalties with positional-dependent weights) and resource consumption costs; (3) Under convex resource allocation, we give a bicriteria analysis for scheduling costs and resource consumption costs; (4) We derive the structural properties of an optimal solution and demonstrate that the problem remains polynomially solvable. This article is organized as follows. Section 2 presents a description of the problem. In Section 3, we first give two lemmas, then we prove that four problems are polynomially solvable. In Section 4, a case study is given. In Section 5, conclusions are presented.

2. Problem Formulation

There are n independent non-preemptive jobs

J = {J_{1}, J_{2}, \dots, J_{n}}

that are to be processed on a single-machine, and all jobs are available at time zero. The machine can only process one job at a time. Let

p_{j}^{A}

be the actual processing time of job

J_{j}

; under a linear resource allocation, we have

\begin{matrix} p_{j}^{A} = {\bar{p}}_{j} - b_{j} u_{j}, \end{matrix}

where

{\bar{p}}_{j}

(resp.

b_{j}

) is the normal processing time (resp. compression rate) of job

J_{j}

u_{j}

is the resource amount of job

J_{j}

0 \leq u_{j} \leq {\bar{u}}_{j} < \frac{{\bar{p}}_{j}}{b_{j}}

and

{\bar{u}}_{j}

is the upper bound of

u_{j}

. For a convex resource allocation, we have

\begin{matrix} p_{j}^{A} = {(\frac{{\bar{p}}_{j}}{u_{j}})}^{k}, \end{matrix}

where

k > 0

is a constant. In this paper, all the jobs are subject to a common due-window

[d_{1}, d_{2}]

(

d_{1} \leq d_{2}

), where

d_{1}

is the due-window starting time and

d_{2}

is the due-window finishing (completion) time; then, the due-window size is

D = d_{2} - d_{1}

. Let

C_{j}

be the completion time of job

J_{j}

, the number of early (

U_{j}

) and tardy jobs (

V_{j}

) that are given as:

U_{j} = \{\begin{matrix} 1, & if d_{1} > C_{j} \\ 0, & otherwise \end{matrix}

and

V_{j} = \{\begin{matrix} 1, & if d_{2} < C_{j} \\ 0, & otherwise . \end{matrix}

Let

E_{j} = max {d_{1} - C_{j}, 0}

(resp.

T_{j} = max {C_{j} - d_{2}, 0}

) be the earliness (tardiness) of job

J_{j}

(

j = 1, 2, \dots, n

). Let

[j]

be the job that is placed in the jth position; the goal is to determine the optimal job sequence

π

d_{1}

and D (such as

d_{2}

) that minimizes the total cost. Formally, the resource consumption (resp. scheduling) cost is

\sum_{j = 1}^{n} v_{j} u_{j}

(

\sum_{j = 1}^{n} (α_{j} U_{[j]} + β_{j} V_{[j]} + η_{j} E_{[j]} + δ_{j} T_{[j]} + θ d_{1} + λ D)

). For the linear (resp. convex) resource allocation, the first (second) problem (denoted by

P_{1}

and

P_{2}

respectively) is to

Z = \sum_{j = 1}^{n} (α_{j} U_{[j]} + β_{j} V_{[j]} + η_{j} E_{[j]} + δ_{j} T_{[j]} + θ d_{1} + λ D) + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]},

(1)

where

α_{j}

β_{j}

η_{j}

and

δ_{j}

are positional-dependent weights (penalty factors) for the one-time penalties for earliness, tardiness and the unit time of earliness, tardiness, respectively;

θ

and

λ

are the unit time weight (penalty) for the due-window starting time

d_{1}

and size D,

ρ

is the weight of resource costs and

v_{j}

is the unit cost of processing job

J_{j}

. For the convex resource allocation, the third (resp. fourth) problem (denoted by

P_{3}

and

P_{4}

, respectively) is to minimize scheduling cost

\sum_{j = 1}^{n} (α_{j} U_{[j]} + β_{j} V_{[j]} + η_{j} E_{[j]} + δ_{j} T_{[j]} + θ d_{1} + λ D)

(resource cost

\sum_{j = 1}^{n} v_{j} u_{j}

) subject to

\sum_{j = 1}^{n} v_{j} u_{j}

\leq \hat{U}

(

\sum_{j = 1}^{n} (α_{j} U_{[j]} + β_{j} V_{[j]} + η_{j} E_{[j]} + δ_{j} T_{[j]} + θ d_{1} + λ D) \leq \hat{V}

), where

\hat{U}

(

\hat{V}

) is a given constant.

Remark 1

(Liu et al. [16]). considered the single machine scheduling with past-sequence-dependent setup times, i.e., for the common due-window assignment; the objective function is to minimize

\sum_{j = 1}^{n} ({\tilde{α}}_{j} U_{j} + {\tilde{β}}_{j} V_{j} + \tilde{η} E_{[j]} + \tilde{δ} T_{[j]} + θ d_{1} + λ D)

, where

{\tilde{α}}_{j}

(

{\tilde{β}}_{j}

) is the weight of job

J_{j}

(i.e., job-dependent weight), and

\tilde{η}

and

\tilde{δ}

are given constants. They also addressed the slack and unrestricted due-window assignments.

3. Method

From Wang et al. [17], for the problems

P_{1}

–

P_{4}

, there exists an optimal sequence

π

such that all jobs are processed at time 0, and there is no idle time during consecutive processing.

Lemma 1.

For the problems

P_{1}

P_{4}

, there exists an optimal sequence π such that

(1) The optimal

d_{1}

is equal to

C_{[m]}

(i.e.,

d_{1} = C_{[m]}

) and the optimal

d_{2}

is equal to

C_{[w]}

(i.e.,

d_{2} = C_{[w]}

), where

1 \leq m \leq w \leq n

;

(2)

d_{1} = C_{[m]}

, where

\sum_{j = 1}^{m - 1} η_{j} \leq n (λ - θ)

d_{2} = C_{[w]}

, where

\sum_{j = w + 1}^{n} η_{j} \leq n λ

Proof.

Similar to the proof of Wang et al. [17]. □

3.1. Problem $P_{1}$

From Lemma 1, for any given sequence,

d_{1}

d_{2}

and D can be calculated as follows:

d_{1} = C_{[m]} = \sum_{l = 1}^{m} p_{[l]}^{A}

d_{2} = C_{[w]} = \sum_{l = 1}^{n} p_{[l]}^{A}

D = C_{[w]} - C_{[m]} = \sum_{l = m + 1}^{w} p_{[l]}^{A}

. Hence, it follows that

\begin{matrix} Z & = & \sum_{j = 1}^{n} (α_{j} U_{[j]} + β_{j} V_{[j]} + η_{j} E_{[j]} + δ_{j} T_{[j]} + θ d_{1} + λ D) + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = & \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{m - 1} η_{j} E_{[j]} + \sum_{j = w + 1}^{n} δ_{j} T_{[j]} + n θ d_{1} + n λ D + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = & \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{m - 1} η_{j} (d_{1} - C_{[j]}) + \sum_{j = w + 1}^{n} δ_{j} (C_{[j]} - d_{2}) + n θ d_{1} + n λ D \\ + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = & \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{m - 1} η_{j} (\sum_{l = 1}^{m} p_{[l]}^{A} - \sum_{l = 1}^{j} p_{[l]}^{A}) + \sum_{j = w + 1}^{n} δ_{j} (\sum_{l = 1}^{j} p_{[l]}^{A} - \sum_{l = 1}^{w} p_{[l]}^{A}) \\ + n θ \sum_{l = 1}^{m} p_{[l]}^{A} + n λ \sum_{l = m + 1}^{w} p_{[l]}^{A} + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = & \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} p_{[j]}^{A} + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]}, \end{matrix}

(2)

where

{\tilde{Φ}}_{j} = \{\begin{matrix} \sum_{l = 1}^{j - 1} η_{l} + n θ, & j = 1, 2, \dots, m, \\ n λ, & j = m + 1, m + 2, \dots, w \\ \sum_{l = j}^{n} δ_{l}, & j = w + 1, w + 2, \dots, n . \end{matrix}

(3)

When

p_{j}^{A} = {\bar{p}}_{j} - b_{j} u_{j}

, we have

\begin{matrix} Z = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} ({\bar{p}}_{[j]} - b_{[j]} u_{[j]}) + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {\bar{p}}_{[j]} + \sum_{j = 1}^{n} (ρ v_{[j]} - b_{[j]} {\tilde{Φ}}_{j}) u_{[j]} \end{matrix}

(4)

Let

u_{[j]}^{*} (1 \leq j \leq n)

be the optimal resource allocation of the jth position; we have

u_{[j]}^{*} = \{\begin{matrix} {\bar{u}}_{[j]}, & ρ v_{[j]} - b_{[j]} {\tilde{Φ}}_{j} < 0 \\ u_{[j]} \in [0, {\bar{u}}_{[j]}], & ρ v_{[j]} - b_{[j]} {\tilde{Φ}}_{j} = 0 \\ 0, & ρ v_{[j]} - b_{[j]} {\tilde{Φ}}_{j} > 0 \end{matrix}

(5)

From (2) and (3), if m and w are given, let

x_{j r} = 1

; if job

J_{j}

is placed in the rth position and

x_{j r} = 0

otherwise, then the optimal job sequence

π

of the problem

P_{1}

can be formulated as the following assignment problem:

min Z = \sum_{j = 1}^{n} \sum_{r = 1}^{n} τ_{j r} x_{j r}

(6)

s . t . \{\begin{matrix} \sum_{j = 1}^{n} x_{j r} = 1, & r = 1, 2, \dots, n, \\ \sum_{r = 1}^{n} x_{j r} = 1, & j = 1, 2, \dots, n, \\ x_{j r} = 0 o r 1, \end{matrix}

(7)

in which

τ_{j r} = \{\begin{matrix} ϖ_{r} + χ_{r} {\bar{p}}_{[j]}, & ρ v_{[j]} - b_{[j]} χ_{r} \geq 0 \\ ϖ_{r} + χ_{r} {\bar{p}}_{[j]} + (ρ v_{[j]} - b_{[j]} χ_{r}) {\bar{u}}_{[j]}, & ρ v_{[j]} - b_{[j]} χ_{r} < 0 \end{matrix}

(8)

χ_{r} = \{\begin{matrix} \sum_{l = 1}^{r - 1} η_{l} + n θ, & r = 1, 2, \dots, m \\ n λ, & r = m + 1, m + 2, \dots, w \\ \sum_{l = r}^{n} δ_{l}, & r = w + 1, w + 2, \dots, n \end{matrix}

(9)

and

ϖ_{r} = \{\begin{matrix} α_{r}, & r = 1, 2, \dots, m - 1 \\ 0, & r = m, m + 1, \dots, w \\ β_{r}, & r = w + 1, w + 2, \dots, n . \end{matrix}

(10)

Based on the above analysis, the problem

P_{1}

can be solved by following Algorithm 1:

Algorithm 1: The problem

P_{1}

Input:

{\bar{p}}_{j}

b_{j}

{\bar{u}}_{j}

v_{j}

α_{j}

β_{j}

η_{j}

δ_{j}

(

j = 1, 2, \dots, n

n, θ, λ, ρ

Output: The optimal sequence

π^{*}

Z^{*}

u_{j}^{*}

{d_{1}}^{*}

D^{*}

Step 1. From Lemma 1, calculate the range of m and w.

Step 2. For each pair of m and w (

m = 1, 2, \dots, n

w = 1, 2, \dots, n

m \leq w

), calculate

τ_{j r}

(see (8)–(10)), to solve the assignment problems (6) and (7).

Step 3. For each pair of m and w, a suboptimal sequence

π (m, w)

and value

Z (m, w)

can be obtained.

Step 4. The global optimal sequence

π^{*}

is the one with the minimum value

Z^{*} = min {Z (m, w),

where

1 \leq m \leq w \leq n}

Step 5. Calculate

u_{j}^{*}

by using Equation (5).

Step 6. Calculate

d_{1}^{*} = C_{[m]}

and

D^{*} = C_{[w]} - C_{[m]}

Theorem 1.

The problem

P_{1}

can be solved by Algorithm 1 in

O (n^{5})

time.

Proof.

The correctness of Algorithm 1 follows the above analysis. Steps 1 and 3–6 need

O (n)

time; in Step 2, for each pair of m and w, solving the assignment problem requires

O (n^{3}

) time; m and w are less than n. Hence, the time complexity of Algorithm 1 is

O (n^{5})

. □

3.2. Problem $P_{2}$

Similar to Section 3.1, for the problem

P_{2}

, under

p_{j}^{A} = {(\frac{{\bar{p}}_{j}}{u_{j}})}^{k}

, we have

Z = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {(\frac{{\bar{p}}_{[j]}}{u_{[j]}})}^{k} + ρ \sum_{j = 1}^{n} v_{[j]} u_{[j]} .

(11)

Lemma 2.

The optimal resource allocation of the problem

P_{2}

is:

u_{[j]}^{*} = \frac{{(k {\tilde{Φ}}_{j})}^{\frac{1}{k + 1}} {({\bar{p}}_{[j]})}^{\frac{k}{k + 1}}}{{(ρ v_{[j]})}^{\frac{k}{k + 1}}}, j = 1, 2, \dots n .

(12)

Proof.

Taking a partial derivative of (11) with respect to

u_{[j]}

and making it equal to 0, we have

ρ v_{[j]} - k {\tilde{Φ}}_{j} \frac{{({\bar{p}}_{[j]})}^{k}}{{(u_{[j]})}^{k + 1}} = 0,

(13)

and the result (12) can be obtained. □

Substituting Equation (12) into Equation (11), we have

Z = (k^{- \frac{k}{k + 1}} + k^{\frac{1}{k + 1}}) ρ^{\frac{k}{k + 1}} \sum_{j = 1}^{n} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {\tilde{Φ}}_{j}^{\frac{1}{k + 1}} + \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} .

(14)

Similarly, from Equation (14), if m and w are given, then the optimal job sequence

π

of the problem

P_{2}

can be formulated as the following assignment problem:

min Z = \sum_{j = 1}^{n} \sum_{r = 1}^{n} τ_{j r} x_{j r}

(15)

s . t . \{\begin{matrix} \sum_{r = 1}^{n} x_{j r} = 1, & j = 1, 2, \dots, n, \\ \sum_{j = 1}^{n} x_{j r} = 1, & r = 1, 2, \dots, n, \\ x_{j r} = 0 o r 1, \end{matrix}

(16)

in which

τ_{j r} = \{\begin{matrix} (k^{- \frac{k}{k + 1}} + k^{\frac{1}{k + 1}}) ρ^{\frac{k}{k + 1}} χ_{r}^{\frac{1}{k + 1}} {(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}} + α_{r}, & r = 1, 2 \dots m - 1 \\ (k^{- \frac{k}{k + 1}} + k^{\frac{1}{k + 1}}) ρ^{\frac{k}{k + 1}} χ_{r}^{\frac{1}{k + 1}} {(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}}, & r = m, m + 1, \dots, w, \\ (k^{- \frac{k}{k + 1}} + k^{\frac{1}{k + 1}}) ρ^{\frac{k}{k + 1}} χ_{r}^{\frac{1}{k + 1}} {(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}} + β_{r}, & r = w + 1, w + 2, \dots, n, \end{matrix}

(17)

and

χ_{r}

is given by Equation (10).

Based on the above analysis, the problem

P_{2}

can be solved by following Algorithm 2:

Algorithm 2: The problem

P_{2}

Input:

{\bar{p}}_{j}

v_{j}

α_{j}

β_{j}

η_{j}

δ_{j}

(

j = 1, 2, \dots, n

n, k, θ, λ, ρ

Output: The optimal sequence

π^{*}

Z^{*}

u_{j}^{*}

{d_{1}}^{*}

D^{*}

Step 1. From Lemma 1, calculate the range of m and w.

Step 2. For each pair of m and w (

m = 1, 2, \dots, n

w = 1, 2, \dots, n

m \leq w

), calculate

τ_{j r}

(see (17)), to solve the assignment problems (15) and (16).

Step 3. For each pair of m and w, a suboptimal sequence

π (m, w)

and value

Z (m, w)

(see (15) and (16) ) can be obtained.

Step 4. The global optimal sequence

π^{*}

is the one with the minimum value

Z^{*} = min {Z (m, w)

, where

1 \leq m \leq w \leq n}

Step 5. Calculate

u_{j}^{*}

by using Equation (12).

Step 6. Calculate

d_{1}^{*} = C_{[m]}

and

D^{*} = C_{[w]} - C_{[m]}

Theorem 2.

The problem

P_{2}

can be solved by Algorithm 2 in

O (n^{5})

time.

3.3. Problem $P_{3}$

In this subsection, we consider the problem

P_{3}

, i.e.,

min Z = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} p_{[j]}^{A} = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {(\frac{{\bar{p}}_{[j]}}{u_{[j]}})}^{k}

(18)

subject to

\sum_{j = 1}^{n} v_{[j]} u_{[j]} \leq \hat{U}

Lemma 3.

The optimal resource allocation of the problem

P_{3}

is:

u_{[j]}^{*} = \frac{{({\tilde{Φ}}_{j} {({\bar{p}}_{[j]})}^{k})}^{\frac{1}{k + 1}}}{{(v_{[j]})}^{\frac{1}{k + 1}} \sum_{j = 1}^{n} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {({\tilde{Φ}}_{j})}^{\frac{1}{k + 1}}} \times \hat{U}, j = 1, 2, \dots, n

(19)

Proof.

For any given job sequence, by the Lagrange multiplier method, we have

\begin{matrix} L (u_{[j]}, μ) = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {(\frac{{\bar{p}}_{j}}{u_{j}})}^{k} + μ (\sum_{j = 1}^{n} v_{[j]} u_{[j]} - \hat{U}), \end{matrix}

(20)

where

μ

is the Lagrangian multiplier,

μ \geq 0

. Taking the partial derivative of

u_{[j]}

and

μ

, we have

\frac{\partial L (u, μ)}{\partial u_{[j]}} = μ v_{[j]} - k {\tilde{Φ}}_{j} \frac{{({\bar{p}}_{[j]})}^{k}}{{(u_{[j]})}^{k + 1}} = 0

(21)

u_{[j]} = \frac{{(k {\tilde{Φ}}_{j} {({\bar{p}}_{[j]})}^{k})}^{1 / k + 1}}{{(μ v_{[j]})}^{1 / k + 1}}

(22)

\frac{\partial L (u, μ)}{\partial μ} = \sum_{j = 1}^{n} v_{[j]} u_{[j]} - \hat{U} = 0

(23)

μ^{1 / k + 1} = \frac{\sum_{j = 1}^{n} {(k {\tilde{Φ}}_{j})}^{1 / k + 1} {(v_{[j]} {\bar{p}}_{[j]})}^{k / k + 1}}{\hat{U}} .

(24)

Substituting Equation (24) into Equation (22), Equation (19) can be obtained. □

Putting

u_{[j]}^{*}

(i.e., Equation (19)) in Equation (18), we have

\begin{matrix} Z = \frac{{(\sum_{j = 1}^{n} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {\tilde{Φ}}_{j}^{\frac{1}{k + 1}})}^{k + 1}}{{\hat{U}}^{k}} + \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} . \end{matrix}

(25)

If m and w are given, then

\sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j}

is a constant number. Hence, it can be obtained that solving problem

P_{3}

is equal to minimizing

\sum_{j = 1}^{n} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {\tilde{Φ}}_{j}^{\frac{1}{k + 1}}

and this term can be minimized by the HLP rule (see Hardy et al. [25]).

Based on the above analysis, the problem

P_{3}

can be solved by following Algorithm 3:

Algorithm 3: The problem

P_{3}

Input:

{\bar{p}}_{j}

v_{j}

α_{j}

β_{j}

η_{j}

δ_{j}

(

j = 1, 2, \dots, n

n, k, θ, λ, \hat{U}

Output: The optimal sequence

π^{*}

Z^{*}

u_{j}^{*}

{d_{1}}^{*}

D^{*}

Step 1. From Lemma 1, calculate the range of m and w.

Step 2. For each pair of m and w, a suboptimal sequence

π (m, w)

and value

Z (m, w)

(see Equation (25)) can be obtained by the HLP rule, i.e., by matching the largest

{(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}}

with the smallest

{\tilde{Φ}}_{j}^{\frac{1}{k + 1}}

, the second largest

{(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}}

matches the second smallest

{\tilde{Φ}}_{j}^{\frac{1}{k + 1}}

, and so on.

Step 3. The global optimal sequence

π^{*}

is the one with the minimum value

Z^{*} = min {Z (m, w),

where

1 \leq m \leq w \leq n}

Step 4. Calculate

u_{j}^{*}

by using Equation (19).

Step 5. Calculate

d_{1}^{*} = C_{[m]}

and

D^{*} = C_{[w]} - C_{[m]}

Theorem 3.

The problem

P_{3}

can be solved by Algorithm 3 in

O (n^{5})

time.

3.4. Problem $P_{4}$

Similar to Section 3.3, we have:

Lemma 4.

The optimal resource allocation of the problem

P_{4}

is:

u_{[j]}^{*} = \frac{{\tilde{Φ}}_{j}^{\frac{1}{k + 1}} {({\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {(\sum_{j = 1}^{n} {({\tilde{Φ}}_{j})}^{\frac{1}{k + 1}} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}})}^{\frac{1}{k}}}{{(v_{[j]})}^{\frac{1}{k + 1}} {(\hat{V} - \sum_{j = 1}^{m - 1} α_{j} - \sum_{j = w + 1}^{n} β_{j})}^{\frac{1}{k}}}, j = 1, 2, \dots, n

(26)

Proof.

For any given job sequence, we have:

\begin{matrix} L (u_{[j]}, μ) = \sum_{j = 1}^{n} v_{[j]} u_{[j]} + μ (\sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {(\frac{{\bar{p}}_{[j]}}{u_{[j]}})}^{k} - \hat{V}), \end{matrix}

(27)

where

μ

is the Lagrangian multiplier,

μ \geq 0

, we have:

\frac{\partial L (u_{[j]}, μ)}{\partial u_{[j]}} = v_{[j]} - μ k {\tilde{Φ}}_{j} \frac{{({\bar{p}}_{[j]})}^{k}}{{(u_{[j]})}^{k + 1}} = 0

(28)

u_{[j]} = \frac{{(μ k {\tilde{Φ}}_{j} {({\bar{p}}_{[j]})}^{k})}^{1 / k + 1}}{{(v_{[j]})}^{\frac{1}{k + 1}}}

(29)

\frac{\partial L (u_{[j]}, μ)}{\partial μ} = \sum_{j = 1}^{m - 1} α_{j} + \sum_{j = w + 1}^{n} β_{j} + \sum_{j = 1}^{n} {\tilde{Φ}}_{j} {(\frac{{\bar{p}}_{[j]}}{u_{[j]}})}^{k} - \hat{V} = 0

(30)

{(k μ)}^{\frac{1}{k + 1}} = \frac{{(\sum_{j = 1}^{n} {({\tilde{Φ}}_{j})}^{\frac{1}{k + 1}} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}})}^{\frac{1}{k}}}{{(\hat{V} - \sum_{j = 1}^{m - 1} α_{j} - \sum_{j = w + 1}^{n} β_{j})}^{\frac{1}{k}}} .

(31)

Substituting Equation (31) into Equation (29), Equation (26) can be obtained. □

Substituting Equation (26) into

\sum_{j = 1}^{n} v_{j} u_{j}

, we have:

\begin{matrix} Z & = & \sum_{j = 1}^{n} v_{[j]} u_{[j]} \\ = & \sum_{j = 1}^{n} v_{[j]} [\frac{{\tilde{Φ}}_{j}^{\frac{1}{k + 1}} {({\bar{p}}_{[j]})}^{\frac{k}{k + 1}} {(\sum_{j = 1}^{n} {({\tilde{Φ}}_{j})}^{\frac{1}{k + 1}} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}})}^{\frac{1}{k}}}{{(v_{[j]})}^{\frac{1}{k + 1}} {(V - \sum_{j = 1}^{m - 1} α_{j} - \sum_{j = w + 1}^{n} β_{j})}^{\frac{1}{k}}}] \\ = & \frac{{(\sum_{j = 1}^{n} {({\tilde{Φ}}_{j})}^{\frac{1}{k + 1}} {(v_{[j]} {\bar{p}}_{[j]})}^{\frac{k}{k + 1}})}^{\frac{1 + k}{k}}}{{(\hat{V} - \sum_{j = 1}^{m - 1} α_{j} - \sum_{j = w + 1}^{n} β_{j})}^{\frac{1}{k}}} . (32) \end{matrix}

Similar to Section 3.3, the problem

P_{4}

can be solved by following Algorithm 4:

Algorithm 4: The problem

P_{4}

Input:

{\bar{p}}_{j}

v_{j}

α_{j}

β_{j}

η_{j}

δ_{j}

(

j = 1, 2, \dots, n

n, k, θ, λ, \hat{V}

Output: The optimal sequence

π^{*}

Z^{*}

u_{j}^{*}

{d_{1}}^{*}

D^{*}

Step 1. From Lemma 1, calculate the range of m and w.

Step 2. For each pair of m and w, a suboptimal sequence

π (m, w)

and value

Z (m, w)

(see Equation (32)) can be obtained by the HLP rule, i.e., by matching the largest

{(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}}

with the smallest

{\tilde{Φ}}_{j}^{\frac{1}{k + 1}}

, the second largest

{(v_{j} {\bar{p}}_{j})}^{\frac{k}{k + 1}}

matches the second smallest

{\tilde{Φ}}_{j}^{\frac{1}{k + 1}}

, and so on.

Step 3. The global optimal sequence

π^{*}

is the one with the minimum value

Z^{*} = min {Z (m, w),

where

1 \leq m \leq w \leq n}

Step 4. Calculate

u_{j}^{*}

by using Equation (26).

Step 5. Calculate

d_{1}^{*} = C_{[m]}

and

D^{*} = C_{[w]} - C_{[m]}

Theorem 4.

The problem

P_{4}

can be solved by Algorithm 4 in

O (n^{5})

time.

4. A Case Study

In this section, we present a case study to illustrate how the proposed algorithms works.

Example 1.

Consider a five-job problem, where

k = 2, θ = 3, λ = 4, ρ = 6

\hat{U} = 100, \hat{V} = 1000

; the parameters of job

J_{j}

(

j = 1, 2, \dots, 5

) are given in Table 1. The parameters of position-dependent weights (i.e.,

α_{j}, β_{j}, η_{j}

and

δ_{j}

, where

j = 1, 2, \dots, 5

) are given in Table 2.

For the problem

P_{1}

, from Algorithm 1,

m = 1, 2

w = 3, 4, 5

, and all results are shown in Table 3. From Table 3, the optimal sequence

π^{*} (2, 3)

J_{3} \to J_{4} \to J_{1} \to J_{2} \to J_{5}

u_{3}^{*} = 3

u_{4}^{*} = 11

u_{1}^{*} = 5

u_{2}^{*} = 4

u_{5}^{*} = 0

Z^{*} (2, 3) = 653

d_{1}^{*} = C_{[2]} = 9

, and

D^{*} = C_{[3]} - C_{[2]} = 3

. When

m = 2

and

w = 3

, the values

{\tilde{Φ}}_{1} = 15

{\tilde{Φ}}_{2} = 17

{\tilde{Φ}}_{3} = 20

{\tilde{Φ}}_{4} = 9

{\tilde{Φ}}_{5} = 6

(see Equation (3)),

τ_{j r}

(see Equation (8)) are given in Table 4, and the optimal solution of the assignment problem is given by Table 4.

Similarly, for the problem

P_{2}

from Algorithm 2, the results are shown in Table 5. From Table 5, the optimal sequence

π^{*} (2, 3)

J_{3} \to J_{1} \to J_{4} \to J_{2} \to J_{5}

u_{3}^{*} = 2.169

u_{1}^{*} = 2.608

u_{4}^{*} = 6.230

u_{2}^{*} = 1.423

u_{5}^{*} = 1.388

Z^{*} (2, 3) = 971.297

d_{1}^{*} = C_{[2]} = 66.505

and

D^{*} = C_{[3]} - C_{[2]} = 5.797

Similarly, for problem

P_{3}

, from Algorithm 3, the results are shown in Table 6. From Table 6, the optimal sequence

π^{*} (2, 3)

J_{3} \to J_{1} \to J_{4} \to J_{2} \to J_{5}

u_{3}^{*} = 5.871

u_{1}^{*} = 6.412

u_{4}^{*} = 10.740

u_{2}^{*} = 4.148

u_{5}^{*} = 4.301

Z^{*} (2, 3) = 375.290

d_{1}^{*} = C_{[2]} = 9.798

, and

D^{*} = C_{[3]} - C_{[2]} = 1.951

Similarly, for problem

P_{4}

, from Algorithm 4, the results are shown in Table 7. From Table 7, the optimal sequence is

π^{*} (2, 3)

J_{3} \to J_{1} \to J_{4} \to J_{2} \to J_{5}

u_{3}^{*} = 3.563

u_{1}^{*} = 3.892

u_{4}^{*} = 6.519

u_{2}^{*} = 2.518

u_{5}^{*} = 2.611

Z^{*} (2, 3) = 60.516

d_{1}^{*} = C_{[2]} = 26.596

, and

D^{*} = C_{[3]} - C_{[2]} = 5.295

5. Conclusions

This article explored resource allocation scheduling with general position-dependent weights and common due-window assignment. The resource allocation can be divided into linear and convex resource allocations. Under some combination of general earliness–tardiness cost and resource consumption cost, we showed that four versions of the problem are polynomially solvable. Future research and investigation will consider the scheduling under flow shop settings, or extending our model from resource allocation to general deterioration effects.

Author Contributions

Conceptualization, Y.-C.W., S.-H.W. and J.-B.W.; methodology, Y.-C.W. and S.-H.W.; investigation, J.-B.W.; writing, review and editing, Y.-C.W., S.-H.W. and J.-B.W. All authors have read and agreed to the published version of the manuscript.

Funding

This Work was supported by LiaoNing Revitalization Talents Program (XLYC2002017).

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

Yin, Y.; Cheng, T.C.E.; Wu, C.-C.; Cheng, S.-R. Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing time. J. Oper. Res. Soc. 2013, 65, 1–13. [Google Scholar] [CrossRef]
Janiak, A.; Janiak, W.A.; Krysia, T.; Kwiatkowski, T. A survey on scheduling problems with due windows. Eur. J. Oper. Res. 2015, 242, 347–357. [Google Scholar] [CrossRef]
Common due-window assignment and group scheduling with position-dependent processing times. Asia-Pac. J. Oper. Res. 2015, 32, 1550045. [CrossRef]
Liu, W.; Hu, X.; Wang, X. Single machine scheduling with slack due dates assignment. Eng. Optim. 2017, 49, 709–717. [Google Scholar] [CrossRef]
Rolim, G.A.; Nagano, M.S. Structural properties and algorithms for earliness and tardiness scheduling against common due dates and windows: A review. Comput. Ind. Eng. 2020, 149, 106803. [Google Scholar] [CrossRef]
Qian, J.; Zhan, Y. The due date assignment scheduling problem with delivery times and truncated sum-of-processing-times-based learning effect. Mathematics 2021, 9, 3085. [Google Scholar] [CrossRef]
Wang, J.-B.; Zhang, B.; Li, L.; Bai, D.; Feng, Y.-B. Due window assignment scheduling problems with position-dependent weights on a single machine. Eng. Optim. 2020, 52, 185–193. [Google Scholar] [CrossRef]
Wang, J.-B.; Cui, B.; Ji, P.; Liu, W.-W. Research on scheduling with position-dependent weights and past-sequence-dependent delivery times. J. Comb. Optim. 2021, 41, 290–303. [Google Scholar] [CrossRef]
Wang, S.-H.; Lv, D.-Y.; Wang, J.-B. Research on position-dependent weights scheduling with delivery times and truncated sum-of-processing-times-based learning effect. J. Ind. Manag. Optim. 2023, 19, 2824–2837. [Google Scholar] [CrossRef]
Sun, X.-Y.; Geng, X.-N.; Liu, T. Due-window assignment scheduling in the proportionate flow shop setting. Ann. Oper. Res. 2020, 292, 113–131. [Google Scholar] [CrossRef]
Qian, J.; Han, H. Improved algorithms for proportionate flow shop scheduling with due-window assignment. Ann. Oper. Res. 2022, 309, 249–258. [Google Scholar] [CrossRef]
Yue, Q.; Zhou, S. Due-window assignment scheduling problem with stochastic processing times. Eur. J. Oper. Res. 2021, 290, 453–468. [Google Scholar] [CrossRef]
Wang, W. Single-machine due-date assignment scheduling with generalized earliness/tardiness penalties including proportional setup times. J. Appl. Math. Comput. 2022, 68, 1013–1031. [Google Scholar] [CrossRef]
Shabtay, D.; Mosheiov, G.; Oron, D. Single machine scheduling with common assignable due date/due window to minimize total weighted early and late work. Eur. J. Oper. Res. 2022, 303, 66–77. [Google Scholar] [CrossRef]
Jiang, Y.; Zhang, Z.; Song, X.; Yin, Y. Seru scheduling problems with multiple due-windows assignment and learning effect. J. Syst. Sci. Syst. Eng. 2022, 31, 480–511. [Google Scholar] [CrossRef]
Liu, W.; Wang, X.; Wang, X.; Zhao, P. Due-window assignment scheduling with past-sequence-dependent setup times. Math. Biosci. Eng. 2022, 19, 3110–3126. [Google Scholar] [CrossRef]
Wang, J.-B.; Wang, S.-H.; Gao, K.; Liu, M.; Jia, X. Due-window assignment methods and scheduling with generalized positional-dependent weights. Asia-Pac. J. Oper. Res. 2022, 39, 2250028. [Google Scholar] [CrossRef]
Shabtay, D.; Steiner, G. A survey of scheduling with controllable processing times. Discret. Appl. Math. 2007, 155, 1643–1666. [Google Scholar] [CrossRef] [Green Version]
Liu, L.; Wang, J.-J.; Liu, F. Single machine due window assignment and resource allocation scheduling problems with learning and general positional effects. J. Manuf. Syst. 2017, 43, 1–14. [Google Scholar] [CrossRef]
Lu, Y.-Y.; Wang, T.-T.; Wang, R.-Q.; Li, Y. A note on due-date assignment scheduling with job-dependent learning effects and convex resource allocation. Eng. Optim. 2021, 53, 1273–1281. [Google Scholar] [CrossRef]
Wang, J.-B.; Lv, D.-Y.; Xu, J.; Ji, P.; Li, F. Bicriterion scheduling with truncated learning effects and convex controllable processing times. Int. Trans. Oper. Res. 2021, 28, 1573–1593. [Google Scholar] [CrossRef]
Zhao, S. Resource allocation flowshop scheduling with learning effect and slack due window assignment. J. Ind. Manag. Optim. 2021, 17, 2817–2835. [Google Scholar] [CrossRef]
Lu, S.; Kong, M.; Zhou, Z.; Liu, X.; Liu, S. A hybrid metaheuristic for a semiconductor production scheduling problem with deterioration efect and resource constraints. Oper. Res. 2022, 22, 5405–5440. [Google Scholar]
Yan, J.-X.; Ren, N.; Bei, H.-B.; Bao, H.; Wang, J.-B. Scheduling with resource allocation, deteriorating effect and group technology to minimize total completion time. Mathematics 2022, 10, 2983. [Google Scholar] [CrossRef]
Hardy, G.-H.; Littlewood, J.-E.; Polya, G. Inequalities, 2nd ed.; Cambridge University Press: Cambridge, UK, 1967. [Google Scholar]

Table 1. Values of job-dependent parameters.

	$j = 1$	$j = 2$	$j = 3$	$j = 4$	$j = 5$
${\bar{p}}_{j}$	13	12	14	15	17
$b_{j}$	2	2	3	1	1
${\bar{u}}_{j}$	5	4	3	11	7
$v_{j}$	3	5	4	1	6

Table 2. Values of position-dependent weights.

	$j = 1$	$j = 2$	$j = 3$	$j = 4$	$j = 5$
$α_{j}$	4	8	7	6	5
$β_{j}$	8	7	3	2	6
$η_{j}$	2	4	6	10	7
$δ_{j}$	2	5	4	3	6

Table 3. Results of problem

P_{1}

(bold numbers are the optimal solution).

Table 3. Results of problem

P_{1}

(bold numbers are the optimal solution).

m	w	$π (m, w)$	$Z (m, w)$
1	3	$(J_{3}, J_{4}, J_{1}, J_{2}, J_{5})$	661
1	4	$(J_{3}, J_{2}, J_{1}, J_{4}, J_{5})$	751
1	5	$(J_{5}, J_{1}, J_{2}, J_{3}, J_{4})$	923
2	3	$(J_{3}, J_{4}, J_{1}, J_{2}, J_{5})$	653
2	4	$(J_{3}, J_{4}, J_{2}, J_{1}, J_{5})$	743
2	5	$(J_{5}, J_{3}, J_{1}, J_{4}, J_{2})$	912

Table 4. Values

τ_{j r}

for

m = 2

and

w = 3

(bold numbers are the optimal solution).

Table 4. Values

τ_{j r}

for

m = 2

and

w = 3

(bold numbers are the optimal solution).

	$r = 1$	$r = 2$	$r = 3$	$r = 4$	$r = 5$
$J_{1}$	139	141	150	119	84
$J_{2}$	184	188	200	110	78
$J_{3}$	151	157	172	119	90
$J_{4}$	130	134	146	104	96
$J_{5}$	259	289	340	155	108

Table 5. Results of problem

P_{2}

(bold numbers are the optimal solution).

Table 5. Results of problem

P_{2}

(bold numbers are the optimal solution).

m	w	$π (m, w)$	$Z (m, w)$
1	3	$(J_{3}, J_{4}, J_{1}, J_{2}, J_{5})$	$977.570$
1	4	$(J_{2}, J_{4}, J_{1}, J_{3}, J_{5})$	$1035.169$
1	5	$(J_{5}, J_{1}, J_{3}, J_{4}, J_{2})$	$1141.332$
2	3	$(J_{3}, J_{1}, J_{4}, J_{2}, J_{5})$	971.297
2	4	$(J_{2}, J_{3}, J_{1}, J_{4}, J_{5})$	$1026.095$
2	5	$(J_{5}, J_{2}, J_{1}, J_{4}, J_{3})$	$1131.643$

Table 6. Results of problem

P_{3}

(bold numbers are the optimal solution).

Table 6. Results of problem

P_{3}

(bold numbers are the optimal solution).

m	w	$π (m, w)$	$Z (m, w)$
1	3	$(J_{3}, J_{4}, J_{1}, J_{2}, J_{5})$	$383.086$
1	4	$(J_{2}, J_{4}, J_{1}, J_{3}, J_{5})$	$454.595$
1	5	$(J_{5}, J_{4}, J_{1}, J_{3}, J_{2})$	$611.830$
2	3	$(J_{3}, J_{1}, J_{4}, J_{2}, J_{5})$	375.290
2	4	$(J_{2}, J_{3}, J_{4}, J_{1}, J_{5})$	$441.715$
2	5	$(J_{5}, J_{2}, J_{4}, J_{1}, J_{3})$	$594.076$

Table 7. Results of problem

P_{4}

(bold numbers are the optimal solution).

Table 7. Results of problem

P_{4}

(bold numbers are the optimal solution).

m	w	$π (m, w)$	$Z (m, w)$
1	3	$(J_{3}, J_{4}, J_{1}, J_{2}, J_{5})$	$61.491$
1	4	$(J_{2}, J_{4}, J_{1}, J_{3}, J_{5})$	$67.180$
1	5	$(J_{5}, J_{4}, J_{1}, J_{3}, J_{2})$	$78.220$
2	3	$(J_{3}, J_{1}, J_{4}, J_{2}, J_{5})$	60.516
2	4	$(J_{2}, J_{3}, J_{4}, J_{1}, J_{5})$	$66.036$
2	5	$(J_{5}, J_{2}, J_{4}, J_{1}, J_{3})$	$76.971$

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Wang, Y.-C.; Wang, S.-H.; Wang, J.-B. Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost. Mathematics 2023, 11, 222. https://doi.org/10.3390/math11010222

AMA Style

Wang Y-C, Wang S-H, Wang J-B. Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost. Mathematics. 2023; 11(1):222. https://doi.org/10.3390/math11010222

Chicago/Turabian Style

Wang, Yi-Chun, Si-Han Wang, and Ji-Bo Wang. 2023. "Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost" Mathematics 11, no. 1: 222. https://doi.org/10.3390/math11010222

APA Style

Wang, Y. -C., Wang, S. -H., & Wang, J. -B. (2023). Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost. Mathematics, 11(1), 222. https://doi.org/10.3390/math11010222

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Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost

Abstract

1. Introduction

2. Problem Formulation

3. Method

3.1. Problem $P_{1}$

3.2. Problem $P_{2}$

3.3. Problem $P_{3}$

3.4. Problem $P_{4}$

4. A Case Study

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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Resource Allocation Scheduling with Position-Dependent Weights and Generalized Earliness–Tardiness Cost

Abstract

1. Introduction

2. Problem Formulation

3. Method

3.1. Problem P 1

3.2. Problem P 2

3.3. Problem P 3

3.4. Problem P 4

4. A Case Study

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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3.1. Problem $P_{1}$

3.2. Problem $P_{2}$

3.3. Problem $P_{3}$

3.4. Problem $P_{4}$