Optimal Volume Planning and Scheduling of Paper Production with Smooth Transitions by Product Grades

Abstract

The article deals with the problem of calculating the volume calendar plan of a paper mill. The presented mathematical model and methods make it possible to schedule paper production orders between several paper machines (PM) to even their loading, devise cutting plans for each winder and arrange the order of their implementation. When forming cutting plans, orders are grouped in accordance with such parameters as grammage, roll diameter, core diameter, product type and number of layers. Deadlines and volumes in customer orders are taken into account. The cutting plans for each winder account for the allowable roll width limits and the maximum number of knives. To find the optimal schedule, a combination of the following criteria is used: minimal trim loss, minimal changes to the knives’ setup and smooth transitions by product grades. Solution algorithms are presented that use a combination of the simplex method, the column generation, the branch and bound methods, the greedy algorithm and the local search procedure. We tested the solution approach on real production data from a paper mill in European Russia and obtained the production sequence that better matches deadlines in customer orders compared to the plan devised manually by production planners.

1. Introduction

Optimal volume planning and scheduling (OVPS) is a necessary component for the success of any company in the paper industry. OVPS allows us to determine the optimal production volumes by product type and distribute them over time, taking into account the constraints and needs of the market. In this way, OVPS helps to harmonize the processes of production, sales and storage of paper goods, which ultimately leads to increased efficiency and successful development of the paper mill in the long term.

An important aspect of OVPS is compliance with constraints on transitions between product types. The manufacturing of each product type requires a specific combination of process parameters, which can cause problems in production planning.

Another important aspect of OVPS is taking into account market demand and consumer expectations. The optimal production volume must match the demand in order to effectively meet the needs of customers and avoid producing goods into the so-called “free stock” (without a specific customer order). An optimal inventory level should also reduce unnecessary costs related to storage and its management.

One of the common problems in production planning is the disparity between the requirements of customer orders and the production plan optimality based on technology and equipment parameters. The solution to this problem requires coordination of various units, including production, technology, logistics and sales departments.

The problem of material cutting and production planning has been well studied by many authors [1,2,3]. However, each enterprise has its own unique specifics, technology and equipment, which must be taken into account when developing a solution to this problem. Therefore, the optimal production planning procedure must be developed by considering the specifics of the materials used, the features of equipment and technological processes, the requirements for quality, quantity and delivery time of finished products [4,5,6,7].

Paper [3] presents an integrated mathematical model of planning the operation of paper-making machines and the multi-step cutting of paper as a linear programming problem of large dimension. The algorithms of constructing the optimal operation schedules, including deadlines of orders and equipment retuning, are provided.

A number of studies are devoted to solving individual subtasks. For example, paper [8] is devoted to minimizing the number of cuts. Other studies [9] suggest splitting the planning period into small time intervals, which can increase the time to find a solution and introduce additional difficulties in balancing the volumes of customer and production orders.

Paper [10] proposes a new variation of particle swarm optimization techniques called “the modified particle swarm optimization”, for finding optimum solutions for single and multi-objective economic load dispatch problems.

In paper [11], a method using an artificial neural network was proposed to estimate the potential result based on the model structure and a given data instance before the start of optimization. Paper [12] proposes a decision support tool for the maintenance capacity planning of complex product systems.

Paper [13] discusses the scheduling of production orders at a packaging mill with parallel machines and sequence-dependent setups aiming to minimize the total tardiness subject to precedence constraints, calendar constraints and also availability constraints. The authors note that for practical applications, the resulting constrained mixed-integer linear programming problem is NP-hard in the strong sense and propose a dedicated heuristic for quick solutions.

In this article, we consider the problem of optimal volume scheduling for paper production and cutting, taking into account the limitations on transitions between product types, as well as a number of conflicting precedence, calendar and other constraints. We will discuss the main methods for solving this problem and give a practical example of successful application of these methods in real production conditions. Similarly to [13], a novel dedicated heuristic is required to ensure quick solutions for this integer linear programming problem of large dimension.

We start by describing the planning challenge and desirable features in Section 2. We then provide the mathematical model and solution algorithms in Section 3. The decision-support software (DSS) implementing these methods is described in Section 4. In Section 5, we provide numerical results for a case study and compare a plan obtained with the DSS to one generated manually by production planners. Finally, Section 6 concludes the paper by discussing the approach and summarizing the results.

2. The Problem Statement

In modern paper production planning, in addition to optimizing trim loss, it is necessary to devise the optimal sequence of cutting with smooth product type changes, taking into account a number of conflicting requirements, including the following:

• Compliance with the deadlines and volumes of customer orders, meaning minimizing both tardiness and premature output.
• Minimization of production without a customer order (to the “free stock”).
• Compliance with the volumes of continuous output of products of each type (e.g., product “C”—between 20 and 45 tons, product “P”—at least 100 tons, etc.).
• Compliance with the rules for transitions between product types (e.g., product “C” may be produced only after product “P”).
• Smooth transitions by grades (for example, by grammages).
• The cutting volume must match the number of rolls and paper layers.
• Minimization of the knives’ setup changes at the winder.

Moreover, the plan should be calculated within a reasonably short time (e.g., 30 s) on a standard personal computer of a planning engineer.

The main purpose of the study is to develop a method for devising a production plan for the paper machine and the winder with smooth product type changes.

The idea of the method is to form a sequence of production and cutting blocks. A set of dedicated heuristic algorithms has been developed that reflect certain specific properties of the industrial partner. The solution process consists of three stages:

8.

Orders are grouped together and distributed among paper machines. Each group includes orders with the same properties (type, grammage, roll diameter, number of layers). For all groups, cutting plans are calculated.

9.

For each paper machine, the production sequence for these cutting plans is determined, taking into account the timing of orders and technology limitations.

10.

The resulting sequences of cutting plans are improved using the local search procedure.

The corresponding mathematical models and solution methods for each stage are provided in the following section.

3. Mathematical Model and Solution Algorithm

3.1. The First Stage

At the first stage, all orders are grouped by the product type, grammage, roll and core diameters, and the number of layers. For each group, cutting plans are calculated.

The main index sets and known parameters are as follows:

• $M$—the set of orders;
• $b_i$—the number of paper rolls of the order $i\in M$;
• $f_i$—the reel width in order $i\in M$ (mm);
• $G$—the set of product types;
• $M_g$—the set of orders of type $g\in G$,

$$M={\bigcup }_{g\in G} M_g.$$

(1)

• $N$—the set of paper-making machines (PMM);
• $L_j^{min}$—the minimum roll width of the PMM $j\in N$ (mm);
• $L_j^{max}$—the maximum roll width of the PMM $j\in N$ (mm);
• $S_{jg}$—the set of all cutting plans of the paper type $g\in G$, produced by PMM $j\in N$. For sets $S_{jg}$, we consider pairwise disjoint, and denote their union as $S$:

$$S={\displaystyle \bigcup }_{j\in N}{\displaystyle \bigcup }_{g\in G}{S}_g.$$

(2)

• $r_s$—the maximum number of rolls in the cutting plan $s\in S$ (determined by the number of knives of the respective winder);
• $w_s$—the weight of one reel segment at the winder for the cutting plan $s$ (tons);
• ${\pi }_{jg}$—the production rate at PMM $j$ for product type $g$ (tons);
• ${\tau }_j$—the minimum required working time of PMM $j\in N$ (hours);
• ${\theta }_j$—the maximum possible working time of PMM $j\in N$ (hours);
• $a_{is}$—the number of reels of order $i\in M$ in the cutting plan $s\in S$;
• ${\Delta }_s$—the trim width in the cutting plan $s\in S$.

Note that the set $S$ of all cutting plans can be very large, but the solution methods include procedures for the automatic generation of its elements.

Unknown parameters:

• $x_s$—the number of reel segments at the winder for the cutting plan $s$.

The objective function minimizes the trim loss:

$${{\displaystyle \sum }}_{j\in N}{{\displaystyle \sum }}_{g\in G}{{\displaystyle \sum }}_{s\in S_{jg}}\frac{w_s{\mathsf{\Delta }}_s}{L_j}{x}_s\to \min .$$

(3)

The next constraints ensure that production of paper rolls is within given volumes:

$${\displaystyle \sum }_{j\in N}{\displaystyle \sum }_{s\in S_{jg}}{a}_{is}{x}_s=b_i,\hspace{1em}i\in M_g,\hspace{1em}g\in G$$

(4)

$${\tau }_j\le {\displaystyle \sum }_{g\in G}{\displaystyle \sum }_{s\in S_{jg}}\frac{w_s}{{\pi }_{jg}}{x}_s\le {\theta }_j,\hspace{1em}j\in N$$

(5)

The next constraint respects cutting width limits:

$$L_j^{min}\le {{\displaystyle \sum }}_{i\in M_g}{a}_{is}{f}_i\le L_j^{max}, s\in S_{jg}, g\in G, j\in N.$$

(6)

The next constraint limits the number of rolls in cutting plans:

$${\displaystyle \sum }_{i\in M_g}{a}_{is}\le r_s, s\in S_{jg},\hspace{1em}g\in G, j\in N.$$

(7)

Then the trim loss in the cutting plan $s$ equals:

$$\begin{gathered} {\Delta }_s=L_j-{\displaystyle \sum }_{i=1}^N{a}_{is}{f}_i \\ {x}_s\ge 0, \mathrm{integer}, s\in S_{jg}, j\in N, g\in G. \end{gathered}$$

(8)

To solve this problem, we use standard procedures that are based on a combination of the branch and bound method to ensure integer solution, and the simplex method with column generation for calculating cutting plans [14,15]. As a result, we obtain the set of cutting plans for the paper rolls produced by each machine.

3.2. The Second Stage

At the second stage, for each paper machine the cutting plans are sorted and the production dates are calculated by considering the customer requirements and respecting the uniformity of volumes by product types. Since the solution is the same for all machines, we provide the mathematical model and the optimization problem for the second stage only for the case of one machine.

Let the set of cutting plans $\Im =\left\{i_1,\dots ,i_N\right\}$ be given. So, for each plan $i\in \Im$, we know:

• $v_i$—the cutting volume (paper weight);
• $d_i$—the output date (the earliest date of the order for the format in the cutting plan);
• ${\delta }_i$—the time for producing the paper on the machine and cutting it at the winder;
• $k_i$—the product type.

Denote the required cutting plan sequence as $I=i\left(1\right),i\left(2\right),\dots ,i\left(N\right)$. Thus, the set $P\left(N\right)$ of all permutations of integers $1,\dots ,N$ forms the set of feasible solutions to the problem, i.e., $I\in P\left(N\right)$.

Let us list the product types in the order of their output at the paper machine as the sequence $J=\kappa \left(1\right),\kappa \left(2\right),\dots ,\kappa \left(M\left(I\right)\right)$, assuming that

$$\kappa \left(j\right)\ne \kappa \left(j+1\right), j=1, \dots , M\left(I\right)-1.$$

(9)

Sequence $J$ may contain the same elements, if some types are produced repeatedly. We call the successive cutting plans of the same product type in the desired sequence of cutting plans a block, with $M\left(\mathrm{I}\right)$ being the number of such blocks.

Let $h\left(j\right)$ be the initial position for the j-th block in the cutting plans sequence, i.e.,

$$k_{i\left(\iota \right)}=\kappa \left(j\right), \iota =h\left(j\right), \dots , h\left(j+1\right)-1, j=1, \dots , M\left(I\right)$$

(10)

Assume that $h\left(M\left(I\right)+1\right)=N+1$.

For each product type $k$, technology determines the minimum and maximum production volumes $b\left(k\right)$ and $B\left(k\right)$ for the block. A block is called feasible, if its production volume falls within these limits.

Denote $D_i$—the date of cutting plan $i$. It depends on the date of the previous cutting plan and the paper production time:

$$\begin{gathered} D_0=0,i\left(0\right)=0 \\ {D}_{i\left(\iota \right)}=D_{i\left(\iota -1\right)}+{\delta }_{i\left(\iota \right)},\hspace{1em}\iota =1,\dots ,M\left(I\right) \end{gathered}$$

(11)

The allowable sequence of cutting plans must respect the following conditions:

11.

The production volume of each block is within the specified boundaries.

12.

Each cutting plan is completed by the required date.

Formally, these conditions are written using the following inequalities:

$$\begin{gathered} D_i\le d_i, \\ b\left(\kappa \left(j\right)\right)\le {\displaystyle \sum }_{\iota =h\left(j\right)}^{h\left(j+1\right)-1}{v}_{i\left(\iota \right)}\le B\left(\kappa \left(j\right)\right),\hspace{1em}j=1,\dots ,M\left(I\right). \end{gathered}$$

(12)

The objective function is the total volume of overdue orders multiplied by the number of delay days:

$$\Delta ={\displaystyle \sum }_{\iota =1}^N{\left(D_i-d_i\right)}^{+}{v}_{i\left(\iota \right)}\to \min ,$$

(13)

where ${\left(D_i-d_i\right)}^{+}$ equals zero, if $D_i-d_i<0$, and equals $\left(D_i-d_i\right)$, if $D_i-d_i\ge 0$.

The number of possible solutions for this combinatorial problem amounts to the factorial of $N$, which makes it impossible to find the optimal solution within a reasonable time. Therefore, we propose to reduce the solution space by introducing the following two rules reflecting the decision-making logic of planning engineers:

Rule 1. 

For each product type, the cutting plans are ordered by date.

Rule 2. 

After block $j$, we proceed to the block containing the unfinished cutting plan with the earliest date.

The cutting plans are placed in the block in the order of their dates until the lower block boundary (determined by the CalcBoundary algorithm) is reached. Then, we continue to fill the block until either its upper bound is reached, or an earlier plan of a different type is encountered. After that, we move on to the next block (the transition rule). When we choose the type of the next block, we need to exclude plans of the previous type from consideration.

We now provide the rules for choosing the next block and calculating the block boundaries.

3.2.1. The Rule for Choosing the Next Block (the Transition Rule)

When the production of a block is completed, it makes sense to move on to the block with the earliest outstanding orders. It also makes sense to start with the product type with a very large number of blocks (if there is such), since otherwise, we might be unable to produce all blocks of this type. Let us consider this in more detail.

The problem of choosing block types can be reduced to the problem of constructing a complete multipartite graph and finding a Hamiltonian path in it. An undirected graph is called multipartite, if its vertex set can be divided into subsets, called parts, so that each edge is incident to vertices from different parts. A multipartite graph is called complete, if any two vertices from different parts are adjacent.

In our case, the vertices of the multipartite graph are blocks, and each part corresponds to a product type. There are no edges inside the parts, as two blocks with the same product type cannot follow one another. The multipartite graph is complete, since transitions between any product types are possible. We need to find the Hamiltonian path, since each block needs to be produced only once.

According to [16], a Hamiltonian cycle in a complete multipartite graph exists, if and only if the number of vertices in each part does not exceed half of the number of graph vertices.

Let $V\left(k\right)$ be the total output volume of product type $k$.

Then, the minimum (maximum) number of blocks for product type $k$ is equal to the ratio of the total block size to the maximum (minimum) block size:

$$n\left(k\right)=\lceil \frac{V\left(k\right)}{B\left(k\right)}\rceil , N\left(k\right)=\lfloor \frac{V\left(k\right)}{b\left(k\right)}\rfloor ,$$

(14)

where the brackets $\lceil ·\rceil$ and $\lfloor ·\rfloor$ mean that the number is rounded up (down).

Let $K$ be the set of all product types. Denote

$$N={\displaystyle \sum }_{k\in K}N\left(k\right).$$

(15)

Then, the necessary condition for devising the acceptable sequence of blocks is as follows.

Statement 1. 

If the given production volumes can be divided into blocks of allowable sizes, then for all $k\in K$, the inequality $n\left(k\right)\le N-N\left(K\right)$ holds.

From Statement 1, it follows that we need to estimate the number of blocks in each type, and if there is a type with at least as many blocks as the total number of blocks of other types, then proceed to the block of this type, otherwise—to the block with the earliest date.

3.2.2. Finding the Block Boundaries (CalcBoundary Algorithm)

We now describe in more detail the auxiliary algorithm for calculating the block boundaries. Assume a set of cutting plans is given, as well as the paper output volume for each of them. The plans are grouped into blocks, inside which the same type of paper is produced. Let us determine whether these blocks can be ordered so that blocks of the same type do not follow each other.

Consider the following statement, which we will use to calculate the boundaries of the current block. Let $V_{min}, V_{max}$ be the minimum and maximum boundaries of the current block.

Statement 2. 

The volume of paper $V$ can be divided into blocks, the size of which falls within the range from $V_{min}$ to $V_{max}$ if and only if $V\ge V_{min}$ and $\lceil \frac{V}{V_{max}}\rceil V_{min}\le V$.

Proof of Statement 2. 

Necessity. Assume the required partition is possible. Let $k$ be the minimum number of blocks into which partitioning is possible. Then $V$ can be represented as a sum $V=V_1+\cdots +V_k$, where $V_{min}\le V_j\le V_{max}$, $j=1,\dots ,k$.

The condition $V\ge V_{min}$ is obvious. Then we have

$$\begin{gathered} V_{min}\le \frac{V_1+\dots +V_k}{k}\le V_{max}, \\ {V}_{min}\le \frac{V}{k}\le V_{max}, \\ k\le \frac{V}{V_{min}},\hspace{1em}k\ge \frac{V}{V_{max}}. \end{gathered}$$

(16)

and since it is impossible to split the volume into ($k-1)$ blocks, then

$$k-1<\frac{V}{V_{max}}.$$

(17)

Hence

$$k=\lceil \frac{V}{V_{max}}\rceil ,$$

(18)

and from (16), it follows that

$$\lceil \frac{V}{V_{max}}\rceil V_{min}\le V.$$

(19)

Sufficiency. Let $V\ge V_{min}$ and $\lceil \frac{V}{V_{max}}\rceil V_{min}\le V$. Denote $k=\lceil \frac{V}{V_{max}}\rceil$, $V_j=\frac{V}{k}$, $j=1,\dots ,k$. Obviously, $V_{min}\le V_j\le V_{max}$ for all $j=1,\dots ,k$.

Statement 2 is proven. □

3.2.3. Algorithm for Calculating the Current Block Boundaries

We now provide the algorithm for calculating feasible block sizes. The input values are the following:

• $V_{min}$—the minimum volume for the block;
• $V_{max}$—the maximum volume for the block;
• $V$—the total production volume.

The output of the algorithm:

• $V_{bottom}$—the minimum volume of the current block;
• $V_{top}$—the maximum volume of the current block.

In this case, the following conditions must be satisfied:

$$V_{min}\le V_{bottom}\le V_{top}\le V_{max},$$

(20)

and if the volume of paper $V$ can be divided into feasible blocks, then for each $\mathrm{v}\in \left[{\mathrm{V}}_{\mathrm{bottom}},{ \mathrm{V}}_{\mathrm{top}}\right]$, the volume $\left(V-v\right)$ can also be divided into feasible blocks.

The solution algorithm is as follows.

Check the Necessary Condition—Is It Possible to Divide into Blocks all Cutting Plans of One Product Type?

$$\mathrm{If} \lceil \frac{V}{V_{max}}\rceil V_{min}>V \mathrm{this} \mathrm{is} \mathrm{impossible}.$$

(21)

$$\mathrm{If} V_{min}\le V\le 2V_{min}\le V_{max}, \mathrm{then} V_{bottom}=V, V_{top}=V.$$

(22)

$$\mathrm{If} 2V_{min}\le V\le V_{max}, \mathrm{then} V_{bottom}=V_{min}, V_{top}=V-V_{min}.$$

(23)

If $V>V_{max}$, then let $k$ be the smallest natural number such that

$$\left(k+1\right)V_{min}\le V\le \left(k+1\right)V_{max}(k=\lceil \frac{V-V_{max}}{V_{max}}\rceil ).$$

(24)

Find the maximum $k^{\prime }\in \left\{0,\dots ,k\right\}$ such that

$$k^{\prime }{V}_{max}<\left(k^{\prime }+1\right)V_{min}\le V\le \left(k+1\right)V_{max}.$$

(25)

If $k^{\prime }<k$, then for any $V$ from interval $[k^{\prime }{V}_{min}, \left(k+1\right)V_{max}],$ it is possible to construct a feasible partition. At the same time, it is obvious that

$$k^{\prime }{V}_{min}\le V-V_{min}\le \left(k+1\right)V_{max}.$$

(26)

Hence

$$\begin{gathered} \left\{\begin{array}{c}{k}^{\prime }{V}_{min}\le V-v\\ V_{min}\le v\le V_{max}\end{array}\right. \\ \left\{\begin{array}{c}v\le V-k^{\prime }{V}_{min}\\ V_{min}\le v\le V_{max}\end{array}\right.. \end{gathered}$$

(27)

Then

$$V_{top}=\min \left\{V_{max}, V-k^{\prime }{V}_{min}\right\}.$$

(28)

If $k^{\prime }=k$, then find

$$\begin{gathered} \left\{\begin{array}{c}k{V}_{min}\le V-v\le k{V}_{max}\\ V_{min}\le v\le V_{max}\end{array}\right. \\ \left\{\begin{array}{c}V-k{V}_{max}\le v\le V-k{V}_{min}\\ V_{min}\le v\le V_{max}\end{array}\right.. \end{gathered}$$

(29)

Then

$$\begin{gathered} V_{bottom}=\max \left\{V_{min}, V-k{V}_{max}\right\}, \\ {V}_{top}=\min \left\{V_{max}, V-k{V}_{min}\right\}. \end{gathered}$$

(30)

The calculated values of $V_{bottom}$ and $V_{top}$ are respectively the lower and upper bound of the block volume.

3.3. The Third Stage

At the third stage, we use an approximate algorithm to minimize the changes of knives’ setups. We provide the model only for the case of one paper machine and one cutting block.

Let $N$ be the number of cutting plans, and $T$ be the time needed to change the position of one knife at the winder.

Define the set of knives’ positions at the winder for the cutting plan $j$, $j=1,\dots ,N$:

$$F\left(j\right)=\left\{{{\displaystyle \sum }}_{t=1}^{i-1}{f}_t{a}_{tj}+f_{i}y |i=1,\dots ,M,y=1,\dots ,a_{ij}\right\}.$$

(31)

Let us order the following:

$$F\left(j\right)=\{r_{j1}<r_{j2}<\dots <r_{jk\left(j\right)}\}, \mathrm{where} k\left(j\right)=\left|F\left(j\right)\right|.$$

(32)

The number of knife position changes during the transition from cutting plan $j_1$ to $j_2$ is

$$c\left(j_1, j_2\right)={\displaystyle \sum }_{l=1}^{\min \left\{k\left(j_1\right),k\left(j_2\right)\right\}}\alpha \left(r_{j_{1}l},r_{j_{2}l}\right)+\left|k\left(j_1\right)-k\left(j_2\right)\right|,$$

(33)

where

$$\alpha \left(x, y\right)=\left\{\begin{array}{c}1,\hspace{1em}\mathrm{if} x\ne y,\hfill \\ 0, \mathrm{otherwise}.\hfill \end{array}\right.$$

(34)

The required object is a permutation of cutting plans $I=i\left(1\right),i\left(2\right),\dots ,i\left(N\right)$.

The objective function is the minimum loss of output due to the knives’ setup changes:

$$\mathrm{T}={\displaystyle \sum }_{j=1}^{N-1}c\left(i\left(j\right), i\left(j+1\right)\right)\to \min ,$$

(35)

We propose to solve the problem by the local search procedure. Within each block, we compare all pairs of cutting plans and attempt to improve the solution by transposing them. Two cutting plans can be interchanged, if the resulting objective function value decreases and the volume of overdue output does not increase.

4. Implementation

The described solution method to the problem of optimal volume planning and scheduling was implemented in the Opti-Paper software, developed by the Russian IT company Opti-Soft together with researchers from Petrozavodsk State University [17,18].

The software is designed for planning and managing the production of paper and cardboard. It integrates the sales and production departments of a pulp-and-paper mill, ensures just-in-time order fulfillment and reduces production costs. The main features include strategic production planning, operational production planning, production scheduling, monitoring and automatic control of process equipment, data collection and storage, product quality management, product tracking, managing the finished goods shipment, production analytics and reporting.

A core component of Opti-Paper is an optimization module taking into account all important features of paper production in Russia and original solution algorithms.

5. Results

We tested the methods and algorithms on a case study from a paper mill in European Russia with a planning horizon up to 9 days. This case study involved the following: 18 production orders, 1 paper machine, 2 winders and 2 product types of 3 different grammages with 1, 2 or 3 layers. The planned production volume amounted to 1030 tons. The maximum roll width for the paper machine was 5600 mm. The paper machine productivity was 7 to 10 tons per hour. The winder productivity was 2 to 4 tons per hour.

The calculation time on a personal computer (Intel Core i7-4710, 2.5 GHz CPU, 16 Gb RAM) was under 5 min. The results are presented in

Table 1. The initial data for planning (tons of paper per day).

Order	Format mm	Type	Grammage g per m2	Layers	Breaches	26.11	27.11	28.11	29.11	30.11	01.12	02.12	03.12	04.12
101	2800	R	15	1							50			33
102	2800	R	15	2										35
103	2800	T	15	1					40	40	40	40	40	25
104	520	T	15	1						21.3
107	2560	T	15	3			26							5
109	980	T	15	3			19.2							9
110	2800	T	16	1		103	4	6		56				9
111	2800	T	16	1					24	30	30	30	30	30
112	2700	T	16	1			69
113	2700	T	16	1		85
115	2560	T	16	2			60
116	410	T	16	2			10			7	7	7	7
117	2800	T	17	1								4	4	4
118	2800	T	15	2

,

Table 2. The manually devised production plan relative to the customer requirements (tons of paper per day).

Order	Format mm	Type	Grammage g per m2	Layers	Breaches	26.11	27.11	28.11	29.11	30.11	01.12	02.12	03.12	04.12
101	2800	R	15	1		0	−3.68	−81.09	−81.09	−81.1	−109	−109	−109	−75.6
102	2800	R	15	2		0	0	0	0	0	−20.4	−36.8	−36.8	−1.8
103	2800	T	15	1		0	−81.09	−126.7	−159.6	−176	−179	−139	−99.4	−74.4
104	520	T	15	1	17.1	0	0	0	−17.88	3.42	3.42	3.42	3.42	3.42
107	2560	T	15	3	39.93	0	26	1.99	1.99	1.99	1.99	1.99	1.99	1.99
109	980	T	15	3	19.2	0	19.2	−8.38	−8.38	−8.38	−8.38	−8.38	−8.38	−3.38
110	2800	T	16	1	866.34	103	88.58	94.58	63.88	101	101	101	101	110
111	2800	T	16	1	594	0	0	0	24	54	84	114	144	174
112	2700	T	16	1	552	0	69	69	69	69	69	69	69	69
113	2700	T	16	1	404.76	85	85	85	66.76	16.6	16.6	16.6	16.6	16.6
115	2560	T	16	2	325.94	0	60	60	60	43.5	25.6	25.6	25.6	25.6
116	410	T	16	2	30	0	10	10	10	−1.43	−14.5	−7.52	−0.52	−0.52
117	2800	T	17	1		0	0	0	−45.09	−49.7	−49.7	−45.7	−41.7	−37.7
118	2800	T	15	2		0	0	0	0	−20.4	−20.4	−20.4	−20.4	−20.4

,

Table 3. The automatically calculated plan relative to the customer requirements (tons of paper per day).

Order	Format mm	Type	Grammage g per m2	Layers	Breaches	26.11	27.11	28.11	29.11	30.11	01.12	02.12	03.12	04.12
101	2800	R	15	1		0	0	0	0	−70.4	−34	−34	−34	−0.96
102	2800	R	15	2		0	0	0	0	−36.8	−36.8	−36.8	−36.8	−1.8
103	2800	T	15	1	0.44	0	0	0	−30.55	−51	−85.9	−64.6	−24.6	0.44
104	520	T	15	1	2.2	0	0	0	−11.92	0.44	0.44	0.44	0.44	0.44
107	2560	T	15	3	63.94	0	26	26	1.99	1.99	1.99	1.99	1.99	1.99
109	980	T	15	3	38.4	0	19.2	19.2	−8.38	−8.38	−8.38	−8.38	−8.38	−3.38
110	2800	T	16	1	103	103	−2.07	−44.18	−65.06	−9.06	−9.06	−9.06	−9.06	−0.06
111	2800	T	16	1	29.44	0	0	0	−0.56	29.4	−35.6	−64.1	−34.1	−4.06
112	2700	T	16	1	73.2	0	69	0.6	0.6	0.6	0.6	0.6	0.6	0.6
113	2700	T	16	1	765	85	85	85	85	85	85	85	85	85
115	2560	T	16	2	239.2	0	60	25.6	25.6	25.6	25.6	25.6	25.6	25.6
116	410	T	16	2	10	0	10	−28.52	−28.52	−21.5	−14.5	−7.52	−0.52	−0.52
117	2800	T	17	1		0	0	0	0	0	0	−8.43	−4.43	−0.43
118	2800	T	15	2		0	0	0	0	0	0	0	0	0

and

Table 4. The knives’ setup in mm (cutting plans).

Formats
2560 + 410 × 7
2560 + 980 × 3
2700 × 2
2800 × 2
520 × 10 + 240

and Figure 1 and Figure 2.

The tables show the initial data for the production (Table 1), the manual plan (Table 2) and the automatic plan developed by the system (Table 3), as well as Gantt charts illustrating the paper production sequences over time for the manual plan (Figure 1) and the automatic plan (Figure 2). The manual plan was devised without Opti-Paper software.

The initial data in Table 1 include order IDs, widths, product types, grammages and number of paper layers, as well as the volumes to be produced by the appropriate date.

Table 2 and Table 3, with manual and automatically calculated plans, show the output volume for each order by dates relative to the customer requirements, where the negative value indicates overproduction, and the positive value indicates underproduction. The “Breaches” column reflects the total volume of discrepancy between production plan and the customer requirements for the order.

The Gantt charts on Figure 1 and Figure 2 illustrate the production sequences of manual and automatically calculated plans (respectively). The “Grade” column combines the “Type”, “Grammage” and “Number of layers” columns from the previous tables, and the colors on the chart correspond to the cutting plans (Table 4):

Therefore, the total volume of violations for the manual and the automatic plans is 2849 t/days and 1325 t/days, respectively. Based on the optimality criterion mentioned earlier, the Opti-Paper system offers the production sequence that better matches deadlines in customer orders compared to the manually devised plan.

6. Conclusions

This paper describes the developed approach to optimizing volume planning and scheduling the production of paper goods with a special emphasis on devising the optimal sequence of cutting with smooth product type changes, taking into account a number of conflicting requirements.

The mathematical model of the problem is formulated and analyzed. The solution process consists of three stages. At the first stage, we solve the integer linear programming problem of large dimension by using a combination of the simplex method with column generation and the branch and bound methods. At the second stage, we apply a greedy algorithm with block size estimation heuristics. At the third stage, we apply the local search procedure.

The proposed mathematical model and numerical methods for solving the described problem were implemented in the Opti-Paper software tool and tested on real production data from a paper mill in European Russia. The Opti-Paper system offered a production sequence that better matches deadlines in customer orders compared to the plan devised manually by production planners.

The solution algorithm for the second (most difficult) stage has polynomial complexity thanks to the greedy approach, which is its advantage. This same approach is also its disadvantage, as in some cases, the optimal solution is obtained with a different block selection sequence. It is possible that an algorithm with partial enumeration of block selection trajectories may be considered to cope with that issue.