# Solving Dual-Channel Supply Chain Pricing Strategy Problem with Multi-Level Programming Based on Improved Simplified Swarm Optimization

^{*}

## Abstract

**:**

## 1. Introduction

- Build an MLPP model to obtain the equilibrium solution of pricing strategy in the dual-channel supply chain system.
- Study and analyze the best decision for the manufacturer on finance strategy.
- Apply the improved, simplified swarm optimization algorithm to multi-level programming problems.

## 2. Literature Review

#### 2.1. Dual-Channel Supply Chain

#### 2.2. Supply Chain Finance

#### 2.3. Game Theory

#### 2.4. Stackelberg Game

#### 2.5. Multi-Level Programming Problem

#### 2.5.1. Bi-Level Programming Problem

**Definition**

**1.1.**

- The problem constraint region,

- 2.
- The follower feasible set for each fixed x,

- 3.
- The follower rational reaction set,

- 4.
- The problem inducible region (IR),

- 5.
- The problem optimal solution set,

**Definition**

**1.2.**

**Definition**

**1.3.**

#### 2.5.2. Multi-Level Programming Problem

**Definition**

**2.1.**

- 6.
- The problem constraint region,

- 7.
- The middle-level follower feasible set for each fixed x,

- 8.
- The bottom-level follower feasible set for each fixed (x, y),

- 9.
- The bottom-level follower rational reaction set,

- 10.
- The middle-level follower rational reaction set,

- 11.
- The problem inducible region,

- 12.
- The problem optimal solution set,

**Assumption**

**2.1.**

**Assumption**

**2.2.**

**Assumption**

**2.3.**

#### 2.6. Improved Simplified Swarm Optimization (iSSO)

## 3. Statement

#### 3.1. Model Description

#### 3.2. Assumptions

- This study constructs a dual-channel supply chain model with three levels of the supply chain (manufacturer → retailer → customer) to profit maximization.
- The manufacturer’s initial capital is zero and must repay the entire capital liability.
- The basic principle of profitability is that the price must be designed to meet the conditions of profitability for all parties.
- In the model, neither the upstream manufacturer nor the downstream manufacturer considers the inventory problem. The upstream manufacturer ships as much product as it makes to the downstream retailer. The downstream manufacturer buys as much as it can and sells it all to the market.

#### 3.3. Notations

#### 3.4. The Mathematical Model Description

#### 3.5. Model Construction

- (a)
- Retailer

- (b)
- Bank

- (c)
- 3rd Party Platform

- (d)
- Constraints of all

## 4. Methodology

#### 4.1. Multi-Level Improved Simplified Swarm Optimization (MLiSSO)

#### 4.1.1. Improved Simplified Swarm Optimization (iSSO)

#### 4.1.2. Fixed-Variables Local Search

#### 4.1.3. Fitness Function

#### 4.1.4. Constraint Handling

#### 4.1.5. Stopping Criteria

- The generation number.
- The maximum iteration.

#### 4.1.6. Level Conversion

#### 4.1.7. Steps of MLiSSO for Solving MLPP

**Main Program: The best solution to solving**

STEP 1-1 | $\mathrm{Maximum}\mathrm{iteration}{T}_{max}$. |

STEP 1-2 | $\mathrm{Set}{T}_{max}$$T=0$. |

STEP 1-3 | $\mathrm{Call}\mathrm{Subprogram}1\mathrm{and}\mathrm{generate}\mathrm{the}\mathrm{initial}\mathrm{solution}\left({X}_{i}^{T},{Y}_{i}^{T}\right)$. |

STEP 1-4 | $\mathrm{Evaluate}F\left({X}_{i}^{T},{Y}_{i}^{T}\right)$$\mathrm{and}\mathrm{let}\left({X}^{*},{Y}^{*}\right)=\left({X}_{i}^{T},{Y}_{i}^{T}\right)$. |

STEP 1-5 | $\mathrm{Fixed}{Y}_{i}^{T}$ to the upper-level programming model. |

STEP 1-6 | $\mathrm{Let}T=T+1$. |

STEP 1-7 | $\mathrm{Call}\mathsf{Subprogram}\mathsf{2}\mathrm{to}\mathrm{generate}{X}_{i}^{T}$. |

STEP 1-8 | $\mathrm{Fixed}\mathrm{the}\mathrm{solution}{X}_{i}^{T}$. |

STEP 1-9 | $\mathrm{Call}\mathsf{Subprogram}\mathsf{2}\mathrm{to}\mathrm{generate}{Y}_{i}^{T}$. |

STEP 1-10 | $\mathrm{Fixed}\left({X}_{i}^{T},{Y}_{i}^{T}\right)$$\mathrm{into}\mathrm{the}\mathrm{objective}\mathrm{function}\mathrm{to}\mathrm{evaluate}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{objective}\mathrm{function}.\mathrm{If}F\left({X}_{i}^{T},{Y}_{i}^{T}\right)F\left({X}^{*},{Y}^{*}\right)$$,\left({X}_{i}^{T},{Y}_{i}^{T}\right)$$\mathrm{it}\mathrm{is}\mathrm{recorded}\mathrm{as}\left({X}^{*},{Y}^{*}\right)$. |

STEP 1-11 | $\mathrm{Stopping}\mathrm{criterion}:\mathrm{if}T\ge Tmax$ go to STEP 1-12; otherwise, go to STEP 1-6. |

STEP 1-12 | $\mathrm{Output}\left({X}^{*},{Y}^{*}\right)$$\mathrm{and}\mathrm{the}\mathrm{objective}\mathrm{function}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{upper}-\mathrm{level}F\left({X}^{*},{Y}^{*}\right)$$\mathrm{and}\mathrm{the}\mathrm{lower}-\mathrm{level}f\left({X}^{*},{Y}^{*}\right)$. |

**Subprogram 1: Solution initialization**

STEP 2-1 | Initiate ${N}_{sol}$, ${N}_{gen}$, ${N}_{var}$, ${C}_{g}$, ${C}_{p}$, and ${C}_{w}$, and the upper and lower bounds of each variable. |

STEP 2-2 | Set ${N}_{gen}$ $t=0$ and $i=1$, where $i=1,2,\dots ,{N}_{sol}$. |

STEP 2-3 | Generate $\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$. Let ${P}_{fi}^{T}=\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$, and calculate $f\left({P}_{fi}^{T}\right)=f\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$ for $i=1,2,\dots ,{N}_{sol}$. And find Gbest such that $f\left({P}_{fG}^{T}\right)$ is the best, and then let $t=1$ and $i=1$. |

STEP 2-4 | Generate $\rho $ and calculate ${u}_{j}$. |

STEP 2-5 | Generate r to update the ${X}_{i}^{Tt}$ and ${Y}_{i}^{Tt}$, and calculate $f\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$. |

STEP 2-6 | If $f\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$ > $f\left({P}_{fi}^{T}\right)$, then ${P}_{fi}^{T}=\left({X}_{i}^{Tt},{Y}_{i}^{Tt}\right)$; Otherwise, go to STEP 2-8. |

STEP 2-7 | If $f\left({P}_{fi}^{T}\right)$ > $f\left({P}_{fG}^{T}\right)$, then ${P}_{fG}^{T}={P}_{fi}^{T}.$ |

STEP 2-8 | If $i\le {N}_{sol}$ then $i=i+1$ and return to STEP 2-4. |

STEP 2-9 | If $t<{N}_{gen}$ then $t=t+1$ and $i=1$, and return to STEP 2-4. Otherwise, go to STEP 2-10. |

STEP 2-10 | Output ${P}_{fG}^{T}=\left({X}_{i}^{T},{Y}_{i}^{T}\right)$. |

**Subprogram 2: Level updating solving**

STEP 3-1 | $\mathrm{Initiate}{N}_{sol}$ for both levels, ${N}_{genl}$ (if updating with upper-level $l=1$; otherwise, $l=2$)$,{N}_{var}$$,{C}_{g}$,${C}_{p}$$,\mathrm{and}{C}_{w}$, and the upper and lower bounds of each variable. |

STEP 3-2 | $\mathrm{Set}{N}_{genl}t=0$$\mathrm{and}i=1$$,\mathrm{where}i=1,2,\dots ,{N}_{sol}$. |

STEP 3-3 | $\mathrm{Generate}{X}_{i}^{Tt}$$\mathrm{or}{Y}_{i}^{Tt}$$.\mathrm{Let}{P}_{Fi}^{T}=\left({X}_{i}^{Tt},{Y}_{i}^{T}\right)$$,{P}_{fi}^{T}=\left({X}_{i}^{T},{Y}_{i}^{Tt}\right)$$,\mathrm{and}\mathrm{calculate}F\left({P}_{Fi}^{T}\right)=F\left({X}_{i}^{Tt},{Y}_{i}^{T}\right),f\left({P}_{fi}^{T}\right)=f\left({X}_{i}^{T},{Y}_{i}^{Tt}\right)$$\mathrm{for}i=1,2,\dots ,{N}_{sol}$$.\mathrm{And}\mathrm{find}\mathrm{Gbest}\mathrm{such}\mathrm{that}F\left({P}_{FG}^{T}\right)$$\mathrm{or}f\left({P}_{fG}^{T}\right)$$\mathrm{is}\mathrm{the}\mathrm{best},\mathrm{and}\mathrm{then}\mathrm{let}t=1$$\mathrm{and}i=1$. |

STEP 3-4 | $\mathrm{Generate}\rho $$\mathrm{and}\mathrm{calculate}{u}_{j}$. |

STEP 3-5 | $\mathrm{Generate}r\mathrm{to}\mathrm{update}{X}_{i}^{Tt}$$\mathrm{and}{Y}_{i}^{Tt}$$\mathrm{and}\mathrm{calculate}F\left({X}_{i}^{Tt}\right)$$\mathrm{and}f\left({Y}_{i}^{Tt}\right)$. |

STEP 3-6 | $\mathrm{For}\mathrm{upper}-\mathrm{level}\mathrm{update},\mathrm{If}F\left({X}_{i}^{Tt},{Y}_{i}^{T}\right)$$F\left({P}_{Fi}^{T}\right)$$,\mathrm{then}{P}_{Fi}^{T}=\left({X}_{i}^{Tt},{Y}_{i}^{T}\right)$$;\mathrm{for}\mathrm{lower}-\mathrm{level}\mathrm{update},\mathrm{if}f\left({X}_{i}^{T},{Y}_{i}^{Tt}\right)$$f\left({P}_{fi}^{T}\right)$$,\mathrm{then}{P}_{fi}^{T}=\left({X}_{i}^{T},{Y}_{i}^{Tt}\right)$; Otherwise, go to STEP 3-8. |

STEP 3-7 | $\mathrm{For}\mathrm{upper}-\mathrm{level}\mathrm{update},\mathrm{if}F\left({P}_{Fi}^{T}\right)$$F\left({P}_{FG}^{T}\right)$$,\mathrm{then}{P}_{FG}^{T}={P}_{Fi}^{T}$$;\mathrm{for}\mathrm{lower}-\mathrm{level}\mathrm{update},\mathrm{if}f\left({P}_{fi}^{T}\right)$$f\left({P}_{fG}^{T}\right)$$,\mathrm{then}{P}_{fG}^{T}={P}_{fi}^{T}.$ |

STEP 3-8 | $\mathrm{If}i\le {N}_{sol}$$,\mathrm{then}i=i+1$ and return to STEP 3-4. |

STEP 3-9 | $\mathrm{If}t{N}_{genl}$$\mathrm{then}t=t+1$$\mathrm{and}i=1$, and return to STEP 3-4. Otherwise, stop. |

STEP 3-10 | $\mathrm{Output}{P}_{FG}^{T}or{P}_{fG}^{T}$. |

## 5. Data Analysis and Results

#### 5.1. Numerical Experiments

#### 5.1.1. Experimental Datasets

#### 5.1.2. Experiments with Orthogonal Arrays

**Dataset: Problem 1**

**Dataset: Problem 2**

**Dataset: Problem 3**

#### 5.1.3. Comparison Experiment Results

**Dataset: Problem 1**

**Dataset: Problem 2**

**Dataset: Problem 3**

#### 5.2. Model Evaluation

## 6. Conclusions

- (1)
- Hybridization of other heuristic mechanisms to improve MLiSSO solving
- (2)
- Consider the dynamical mechanism for adjusting the upper and lower terms in terms of the turbulence of the update mechanism to improve the generated solutions towards the desired optimal solution to improve the efficiency and quality of the solutions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Relationship between participants in the Stackelberg game. (

**a**) represents a single leader and follower, (

**b**) represents a single leader and multiple followers, (

**c**) represents multiple leaders and single followers, and (

**d**) represents multiple leaders with multiple followers.

Type | Symbol | Description |
---|---|---|

parameter | a | The total potential market size. |

$\lambda $ | $\mathrm{The}\mathrm{underlying}\mathrm{market}\mathrm{share}\mathrm{of}\mathrm{the}\mathrm{retailer}\mathrm{for}\mathrm{the}\mathrm{manufacturer}\mathrm{is}\left(1-\lambda \right)$$.0\le \lambda \le 1$. | |

b | $\mathrm{Demand}\mathrm{sensitivity}\mathrm{to}\mathrm{its}\mathrm{selling}/\mathrm{retail}\mathrm{price}.0b\le 1.$ | |

d | $\mathrm{The}\mathrm{coefficient}\mathrm{of}\mathrm{cross}-\mathrm{price}\mathrm{sensitivity}.0d\le 1.$ | |

c | Product production cost. | |

$\eta $ | $\mathrm{Revenue}\mathrm{sharing}\mathrm{of}3\mathrm{rd}\mathrm{party}\mathrm{platform}.0\eta \le 1.$ | |

$i$ | Finance strategy for: $i=\{\begin{array}{l}B,bankfinancestrategy.\\ T,3rdplatformfinancestrategy.\\ R,retailerfinancestrategy.\end{array}$ | |

variables | ${w}^{i}$ | $\mathrm{Wholesale}\mathrm{price},\mathrm{for}i=\{\begin{array}{c}\begin{array}{c}B\\ T\\ R\end{array}\\ None\end{array},{w}^{i}\ge 0$. |

${P}_{R}^{i}$ | $\mathrm{Retailer}\u2019\mathrm{s}\mathrm{retail}\mathrm{channel}\mathrm{retail}\mathrm{price},\mathrm{with}\mathrm{finance}\mathrm{strategy}\mathrm{for}i=B,TorR.$${P}_{R}^{i}\ge 0.$ | |

${P}_{M}^{i}$ | $\mathrm{Manufacturer}\u2019\mathrm{s}\mathrm{direct}\mathrm{channel}\mathrm{selling}\mathrm{price}\mathrm{through}3\mathrm{rd}\mathrm{party}\mathrm{platform},\mathrm{with}\mathrm{finance}\mathrm{strategy}\mathrm{for}i=B,TorR.$${P}_{M}^{i}\ge 0.$ | |

${q}_{R}^{i}$ | $\mathrm{Retail}\mathrm{channel}\mathrm{demand},\mathrm{with}\mathrm{finance}\mathrm{strategy}\mathrm{for}i=B,TorR.$ | |

${q}_{M}^{i}$ | $\mathrm{Direct}\mathrm{channel}\mathrm{demand},\mathrm{with}\mathrm{finance}\mathrm{strategy}\mathrm{for}i=B,TorR.$ | |

${r}^{i}$ | $\mathrm{Revenue}\mathrm{sharing}\mathrm{rate},\mathrm{with}\mathrm{finance}\mathrm{strategy}\mathrm{for}i=B,TorR.0{r}^{i}\le 1.$ |

Formula | Description | |
---|---|---|

Standard deviation (SD) | $\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{R}{\left({F}_{Mi}^{*}-{F}_{MA}^{*}\right)}^{2}}{R}}$ $\mathrm{where}i=1,2,\dots ,R$. R = 30 in this paper. | ${F}_{Mi}^{*}$= The optimal solution for MLiSSO in ith run. ${F}_{MA}^{*}$= The average of R optimal solutions for MLiSSO. |

No. | Problem Functions |
---|---|

Problem 1 [62] | $MaxF=8{x}_{1}+4{x}_{2}-4{y}_{1}+40{y}_{2}+4{y}_{3},$ $\mathrm{where}\left({y}_{1},{y}_{2},{y}_{3}\right)$ solves, $Maxf=-{x}_{1}-2{x}_{2}-{y}_{1}-{y}_{2}-2{y}_{3}$ s.t. ${y}_{1}-{y}_{2}-{y}_{3}\ge -1$ $-2{x}_{1}+{y}_{1}-2{y}_{2}+0.5{y}_{3}\ge -1$ $-2{x}_{2}-2{y}_{1}+{y}_{2}+0.5{y}_{3}\ge -1$ ${x}_{1},{x}_{2},{y}_{1},{y}_{2},{y}_{3}\ge 0$ |

No. | Problem Functions |
---|---|

Problem 2 [63] | $MinF=-{x}_{1}{}^{2}-3{x}_{2}{}^{2}-4{\mathrm{y}}_{1}+{y}_{2}{}^{2}$$,\mathrm{where}\left({y}_{1},{y}_{2}\right)$ solves, s.t. ${x}_{1}{}^{2}+2{x}_{2}\le 4$ ${x}_{1},{x}_{2}\ge 0$ $Minf=2{x}_{1}{}^{2}+{y}_{1}{}^{2}-5{y}_{2}$ s.t. ${x}_{1}{}^{2}-2{x}_{1}+{x}_{2}{}^{2}-2{y}_{1}+{y}_{2}\ge -3$ $4{x}_{2}+3{y}_{1}-4{y}_{2}\ge 4$ ${y}_{1},{y}_{2}\ge 0$ |

No. | Problem Functions |
---|---|

Problem 3 [6] | $MinF={x}^{2}+{\left(y-10\right)}^{2}$$,\mathrm{where}y$ solves, s.t. $x+2y-6\le 0,$ $-x\le 0$ $Minf={x}^{3}-2{y}^{3}+x-2y-{x}^{2}$ s.t. $-x+2y-3\le 0,$ $-y\le 0$ |

Level\Factor | $\mathbf{Parameter}\mathit{C}\mathit{p}$ | $\mathit{u}$ Value Setting |
---|---|---|

1 | Without | Constant |

2 | Add-in | Dynamic |

Setting\Factor | $\mathbf{Parameter}\mathit{C}\mathit{p}$ | $\mathit{u}$ Value Setting |
---|---|---|

1 | Without | Constant |

2 | Without | Dynamic |

3 | Add-in | Constant |

4 | Add-in | Dynamic |

Source | DF | SS | MS | F-Value | p-Value |
---|---|---|---|---|---|

A | 1 | 23.401 | 23.4015 | 4.53 | 0.036 |

B | 1 | 0.413 | 0.4126 | 0.08 | 0.778 |

Error | 76 | 392.193 | 5.1604 | ||

Total | 79 | 427.444 | 0.000 | ||

S = 2.27166 | R − Sq = 8.25% | R − Sq (adj) = 4.63% |

Level | A | B |
---|---|---|

1 | 27.5 | 27.0 |

2 | 28.0 | 26.0 |

Delta | 0.5 | 1.0 |

Rank | 2 | 1 |

Source | DF | SS | MS | F-Value | p-Value |
---|---|---|---|---|---|

A | 1 | 0.02050 | 0.020505 | 5.13 | 0.026 |

B | 1 | 0.04019 | 0.040187 | 10.06 | 0.002 |

Error | 76 | 0.30370 | 0.003996 | ||

Total | 79 | 0.45029 | 0.000000 | ||

S = 0.0632139 | R − Sq = 32.56% | R − Sq (adj) = 29.89% |

Level | A | B |
---|---|---|

1 | 18.59 | 18.63 |

2 | 18.625 | 18.585 |

Delta | 0.035 | 0.045 |

Rank | 2 | 1 |

Source | DF | SS | MS | F-Value | p-Value |
---|---|---|---|---|---|

A | 1 | 3.1344 | 3.1344 | 3.24 | 0.076 |

B | 1 | 0.6697 | 0.6697 | 0.69 | 0.408 |

Error | 76 | 73.4531 | 0.9665 | ||

Total | 79 | 78.8582 | 0.0000 | ||

S = 0.983102 | R−Sq = 6.85% | R−Sq (adj) = 3.18% |

Level | A | B |
---|---|---|

1 | −67.5 | −67.2 |

2 | −67.1 | −67.4 |

Delta | 0.4 | 0.2 |

Rank | 1 | 2 |

Setting | ${\mathit{F}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

1 | 27.1654863 | 2.7327978 | 2.7327978 | 17.3713556 |

2 | 26.2656556 | 3.0503393 | 3.0503393 | 20.0050241 |

3 | 27.4909892 | 1.7662205 | 1.7662205 | 21.7917118 |

4 | 28.1035510 | 0.8657080 | 0.8657080 | 26.4103750 |

Setting | ${\mathit{F}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

1 | 18.6020324 | 0.0710975 | 18.7231808 | 18.4959183 |

2 | 18.5813208 | 0.0616991 | 18.7136139 | 18.4883674 |

3 | 18.5685148 | 0.0636811 | 18.7202904 | 18.4420441 |

4 | 18.6788774 | 0.0553813 | 18.8334935 | 18.6205643 |

Setting | ${\mathit{F}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{d}\mathit{e}\mathit{v}}$ | ${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

1 | 67.24701323 | 0.977282509 | 68.93931587 | 65.6916659 |

2 | 67.71292941 | 1.031981155 | 69.28061716 | 65.54469136 |

3 | 67.13405382 | 0.969031058 | 69.17334034 | 65.33748003 |

4 | 67.03412608 | 0.952295648 | 67.03412608 | 65.23260349 |

GA [64] | PSO [11] | MLiSSO |
---|---|---|

opulation: 20, Crossover rate: 0.9, Mutation rate: 0.1, Iterations: N/A | Population: 20, Vmax: 10, Inertial weight: 1.2–0.2, Iterations: 150 | Population: 20, Cg: 0.3, Cp: 0.6, Cw: 0.8, Generations:100/150, Iterations: 500/150 |

GA | PSO | MLiSSO (500) | MLiSSO (As Literature/150) | |
---|---|---|---|---|

${x}_{1}$ | 0.000 | 0.0004 | 0.0002 | 0.0266 |

${x}_{2}$ | 0.898 | 0.8996 | 0.8991 | 0.0205 |

${y}_{1}$ | 0.000 | 0.0000 | 0.0000 | 0.7969 |

${y}_{2}$ | 0.599 | 0.5995 | 0.5993 | 0.7944 |

${y}_{3}$ | 0.399 | 0.3993 | 0.3986 | 0.1503 |

F | 29.1480 | 29.1788 | 29.6631 | 29.4853 |

$f$ | −3.193 | −3.1977 | −3.1948 | −1.9594 |

Runtime(s) | N/A | N/A | 35 | 32 |

GA | PSO | MLiSSO (500) | MLiSSO (As Literature/150) | |
---|---|---|---|---|

${x}_{1}$ | 0.15705 | 0.02192 | 0.00078 | 0.01579 |

${x}_{2}$ | 0.86495 | 0.86693 | 0.89607 | 0.18669 |

${y}_{1}$ | 0.00000 | 0.00000 | 0.00000 | 0.41225 |

${y}_{2}$ | 0.47192 | 0.56335 | 0.59701 | 0.66371 |

${y}_{3}$ | 0.51592 | 0.34108 | 0.39351 | 0.19149 |

F | 21.52948 | 24.81256 | 29.04494 | 26.53842 |

$f$ | −3.39072 | −3.1977 | −3.17696 | −1.84811 |

$F$ stdev | 3.14432 | 1.55374 | 0.10689 | 2.23245 |

Runtime(s) | N/A | N/A | 45 | 35 |

EA [65] | PSO-CST [66] | MLiSSO |
---|---|---|

Population: 30, Crossover rate: 0.8, Mutation rate: 0.2, Iterations: 100 | $\mathrm{Population}:45,\mathrm{Numbers}\mathrm{of}\mathrm{particles}:\mathrm{m}=40(\mathrm{first}\mathrm{update}),\mathrm{n}=5(\mathrm{CST}\mathrm{particles}),{V}_{max}=2$$,{c}_{1}={c}_{2}=2,$ Iterations: 8 | Population: 20, Cg: 0.2, Cp: 0.3, Cw: 0.5, Generations: 100, Iterations: 100 |

EA [65] | PSO-CST [66] | MLiSSO | |
---|---|---|---|

${x}_{1}$ | 0.00000044 | 0.3844 | 0.0115 |

${x}_{2}$ | 2 | 1.6124 | 1.9765 |

${y}_{1}$ | 1.875 | 1.8690 | 1.8466 |

${y}_{2}$ | 0.9063 | 0.8041 | 0.7988 |

F | −12.68 | −14.7772 | −18.4633 |

$f$ | −1.016 | −0.2316 | −6.1174 |

$F$ stdev | N/A | N/A | 0.1396 |

$F$ avg | N/A | N/A | −18.4566 |

Runtime(s) | N/A | N/A | 5 |

HPSOBLP [67] | IBPSO [6] | MLiSSO |
---|---|---|

$\mathrm{Population}:{N}_{max}=$$20,40,{c}_{1}={c}_{2}=2$$,{V}_{max}=$$\mathrm{bounds},w=$ decrease linearly from 1.2 to 0.1, Iterations: 120, 30 | $\mathrm{Population}:{N}_{1}={N}_{2}=$$20,{V}_{max}=10$$,{c}_{1}={c}_{2}=2$$.5,\mathrm{Iteration}:{T}_{1}={T}_{2}=$100 | Population: 20, Cg: 0.2, Cp: 0.3, Cw: 0.5, Generations: 100, Iterations: 100 |

HPSOBLP | IBPSO | MLiSSO | |
---|---|---|---|

$x$ | N/A | 0.4960 | 1.0186 |

$y$ | N/A | 1.7356 | 1.9753 |

F | 88.77571 | 68.5459 | 65.8663 |

$f$ | −0.7698 | −13.5561 | −18.9133 |

Runtime(s) | N/A | N/A | 5 |

HPSOBLP | IBPSO | MLiSSO | |
---|---|---|---|

$x$ | N/A | 1.1985 | 1.1036 |

$y$ | N/A | 1.7791 | 1.8756 |

F | 88.7835 | 69.0192 | 67.4949 |

$f$ | N/A | −13.3375 | −15.8649 |

$F$ stdev | 0.0016 | N/A | 1.0366 |

Runtime(s) | N/A | N/A | 5 |

Parameter | a | λ | b | d | c | η |
---|---|---|---|---|---|---|

Setup | 1 | 0.4 | 1 | 0.5 | 0.4 | 0.15 |

MLiSSO |
---|

Population: 20 Cg: 0.2 Cp: 0.3 Cw: 0.5 Generations:100 Iterations: 500 |

RF | BF | 3PF | |
---|---|---|---|

$w$ | 0.64952 | 0.40506 | 0.40506 |

${P}_{M}$ | 0.83284 | 0.70765 | 0.70765 |

${P}_{R}$ | 0.71113 | 0.62951 | 0.62951 |

${q}_{M}$ | 0.10529 | 0.12432 | 0.12432 |

${q}_{R}$ | 0.12272 | 0.20710 | 0.20710 |

r | 0.34810 | 0.01265 | 0.01265 |

${f}_{1}$ | 0.03231 | 0.04068 | 0.04068 |

${f}_{2}$ | 0.00756 | 0.00284 | 0.00284 |

${f}_{3}$ | N/A | 0.00168 | 0.01487 |

RF | BF | 3PF | |
---|---|---|---|

$w$ | 0.73053 | 0.40495 | 0.40495 |

${P}_{M}$ | 0.88513 | 0.71293 | 0.71293 |

${P}_{R}$ | 0.76135 | 0.61588 | 0.61588 |

${q}_{M}$ | 0.08122 | 0.14058 | 0.14058 |

${q}_{R}$ | 0.09554 | 0.19501 | 0.19501 |

r | 0.53832 | 0.01237 | 0.01237 |

${f}_{1}$ | 0.02197 | 0.03898 | 0.03898 |

${f}_{2}$ | 0.00355 | 0.00271 | 0.00271 |

${f}_{3}$ | N/A | 0.00161 | 0.01664 |

${f}_{1}$ stdev | 0.00524 | 0.00068 | 0.00068 |

${f}_{2}$ stdev | 0.00434 | 0.00265 | 0.00265 |

${f}_{3}$ stdev | N/A | 0.00153 | 0.00106 |

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## Share and Cite

**MDPI and ACS Style**

Yeh, W.-C.; Liu, Z.; Yang, Y.-C.; Tan, S.-Y.
Solving Dual-Channel Supply Chain Pricing Strategy Problem with Multi-Level Programming Based on Improved Simplified Swarm Optimization. *Technologies* **2022**, *10*, 73.
https://doi.org/10.3390/technologies10030073

**AMA Style**

Yeh W-C, Liu Z, Yang Y-C, Tan S-Y.
Solving Dual-Channel Supply Chain Pricing Strategy Problem with Multi-Level Programming Based on Improved Simplified Swarm Optimization. *Technologies*. 2022; 10(3):73.
https://doi.org/10.3390/technologies10030073

**Chicago/Turabian Style**

Yeh, Wei-Chang, Zhenyao Liu, Yu-Cheng Yang, and Shi-Yi Tan.
2022. "Solving Dual-Channel Supply Chain Pricing Strategy Problem with Multi-Level Programming Based on Improved Simplified Swarm Optimization" *Technologies* 10, no. 3: 73.
https://doi.org/10.3390/technologies10030073