# Optimization and Simulation of Dynamic Performance of Production–Inventory Systems with Multivariable Controls

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## Abstract

**:**

## 1. Introduction

^{®}3 [3]. Either way, the production–inventory system would not be operating at a desired status for both profit generation and customer satisfaction.

## 2. Background Information of the Control Models for Production-Inventory Systems

#### 2.1. A Beief Review of the the APIOPBCS and 2APIOPBCS Models

- The forecasting mechanism is a feed-forward loop designed to provide the estimated average sales (AV
_{CONS}) and to set the desired work-in-process (WIP) level (D_{WIP}). CONS represents the sales or consumptions. The feed forward gain (${\tilde{T}}_{p}$) works as a safety factor to compensate the production delay and equals the production lead-time (T_{p}). The estimated average sales (AV_{CONS}) is commonly used to control the inventory steady-state error. Exponential smoothing with time constant (T_{a}) representing the average age of the data is a forecasting method commonly used to smooth the demand because of its simplicity and comprehensibility in mathematics for practitioners. The D_{WIP}is obtained from multiplying the AV_{CONS}by feed forward gain ${\tilde{T}}_{p}$. - The production lead time represents the total time required between placing an order and receiving the product as a finished item in the inventory. The controller designer cannot manipulate the lead time as it is considered as a characteristic of the system. The production lead time in the production–inventory control system is modelled as a first order lag with time constant T
_{p}that responds to a sudden change in the demand. - The controller strategy utilizes the forward and feedback information to generate a sophisticated decision to determine the manufacturing rate for the production–inventory system. In the APIOBPCS model, a production policy based on the pipeline output where the completion rate (COMRATE) is compared to the averaged demand AV
_{CONS}and their difference is fed back to the controller. T_{i}is an inventory order constant time for proportional control.

_{CONS}based on exponential smoothing forecasts with time constant T

_{a}is the forward control policy. The feedback consists of two control polices, the fraction 1/T

_{i}of the difference between the desired inventory D

_{inv}and the actual inventory A

_{inv}and the fraction 1/T

_{W}of the difference between the desired WIP D

_{WIP}and the actual WIP A

_{WIP}.

_{c}of the difference between the desired completion production rates D

_{COMP}and the actual completion production rates A

_{COMP}as an extra control for the production–inventory system. T

_{c}is a constant time for the completion rate (COMP) for proportional control.

#### 2.2. Mathematical Formulation of the Two APIOPBCS Models in the State Space

_{1}denotes the inventory level; x

_{2}is the items in the production process that are not finished yet (WIP); x

_{3}represents the consumption or sales state. The derivative state of the APIOBPCS model is:

_{inv}and the order rate ORATE defined by

#### 2.3. Performance Metrics

_{ar}) between the order rate and the consumption or the sales defined in Equation (12).

_{ar}index is used as a metric to measure bullwhip effect. In this criterion, there is zero bullwhip if V

_{ar}= 1; the system is amplified if V

_{ar}> 1; the system is smoothed if V

_{ar}< 1.

_{inv}level from the D

_{inv}level. The IAE measures positive and negative errors equally. A lower IAE indicates that the system has a better customer service level (CSL). The bullwhip effect and inventory responsiveness are two objectives that have direct impacts on the nature of the basic trade-offs between maintaining the order rates at the optimal performance, in order to avoid the impact of high amplification of orders, and maintaining stocks at a desired level to improve CSL.

## 3. Simulation Procedure and Experimental Considerations

#### 3.1. Simulation Procedure

_{i}, T

_{w}and T

_{a}), in which each set can achieve the best balance between the variance ratio (V

_{ar}) and the integral of absolute error (IAE), in other words, between the bullwhip effect (cost-effectiveness for the industry) and CSL or customer satisfaction. These sets of control parameters are usually presented as a Pareto optimality curve, in which the system manager can choose any desired set on the curve as the best control parameters.

_{inv}, which would help the manager to adjust the stock level near the optimal status.

#### 3.2. Experimental Considerations

- The period of physical production lead time is four units of time (T
_{p}= 4). Of course, this can be assigned to different values but it would not largely alter the general trend. - Backorders (negative inventory) are permitted.
- The desired inventory is set to zero (D
_{inv}= 0). - Day is the basic time unit in the model.
- The simulation was run for 180 days for each scenario.
- The production process can only produce a single unit at a time.

_{1}= {T

_{a}, T

_{i}, T

_{w}} for the APIOBPCS model. As the completion rate T

_{c}for the 2APIOBPCS model is in practice a certain value not to be optimized, it is simply added to the vector p

_{1}to form the control for the 2APIOBPCS model as p

_{2}= {T

_{a}, T

_{i}, T

_{w}, T

_{c}}.

- The maximum number of iterations was set to 100.
- The number of particles in the swarm was set to 50.
- The learning coefficients for local and global searches were both set to 2.
- The inertia weight was set as 0.6.
- The size of the archive was set to 20.

- Matched lead time means that the actual lead time and the estimated lead time are assumed to be matched during the operation, or in other words, the ordered amount of product should be delivered by the manufacturer to the retailer on time.
- Mismatched lead time means that there is a delay of the ordered product from the manufacturer to the retailer. This may be caused by machine breakdowns and/or material shortages to the manufacturer. In such a situation, a longer lead time is expected. The mismatched scenarios evaluate the robustness of the two models by measuring how the systems can recover from such disruptions and get back to the normal level. Such a simulation is represented by a lead time starting at the nominal value T
_{p}= 4, then to T_{p}= 6 for a period, and back to the normal T_{p}= 4. - Flexible production capacity means that the manufacturer has no problem to produce the ordered product on time. Even if there is a disruption during production, the manufacturer is able to mitigate the negative impact without delaying the delivery of the ordered product.
- Fixed production capacity means that there is a limit for the manufacturer to produce the ordered product within the timeframe. In a normal scenario, the ordered amount would match the top limit of the manufacturer’s capacity. However, the situation is prone to any disruption to the production caused by machine breakdowns and/or material shortages. In such a situation, when the production capacity in a period is insufficient to complete the production for an order, the capacity of the next period is used to continue the production of this order. The order in the affected period is capped by a constant C, i.e.,

## 4. Simulation Results and Discussion

_{1}= {T

_{a}, T

_{i}, T

_{w}} represents a Pareto curve of infinitely many combinations that can lead to optimal performance, and each combination can result in a set of simulation results. In this work, we selected three sets to simulate the two models separately (Table 1). These three selections correspond to three different ranges of the bullwhip effect as follows.

- Set 1: Bullwhip smoothing in range 0.8 < V
_{ar}< 1 - Set 2: Bullwhip avoidance where V
_{ar}= 1 - Set 3: Small bullwhip in range 1 < V
_{ar}< 1.3

_{p}) that are the same for the two models in the same range of bullwhip effect simulated in this study. As a result, the values of the control parameters for the two models in the same range are the same. The completion rate for the 2APIOBPCS model is chosen in proportion to each set accordingly.

#### 4.1. Case 1: Matched Lead Time with Flexible Production Capacity

#### 4.2. Case 2: Matched Lead Time with Fixed Production Capacity

_{ar}. As the system waits for the slow production, the actual IAE is the same to the normal scenario in Case 1. This is in line with the observation reported in [18].

#### 4.3. Case 3: Mismatched Lead Time with Flexible Production Capacity

#### 4.4. Case 4: Mismatched Lead Time with Fixed Production Capacity

_{ar}. As the system waits for the slow production, the actual IAE is the same to Case 3.

## 5. Conclusions

- Both models can produce the situational best possible balance between the order rate and inventory level under the same bullwhip effort if the production lead time is matched, regardless of the production capacity. However, the 2APIOBPCS model seemed able to improve the inventory responsiveness by a few percentages compared to the APIOBPCS model.
- The 2APIOBPCS model seemed able to improve the inventory responsiveness by more than 10% compared to the APIOBPCS model under the same bullwhip effect if the production lead time is mismatched.
- By imposing a constraint to the production capacity, the bullwhip effect for both models seemed reduced but the inventory responsiveness kept the same.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Block diagram of the automatic pipeline, inventory, and order based production control system (APIOBPCS) model.

**Figure 5.**Simulation of Case 1 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (

**top**) and inventory level (

**bottom**).

**Figure 6.**Simulation of Case 2 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (

**top**) and inventory level (

**bottom**).

**Figure 7.**Simulation of Case 3 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (

**top**) and inventory level (

**bottom**).

**Figure 8.**Simulation of Case 4 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (

**top**) and inventory level (

**bottom**).

Set | V_{ar} | T_{i} | T_{w} | T_{a} | T_{c} |
---|---|---|---|---|---|

Set 1 | 0.8–1.0 | 3.91 | 0.55 | 6.71 | 10 |

Set 2 | 1.0 | 5 | 1.13 | 5.66 | 21.9 |

Set 3 | 1.0–1.3 | 3.5 | 1.2 | 6 | 2 |

Matched Lead Time with Fixed Production Capacity | |||||
---|---|---|---|---|---|

APIOBPCS | 2APIOBPCS | ||||

V_{ar} | IAE | V_{ar} | IAE | IIR | |

0.85 | 3.57 × 10^{4} | 0.85 | 3.46 × 10^{4} | 3% | |

1.0 | 3.07 × 10^{4} | 1.0 | 2.97 × 10^{4} | 3% | |

1.25 | 2.84 × 10^{4} | 1.25 | 2.59 × 10^{4} | 9% |

Matched Lead Time with Fixed Production Capacity | ||||
---|---|---|---|---|

APIOBPCS | 2APIOBPCS | |||

V_{ar} | IAE | V_{ar} | IAE | IIR |

0.38 | 3.57 × 10^{4} | 0.38 | 3.46 × 10^{4} | 3% |

0.47 | 3.07 × 10^{4} | 0.45 | 2.97 × 10^{4} | 3% |

0.54 | 2.84 × 10^{4} | 0.54 | 2.59 × 10^{4} | 9% |

Mismatched Lead Time with Flexible Production Capacity | ||||
---|---|---|---|---|

APIOBPCS | 2APIOBPCS | |||

V_{ar} | IAE | V_{ar} | IAE | IIR |

0.85 | 7.60 × 10^{4} | 0.85 | 6.32 × 10^{4} | 17% |

1 | 5.56 × 10^{4} | 1 | 4.76 × 10^{4} | 14% |

1.23 | 4.44 × 10^{4} | 1.23 | 3.93 × 10^{4} | 12% |

Mismatched Lead Time with Fixed Production Capacity | ||||
---|---|---|---|---|

APIOBPCS | 2APIOBPCS | |||

V_{ar} | IAE | V_{ar} | IAE | IIR |

0.38 | 7.60 × 10^{4} | 0.38 | 6.32 × 10^{4} | 17% |

0.47 | 5.56 × 10^{4} | 0.45 | 4.76 × 10^{4} | 14% |

0.54 | 4.44 × 10^{4} | 0.54 | 3.93 × 10^{4} | 12% |

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**MDPI and ACS Style**

AL-Khazraji, H.; Cole, C.; Guo, W. Optimization and Simulation of Dynamic Performance of Production–Inventory Systems with Multivariable Controls. *Mathematics* **2021**, *9*, 568.
https://doi.org/10.3390/math9050568

**AMA Style**

AL-Khazraji H, Cole C, Guo W. Optimization and Simulation of Dynamic Performance of Production–Inventory Systems with Multivariable Controls. *Mathematics*. 2021; 9(5):568.
https://doi.org/10.3390/math9050568

**Chicago/Turabian Style**

AL-Khazraji, Huthaifa, Colin Cole, and William Guo. 2021. "Optimization and Simulation of Dynamic Performance of Production–Inventory Systems with Multivariable Controls" *Mathematics* 9, no. 5: 568.
https://doi.org/10.3390/math9050568