A design of experiments based on the Normal‐boundary‐intersection method to identify optimum machine settings in manufacturing processes

Finding the appropriate machine settings for a given manufacturing process is an important issue in industrial production. A set of minimum and maximum machine settings correspond to the lower and upper quality limits that are specified for the produced product, and by this define the boundaries of all appropriate machine settings. This paper shows that these boundaries are the solution of a multi‐objective optimisation problem, which is called the optimum machine settings problem. However, for most processes there is no mathematical model of the manufacturing process available, which maps the setting parameters on the quality key figures in a way that allows to compute the optimisation problem. In this case, experiments may provide the required empirical data simultaneously while executing the optimisation procedure. Using a case study on heat sealing in industrial packaging, the paper shows, how to develop a design of experiments based on the Normal‐boundary‐intersection method (NBI), and how to generate the Pareto‐frontier by executing test according to this test plan. It addresses the specific limitations inherent in solving an optimisation problem by experiments. The behaviour of the method towards discrete and binary objectives and constraints is discussed.


INTRODUCTION
Automated industrial production supplies a majority of consumer products, including food, cosmetics, and pharmaceuticals.Production plants usually consist of a number of linked machines each providing a specific process step in the production chain.It is of paramount importance for the quality of the products and the efficiency of the production to adjust these processing machines appropriately.Sections 2 and 3 give a formal description of this problem.However, there is no quantitative model available for most manufacturing processes that would allow to calculate the appropriate machine settings in advance.For this reason, empirically testing a machine under production conditions is the standard approach to identify the machine settings.The most popular method is statistical design of experiments [1].Since this method has some substantial disadvantages regarding approximation errors, this paper presents a deterministic alternative based on multi-objective optimisation.The new approach is elaborated and evaluated in Section 4 using a case study on heat sealing of flexible packaging.
F I G U R E 1 Robustness plot of a scalar processing function with  = (, ), and different settings  1 ,  2 , and  3 .

Processing function
Let  = [ 1 ,  2 , … ,   ] ⊤ ∈ ℜ  be the vector of quality key figures, which characterise the product to be produced.Let further  = [ 1 ,  2 , … ,   ] ⊤ ∈ ℜ  be the vector of setting parameters to adjust the production process.Finally, let  = [ 1 ,  2 , … ,   ] ⊤ ∈ ℜ  denote the vector of uncertain, or unknown influencing factors.Then the processing function is with  ∶ {ℜ  , ℜ  } → ℜ  , which is usually unknown, but assumed to be continuous for most processes.

Product quality and process robustness
Since Shewhard's fundamental work on quality management in industrial production [2], quality is seen as a matter of fulfilling the requirements of the product.Formally, one lower and one upper limit for each quality key figure, called quality criteria  low Fluctuating values of the influencing factors  cause unintended deviations of the quality key figures .In the event of certain amplitudes of the influencing factors, the quality criteria may be violated and the product is rejected, compare ref. [3].Thus, the robustness criteria  low  and  in the event that both options are possible.A process is called robust, if it does not violate the limits specified by the critical requirements.

Feasible and appropriate machine settings
If a machine performs a robust process or not, eventually depends on the machine settings to adjust.In this case, feasible and appropriate machine settings shall be distinguished as Figure 2 shows for an example with two setting parameters: Feasible machine settings are those which the operator can choose from, and appropriate settings are those which guarantee quality products and process robustness, compare Figures 1 and 2. In order to operate the machines under various circumstances, knowing all appropriate machine settings is preferred over selecting only one favourable working point.In this context, adjusting a processing machine requires the identification of the boundaries of the set that includes all feasible and appropriate machine settings  f as ∈  ∩ .This task may be equivalent to solving a multi-objective optimisation problem.
In addition, the term Pareto optimal is used, if not explicitly referring to the objective or decision space, compare [5].In this case, the solution of a multi-objective optimisation problem consists of a set of equally optimal solutions: Definition 3.2.The set  * with  * = { * ∈ } is called the Pareto frontier, and includes all non-dominated points.
The Pareto frontier therefore represents the solution of a multi-objective optimisation problem in the objective space, and solving a multi-objective optimisation problems means generating the Pareto frontier.

Defining the optimisation problem
When adjusting a processing machine, the quality key figures  = [ 1 , Similarly, maximising  = (, ) and using   ≤  upp  gives all Pareto optimal solutions for the upper quality criteria.Assuming continuity, the efficient settings represent the boundaries of the set of the appropriate machine settings with  * ∈ .Solving this optimisation problem by tests on the processing machine may provide the set of feasible and appropriate machine settings.The next section implements this approach for an exemplary manufacturing process.

Heat sealing in industrial packaging processes
Heat sealing of polymer films is widely used in packaging machines for closing packages of any kind of consumer products [6].By pressing a hot sealing jaw against the packaging films for a certain time, the film layers fuse and form a tight bond (see Figure 3).The sealed seam is a major focus of quality management, because it determines the resistance of the package towards stresses from transportation, barrier against migration of contaminants, and accessibility of the product.For this reason, knowing the appropriate settings of a heat sealing process is a key factor for operating a packaging machine.] ⊤ are the setting parameters to adjust.The most critical influencing factor is the overall thickness  of the packaging film layers to seal, which determines robustness of the heat sealing process by  s = {| low ≤  ≤  upp }, compare ref. [7].Thus, the sealing function

The sealing function
F I G U R E 4 Appropriate settings of a heat sealing process.
formally describes the heat sealing process, compare ref. [8].Although, there is no comprehensive model of the underlying mechanisms of heat conduction [6] and molecular diffusion [8], it is generally accepted that increasing sealing temperature and dwell time increases seal strength, and that thicker films require higher sealing temperatures and longer dwell times [7].Formally, this means that the sealing function is continuous and monotonic.

]
. Since the sealing function ( 4) is assumed to be continuous and monotonic, the feasible and appropriate settings of a heat sealing process  f as s ∈  s ∩  s are located between two convex boundaries (Figure 4), which are the solution of the following optimisation problem.

The optimum machine settings problem of heat sealing
As a consequence of continuity and monotony, the optimality condition from Definition 3.
where the quality key figure is no longer an objective, but defines a constraint of the optimisation problem.Continuity and monotony also allow the application of this problem to both the lower and the upper implicit boundary.

Selection of the optimisation method
Solving the optimum heat sealing settings problem by executing experiments on a packaging machine makes special requirements to the applied optimisation method: (1) The experimenter may be incapable of controlling more than one parameter at the same time, and of observing multiple quality key figures.An optimisation method must therefore provide a unique, predefined instruction of how to vary the setting parameters.( 2) Experiments generate costly machine downtime, and therefore should consume as little time as possible.An optimisation method should therefore require as few test points as possible, and avoid iteration loops.(3) To know the feasible and appropriate settings, it is important to identify the complete implicit boundary.Thus, generating a complete and even Pareto frontier including also weakly Pareto optimal solutions is preferred.Optimisation methods based on scalarisation convert a multi-objective optimisation problem into a sequence of singleobjective problems, which are special cases of the Pascoletti-Serafini problem [9].The experimenter must then minimise only one scalar  according to a certain algorithm instead of varying dwell time  d and sealing temperature  s iteratively.Thus, an optimisation method fulfilling the first and second requirement must be from this class of approaches.
The Normal-boundary-intersection method (NBI) is a scalarisation method, which generates an evenly distributed set of points on the Pareto frontier including weakly Pareto optimal points [10].Since alternative scalarisation methods either do not provide weakly Pareto optimal points or must be adapted by mathematically complex parameter control to generate evenly distributed points on the Pareto frontier, compar ref. [9], the NBI method is a promising candidate to build a design of experiments on.

Adapting the NBI method to design of experiments
The following design of experiments gives an unique instruction to generate the Pareto frontier of the optimum heat sealing settings problem (5), which represents one implicit boundary.It is a simplified adaption of the NBI method from [10] for the bi-objective case extended by an algorithm defining the initial test points of the optimisation.Figure 5 illustrates the test plan.
where  t * s +  is the starting point of a test series, and n is the direction of the tests to execute.Output: Each solution according to optimisation problem (10) is an element of the Pareto frontier of the optimum heat sealing settings problem (5) and the boundary of the feasible and appropriate heat sealing settings.

Case study
A typical application of heat sealing in packaging machines is making the transversal seam of a pouch bag, see Figure 3.This seam should be strong and leak-tight for safe transport and distribution, which requires a seam strength of minimum  low s = 5N∕15mm, and should enable easy opening for older and younger people, who may not apply more than    6 shows the test plan for the lower boundary according to Algorithm 4.1.Figure 7 gives the corresponding seam strengths, and Figure 8 the resulting appropriate settings for this heat sealing process.Refer to ref. [7] for a detailed description of this case study and interpretation of the results.

Limitations of the Pareto-based approach
Although the presented design of experiments reliably identifies the set of feasible and appropriate machine settings, it has two limitations.Firstly, the NBI method may not find all Pareto optimal points for  > 2 setting parameters [5].While this effect does not appear for the presented case study, it may cause problems for other applications.In this case, an alternative scalarisation method may provide a better base for a design of experiments.Secondly, discrete setting parameters and binary quality key figures restrict the test series to execute, and induce inaccuracy of the Pareto optimal points.The first effect is apparent in Figure 6 from the discrepancy between calculated test series and executed test points rounded to the closest increment Δ d = 0.1  and Δ s = 1 .This effect also causes that for the upper boundary in Figure 8 any additional test series between the anchor points would increase the number of test points, but would not provide a more accurate solution.Furthermore, the authors showed in ref. [11] that binary quality key figures lead to significantly inaccurate Pareto frontiers.However, this limitation is due to the setting parameters and quality key figures not being real numbers, but the multiple  of an integer, for example, (1∕)  ∈ ℤ, and may therefore occur with any other approach, as well.

CONCLUSION
This paper has addressed the problem of adjusting industrial production processes.Based on standard terms for product quality, robustness analysis, and multi-objective optimisation, a general formulation for the machine settings problem was defined.A deterministic design of experiments based on the NBI method to identify the optimal settings of a production process was proposed for the heat sealing process.It was shown that the formal description of production processes is qualified for practical applications, and the Pareto-based design of experiments reliably identifies the appropriate sealing settings for a required product quality and process robustness.Further research may address machine settings problems with more than two setting parameters, investigate deterministic design of experiments for non-convex and non-monotonic processing functions, and deal with discrete setting parameters and binary quality key figures.Besides continuous methods, as presented in this paper, approaches from integer programming may also provide a suitable base for design of experiments.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

Definition 2 . 1 .
the requirements.Two types may be distinguished: the two-sided quality criteria, where both limits are real numbers { low  , upp  } ∈ ℜ, and the one-sided quality criterion, where either  low  A product is called a quality product, if  qp ∈  with  = {| low  ≤   ≤  upp  ,  = 1, 2, .., }.

Definition 2 . 2 .Definition 2 . 3 .
upp  specify a required range for each influencing factor   , within which the process shall fulfil the quality criteria  low  and  upp  .See Figure 1 for a unidimensional example with {, , } = 1.Process robustness is defined as the set  = {| low  ≤   ≤  upp  ,  = 1, 2, .., }.The pairwise combinations of quality and robustness criteria  , = [ for the manufacturing process, where the notation  crit  denotes  low  or  upp

F I G U R E 3
Heat sealing station and pouch bag.

Definitions 2 .≤≤
3 to 2.5 apply for the heat sealing process by the following, compare ref.[7]: the technical restrictions of the packaging machine and the material properties of the packaging film define the set of the feasible settings with s = { s | min d  d ≤  max d , min s  s ≤  max s }.The set of the appropriate settings is given by  s = { s | s =  s ( d ,  s , ),  s ∈  s ,  ∈  s } in respect of the quality and robustness criteria.The critical requirements are  , = [

F I G U R E 5
Design of experiments identifying the optimum sealing parameters, which constitute one boundary  of the set of feasible and appropriate sealing parameters.

Algorithm 4 . 1 .≤≤
(Design of experiments based on the NBI method) Input: Set  min s  s ≤  max s and  min d  d ≤  max d to define the feasible setting parameters  s .Define the quality and the robustness criterion for the implicit boundary to generate by: If the minimum boundary is required, set  crit s =  low s and  crit according to the critical requirements.Else set  crit s =  upp sand  crit according to the critical requirements.Step 1: Execute the initial test points to define the initial direction of optimisation by: For  = t, T execute tests  , which represent the individual optimum of  d and  s , by: to the seam.A quality seam is therefore defined by  s = { s |5N∕15mm ≤  s ≤ 25N∕15mm}.The seam features sections of two and four film layers to seal through, so that  = {|2 lay ≤  ≤ 4 lay }.Therefore, the critical requirements of the process are  s = [  s low  upp  s peel  low ].The minimum dwell time is restricted by the sealing station to  min d = 0.2 , and the maximum dwell time is  max d = 0.8  due to productivity issues.The thermal properties of the packaging determine the minimum and maximum sealing temperature by  min s = 100 •  and  max s = 180 • .Thus the set of feasible sealing settings is  s = { s |0.2  ≤  d ≤ 0.8 , 100 •  ≤  s ≤ 180 • }.

F I G U R E 6 F I R E 7 I G U R E 8
Rounded test points for the lower boundary.Results of for each test series and corresponding Pareto points.Resulting boundaries and appropriate settings of the case study.
2 , … ,   ] ⊤ can be seen as objective variables, and the setting parameters  = [ 1 ,  2 , … ,   ] ⊤ as decision variables, of a multi-objective optimisation problem.Definitions 2.1 to 2.4 serve as constraints of this optimisation problem with  ∈ ,  ∈  , and  ∈  .The following optimum machine settings problem gives the Pareto frontier for the lower quality criteria of the process, which includes all non-dominated values  * and the corresponding efficient settings * : min   = (, ) Seal strength  s is the most frequently mentioned quality key figure for heat sealing processes, and a quality seam is usually given by  Sealing temperature  s and dwell time  d have the strongest effect on seal strength, so that  s = [ d ,  s qs s ∈  s with  s = { s | low s ≤  s ≤  upp s }.