# Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Planning Layer Model

#### 2.1.1. Deterministic Model

#### 2.1.2. Chance Constrained Uncertain Model

**Theorem**

**1.**

**Theorem**

**2.**

#### 2.2. Scheduling Layer Model

#### 2.2.1. Deterministic Model

#### 2.2.2. Fuzzy Uncertain Model

## 3. Integrated Model and Solving Strategy

## 4. Case Study

#### 4.1. The Influence of the Planning Layer Confidence Factor on the Optimization Results

#### 4.2. The Influence of Scheduling Layer Fuzzy Weight on the Optimization Results

#### 4.3. Integrated Model Simulation Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

(1) Production planning layer | |

a. Indices: | |

i | Material status |

k | Planning period |

b. Sets: | |

${G}_{all}$ | Raw material, intermediate product, final product material status set |

${G}_{ip}$ | Intermediate product and final product material status set |

${G}_{p}$ | Final product material status set |

${N}_{i}$ | The downstream material status set of material status $i$ |

c. Parameters: | |

$Pric{e}_{i}$ | The market price of product $i$, $\$\cdot k{g}^{-1}$ |

$\alpha $ | The penalty weight of backlog order item, which is 100 in this paper |

$\beta $ | The penalty weight of productivity fluctuation item, which is 0.0001 in this paper |

$\theta $ | The penalty weight of inventory limit item, which is 0.0001 in this paper |

${S}_{i}^{ini}$ | Initial inventory of material status $i$ |

${c}_{ij}$ | Conversion coefficient between material status $i$ and material status $j$ |

$Or{d}_{ik}$ | Order quantity for product $i$ in period $k$ |

${p}_{i}^{\mathrm{min}}$ | Minimum production capacity of material status $i$ |

${p}_{i}^{\mathrm{max}}$ | Maximum production capacity of material status $i$ |

${s}_{i}^{\mathrm{min}}$ | Inventory lower limit of material status $i$ |

${s}_{i}^{\mathrm{max}}$ | Inventory upper limit of material status $i$ |

${s}_{i}^{Ll}$ | Inventory reference lower limit of material status $i$ |

${s}_{i}^{Ul}$ | Inventory reference upper limit of material status $i$ |

$t{d}_{i,k-1}$ | Backlog order quantity from material status $i$ of the scheduling layer |

$Pr\{\cdot \}$ | Probability computing operator |

$\lambda $ | Confidence factor |

d. Variables: | |

$S(i,k)$ | Inventory of material status $i$ in planning period $k$ |

$P\left(i,k\right)$ | Amount of material status $i$ produced in planning period $k$ |

$D(i,k)$ | Delivery of material status $i$ in planning period $k$ |

$Td(i,k)$ | Backlog order quantity of material status $i$ in planning period $k$ |

$Flu(i,k)$ | Auxiliary variable of productivity fluctuation |

$Inv(i,k)$ | Auxiliary variable of inventory limit |

(2) Production scheduling layer | |

a. Indices: | |

$i,{i}^{\prime}$ | Task |

$r$ | Resource |

$n$ | Event point |

$u$ | Utility |

b. Sets: | |

${I}_{r}$ | A set of tasks that consume material resources |

${I}_{u}$ | A set of tasks that consume utilities |

$R\text{},{R}^{S}\text{},{R}^{J}$ | Resource set, Material resource set, Device resource set |

$N$ | Event point set |

$H$ | Scheduling horizon |

c. Parameters: | |

$De{f}_{r,k}$ | The amount of reference tasks assigned at the end of the planning period $k$ |

${E}_{ini}$ | Initial resource quantity |

$us{e}_{i}$ | Consumption amount of utility in the planning period $i$$(i=1\cdots 5)$ |

$us{e}_{all}$ | Total usage amount of utility |

${\mu}_{ri}^{p}\text{},{\mu}_{ri}^{c}$ | Production and consumption coefficients of device resources involved in task $i$; ${\mu}_{ri}^{p}\ge 0{\mu}_{ri}^{c}\le 0$ |

${\rho}_{ri}^{p}\text{},{\rho}_{ri}^{c}$ | Production and consumption coefficients of material resources involved in task $i$; ${\rho}_{ri}^{p}\ge 0{\rho}_{ri}^{c}\le 0$ |

${E}_{r}^{\mathrm{min}}$ | The lower limit of resource $r$ |

${E}_{r}^{\mathrm{max}}$ | The upper limit of resource $r$ |

${B}_{i}^{\mathrm{min}}$ | Lower limit of material handing amount of task $i$ |

${B}_{i}^{\mathrm{max}}$ | Upper limit of material handing amount of task $i$ |

$Sla$ | Large positive number in relaxation constraints. |

${w}_{1}\text{},{w}_{2}\text{},{w}_{3}$ | Weights |

${U}_{1\delta}^{\mathrm{max}}\text{},{U}_{2\delta}^{\mathrm{max}}\text{},{U}_{3\delta}^{\mathrm{max}}$ | The boundary points of the triangular membership function under the cut set $\delta $ |

${T}_{ri}(u)$ | Triangle membership function of ${U}_{u}^{\mathrm{max}}$ |

d. Variables: | |

$d(r,n)$ | The delivery of material resources $r$ at the event point $n$ |

$E(r,n)$ | The amount of available resources for resource $r$ at the event point $n$ |

$w(i,n)$ | Binary variable for task $i$ active at event $n$ |

$b(i,n)$ | The amount of resources processed by task $i$ at event point $n$ |

$td(r,k)$ | The backlog order quantity of product in the period $k$ |

${T}^{s}(i,n)$ | Start time of task $i$ at event point $n$ |

${T}^{f}(i,n)$ | End time of task $i$ at event point $n$ |

${T}_{u}^{s}(u,n)$ | The start time of the utility $u$ consumed at event point $n$ |

${\alpha}_{i}\cdot w(i,n)$ | Constant term of duration time |

${\beta}_{i}\cdot b(i,n)$ | Variable term of duration time |

${\lambda}_{iu}\cdot w(i,n)$ | Constant term of utility consumption |

${\phi}_{iu}\cdot b(i,n)$ | Variable term of utility consumption |

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Task (i) | Unit (j) | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{B}}_{\mathit{i}}^{\mathbf{min}}$ | ${\mathit{B}}_{\mathit{i}}^{\mathbf{max}}$ |
---|---|---|---|---|---|

Heating (i = 1) | Heater | 0.667 | 0.00667 | - | 100 |

Reaction 1 (i = 2) | Reactor 1 | 1.334 | 0.02664 | - | 50 |

Reaction 1 (i = 3) | Reactor 2 | 1.334 | 0.01665 | - | 80 |

Reaction 2 (i = 4) | Reactor 1 | 1.334 | 0.02664 | - | 50 |

Reaction 2 (i = 5) | Reactor 2 | 1.334 | 0.01665 | - | 80 |

Reaction 3 (i = 6) | Reactor 1 | 0.667 | 0.01332 | - | 50 |

Reaction 3 (i = 7) | Reactor 2 | 0.667 | 0.008325 | - | 80 |

Separation (i = 8) | Separator | 1.3342 | 0.00666 | - | 200 |

Resource | Capacity (kg) | Initial (kg) | Price ($/kg) |
---|---|---|---|

S1 | UL | AA | 0 |

S2 | UL | AA | 0 |

S3 | UL | AA | 0 |

S4 | 100 | 0 | 0 |

S5 | 200 | 0 | 0 |

S6 | 150 | 0 | 0 |

S7 | 200 | 0 | 0 |

S8 | UL | 0 | 40 |

S9 | UL | 0 | 30 |

Heater | 1 | 1 | - |

Reactor1 | 1 | 1 | - |

Reactor2 | 1 | 1 | - |

Separator | 1 | 1 | - |

Task (i) | ${\mathit{\lambda}}_{\mathit{i}\mathit{H}\mathit{S}}$ $(\mathit{k}\mathit{g}\cdot {\mathbf{min}}^{-1})$ | ${\mathit{\phi}}_{\mathit{i}\mathit{H}\mathit{S}}$ $(\mathit{k}\mathit{g}\cdot {\mathbf{min}}^{-1})$ | ${\mathit{\lambda}}_{\mathit{i}\mathit{C}\mathit{W}}$ $(\mathit{k}\mathit{g}\cdot {\mathbf{min}}^{-1})$ | ${\mathit{\varphi}}_{\mathit{i}\mathit{C}\mathit{W}}$ $(\mathit{k}\mathit{g}\cdot {\mathbf{min}}^{-1})$ |
---|---|---|---|---|

i = 1 | 6 | 0.25 | - | - |

i = 2 | - | - | 4 | 0.25 |

i = 3 | 5 | 0.25 | - | - |

i = 4 | - | - | 4 | 0.3 |

i = 5 | 4 | 0.5 | - | - |

i = 6 | - | - | 3 | 0.3 |

i = 7 | 4 | 0.2 | - | - |

i = 8 | - | - | 6 | 0.35 |

Confidence Factor | Inverse Function | Objective Function ($) |
---|---|---|

0.6 | −0.25 | 99,206.36 |

0.65 | −0.39 | 98,945.22 |

0.7 | −0.52 | 98,702.73 |

0.75 | −0.67 | 98,422.94 |

0.8 | −0.84 | 98,105.84 |

0.85 | −1.04 | 97,732.78 |

0.9 | −1.28 | 97,285.11 |

0.95 | −1.65 | 96,594.95 |

${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | Objective Function (kg) |
---|---|---|---|

0.1 | 0.1 | 0.8 | 67.59 |

0.1 | 0.2 | 0.7 | 72.51 |

0.1 | 0.3 | 0.6 | 77.43 |

0.1 | 0.4 | 0.5 | 82.35 |

0.1 | 0.5 | 0.4 | 87.27 |

0.1 | 0.6 | 0.3 | 92.19 |

0.1 | 0.7 | 0.2 | 97.11 |

0.1 | 0.8 | 0.1 | 102.03 |

Modeling Method | Objective Function ($) | ${\mathit{S}}_{8}^{\mathit{p}\mathit{l}\mathit{a}\mathit{n}}(\mathit{k}\mathit{g})$ | ${\mathit{S}}_{9}^{\mathit{p}\mathit{l}\mathit{a}\mathit{n}}(\mathit{k}\mathit{g})$ |
---|---|---|---|

Deterministic method | 99,688.52 | 1330 | 1550 |

Uncertain method | 97,285.11 | 1296.998 | 1514.416 |

Modeling Method | Objective Function ($) | ${\mathit{S}}_{8}^{\mathit{p}\mathit{l}\mathit{a}\mathit{n}}(\mathit{k}\mathit{g})$ | ${\mathit{S}}_{9}^{\mathit{p}\mathit{l}\mathit{a}\mathit{n}}(\mathit{k}\mathit{g})$ |
---|---|---|---|

Deterministic method | 128.9 | 1201.1 | 1550 |

Uncertain method | 87.277 | 1209.721 | 1514.416 |

$\mathit{u}\mathit{t}\mathit{i}\mathit{l}\mathit{i}\mathit{t}\mathit{y}$ | $\mathit{u}\mathit{s}{\mathit{e}}_{1}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{2}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{3}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{4}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{5}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{\mathit{a}\mathit{l}\mathit{l}}(\mathit{k}\mathit{g})$ |
---|---|---|---|---|---|---|

HS | 352.31419 | 345.83433 | 383.03704 | 370.31418 | 353.44445 | 1804.94419 |

CW | 337.26051 | 339.90991 | 357.33333 | 322.22298 | 328.33334 | 1685.06007 |

$\mathit{u}\mathit{t}\mathit{i}\mathit{l}\mathit{i}\mathit{t}\mathit{y}$ | $\mathit{u}\mathit{s}{\mathit{e}}_{1}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{2}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{3}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{4}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{5}(\mathit{k}\mathit{g})$ | $\mathit{u}\mathit{s}{\mathit{e}}_{\mathit{a}\mathit{l}\mathit{l}}(\mathit{k}\mathit{g})$ |
---|---|---|---|---|---|---|

HS | 339.07849 | 371.35303 | 340.01582 | 343.52171 | 341.69782 | 1735.66687 |

CW | 340.84803 | 359.11669 | 372.16621 | 330.04896 | 330.19241 | 1732.3723 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, T.; Wang, Y.; Jin, X.; Lu, S.
Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties. *Algorithms* **2019**, *12*, 120.
https://doi.org/10.3390/a12060120

**AMA Style**

Zhang T, Wang Y, Jin X, Lu S.
Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties. *Algorithms*. 2019; 12(6):120.
https://doi.org/10.3390/a12060120

**Chicago/Turabian Style**

Zhang, Tao, Yue Wang, Xin Jin, and Shan Lu.
2019. "Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties" *Algorithms* 12, no. 6: 120.
https://doi.org/10.3390/a12060120