Optimization of Semi-Finished Inventory Management in Process Manufacturing: A Multi-Period Delayed Production Model

Abstract

This study investigates how process manufacturing enterprises can optimize semi-finished inventory (SFI) distribution in delayed production models, with particular attention to differences in cost volatility between single- and multi-period planning scenarios. To address this research gap, we develop a mixed-integer programming model that determines optimal customer order decoupling point (CODP)/product differentiation point (PDP) positions and SFI quantities (both generic and dedicated) for each production period, employing particle swarm optimization for solution derivation and validating findings through a comprehensive case study of a steel manufacturer with characteristic long-period production processes. The analysis yields two significant findings: (1) single-period operations demonstrate marked cost sensitivity to service level requirements and delay penalties, necessitating end-stage inventory buffers, and (2) multi-period optimization generates a distinctive cost-smoothing effect through strategic order deferrals and cross-period inventory reuse, resulting in remarkably stable total costs (≤2% variation observed). The study makes seminal theoretical contributions by revealing the convex cost sensitivity of short-term inventory decisions versus the near-flat cost trajectories achievable through multi-period planning, while establishing practical guidelines for process industries through its empirically validated two-period threshold for optimal order deferral and inventory positioning strategies.

1. Introduction

Steel companies commonly adopt multi-period long-process production techniques, where cyclical fluctuations significantly impact the delayed steel production model [1]. As a manufacturing approach that enables large-scale, multi-variety, and customized production, the delayed production model is increasingly being adopted by steel companies [2]. However, optimizing the delayed production strategy under multi-period conditions, particularly coordinating SFI management with the positioning of key decision points (CODP/PDP), remains a critical challenge.

Current quantitative research on delayed production models has primarily focused on discrete manufacturing industries, examining the optimal positioning of CODP [3,4,5] or PDP [6,7]. A limited number of studies have explored the impact of SFI on delayed production [8,9], yet most of these studies are confined to single-period scenarios [10,11], with a few addressing multi-period optimization [12,13]. Moreover, multi-period research often relies on probability-based simulations [14,15] or empirical enterprise data modeling [16], without fully accounting for the dynamic management of SFI and the synergistic effects across periods.

Existing research has yet to fully address the dynamic characteristics of process industries, the co-optimization of CODP, PDP, and SFI, as well as cross-period coordination mechanisms. To address these gaps, this study focuses on a typical process manufacturing industry (the steel sector) and develops a multi-period delayed production optimization model based on mixed-integer programming. The model dynamically determines CODP/PDP positioning and SFI allocation (generic vs. dedicated) while ensuring compliance with minimum service level requirements in each period and minimizing total costs across all periods. A particle swarm optimization algorithm is employed to solve the model, and the effectiveness of the optimization strategy is validated through a real-world case study of a steel company.

This study makes contributions by addressing key challenges in process industries, particularly steel manufacturing, through a tailored optimization framework. First, it introduces a novel multi-period production model that integrates CODP-PDP positioning with generic/dedicated SFI allocation, enabling joint cost-service trade-off optimization. Second, it advances multi-period dynamic modeling by explicitly accounting for SFI transitions and order propagation, empirically determining that a 2-period carryover policy optimizes dedicated SFI and order management. Finally, the study validates the model using real production and order data, with sensitivity analyses providing practical insights into service level adjustments and postponement penalty impacts, offering actionable guidance for inventory policy design in process industries.

The remainder of this paper is structured as follows: Section 2 reviews the relevant literature; Section 3 formulates the research problem and develops the quantitative optimization model; Section 4 presents the particle swarm optimization (PSO) algorithm for solving the model and demonstrates its application through a numerical case study; Section 5 conducts a multi-scenario analysis to discuss key insights; and Section 6 concludes the study.

2. Literature Review

2.1. Optimization Research on Delayed Production Mode

(1) Existing research on delayed production (the strategic postponement of customization in later stages of manufacturing, such as rolling or heat treatment, while maintaining continuous upstream production to optimize flexibility and inventory efficiency) and SFI management (intermediate inventories comprising generic stocks for multi-product sharing and dedicated stocks for rapid order fulfillment) can be categorized by manufacturing type (discrete vs. process) and research focus, yet collectively overlooks critical multi-period dynamics in process industries. Studies in discrete manufacturing either prioritize efficiency-driven CODP/PDP positioning—where CODP (the critical transition point from generic to order-specific production, shaping dedicated SFI storage and cost efficiency) determines customization timing, while PDP (where generic SFI buffers demand variability within product families, trading off flexibility against lead times) regulates postponement flexibility [17,18,19,20]—or emphasize customization agility [21,22,23,24]. However, their models assume discrete production stages, neglecting process-industry constraints like continuous material flows. For process manufacturing, research either develops CODP frameworks [25,26,27] or optimizes production planning [3], yet these treat SFI as static, ignoring cross-period cost volatility and inventory reuse potential—key gaps our study bridges via dynamic modeling validated in steel production.

2.2. Research on Delayed Production Models Based on Periodic Variations

Existing literature falls into three temporal categories: (1) Single-period studies optimize basic modes (e.g., Make-to-Order vs. Make-in-Advance efficiency [28] or hybrid system allocation [29] but ignore cross-period dynamics. (2) Multi-period impact studies model cyclical variations via statistical/simulation approaches [16,30,31] or address hybrid systems/CODP adjustments [21,32], yet rely on hypothetical parameters and underrepresent continuous production constraints. (3) Multi-period empirical works span production-inventory coordination [33], hierarchical planning [34], and defect control [35], but statically treat inventory—even in multi-item models [36,37]—neglecting SFI’s dynamic reuse and cost transmission. Key gaps persist: (1) single-period models miss long-term cost smoothing; (2) SFI’s (generic/specialized) cross-period reallocation is overlooked; (3) probabilistic models lack validation for continuous flows (e.g., steel). These limitations underscore this study’s potential in dynamic SFI allocation and cross-period cost optimization.

2.3. Research Gap

There are four main shortcomings in current research: (1) Narrow Scope of Application: Prior works predominantly target discrete manufacturing, neglecting process industries’ unique characteristics (e.g., continuous production, product variety, complex processes) in postponement strategy optimization. (2) Incomplete Decision Framework: Most literature examines either CODP or PDP positioning in isolation, with scant studies integrating both. Crucially, the interplay between positioning decisions and SFI levels remains underexplored. (3) Oversimplified Temporal Dynamics: Existing models are largely single-period or treat multi-period problems as repeated single period. Even limited multi-period studies fail to rigorously account for SFI carryover and cross-period order propagation effects. (4) Overly Simplistic Modeling: Prevailing approaches rely on restrictive assumptions (e.g., static costs, single objectives) or simulation-based methods, which inadequately capture real-world complexities like dynamic SFI redistribution or offer actionable optimization guidance.

3. Problem Analysis and Modeling

3.1. Problem Analysis

3.1.1. Process Manufacturing with Delayed Production Model

In the production of long-process steel enterprises, the blast furnace serves as the starting point of the product manufacturing process, where the molten iron produced represents the initial production capacity and the source for subsequent storage of semi-finished products at various stages.

The majority of the molten iron produced by the blast furnace directly enters different downstream production processes, including basic oxygen furnace steelmaking, hot rolling, coil dividing, leveling production, cold rolling, pickling, temper rolling, cutting, and various deep processing operations. The remaining capacity is stored at different production stages as either generic semi-finished inventory (PDP) or customized semi-finished inventory (CODP). These include: pig iron (PDP) stored from molten iron, steel billets (PDP) stored from liquid steel in the converter, hot-rolled coils (PDP) stored after hot rolling, divided coils (CODP) stored after coil dividing, leveled sheets (CODP) stored after leveling, and deep-processed materials (CODP) stored at various deep processing stages. These semi-finished inventories form the foundation for subsequent postponed production, enabling the response to market fluctuations, production variability, and the fulfillment of customized customer demand. The delayed production model in process manufacturing is illustrated in Figure 1.

3.1.2. Single- Period Optimization Analysis

In the case of single-period production, when a steel enterprise adopts a delayed production mode, it is necessary to simultaneously consider the impact of dedicated SFI (corresponding to CODP) and generic SFI (corresponding to PDP). Additionally, due to the diversity of production processes and product variations, it is essential to consider multiple CODPs and PDPs. The specific production process is illustrated in Figure 2.

The single- Period delayed production optimization minimizes total cost by optimizing PDP/CODP positions and inventory levels under two constraints: (1) meeting minimum service level requirements, and (2) staying within SFI capacity limits. The objective of building the model is to minimize the total cost ($\mathrm{C}={\mathrm{C}}_{\mathrm{h}}$ + ${\mathrm{C}}_{\mathrm{r}}$ + ${\mathrm{C}}_{\mathrm{d}}$), where: holding cost (${\mathrm{C}}_{\mathrm{h}}$) covers SFI in production & non-production stocks; backordering cost (${\mathrm{C}}_{\mathrm{r}}$) includes logistics/processing costs for SFI re-entry; delay cost (${\mathrm{C}}_{\mathrm{d}}$) arises from SFI storage variations leading to order fulfillment delays.

The model enforces the constraint ${\mathrm{R}}_{\mathrm{p}\mathrm{r}}\ge {\mathrm{R}}_{\mathrm{s}\mathrm{l}}$, where ${\mathrm{R}}_{\mathrm{p}\mathrm{r}}$ denotes the average on-time fulfillment rate across all products and ${\mathrm{R}}_{\mathrm{s}\mathrm{l}}$ is the minimum required service level. Each product has dynamic order attributes (quantity, production start time, due date, delay penalty) during the study period. On-time fulfillment requires production to start after the specified start time and finish before the due date.

Based on the above analysis, the relevant parameters and decision variables of the model are provided in attachments 4. The model and constraints are outlined as follows:

Objective Function:

$$MinZ=C_h+C_r+C_d$$

(1)

$$\begin{array}{l}{C}_h={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast C_{hmnj}\ast {\displaystyle \sum _{l=1}^{p\left(H_{mnj}\right)}\left(T_{emnl}-T_s\right)}\ast Q_{mnl}\right)}}}\\ +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast C_{hmnj}\ast T\ast \left(H_{mnj}-{\displaystyle \sum _{l=1}^{p\left(H_{mnj}\right)}{Q}_{mnl}}\right)\right)}}}\\ +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N\left({\displaystyle \sum _{k=1}^K\left(X_{mk}\ast C_{hmk}\right)\ast {\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast \left({\displaystyle \sum _{L_{mn}=p\left(H_{mnj}\right)+1}^{L_{mn}}\left(T_{emnl}-T_s\right)}\ast Q_{mnl}\right)\right)}}\right)}}\\ +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{k=1}^K\left(\left(X_{mk}\ast C_{hmk}\right)\ast T\ast {\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast \left(H_{mk}-{\displaystyle \sum _{l_{mn}=p\left(H_{mnj}\right)+1}^{L_{mn}}{Q}_{mnl}}\right)\right)}\right)}}}\end{array}$$

(2)

$$\begin{array}{l}{C}_r={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast C_{rmnj}\ast {\displaystyle \sum _{l=1}^{p\left(H_{mnj}\right)}{Q}_{mnl}}\right)}}}\\ +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{k=1}^K\left(X_{mk}\ast C_{rmk}\ast {\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast {\displaystyle \sum _{l_{mn}=p\left(H_{mnj}\right)+1}^{L_{mn}}{Q}_{mnl}}\right)}\right)}}}\end{array}$$

(3)

$$\begin{array}{l}{C}_d={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast {\displaystyle \sum _{l=1}^{p\left(H_{mnj}\right)}{Q}_{mnl}}\ast \left(D_{smnl}+Y_{mnj}\ast T\left(j_{mn},l_{mn}\right)-T_{dmnl}\right)\ast {\theta }_{mnl}\ast V\left(j_{mn},l_{mn}\right)\right)}}}\\ +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{k=1}^K\left({\displaystyle \sum _{j=1}^J\left(Y_{mnj}\ast X_{mk}\ast {\displaystyle \sum _{l_{mn}=p\left(H_{mnj}\right)+1}^{L_{mn}}{Q}_{mnl}\ast \left(T_{smnl}+X_{mk}\ast T\left(k_m,l_{mn}\right)-T_{dmnl}\right)\ast {\theta }_{mnl}\ast U\left(j_{mn},l_{mn}\right)}\right)}\right)}}}\end{array}$$

(4)

Constraints:

$${ Y}_{mnj}\in \left\{\mathrm{0,1}\right\}$$

(5)

$$X_{mk}\in \left\{\mathrm{0,1}\right\}$$

(6)

$${\displaystyle \sum _{j=1}^J{Y}_{mnj}=1}$$

(7)

$${\displaystyle \sum _{k=1}^K{X}_{mk}=1}$$

(8)

$$0\le H_{mnj}\le Q_{cmnj}$$

(9)

$$0\le H_{mk}\le Q_{cmk}$$

(10)

$$R_p\ge R_s$$

(11)

where, Objective function (1) aims to minimize the total cost.

Equation (2) represents the holding cost, while Equation (3) denotes the setup cost. Equation (4) represents the delay cost.

Constraint (5) specifies that the decision variable for selecting CODP is a binary variable (0–1).

Constraint (6) indicates that the decision variable for selecting PDP is a binary variable (0–1).

Constraint (7) ensures that each product has exactly one CODP.

Constraint (8) ensures that each product category has exactly one PDP.

Constraints (9) and (10) restrict the inventory levels of “specialized semi-finished products” and “universal semi-finished products,” respectively, to be less than or equal to the inventory capacity.

Constraint (11) specifies that the average on-time fulfillment rate of customer orders must surpass the enterprise’s minimum requirement.

3.1.3. Multi-Period Optimization Model

Multi-period production extends the single-period framework, where production in one period affects subsequent periods. Its dynamics manifest in two aspects: (1) The distribution of SFI shifts across periods, with carryover effects from both inventory and backlogged orders; (2) Costs, production lead times, and minimum service levels vary temporally, altering inventory allocation. Such ripple effects create intertemporal dependencies in the production process.

In the multi-period delayed production of steel, the storage and utilization process of semi-finished goods inventory is illustrated in Figure 3.

The initial production capacity is transformed into generic semi-finished goods inventory corresponding to PDP (depicted by solid yellow circles in the figure) and specialized semi-finished goods inventory corresponding to CODP (depicted by solid yellow diamonds in the figure). In the second period, the PDP positioning for each product category may remain the same as the previous period (depicted by solid red circles in the figure), or it may change compared to the previous period (depicted by solid green circles in the figure). Similarly, the CODP positioning for each product may either stay the same as the previous period (depicted by solid red diamonds in the figure) or differ from the previous period (depicted by solid green diamonds in the figure).

Order fulfillment prioritizes CODP-aligned SFI, utilizing PDP-aligned SFI for shortages, with remaining inventory and backlogs carried forward. In the second period, PDP and CODP positions may remain unchanged (red markers) or shift (green markers). Order fulfillment again prioritizes prior-period specialized SFI, then current-period specialized SFI if needed, followed by generic SFI (past or current). Due to this hierarchy, cross-period transfers rarely extend beyond two periods.

3.2. Model Assumptions and Parameters

This section formulates the optimization problem for multi-period delayed production, focusing on PDP/CODP location selection (by product category/type) and their corresponding generic/dedicated intermediate inventory levels. Leveraging the analytical structure, we model the multi-period problem as a two-period framework for tractability.

3.2.1. Model Assumptions

Within each period, capacity constraints, optional intermediate inventory locations (with fixed unit costs, production times, and processes), and the number of candidate CODP/PDP locations (as well as product categories/varieties) remain static. Available capacity and period duration are known a priori, while stochastic customer orders (modeled via expected values) must satisfy a minimum service-level constraint.

3.2.2. Model Parameters

The model constructed in this paper shares similarities with the single-period model. By introducing parameters expressing the period denoted as “t” and adding period subscripts to relevant parameters from the single-period model, various parameters for multi-period production can be expressed. The newly added cyclical parameters are presented in

Table 1. Parameters in the Multi- period Optimization Model.

Parameters	Meaning of Parameters
$t$	Denoting the t-th period
$T_{pres}^t$	The start time of the production preparation phase in the t-th period
$T_{pred}^t$	The end time of the production preparation phase in the t-th period
$T_{pre}^t$	The duration of the production preparation phase in the t-th period, $T_{pre}^t=T_{pred}^t-T_{pres}^t$
$T_{imps}^t$	The start time of the order fulfillment phase in the t-th period
$T_{impd}^t$	The end time of the order fulfillment phase in the t-th period.
$T_{imp}^t$	The duration of the order fulfillment phase in the t-th period, $T_{imp}^t={T_{impd}^t-T}_{imps}^t$
$T^t$	The overall duration of the t-th period, $T^t{=T}_{impd}^t-T_{pres}^t$

:

The decision variables are presented in

Table 2. Decision Variables in the Multi-Period Optimization Model.

Parameters	Meaning of Parameters
$X_{mk}^t$	Whether to choose $k_m$ as the PDP for the $m$-th product category in the t-th period, where 1 represents selection, and 0 represents non-selection
$H_{mk}^t$	The corresponding generic intermediate inventory level when the $m$-th product category selects the $k_m$ candidate location as the PDP in the t-th period
$Y_{mnj}^t$	Whether to choose $j_{mn}$ as the CODP for the $n_m$ product within the t-th period, where 1 represents selection, and 0 represents non-selection.
$H_{mnj}^t$	The corresponding dedicated intermediate inventory level when the $n_m$ product selects the $j_{mn}$ candidate location as the CODP in the t-th period

.

3.2.3. Analysis of Costs in Different Stages

Drawing upon the foundational assumptions, parameters, and decision variables within this paper, we will provide a comprehensive delineation of the cost structure across various stages of the model. The scope of delayed production for each period ‘t’ encompasses two distinct phases: the “Production Preparation Phase” and the “Order Fulfillment Phase. In the Production Preparation Phase during period ‘t’, it’s important to note that only the SFI generated in the preceding period, ‘t−1’, is available for production. Consequently, we can get $C_{pre}^t$ = $C_{pre}^{(t-1)\to t}$; Conversely, during the Order Fulfillment Phase in period ‘t’, manufacturers have the flexibility to utilize both the SFI produced in period ‘t−1’ and the inventory generated within the current period. This dual sourcing leads to $C_{imp}^t$ = $C_{imp}^{(t-1)\to t}$ + $C_{imp}^{t\to t}$. Furthermore, in each stage of the production process, there are specific components to consider, including setup costs, holding costs, and delayed penalty costs, as illustrated in

Table 3. Parameters of Cost Calculation Composition in Different Stages.

Parameters	Meaning of Parameters
$C^t$	All costs incurred within the t-th period. Delayed production encompasses both the ‘production preparation phase’ and the ‘order fulfillment phase,’ each of which incurs respective costs. In this context, $C^t$= $C_{pre}^t$+$C_{imp}^t$.
$C_{pre}^t$	The total cost incurred during the production preparation phase of the t-th period.
$C_{pre}^{(t-1)\to t}$	In the t-th period, the total cost generated by utilizing the SFI produced in the previous period (t−1 period) for production.
$C_{pre\to r}^{(t-1)\to t}$	The setup cost incurred during the production preparation phase of the t-th period when using the SFI passed over from the (t−1) period.
$C_{pre\to ph}^{(t-1)\to t}$	The holding cost generated during the production preparation phase of the t-th period when incorporating the SFI passed on from the (t−1) period into the production process.
$C_{pre\to wh}^{(t-1)\to t}$	The holding cost incurred during the production preparation phase of the t-th period when the SFI passed on from the (t−1) period is not utilized in the production process.
$C_{imp}^t$	All costs generated during the order fulfillment phase in the t-th period.
$C_{imp\to r}^{(t-1)\to t}$	The setup cost incurred during the order fulfillment phase of the t-th period when utilizing the SFI passed over from the (t−1) period in production.
$C_{imp\to ph}^{(t-1)\to t}$	The holding cost generated during the order fulfillment phase of the t-th period when incorporating the SFI passed on from the (t−1) period into the production process.
$C_{imp\to wh}^{(t-1)\to t}$	The holding cost incurred during the order fulfillment phase of the t-th period when the SFI passed on from the (t−1) period is not utilized in the production process.
$C_{imp\to d}^{(t-1)\to t}$	The delayed penalty cost incurred during the order fulfillment phase of the t-th period when utilizing the SFI passed on from the (t−1) period in the production process.
$C_{imp\to r}^{t\to t}$	The setup cost incurred during the order fulfillment phase of the t-th period when utilizing the SFI passed over from the t-th period in the production process.
$C_{imp\to ph}^{t\to t}$	The holding cost generated during the order fulfillment phase of the t-th period when incorporating the SFI passed on from the t-th period into the production process.
$C_{imp\to wh}^{t\to t}$	The holding cost incurred during the order fulfillment phase of the t-th period when the SFI passed over from the t-th period is not utilized in the production process.
$C_{imp\to d}^{t\to t}$	The delayed penalty cost incurred during the order fulfillment phase of the t-th period when utilizing the SFI passed over from the t-th period in the production process.

.

3.3. Model Construction

Based on the formulations described above, we can express the total cost within each period as follows:

3.3.1. The Total Cost in the First Period

$$C^1=C_{pre}^1+C_{imp}^1$$

(12)

The total cost for the first period is denoted as $C^1$ comprising two cost components: production preparation cost $C_{pre}^1$ and order fulfillment cost $C_{imp}^1$. The production preparation stage cost $C_{pre}^1$ includes the cost $C_{pre}^{0\to 1}$ associated with the transfer of semi-finished goods from the previous period to the current one, incurred due to storage and participation in production. The order fulfillment stage cost $C_{imp}^1$ includes the costs $C_{imp}^{0\to 1}$ for semi-finished goods transferred from the previous period and $C_{imp}^{1\to 1}$ for semi-finished goods produced in the current period, both costs arising from storage and production.

$$C_{pre}^1=C_{pre}^{0\to 1}$$

(13)

$$C_{imp}^1=C_{imp}^{0\to 1}+C_{imp}^{1\to 1}$$

(14)

In the production preparation stage of each period, there will be incurrence of setup costs and holding costs, thus

$$C_{pre}^{0\to 1}=C_{pre\to r}^{0\to 1}+C_{pre\to ph}^{0\to 1}+C_{pre\to wh}^{0\to 1}$$

(15)

In each period’s order fulfillment stage, there will be incurrence of setup costs, holding costs, and delay penalty costs, thus

$${C_{imp}^{0\to 1}=C}_{imp\to r}^{0\to 1}+C_{imp\to ph}^{0\to 1}+C_{imp\to wh}^{0\to 1}+C_{imp\to d}^{0\to 1}$$

(16)

$$C_{imp}^{1\to 1}=C_{imp\to r}^{1\to 1}+C_{imp\to ph}^{1\to 1}+C_{imp\to wh}^{1\to 1}+C_{imp\to d}^{1\to 1}$$

(17)

Therefore, the total cost for the first period is:

$$\begin{array}{c}{C}^1=C_{pre}^1+C_{imp}^1=C_{pre}^{0\to 1}+C_{imp}^{0\to 1}+C_{imp}^{1\to 1}\hfill \\ {=C}_{pre\to r}^{0\to 1}+C_{pre\to ph}^{0\to 1}+C_{pre\to wh}^{0\to 1}\hfill \\ {+C}_{imp\to r}^{0\to 1}+C_{imp\to ph}^{0\to 1}+C_{imp\to wh}^{0\to 1}+C_{imp\to d}^{0\to 1}\hfill \\ +C_{imp\to ph}^{1\to 1}+C_{imp\to wh}^{1\to 1}+C_{imp\to d}^{1\to 1}\hfill \end{array}$$

(18)

For the first period, the decision variables include $X_{mk}^1,Y_{mnj}^1$, $H_{mk}^1$, $H_{mnj}^1$, representing the position of PDP, the position of CODP, the inventory level of PDP, and the inventory level of CODP within this period.

3.3.2. The Total Cost in the Second Period

$$C^2=C_{pre}^2+C_{imp}^2$$

(19)

The total cost for the second period, denoted as $C^2$, comprises two components: the production preparation cost $C_{pre}^2$ and the order fulfillment cost $C_{imp}^2$. Similar to the first period,

$$C_{pre}^2=C_{pre}^{1\to 2}$$

(20)

$$C_{imp}^2=C_{imp}^{1\to 2}+C_{imp}^{2\to 2}$$

(21)

In the production preparation stage of the second period, costs associated with backtracking and holding inventory will be incurred. Therefore,

$$C_{pre}^{1\to 2}=C_{pre\to r}^{1\to 2}+C_{pre\to ph}^{1\to 2}+C_{pre\to wh}^{1\to 2}$$

(22)

In the production preparation stage of the second period, costs related to backtracking, inventory holding, and delay penalty will be incurred. Therefore,

$${C_{imp}^{1\to 2}=C}_{imp\to r}^{1\to 2}+C_{imp\to ph}^{1\to 2}+C_{imp\to wh}^{1\to 2}+C_{imp\to d}^{1\to 2}$$

(23)

$$C_{imp}^{2\to 2}={C_{imp\to r}^{2\to 2}+C}_{imp\to ph}^{2\to 2}+C_{imp\to wh}^{2\to 2}+C_{imp\to d}^{2\to 2}$$

(24)

Finally, the total cost for the second period is as follows:

$$\begin{array}{c}{C}^2=C_{pre}^2+C_{imp}^2=C_{pre}^{1\to 2}+C_{imp}^{1\to 2}+C_{imp}^{2\to 2}\hfill \\ =C_{pre\to r}^{1\to 2}+C_{pre\to ph}^{1\to 2}+C_{pre\to wh}^{1\to 2}\hfill \\ +C_{imp\to r}^{1\to 2}+C_{imp\to ph}^{1\to 2}+C_{imp\to wh}^{1\to 2}+C_{imp\to d}^{1\to 2}\hfill \\ +C_{imp\to r}^{2\to 2}+C_{imp\to ph}^{2\to 2}+C_{imp\to wh}^{2\to 2}+C_{imp\to d}^{2\to 2}\hfill \end{array}$$

(25)

The situation for subsequent periods following the second period is similar and is not repeated. Based on the above analysis, the production cost model under delayed production mode is constructed as follows:

The objective function is given by:

$$MinZ={\displaystyle \sum _{t=1}^T{C}^t}=C^1+C^2+\dots +C^t+\dots$$

(26)

where, Formula (26) represents the objective function aimed at minimizing the sum of costs across all periods. The meanings of the remaining constraint conditions are consistent with the descriptions provided earlier.

4. Algorithm Design and Case Analysis

4.1. Algorithm Design

In addressing the aforementioned issue, this paper formulated a non-linear mixed-integer programming model. Given that the model falls under the category of NP-Hard problems, a particle swarm optimization algorithm was employed for its solution. The multi-period postponement problem in process industries involves continuous variables (e.g., production rates, inventory levels) and dynamic constraints (e.g., postponement penalty functions). PSO is particularly effective for such continuous-space optimization.

4.1.1. Algorithm Steps

The main steps of the model-solving algorithm include:

Step 1: Initialize the storage quantities for all periods and storage locations.

Step 2: Read the parameters, objective function, and constraints of the model. Set the classification, constraints, and function expressions for each particle. Specify the population size and iteration count for each period.

Step 3: Import the costs and parameters for the current period.

Step 4: Consume inventory from the previous period, fulfill customer orders from the previous period, calculate costs, and update the current inventory.

Step 5: Randomly assign initial positions and velocities to each particle.

Step 6: Perform position operations for each particle in the population.

Step 7: Calculate the fitness of each individual on the objective function.

Step 8: Update the population and check if the evolution generation is reached. If “yes,” proceed to the next steps; if “no,” go back to the “randomly assign initial positions and velocities for each particle” step.

Step 9: Output the solution for this stage and check if the dedicated SFI can meet customer order demands. If “yes,” proceed to the next steps; if “no,” go back to the “randomly assign initial positions and velocities for each particle” step.

Step 10: Output the optimal solution for the current period, record the historical optimal solution, and check if all periods have been traversed. If “yes,” complete the computation; if “no,” go back to the “reset the population size and iteration count for each period” step.

In the algorithm steps of this paper, the main difference lies in setting different population sizes and iteration counts for particles in different periods. Additionally, multiple traversal loops are required based on the number of periods. Specifically, as illustrated in Figure 4.

4.1.2. Algorithm Effectiveness Analysis

For the constructed model, precise algorithms and particle swarm algorithms were applied and compared on medium and small-scale instances. Algorithmic code excerpts can be found in the Appendix A. The results are as follows:

As shown in

Table 4. Comparison Between Particle Swarm Optimization Heuristic Algorithm and Exact Algorithm.

Number of Periods	The Difference in Solution Results Between Particle Swarm Optimization Algorithm and Exact Algorithm	The Computation Time of the Exact Algorithm	The Computation Time of the Particle Swarm Algorithm
6	1.5%	6 min	0.5 min
8	1.8%	21 min	1.1 min
12	2.4%	1.5 h	1.5 min
24	4.8%	4.3 h	10 min

, with the increase in the number of model periods, the computation time of the precise algorithm gradually increases. When the number of periods reaches 12, the computation time suddenly jumps to 1.5 h. At the same time, the computation time of the particle swarm algorithm also increases gradually. Additionally, as the number of periods increases, the difference in results between the particle swarm algorithm and the precise algorithm also gradually increases, with a maximum difference within 5%, which is within an acceptable range.

The minor PSO errors primarily stem from the relaxation of continuous variables in production scheduling (e.g., backtracking production adjustments), whereas discrete decisions (e.g., operation sequencing) maintain precision through hierarchical optimization. This trade-off avoids combinatorial explosion while aligning with flexible production requirements in most practical applications.

4.2. Empirical Analysis

4.2.1. Data Collection and Organization

In this section, we delve into a case study of a steel company, exploring delayed production strategies across three key product categories: Coil Products, Plate Products, and Tube Products. Within each category, three specific products are selected for a detailed analysis. Relevant production data, order information, and period variations can be found in attachments 5 and 6. General information includes a 30-day production period, an initial capacity of 60,000 units, CODP-associated inventory capped at 5000 units, PDP-associated inventory capped at 10,000 units, and a minimum customer service level set at 0.85.

This company represents a typical long-process integrated steel producer (blast furnace—basic oxygen furnace—rolling mill) with complex production sequences and continuous operations. The primary production facilities comprise: Multiple large-scale blast furnaces, each with a daily hot metal production capacity of 8800 tonnes; Several heterogeneous converters, each capable of producing 10,000 tonnes of crude steel per day; Integrated rolling mills, including hot-rolling, cold-rolling, and long product rolling lines; Complementary finishing facilities for downstream processing, such as coil leveling, plate cutting, and specialized finishing operations.

4.2.2. Model Solution and Results

The multi-period delayed production model belongs to the NP-Hard problem, making it challenging to solve and analyze using exact algorithms. Therefore, this paper employs the heuristic particle swarm optimization algorithm for its solution. Additionally, utilizing real operational data from the enterprise and based on the constructed quantitative model, the multi-period model is solved using the heuristic particle swarm optimization algorithm (refer to the Algorithm A1 section in the Appendix A). The decision results for the first period are presented in

Table 5. Calculation Results of the First Period of Optimization Model.

	Product Category	PDP1	PDP2	PDP3	PDP4	PDP5	Product Number	CODP1	CODP2	CODP3	CODP4	CODP5	CODP6	Order Quantity
The initial production quantity for the first period is 39,000	1				3500		1						4000	3800
							2					4000		4000
							3		1500					4300
	2	1500					1	4000						4000
							2						4000	4000
							3					4000		4000
	3					500	1						4000	4100
							2						4000	3900
							3						4000	4000
The remaining quantity at the end of the first period	1				700		1						200
							2						0
							3		0
	2	1500					1	0
							2						0
							3					0
	3					400	1						0
							2						100
							3						0

.

The optimal decisions for the first product category in the initial period involve placing PDP at position 4 with an optimal inventory of 3500 for the corresponding semi-finished product. Additionally, for the first product category, the optimal positions for CODP are 6, 5, and 2, with corresponding optimal inventory levels of 4000 and 1500. Similar decisions are applied to other product categories. At the end of the first period, specific remaining inventories include 700 semi-finished products at PDP position 4, 200 at CODP position 6 for the first product in the first category, 1500 at PDP position 1 for the second product category, 400 at PDP position 5 for the third product category, and 100 at CODP position 6 for the second product in the first category.

The decision outcomes for the second period are presented in

Table 6. Calculation Results for the Second Period of Optimization Mode.

		PDP1	PDP2	PDP3	PDP4	PDP5	Product Number	CODP1	CODP2	CODP3	CODP4	CODP5	CODP6	Order Quantity
The remaining inventory at the end of the first period	1				700		1						200
							2						0
							3		0
	2	1500					1	0
							2						0
							3					0
	3					400	1						0
							2						100
							3						0
The initial production quantity for the second period is 37,000	1				700	1500	1					3800	200	4300
							2					4000		4000
							3	2500						4000
	2	1500 + 700					1		4000					3900
							2						4000	4200
							3					4000		4000
	3					400 + 800	1					4000		4000
							2						100+ 3900	4000
							3						4000	3800
The remaining inventory at the end of the second period	1					400	1						0
							2						0
							3	0
	2	2000					1		100
							2						0
							3					0
	3					1200	1					0
							2						0
							3						200

.

The results show that expected order values strongly influence SFI allocation. Dedicated inventory typically matches expected orders, with deviations occurring only when generic inventory is produced in cost-minimizing periods. If actual orders meet expectations, no surplus dedicated inventory remains; if demand exceeds expectations, generic inventory is used after dedicated stock depletion, still avoiding surplus. Only when actual orders fall short does dedicated inventory carry over to the next period. Additionally, generic inventory is placed at the lowest holding-cost location.

5. Discussion

Using the multi-period optimization model, this section analyzes the impact of customer service levels, delay penalty coefficients, etc., on the optimal positions of CODP and PDP, taking the second period as an example.

5.1. The Impact of Customer Service Levels on Delayed Production Model

5.1.1. The Impact of Customer Service Levels on the Optimal Position of CODP

From Figure 5, it can be observed that as the customer service level increases, the CODP positions of the three products gradually move towards the end of production. Specifically, the first product moves from position 1 to position 6, the second product from position 1 to position 5, and the third product from position 1 to position 2. This indicates that, within a multi-period single period, to meet the improvement in customer service levels, it is necessary to reduce the production time for products, requiring the enterprise to store more advanced specialized semi-finished inventories. This leads to the optimal CODP positions for each product moving towards the end of production.

5.1.2. The Impact of Customer Service Level on the Optimal Positions of PDP

As shown in Figure 6, as the customer service level multiplies from small to large, the optimal PDP positions for the two product categories gradually shift towards the production end, although the shift is not pronounced. Specifically, the first product category transitions from position 1 to position 4, and the second product category from position 1 to position 2. This indicates that, within a single period of a multi-period framework, meeting the continuously increasing customer service levels requires an enhancement in the processing degree of involved general semi-finished inventories, consequently leading to a shift in the optimal PDP positions towards the production end for product categories.

5.2. The Influence of Delay Penalty Coefficients on Delayed Production Model

5.2.1. The Impact of Delay Penalty Coefficient on the Optimal Positions of CODP

As observed in Figure 7, with an increase in the delay penalty coefficient, the positions for the three product types gradually shift towards the end of the production process. Specifically, the first product type transitions from position 1 to position 6, the second product type from position 4 to position 5, and the third product type from position 1 to position 4. This indicates that within a single period of a multi-period production system, the elevation of the delay penalty coefficient leads to an increase in delay costs. To mitigate these costs, enterprises tend to store more highly processed specialized semi-finished products. This strategy aims to reduce product production times, thereby causing the optimal CODP positions for products to shift towards the end of the production process.

5.2.2. The Impact of Delay Penalty Coefficient on Optimal Positions of PDP

As illustrated in Figure 8, there is minimal variation in PDP positions for the two product categories as the delay penalty coefficient increases. Specifically, the first product category moves from position 4 to position 5, while the second product category remains unchanged. This indicates that within a single period of a multi-period production system, the endeavor to reduce delay costs prompts an enhancement in the processing degree of involved generic semi-finished products. Consequently, this leads to a movement of optimal PDP positions towards the end of the production process for product categories.

The observed weaker sensitivity of PDP positioning to penalty cost variations compared to CODP stems from three key factors in steel production: upstream process rigidity with high restart costs, inherent buffering capacity of semi-finished inventories at PDP, and its strategic decision-making level involving capacity planning. This pattern exhibits domain-specific characteristics—while particularly applicable to capital-intensive process industries (e.g., steel, chemicals) where upstream flexibility is constrained, discrete manufacturing (e.g., automotive) may demonstrate stronger PDP responsiveness due to modular production systems. From a managerial perspective, the trade-off suggests maintaining stable PDP positions when penalty costs are tolerable (leveraging CODP adjustments), but proactively relocating PDP when penalties escalate critically, reflecting the fundamental balance between production stability assurance (via PDP) and market responsiveness (via CODP) in delayed differentiation strategies.

5.3. Total Cost Dynamics Analysis

5.3.1. High Cost Sensitivity in Single-Period Planning: Rigid Short-Term Constraints Dominate

Under a single-period model, firms must complete production within a fixed time window without cross-period inventory adjustments. Postponing the customer order decoupling point (CODP) shortens lead times but increases in-period inventory holding and production switching costs. Since delay penalties impact only immediate orders, firms often resort to overtime or emergency procurement, causing steep marginal cost escalation. Due to rigid resource constraints, minor CODP/PDP adjustments amplify cost fluctuations, resulting in total cost variations of 5–8%: a 4.8% increase from1.65 M to 1.73 M with higher service levels, and a 7.9% rise to 1.78 M under elevated delay penalty coefficients.

5.3.2. Multi-Period Cost Smoothing: Dynamic Optimization and Cross- Period Buffering

In multi-period production, cross-period resource allocation and dynamic inventory management mitigate cost volatility. Order deferral creates time buffers, while adaptive CODP/PDP positioning absorbs demand variability. Learning effects and economies of scale emerge—though processing more generic semi-finished products (PDPs) raises near-term costs, their reuse across periods reduces long-term unit costs. For high delay penalties, advancing the CODP to stock lower-processed PDPs distributes risks. These strategies limit total cost fluctuations to ≤2%: service-level improvements increase costs only from 3.30 M to 3.35 M, and penalty coefficient hikes yield a marginal uptick to 3.36 M.

5.4. Managerial Implications

(1)

Adopt Cross- Period Inventory Coordination. Multi-period production benefits from order postponement and inventory reuse (especially generic SFI), reducing total cost volatility to ≤2%. For industries like steel, a two-period optimization framework improves decision efficiency.

(2)

Dynamically Adjust CODP/PDP for Demand Variability. Under high-service-level or high-delay-penalty conditions, firms should shift CODP downstream (storing highly processed SFI) to shorten lead times, while PDP positioning is cost-driven and less sensitive to delays.

(3)

Enhance Demand Forecasting to Minimize Mismatches. Optimal cost occurs when SFI aligns with actual orders; forecast errors lead to cross-period transfers or costly adjustments—requiring real-time demand sensing and dynamic safety stock policies.

(4)

Differentiate Single- vs. Multi-Period Strategies. Single-period planning relies on end-stage inventory buffers (cost fluctuations: 5–8%), whereas multi-period optimization leverages time buffers and resource sharing, necessitating distinct models.

(5)

Extend the Two-Period SFI Transfer Framework. The two-period SFI transfer rule, validated in steel, applies to process industries with long periods (e.g., chemicals). Firms should integrate algorithmic decision modules (e.g., MILP, PSO) into production systems.

(6)

To mitigate order forecast errors, measures such as dynamically designing generic/dedicated SFI buffer stocks, implementing delayed option contracts to enhance forecast accuracy, and adopting rolling production scheduling to differentiate firm/flexible orders can be implemented.

6. Conclusions

The research indicated that in the context of multi-period delayed production in the steel industry, the assumption of a two-period transfer of SFI and customer orders was more advantageous for production organization, with minimal impact on total costs. Additionally, for the purpose of reducing overall costs and meeting customer order production time requirements, it was common for businesses to store a predictable amount of SFI corresponding to customer orders. Therefore, accurate prediction of customer order quantities was crucial. The study revealed:

(1)

In the delayed production of steel, the transfer of SFI and customer orders typically does not exceed two periods. Therefore, the assumption of a two-period transfer can effectively capture multi-period production scenarios and simplify the optimization model. Furthermore, solving the model through programming also confirms the viewpoint that SFI usually does not extend beyond two periods.

(2)

In the multi-period delayed production of steel enterprises, generic SFI usually does not participate in production during the period of its generation. The optimal storage position (PDP) for generic inventory is primarily influenced by the unit holding cost.

(3)

In multi-period delayed production, the impact of customer service levels, delay penalty coefficients, etc., on the optimal position of SFI is similar to the single-period scenario.

(4)

The accuracy of customer order quantity forecasting is crucial for the distribution of SFI in delayed steel production. Specifically, when the quantity of SFI matches the forecasted quantity of customer orders, there is typically no cross-period transfer of inventory, leading to the lowest total cost. If the forecasted quantity exceeds the actual customer orders, the remaining orders are transferred to the next period. When the forecasted quantity is less than the actual customer orders, generic SFI may be used in production during its generation period, but this often results in additional delay penalty costs.

Future research could advance in three directions: Establishing a rolling-horizon optimization framework to dynamically adjust planning cycles for balancing decision granularity and computational efficiency; Incorporating stochastic programming methods to address multi-period robust optimization under demand uncertainty; Exploring reinforcement learning-based adaptive strategies to achieve co-optimization of postponement decisions and inventory allocation. These extensions would further enhance the model’s applicability in dynamic market environments and provide more flexible decision support for process manufacturing enterprises.