A multi-objective production planning optimization method with uncertain demand and delay costs

Abstract

This invention discloses a multi-objective production planning optimization method with uncertain demand and delay costs. Based on existing production planning, it analyzes uncertainties caused by internal and external factors, considering uncertainties including product demand and late delivery penalty cost coefficients. By constructing multiple constraints such as inventory, materials, and capacity, it establishes a production planning model with the objectives of minimizing delay costs and minimizing inventory costs, addressing the uncertainty of demand and delay costs. The quantum behavior particle swarm optimization algorithm is used to effectively solve the multi-objective optimization problem, improving the accuracy and robustness of the production plan. To overcome the problem that master production scheduling is affected by uncertainties and lacks optimization mechanisms due to material and production capacity limitations, compared with existing methods, this invention's method has advantages such as high solution accuracy and strong robustness, making it very suitable for solving complex function optimization problems under uncertain conditions.

Description

Technical Field

[0001] This invention relates to the field of production planning and control technology for manufacturing enterprises, and in particular to a multi-objective production planning optimization method with uncertain demand and delay costs. Background Art

[0002] Master Production Scheduling (MPS) plays a crucial role in modern manufacturing enterprises. It is not only a core component of Enterprise Resource Planning (ERP) systems but also serves as a bridge between market demand and production capacity, effectively coordinating production processes and resource allocation. With increasingly dynamic market demands, traditional MPS methods face numerous challenges, such as demand fluctuations, insufficient production capacity, and material shortages. Therefore, developing an efficient and flexible MPS methodology is of great significance for improving production response speed, optimizing resource utilization, and enhancing enterprise competitiveness.

[0003] Traditional master production scheduling methods suffer from two main shortcomings: First, most master production schedules are designed for production under deterministic conditions, relying heavily on experience or mathematical methods and employing a trial-and-error approach, neglecting potential uncertainties in the production process. In actual production, the manufacturing system must consider both external uncertainties related to product demand and raw material supply, as well as fluctuations in internal production conditions. Second, they fail to account for capacity constraints. The planning process is initially based on the assumption of unlimited capacity, requiring repeated coordination and balancing between the production plan and actual capacity. Lacking optimization tools, these methods are ineffective in comprehensively resolving capacity conflicts when there are numerous production tasks and severe capacity conflicts. Summary of the Invention

[0004] To address the aforementioned technical problems, the present invention aims to provide a multi-objective production planning optimization method with uncertainties in demand and delay costs. This method not only considers the impact of uncertainties caused by internal and external factors on production planning, but also uses materials and capacity as constraints in the production planning model, and delay costs and inventory costs as objectives, thereby improving the accuracy and robustness of production planning.

[0005] The technical solution adopted by this invention to solve the above-mentioned technical problems is: a multi-objective production planning optimization method with uncertain demand and delay costs, specifically including the following steps:

[0006] Step 1: Construct a multi-objective production planning model based on uncertain product demand and delay costs, specifically as follows:

[0007] Step 1.1: Define the uncertain variable;

[0008] Product demand and delay costs are set as uncertain variables. Product demand is defined as an uncertain set, and the delay penalty cost coefficient is expressed as a random variable and follows a normal distribution.

[0009] Step 1.2: Define the objective function:

[0010] The model objective is to minimize the total cost during the production process, which includes both deferred costs and inventory costs.

[0011] Step 1.3: Set model constraints;

[0012] Step 1.3a: Set the material constraints as follows: neither intermediate materials nor raw materials can be out of stock;

[0013] Step 1.3b: Set the capacity constraint as follows: the planned output of the product in each period cannot exceed the available normal production capacity for that period;

[0014] Step 1.3c: Set the product inventory constraint as follows: the ending inventory of the product in each period must not be less than the safety stock and must not exceed the maximum allowable inventory.

[0015] Step 1.3d: Set non-negativity and integer constraints for variables;

[0016] Step 1.4: Analyze the uncertainties caused by the internal and external factors of the system, including uncertainties such as product demand and late delivery penalty cost coefficient, and obtain a multi-objective production planning model with the objectives of minimizing late delivery costs and minimizing inventory costs under multiple constraints of non-negative and integer product materials, capacity, inventory and planned output.

[0017] Step 2: Transform the uncertain multi-objective production planning model into a deterministic model. The uncertain part, defined as an uncertain set, is optimized using a robust optimization method, while the part defined as random variables is optimized using a chance-constrained programming method. The model constructed in Step 1 is then transformed into an equivalent deterministic model using the integral method.

[0018] Step 3: Solve the determined model using the quantum behavior particle swarm optimization algorithm to obtain an optimized production plan. Specifically, the encoding method is set so that one particle represents a production operation plan scheme, the dimension represents the number of planning cycles, the position information of each dimension represents the planned output of each cycle, and the overdue cost and inventory cost are used as the fitness function of the quantum behavior particle swarm optimization algorithm to solve the multi-objective production planning model.

[0019] Step 3.1: Set the number of particles and dimension, initialize the population according to the set number of particles and dimension, set the number of iterations G, and set the upper and lower bounds of the search space;

[0020] Step 3.2: Calculate the fitness value of each particle, i.e., the objective function value Z. The optimal position Pn for an individual is the individual with the best fitness value among the initial positions X. The optimal position Gn for the population is the individual with the best fitness value among all particles.

[0021] Step 3.3: Update the position and velocity of the particle using a quantum random walk. The updated position must satisfy the upper and lower bounds of the search space.

[0022] Step 3.4: Calculate the fitness value Z of each particle after the update. i Update the individual optimal position and the group optimal position;

[0023] Step 3.5: Iterative optimization. Determine if the current iteration is the maximum number of iterations. If so, output the optimal solution set and obtain the optimized individual; otherwise, continue iterating and repeat steps 3.2 to 3.4 for the new population.

[0024] Step 3.6: Output the optimized individual to obtain the optimal production plan.

[0025] Specifically, in step 1.1

[0026] Define an uncertain variable:

[0027] Step 1.1a: For uncertain product demand Assumption Belongs to the following uncertain set U t

[0028]

[0029] in, Let θ be the median of the demand interval for the product during period t, and R be the set of product demands; t It is the level of demand uncertainty, i.e., the maximum percentage deviation of actual demand from the median, θ t The larger the value, the higher the level of demand uncertainty.

[0030] Step 1.1b: Delay Penalty Cost Coefficient Expressed as a random variable and following a normal distribution, where, This is the penalty cost coefficient for a unit quantity of products failing to meet demand per unit period. Parameters marked with "~" are uncertain parameters.

[0031] In step 1.2

[0032] The objective function of the model is

[0033] min z = (z1, z2)

[0034] The model aims to minimize the total cost z, which includes delinquency costs z1 and inventory costs z2. Their respective calculation formulas are as follows:

[0035]

[0036] Where: t is the planning period, and T is the total set of planning periods. S is the penalty cost coefficient for unmet demand per unit quantity of products per unit period. t This represents the quantity of unmet demand for products in period t.

[0037]

[0038] Where: H is the inventory holding cost coefficient per unit quantity of product per unit period, GI t Let be the average inventory quantity of the product in period t, and its value is equal to the initial inventory quantity BI of the product in period t. t Ending inventory quantity EI t Half of the sum;

[0039] In step 1.3;

[0040] Step 1.3a: Establish material constraints: During the production process, neither intermediate materials nor raw materials can be out of stock.

[0041]

[0042] Among them, BI mt Let R be the initial inventory quantity of intermediate material m in period t. mt Let BI be the quantity of intermediate material m required in period t. nt Let R be the initial inventory quantity of raw material n in period t. nt Let M be the quantity of raw material n required in period t, M be the set of intermediate materials, and RM be the set of raw materials.

[0043] Step 1.3b: Establish capacity constraints. The planned output for each period cannot exceed the available normal production capacity, i.e., the planned output Q arranged in period t. t It must be less than or equal to the available normal production capacity C for the current period. t .

[0044] Step 1.3c: Establish product inventory constraints. The ending inventory of each product must be greater than or equal to the safety stock and less than or equal to the maximum allowable inventory.

[0045]

[0046] Among them, SS t Let EI be the safety stock quantity of the product in period t. t Let MI be the ending inventory of the product in period t. tLet be the maximum allowable inventory quantity of the product in period t;

[0047] In the above formula: for the ending inventory quantity EI t When t=1, EI t = Initial inventory quantity of the product OI + Planned production quantity of the product Q t - Quantity required for the product When the value is less than or equal to zero, the ending inventory is zero; t>1, EI t =Initial inventory quantity of products in the current period BI t +Planned production quantity of products for the current period Q t - Quantity of products required in the current period - The cumulative number of products whose demand was not met in the previous period (S) t-1 When the value is less than or equal to zero, the ending inventory is zero.

[0048] Among them, the initial inventory quantity BI of the product in period t t When t=1, BI t = equals the given initial inventory quantity OI, when t>1, BI t =Ending inventory of products from the previous period (EI) t-1 ;

[0049] Step 1.3d: Establish non-negativity and integer constraints: the planned output Q t Quantity S of products not meeting demand t Initial inventory quantity BI t Ending inventory quantity EI t The required quantity R of intermediate material m mt The required quantity R of raw material n nt All must satisfy the non-negativity and integer constraints.

[0050] Compared with existing technologies, the advantages of this invention lie in its analysis of uncertainties caused by internal and external factors within the system, considering uncertainties including product demand and late delivery penalty cost coefficients. It constructs a production planning model with constraints such as materials, capacity, and inventory, aiming to minimize late delivery costs and inventory costs, while addressing the uncertainty of demand and late delivery costs. Furthermore, it utilizes the quantum behavior particle swarm optimization algorithm to effectively solve the multi-objective optimization problem, improving the accuracy and robustness of the production plan. To overcome the problem of master production scheduling being affected by uncertainties and lacking optimization mechanisms due to material and production capacity limitations, this invention offers advantages such as high solution accuracy and strong robustness compared to existing methods, making it highly suitable for solving complex function optimization problems under uncertain conditions. Attached Figure Description

[0051] Figure 1 This is a flowchart of the multi-objective production planning optimization method in the embodiments of this specification;

[0052] Figure 2 This is a flowchart of the quantum behavior particle swarm algorithm in the embodiments of this specification. Detailed Implementation

[0053] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings and examples.

[0054] like Figure 1 As shown, this invention proposes a multi-objective production planning optimization method with uncertain demand and delay costs, mainly comprising the following steps:

[0055] Step 1: Construct a multi-objective production planning model based on uncertain product demand and delay costs, specifically as follows:

[0056] Step 1.1: Define the uncertain variable;

[0057] Step 1.1a: For uncertain product demand Assumption Belongs to the following uncertain set U t

[0058]

[0059] Let θ be the median of the demand interval for the product during period t, and R be the set of product demands; t It is the level of demand uncertainty, i.e., the maximum percentage deviation of actual demand from the median, θ t The larger the value, the higher the level of demand uncertainty.

[0060] Step 1.1b: Delay Penalty Cost Coefficient It is expressed as a random variable and follows a normal distribution; The penalty cost coefficient for unmet demand per unit quantity of product per unit period. Parameters marked with "~" are uncertain parameters.

[0061] Step 1.2: Define the model objective as minimizing total cost, including deferred costs and inventory costs:

[0062] The objective function of the model is

[0063] min z = (z1, z2)

[0064] The goal of the model is to minimize the total cost z, which includes the delinquency cost z1 and the inventory cost z2.

[0065] in,

[0066]

[0067] Where: t is the planning period, and T is the total set of planning periods. S is the penalty cost coefficient for unmet demand per unit quantity of products per unit period. t This represents the quantity of unmet demand for products in period t.

[0068]

[0069] Where: H is the inventory holding cost coefficient per unit quantity of product per unit period, GI t Let be the average inventory quantity of the product in period t, and its value is equal to the initial inventory quantity BI of the product in period t. t Ending inventory quantity EI t Half of the sum;

[0070] Step 1.3: Set constraints;

[0071] Step 1.3a: Establish material constraints: During the production process, neither intermediate materials nor raw materials can be out of stock.

[0072]

[0073] Among them, BI mt Let R be the initial inventory quantity of intermediate material m in period t. mt Let BI be the quantity of intermediate material m required in period t. nt Let R be the initial inventory quantity of raw material n in period t. nt Let M be the quantity of raw material n required in period t, M be the set of intermediate materials, and RM be the set of raw materials.

[0074] Where: the demand quantity R of intermediate material m in period t mt =All intermediate materials at t+LT m The planned completion quantity for the period The quantity B of intermediate materials m required per unit of product production m The sum of products; LT m The minimum production lead time for intermediate material m required to produce the product.

[0075] The quantity of raw material n required in period t is R. nt =All raw materials at t+LT n The planned completion quantity for the period The quantity B of raw materials n required to produce one unit of product n The sum of their products;

[0076] Step 1.3b: Establish capacity constraints. The planned output for each period cannot exceed the available normal production capacity, i.e., the planned output Q arranged in period t. t ≤Current available normal production capacity Ct .

[0077] Step 1.3c: Establish product inventory constraints. The ending inventory of each product must be greater than or equal to the safety stock and less than or equal to the maximum allowable inventory.

[0078]

[0079] Among them, SS t Let EI be the safety stock quantity of the product in period t. t Let MI be the ending inventory of the product in period t. t Let be the maximum allowable inventory quantity of the product in period t;

[0080] In the above formula: for the ending inventory quantity EI t When t=1, EI t = Initial inventory quantity of the product OI + Planned production quantity of the product Q t - Quantity required for the product When the value is less than or equal to zero, the ending inventory is zero; t>1, EI t =Initial inventory quantity of products in the current period BI t +Planned production quantity of products for the current period Q t - Quantity of products required in the current period - The cumulative number of products whose demand was not met in the previous period (S) t-1 When the value is less than or equal to zero, the ending inventory quantity EI t is zero.

[0081] The cumulative number S of products whose demand was not met in period t. t The calculation is as follows:

[0082] At t=1, the cumulative number S of products whose demand has not been met t =Product demand quantity - Planned production quantity Q t - The initial inventory quantity OI of a product. When this value is less than or equal to zero, the cumulative quantity of products with unmet demand at the beginning of the planning period is zero, indicating that there are no products with unmet demand; when t>1, the cumulative quantity S of products with unmet demand. t = Cumulative number of products whose demand was not met in the previous period (S) t-1 +Quantity of products needed this period - Planned production quantity Q for this period t - Initial inventory quantity of products in this period (BI) t When this value is less than or equal to zero, the cumulative number of products with unmet demand is zero, indicating that there are no products with unmet demand.

[0083] Among them, the initial inventory quantity BI of the product in period t tWhen t=1, BI t = equals the given initial inventory quantity OI, when t>1, BI t =Ending inventory of products from the previous period (EI) t-1 ;

[0084] Step 1.3d: Establish non-negativity and integer constraints: the planned output Q t Quantity S of products not meeting demand t Initial inventory quantity BI t Ending inventory quantity EI t The required quantity R of intermediate material m mt The required quantity R of raw material n nt All must satisfy the non-negativity and integer constraints.

[0085] Step 1.4: Analyze the uncertainties caused by internal and external factors of the system, including uncertainties such as product demand and late delivery penalty cost coefficient, and obtain a multi-objective production planning model with the objectives of minimizing late delivery costs and minimizing inventory costs under multiple constraints of non-negative and integer product materials, capacity, inventory and planned output.

[0086] Step 2: Transform the uncertain multi-objective production planning model into a deterministic model. The uncertain part, defined as an uncertain set, is optimized using a robust optimization method, while the part defined as random variables is converted using a chance-constrained programming method, thus transforming the model constructed in Step 1 into a deterministic model.

[0087] Step 2.1: Opportunity-Constrained Programming Transformation: In the production planning model constructed above, the uncertain variables... The random variable z1 is expressed as a random variable that follows a normal distribution and is included in the objective function. The chance-constrained programming method is used to express the model as a stochastic programming model, and then further transformed into an equivalent deterministic model through integration, that is, the random variable is transformed into a deterministic variable.

[0088] In the optimization model, the deterministic part of the objective function remains unchanged, while the part containing uncertainty coefficients... This can be expressed as a chance-constrained programming problem in the following form:

[0089]

[0090] Where Pr{·} is the probability calculation function, and α1∈[0,1] represents the confidence level of the objective function z1. The larger the value, the smaller the risk of violation. Correspondingly, a larger value needs to be taken for z1, which increases the cost, but the optimization result will be more reliable.

[0091] Assuming that all random variables in the model follow a normal distribution, the chance-constrained programming model is transformed into a deterministic model using integration. Based on the probability distribution function of each random variable, its corresponding equivalent deterministic variable is obtained, and indirect integration is used. Through model transformation, it can be equivalently converted to:

[0092]

[0093] Where E() and σ() represent the expected value and standard deviation of the random variable, respectively. It represents the inverse function of the probability distribution function of a random variable. This indicates the confidence level of the above inequality constraints.

[0094] Step 2.2: Robust conversion optimization: For uncertain product demand The original optimization problem is transformed into a robust equivalence model. To ensure that the constraints are met even in the worst case, the ending inventory quantity EI of the product in period t is included in the constraints. t Transform into robust form:

[0095]

[0096] Quantity S out of stock t Transform into robust form:

[0097]

[0098] Step 3: Solve the determined model using the quantum behavior particle swarm optimization algorithm to obtain an optimized production plan. Specifically, the encoding method is set so that one particle represents a production operation plan scheme, the dimension represents the number of planning cycles, and the position information of each dimension represents the planned output for each cycle. The delay cost and inventory cost are used as the fitness function of the quantum behavior particle swarm optimization algorithm to solve the multi-objective production planning model. Specifically, as follows... Figure 2 As shown.

[0099] Step 3.1: Set the number of particles and dimension, initialize the population according to the set number of particles and dimension, set the number of iterations G, and set the upper and lower bounds of the search space;

[0100] Step 3.2: Calculate the fitness value of each particle, i.e., the objective function value Z. The optimal position Pn for an individual is the individual with the best fitness value among the initial positions X. The optimal position Gn for the population is the individual with the best fitness value among all particles.

[0101] Step 3.3: Update the position and velocity of the particle using a quantum random walk. The updated position must satisfy the upper and lower bounds of the search space.

[0102] Step 3.4: Calculate the fitness value Z of each particle after the update. i Update the individual optimal position and the group optimal position;

[0103] Step 3.5: Iterative optimization. Determine if the current iteration is the maximum number of iterations. If so, output the optimal solution set and obtain the optimized individual; otherwise, continue iterating and repeat steps 3.2 to 3.4 for the new population.

[0104] Step 3.6: Output the optimized individual to obtain the optimal production plan.

[0105] Compared with existing technologies, the advantages of this invention lie in its analysis of uncertainties caused by internal and external factors within the system, considering uncertainties including product demand and late delivery penalty cost coefficients. It constructs a production planning model with constraints such as materials, capacity, and inventory, aiming to minimize late delivery costs and inventory costs, while addressing the uncertainty of demand and late delivery costs. Furthermore, it utilizes the quantum behavior particle swarm optimization algorithm to effectively solve the multi-objective optimization problem, improving the accuracy and robustness of the production plan. To overcome the problem of master production scheduling being affected by uncertainties and lacking optimization mechanisms due to material and production capacity limitations, this invention offers advantages such as high solution accuracy and strong robustness compared to existing methods, making it highly suitable for solving complex function optimization problems under uncertain conditions.

[0106] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims of this application.

[0107] The above are merely specific embodiments of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A multi-objective production planning optimization method with uncertain demand and delay costs, characterized in that, Specifically, the following steps are included: Step 1: Construct a multi-objective production planning model based on uncertain product demand and delay costs; Step 2: Transform the uncertain multi-objective production planning model into a deterministic model; Step 3: Solve the determined model based on the quantum behavior particle swarm optimization algorithm to obtain the optimized production plan; Step 1 includes the following steps: Step 1.1: Define the uncertain variables, which include the uncertain product demand set. and the cost coefficient of penalties for delays Step 1.2: Define the objective function of the uncertain multi-objective production planning model; Step 1.3: Set model constraints; Step 1.4: Based on the aforementioned uncertain variables, objective function, and constraints, obtain a multi-objective production planning model for demand and uncertainties of delay costs, under multiple constraints of non-negative and integer factors related to product materials, capacity, inventory, and planned output; The definition of the uncertain variable in step 1.1 includes the following steps: Step 1.1a: The set of uncertain product demands Belongs to the following uncertain set U t : in, Let θ be the median of the demand interval for the product during period t, and R be the set of product demands; t It is the level of demand uncertainty, i.e., the maximum percentage deviation of actual demand from the median, θ t The larger the value, the higher the level of demand uncertainty; Step 1.1b: The delay penalty cost coefficient Expressed as a random variable and following a normal distribution, where, The penalty cost coefficient for unmet demand per unit quantity of products per unit period; parameters marked with "~" are uncertain parameters. The objective function of the model in step 1.2 is: min z = (z1, z2) The goal of the model is to minimize the total cost z, which includes the delinquency cost z1 and the inventory cost z2. in, In the formula: t is the planning period, and T is the total set of planning periods. S is the penalty cost coefficient for unmet demand per unit quantity of products per unit period. t This represents the quantity of unmet demand for products in period t. In the formula: H is the inventory occupancy cost coefficient per unit quantity of product per unit period, GI t Let be the average inventory quantity of the product in period t, and its value is equal to the initial inventory quantity BI of the product in period t. t Ending inventory quantity EI t Half of the sum; Step 2 specifically includes: Step 2.1: Opportunity-Constrained Programming Transformation: In the production planning model constructed in Step 1 above, the delay penalty cost coefficient... The random variable z1 is expressed as a random variable that follows a normal distribution and is included in the objective function. The chance-constrained programming method is used to express the model as a stochastic programming model, and then further transformed into an equivalent deterministic model through integration, that is, the random variable is transformed into a deterministic variable. In the optimization model, the deterministic part of the objective function remains unchanged, while the part containing uncertainty coefficients... This can be expressed as a chance-constrained programming problem in the following form: Where Pr{·} is the probability calculation function, and α1∈[0,1] represents the confidence level of the objective function z1. The larger the value, the smaller the risk of violation. Correspondingly, a larger value needs to be taken for z1, which increases the cost, but the optimization result will be more reliable. Assuming that all random variables in the model follow a normal distribution, the chance-constrained programming model is transformed into a deterministic model using integration. Based on the probability distribution functions of each random variable, their corresponding equivalent deterministic variables are obtained, and indirect integration is used. Through this model transformation, it can be equivalently converted to: Where E() and σ() represent the expected value and standard deviation of the random variable, respectively. It represents the inverse function of the probability distribution function of a random variable. This indicates the confidence level of the above inequality constraints; Step 2.2: Robust conversion optimization: For uncertain product demand The original optimization problem is transformed into a robust equivalence model. To ensure that the constraints are met even in the worst case, the ending inventory quantity EI of the product in period t is included in the constraints. t Transform into robust form: Quantity S out of stock t Transform into robust form:

2. The multi-objective production planning optimization method with uncertain demand and delay costs as described in claim 1, characterized in that, Step 1.3 specifically includes: Step 1.3a: Set the material constraints as follows: neither intermediate materials nor raw materials can be out of stock; Step 1.3b: Set the capacity constraint as follows: the planned output of the product in each period cannot exceed the available normal production capacity for that period; Step 1.3c: Set the product inventory constraint as follows: the ending inventory of the product in each period must not be less than the safety stock and must not exceed the maximum allowable inventory. Step 1.3d: Set non-negativity and integer constraints for variables.

3. The multi-objective production planning optimization method with uncertain demand and delay costs as described in claim 2, characterized in that, Step 1.3a specifically includes: Among them, BI mt Let R be the initial inventory quantity of intermediate material m in period t. mt Let BI be the quantity of intermediate material m required in period t. nt Let R be the initial inventory quantity of raw material n in period t. nt Let M be the quantity of raw material n required in period t, M be the set of intermediate materials, and RM be the set of raw materials.

4. The multi-objective production planning optimization method with uncertain demand and delay costs as described in claim 3, characterized in that, Step 1.3c specifically includes: Among them, SS t Let EI be the safety stock quantity of the product in period t. t Let MI be the ending inventory of the product in period t. t Let be the maximum allowable inventory quantity of the product in period t; In the above formula: for the ending inventory quantity EI t When t=1, EI t = Initial inventory quantity OI of the product + Planned production quantity Q of the product t - Quantity required for the product When the value is less than or equal to zero, the ending inventory is zero; t>1, EI t =Initial inventory quantity of products in the current period BI t +Planned production quantity of products for the current period Q t - Quantity of products required in the current period - The cumulative number of products whose demand was not met in the previous period (S) t-1 When the value is less than or equal to zero, the ending inventory is zero; Wherein, the initial inventory quantity BI of the product in period t. t The value is: when t=1, BI t The initial value OI is equal to the given value; when t>1, BI t Equal to the ending inventory of the previous period, EI t-1 .

5. The multi-objective production planning optimization method with uncertain demand and delay costs as described in claim 1, characterized in that, Step 3 specifically includes the following steps: Step 3.1: Set the number of particles and dimension, initialize the population according to the set number of particles and dimension, set the number of iterations G, and set the upper and lower bounds of the search space; Step 3.2: Calculate the fitness value of each particle, i.e., the objective function value Z. The individual optimal position Pn is the individual with the best fitness value among the initial positions X; the group optimal position Gn is the individual with the best fitness value among all particles. Step 3.3: Update the position and velocity of the particle using a quantum random walk. The updated position must satisfy the upper and lower bounds of the search space. Step 3.4: Calculate the fitness value Z of each particle after the update. i Update the individual optimal position and the group optimal position; Step 3.5: Iterative optimization. Determine if the current iteration is the maximum number of iterations. If so, output the optimal solution set to obtain the optimized individual. Otherwise, continue iterating, repeating steps 3.2 to 3.4 for the new population; Step 3.6: Output the optimized individual to obtain the optimal production plan.

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