## Abstract

In engineering-design problems, usually, there are multiple goals with different units, continuous and discrete variables, nonlinear equations, nonconvex equations, and coupled decisions. Ideally, all goals’ target are reached simultaneously within the feasible space. However, the optimal solution may not be available. To deal with all those complexities, a modeling strategy named “satisficing” was proposed in the 1980s. The satisficing strategy allows designers to find “good enough” but may not be optimal solutions. In this paper, we review the publications applying the satisficing strategy on engineering-design problems, and categorize the methods regarding the design stages they manage. We define the methods dealing with all four design stages—formulation, approximation, solution, and evaluation—as the whole process satisficing methods. We review the publications using the whole process satisficing strategy in great detail. In the past 30 years, the whole process satisficing strategy has been improved and applied to a wide variety of engineering-design problems, based on which derived methods, concepts, and platforms are developed. We generalize the specialties, advantages, and scope of applications of the methods in the whole process satisficing strategy. We expect this paper provides information on when and how designers may apply satisficing for their problems.

## 1 Special Requirements for Methods in Engineering Design

Engineering design is a task that involves a group of designers with different interests and knowledge to make decisions that comply with their mutual requirements [1]. As decision-making requires designers to select the desired solution among several alternatives regarding the output performance, one or more objective functions are usually consciously or subconsciously applied by the designers to measure the output performance. Designers can define an objective function through different methods, e.g., through utility theory [2], game theory [3], analytic hierarchy process [4], Pareto comparisons [5], etc. However, regardless of the seeming rationale that those methods present, the use of utility theory may result in poor decision-making and may not be appropriate for engineering design [6]. In addition, it is difficult for designers to identify and maintain the optimal solution using optimization methods, given that there are various complexities and uncertainties underlying an engineering-design problem. Moreover, the mission of engineering design is more than obtaining optimal solutions. There are other tasks and foci, such as acquiring knowledge on the output performance in a wide range of solution space (not only the near-optimal area), improving the robustness of the decision model, coupling the continuous and discrete decisions, connecting sub-models with different levels of refinement or fidelity, and so on.

### 1.1 Designers Accept Problematic Assumptions When Applying Utility Theory.

For optimization problems that minimize or maximize a mathematical objective, one can define that such problems are based on the maximization of expected utility, which enables designers to select the optimal solution among alternatives. Nevertheless, not all design problems can be solved using optimization. Even if the optimal solution to the decision model is acquired, it may not be optimal or even feasible for the physical problem, as the optimality conditions can be easily broken [7]. This often results from the incapability of the decision model to capture all information and requirements [8] of the design. One source of the incompleteness and inaccuracy of the model is that by modeling the objective function using utility theory, designers naturally accept the assumption that all decision-makers are rational and every detail in the decision model perfectly maps the physical world to the model abstraction. Such an assumption is often wrong, especially in engineering design.

### 1.2 Complexities Incorporated in Engineering Design Cause Convergence Failure.

Engineering-design problems encounter multiple complexities [9]; therefore, the decision models may have nonlinear, nonconvex equations, multiple objectives with different units and scales, and objectives with different levels of achievability. These complexities sometimes lead to immature convergence or no convergence.

### 1.3 Multiple Sources of Variation Bring Uncertainties.

How do uncertainties take place? From the view of the source of uncertainties, there are variations (i) in environmental or other noise-noise factors [10], (ii) in design (or decision) variables—control factors [11], (iii) brought by modeling methods [12,13], and (iv) brought by the process of managing the previous three types of variation [14]. How do uncertainties affect the design? From the perspective of optimality conditions, uncertainties that break the equilibrium of any Karush–Kuhn–Tucker (KKT) conditions destroy the optimality of a solution, thereby making an optimal solution infeasible or useless [15].

### 1.4 The Tasks of Engineering Design Include Way More Than Finding the Optimal Solution.

The demand for case-by-case, general, or reusable knowledge and insight through the designing process entails design methods to enable learning model behaviors [16], exploring the solution space (ESS) to satisfy a variety of design scenarios [17,18], determining the refinement of a model and its sub-models [19,20], and choosing the proper set of solutions as the input of the previous stage of design [21–23]. Although optimization methods sometimes work efficiently in returning optimal solutions to mathematical models, they do not support knowledge attainment.

Therefore, we need a design strategy to overcome the shortcomings in optimization methods, including avoiding the assumption in utility theory-based functions, managing the typical complexities and uncertainties in engineering-design problems, and supporting knowledge learning through post-solution analysis.

In Sec. 2, we summarize the satisficing methods in the literature and identify the whole process of satisficing in engineering design. In Sec. 3, we introduce why the whole process satisficing works as an alternative and effective in managing design problems. We describe a packet of methods that realize the whole process satisficing strategy. In Sec. 4, we summarize the research topics, derived work (new features and modules, tools, methods, theories, concepts, frameworks, and platforms), applications, and contributions to the whole process satisficing strategy. In Sec. 5, we anticipate the way forward of whole process satisficing strategy regarding the ideas, functions, and applications. In Sec. 6, we conclude the contributions and limitations of this paper.

## 2 Literature on Satisficing Strategy

An alternative strategy to optimizing is called “satisficing.” The satisficing strategy provides “good enough” but not optimal solutions to complex problems. The term satisficing is a portmanteau of satisfy and suffice [24]. The criteria of “good enough” vary with context. There are a variety of methods and techniques to reach satisficing in literature. To gain a systematic understanding of “what is satisficing,” “why and when do designers use satisficing strategy,” “how does satisficing strategy work,” and “what are the classic satisficing methods,” we first generally review the literature using satisficing strategy to manage engineering-design problems, and then choose one packet of methods that realize the “whole process satisficing” for a deep review.

Using the Web of Science, by searching “satisficing” and “engineering design” in topic, we obtained 189 publications. By manually selecting the relevant journal articles and research articles that propose new methods, we end up with 50 publications. We review those selected articles regarding the proposed methods, complexities being managed, intellectual merits, foundational theory, application field, and limitations. Among the 50 publications, we select 33 representative publications, and summarize the focus and technique of the proposed methods. In Table 1, “Representative” papers are those that fall into one or more of the following categories:

Articles that manage engineering-design problems with one or more typical features identified in Sec. 1.

The baseline methods or foundational theories in those articles are relatively well-known or generally used.

The most cited articles or the articles with core intellectual merit of an author or a group of authors who have published a series of papers on the topic.

Stage | Formulation | Approximation | Solution | Evaluation | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Focus/Means | Fuzzy formulation | Probability/chance optimization, stochastic programming | Formulation improvement through iterations or gaming | Change the format of objectives or add objectives | Integrate or couple models, concurrent design models | Linearization | Relaxation (change the model) | Use other techniques to approximate the model, (surrogate model, neural network, Bayesian etc.) | Interior-point searching | Forego optimality (relaxation without changing the model) | Combining interior-point searching with simplex algorithm | Tradeoff analysis and discussion | Use information in evaluation to modify formulation | Sensitivity analysis, loss-gain analysis | Develop evaluation functions or judgment parameter |

Example papers | [25–31] | [32–36] | [26,30,37–39] | [22,27,32,38,40–47] | [22,44,48] | [23,42,44] | [25] | [22,23,31,35,36,39,47] | [25,31,33,43,49,50] | [48] | [22,23,42,44] | [26,47,49,51,52] | [22,23,27,38,39,45,46,53] | [31,32,35] | [22,23,28,29,33,35–37,41–44,53,54] |

Stage | Formulation | Approximation | Solution | Evaluation | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Focus/Means | Fuzzy formulation | Probability/chance optimization, stochastic programming | Formulation improvement through iterations or gaming | Change the format of objectives or add objectives | Integrate or couple models, concurrent design models | Linearization | Relaxation (change the model) | Use other techniques to approximate the model, (surrogate model, neural network, Bayesian etc.) | Interior-point searching | Forego optimality (relaxation without changing the model) | Combining interior-point searching with simplex algorithm | Tradeoff analysis and discussion | Use information in evaluation to modify formulation | Sensitivity analysis, loss-gain analysis | Develop evaluation functions or judgment parameter |

Example papers | [25–31] | [32–36] | [26,30,37–39] | [22,27,32,38,40–47] | [22,44,48] | [23,42,44] | [25] | [22,23,31,35,36,39,47] | [25,31,33,43,49,50] | [48] | [22,23,42,44] | [26,47,49,51,52] | [22,23,27,38,39,45,46,53] | [31,32,35] | [22,23,28,29,33,35–37,41–44,53,54] |

During the literature review, we find that so far, there has not been a review article that systematically categorizes the satisficing methods based on different stages in engineering design, specific techniques, and the complexities of their problems. Such categorization is necessary because it may provide guidance to users in selecting appropriate satisficing methods. Therefore, we provide such a categorization in Table 1.

We define, in engineering-design problems, there are four stages, model formulation or formulation modification (F), model approximation (A), solution searching (S), and evaluation (E). Each method proposed in the literature may deal with one or more stages through different means, and we mark the categories of the 33 representative papers in Table 1. Although researchers reach satisficing using various modeling and measurement methods, they all agree on the idea that “good enough” but not necessarily optimal solutions well balance the performance of their designs and the workload of finding the solutions.

To show the details about the problems, methods, and contributions of the selected publications for readers to further explore each division of the topic, we select 13 milestone papers from the 33 and summarize their information in Appendix. The selection of the milestone articles is based on our knowledge and experience about the effectiveness of the methods and the typicalness of engineering-design problems used in the papers.

Among those selected publications, only Mistree’s research team proposes methods that deal with all four stages in engineering design. In this paper, we name it “whole process satisficing,” which is also the reviewing focus of this paper. We find out the publications in whole process satisficing in engineering design, analyze what specialties in the method that enable designers to acquire satisficing (Sec. 3), categorize the application fields (Sec. 4), and anticipate the way forward (Sec. 5).

## 3 Whole Process Satisficing and Why

Modeling methods for engineering-design problems fall into two categories, (i) formulating a complex problem exactly and searching for the optimal solution or near-optimal solutions, and (ii) approximating a complex problem and finding a set of solutions that output acceptable performance. We name the first strategy “optimizing” and the second one “satisficing.” The major differences between the two strategies are represented in three aspects: model formulation, solution algorithms, and post-solution analysis [8,15]. In this section, we describe the typical features of the satisficing strategy regarding those three aspects.

The concept of satisficing solution or satisficing searching is proposed by Simon [55–57] as “good enough” solutions that make the design output acceptable but maybe not optimal performances. Later, the idea of satisficing is put into practice by several researchers. In the past three decades, Mistree and his research team [8,58–60] realized the whole process satisficing strategy in engineering design using the compromise decision support problem (cDSP) as the formulation construct, the adaptive linear programming algorithm (ALP) as the solution algorithm, and the decision support in the design of engineering systems (DSIDES) as a platform and solver to formulate and solve a cDSP with the ALP. In this paper, when referring to the whole process satisficing, we specifically mean using the cDSP, ALP, and DSIDES to realize satisficing in each of the four stages of engineering design. For decades, the cDSP-ALP-DSDIES community has systematically developed and derived a packet of design methods along the satisficing strategy, especially for engineering designs. In this paper, we discuss the whole process satisficing strategy mainly by reviewing this community’s representative works. In the context of using the cDSP-ALP-DSDIES packet, we define satisficing solutions as solutions that only meet the necessary KKT conditions.

### 3.1 The Specialty in the Formulation.

Examples of the formulations using the optimizing strategy include mathematical optimization and its variants, such as goal programming. Solutions are usually obtained by identifying the Pareto front consisting of non-dominated or near-optimal solutions using optimization solution algorithms. The formulation of design problems using satisficing strategy, namely, the cDSP, has the key features that allow designers to identify satisficing solutions that meet the necessary KKT condition but not the sufficient condition.

The format of a nonlinear optimization problem is like this: For a given objective function *f*(*x*), Euler and Lagrange develop the Euler–Lagrange equation forming the second-order ordinary differential equations $\u2207xx2f(x)$ to facilitate finding the stationary solutions. The value of the variables that maximize *f*(*x*) within the feasible set $F$ is the solution to the optimization problem, where $F$ is the set bounded by constraints and bounds. The format of an optimization problem $O$ can be represented as follows. *x* is the vector of decision variables as real numbers. *g*_{i}(*x*) is the *i*th inequality constraint, and *h*_{j}(*x*) is the *i*th equality constraint. Any point *x* that is a local extrema of the set mapped by multiplying active equations with a non-negative vector, is a local optimum of $O$ [61], denoted as $x*$. The elements of such a non-negative vector are Lagrange multipliers, *μ* and *λ*.

** The format of an optimization problem**$O$

*:*

*Given*

*Find*One variant of optimization is goal programming. The format of a goal programming problem $Ogoal$ is represented as follows. A target value *T* is predefined for the objective function *f*(*x*) as the right-hand side value, so the objective becomes an equation, and we call it a goal. *d*^{−} and *d*^{+} are deviation variables measuring the underachievement and over-achievement of the goal toward its target. The problem is solved by minimizing the deviation variables, which is minimizing the difference between *f*(*x*) and *T*. In other words, goal programming is aimed at finding *T*’s closest projection on $F$.

** The format of a goal programming problem**$Ogoal$

*:*

*Given*

*Find*In the cDSP, elements of mathematical programming and goal programming are combined. A cDSP ℂ is represented as follows. For a nonlinear cDSP, we first linearize the nonlinear equations, including nonlinear constraints and nonlinear goals. Therefore, the nonlinear cDSP first becomes a linear problem with a linear goal *Goal*^{linear}, a linear feasible space $Fli$ bounded by linear constraints *g*(*x*)^{li} ≥ 0 and *h*(*x*)^{li} = 0. Thus, using a cDSP, we seek the closest projection from the linear goal set onto a linear feasible set. We define the solution as a satisficing solution, and we use *x*^{s} to denote it.

** The format of a cDSP** ℂ

*:*

*Given*

*Find*The difference between $x*$ and *x*^{s} is that $x*$ conforms both the necessary (first-order) and sufficient (second-order) KKT conditions, whereas *x*^{s} conforms the necessary KKT condition but may not conforms the sufficient KKT condition. This is because the second derivative of the linear equations, *Goal*^{li}, *g*_{i}(*x*)^{li} ≥ 0, and *h*_{j}(*x*)^{li} = 0, degenerates—as a result, no uncertainty may affect the feasibility of *x*^{s} since no uncertainty gets a chance to break the equilibrium of the second-order Lagrange equation. In addition, when the convexity of $F$ is greater than the convexity of the *f*(*x*), $x*$ may not be identified as the second-order Lagrange equation has no solution, but *x*^{s} is obtainable due to its irrelevancy with the second-order KKT condition. The details of the mathematical demonstration and illustration using example problems are in Chap. 2 of Ref. [7].

### 3.2 The Specialty in the Solution Algorithm.

There are different ways of categorizing the solution algorithms. Sharma and Kumar [62] proposed that all solution algorithms fall into two categories, classical methods and metaheuristics algorithms. Classical methods are based on vertex searching or solving equation set to meet the KKT conditions, (e.g., Lagrange’s multiplier method, branching, etc.) Metaheuristics algorithms include deterministic searching methods (Tabu search and local search) and stochastic searching-based methods (simulated annealing, Hill climbing, ant colony algorithm, evolutionary algorithm, etc.). Abualigah et al. [63] classified solution algorithms into two types—metaheuristics algorithms and heuristics algorithms. The former contains three sub-categories, local-based algorithms, evolutionary algorithms, and swarm-based algorithms. The latter specifically refers to vertex searching algorithms such as the simplex algorithm. Daoud et al. [64] refined the definition by identifying four sub-categories in the metaheuristic algorithms: physics-based, population-based, human-based, and evolutionary-based algorithms; they define the local searching algorithms as heuristic algorithms.

In this paper, not only do we care about the way of solving a problem, but we also discuss whether the solution is optimal and whether a set of non-optimal but good enough solutions may provide designers with more options and relevant knowledge in the design space. We use a matrix to categorize the solutions algorithms, as Table 2 shows. The two dimensions of the matrix are the optimality of the converged solutions and the searching conditions. No interior-point searching algorithms guarantee the optimality unless KKT conditions are applied to facilitate the examination of the optimality. Such an examination is a posteriori supplementary method; therefore, we define all interior-point searching algorithms in the category of converging near-optimal solutions.^{2}

Converge the optimal solution | Type of converged solutions | Strength | Weakness | ||
---|---|---|---|---|---|

Near-optimal | Satisficing | ||||

Vertex search | Simplex, dual simplex, Lagrange’s multipliers | slp, sql | ALP | Requiring relatively low computing power; may give model formulation insight | May not work effectively for nonconvex problems |

Interior-point search | Metaheuristic interior-point searching algorithms (non-dominated sorting genetic Algorithm (NSGA), gradient-based optimizer (GBO), etc.) | Easy to deal with nonlinear, high-dimension problems | Requiring large computing power; no insight of formulation modification |

Converge the optimal solution | Type of converged solutions | Strength | Weakness | ||
---|---|---|---|---|---|

Near-optimal | Satisficing | ||||

Vertex search | Simplex, dual simplex, Lagrange’s multipliers | slp, sql | ALP | Requiring relatively low computing power; may give model formulation insight | May not work effectively for nonconvex problems |

Interior-point search | Metaheuristic interior-point searching algorithms (non-dominated sorting genetic Algorithm (NSGA), gradient-based optimizer (GBO), etc.) | Easy to deal with nonlinear, high-dimension problems | Requiring large computing power; no insight of formulation modification |

The ALP algorithm performs both interior-point searching and vertex searching to maintain a balance between exploitation and exploration. With the module “XPLORE” in DSIDES, a wide-spread interior-point random search is performed to select a good starting point (see Fig. 1), then the problem is linearized around the starting point using the second-order sequential linear programming algorithm (SLIP2), and then the linear problem is solved using dual simplex. The linearization can go through a number of iterations to make sure the nonlinear problem is linearized sufficiently and accurately at different local optimums. As there are often more constraints than variables in the engineering-design problems, using the dual simplex is easier than using the simplex. In this way, the solution is on the vertex of the linearized problem and meets the necessary KKT condition to ensure near optimality. The difference between a satisficing solution, a near-optimal solution, and the optimum is illustrated in Fig. 2.

With the satisficing solution algorithm—the ALP, why the nonconvex problem can be dealt with, and why is the linearization relatively accurate? Two mechanisms in the ALP make it possible to linearize the nonconvex—using the SLIP2 algorithm and the accumulation of the linear constraints. The SLIP2 first uses the parabola (the one that passes through the starting point and the two intersection points of the surface and *x*1–*x*2 plane) to approximate the nonlinear surface, then linearizes the parabola into a plane. And the linearized planes from multiple iterations are accumulated as the linear constraints to replace the nonlinear constraint. In this way, the nonconvex equations can be linearized, and the linearized constraints are more accurate. The detailed mathematical descriptions and illustrations are given in Ref. [65].

### 3.3 Features That Support Knowledge Attainment and Coupled Decision-Making.

Managing engineering-design problems requires designers to find a set of satisficing solutions and way more than that—gain knowledge about the design space, make coupled decisions, and tailor the methods to engineering designs in different fields. The satisficing strategy provides a good construct to complete those tasks. For example, DSIDES provides the formulation construct and solvers for cDSP, selection DSP (sDSP), and any sequences or combinations of the compromise and selection DSPs, which support the mixture of continuous and discrete decision-making.

We explain how the satisficing strategy realized using cDSP-ALP-DSIDES supports various tasks in engineering design by guiding readers through the relevant publications in the past three decades in Sec. 3.

### 3.4 Scope of Applications and Limitations.

Satisficing strategy facilitates designers to deal with engineering problems with multiple goals, goals with various units and achievability, nonlinear features, continuous and discrete variables, evolving design preferences, and coupled decision-making. However, for problems that do not contain the features above, the satisficing strategy may not outperform optimization methods.

In addition, to formulate a problem into a cDSP, a target for each goal should be defined. For an objective that has no target value, or when designers have no idea about the target value, the satisficing strategy may not be the best option.

## 4 Applications and Evolution of Satisficing Strategy

There are seven major contributions (subjects) in the satisficing engineering designs in the past three decades: (i) robust design methods, (ii) multiscale and microstructure design, (iii) multistage design and concurrent design, (iv) coupled decision-making, (v) exploration of the solution space and design space, (vi) multigoal (multi-objective) or multidisciplinary design, and (vii) knowledge-based design and platformation. There are overlaps among those subjects, as some applications in one subject are the verifications of the methods in other subjects. In this section, we select the representative work in each subject based on the utility, popularity, and extendibility of the publications and summarize the typical features, contributions, and scope of applications.

### 4.1 Robust Design Managing Four Types of Uncertainty.

The idea of robust design using the satisficing strategy is to make the design relatively insensitive to different types of uncertainty [12,14]. Uncertainties are classified into four types based on their sources [12]: Type I—the uncertainty brought by the variation in environmental noise or other noise factors [10], Type II—the uncertainty brought by the variation in design variables of control factors [11], Type III—the uncertainty introduced by modeling methods [12,13], and Type IV—the uncertainty introduced during the management of the previous three types of uncertainty [14]. We give an example of the four types of uncertainty in supply chains (SCs) in Table 3. There are four types of robust design using the satisficing strategy, which facilitates the designers to manger the four types of uncertainty.

Types of robust design | Types of uncertainty | Example in SC | Quantification |
---|---|---|---|

I | Uncertainty in parameter | Uncertainty in the demand side, such as unpredictable order | Type I, II: EMI, Monte Carlo simulation, Latin hypercube, first/second moment method, etc. |

II | Variable uncertainty | Uncertainty in the supply side, such as variation in productivity | |

III | Uncertainty in model structure | Supply chain broken due to unexpected disasters (e.g., the COVID-19 pandemic) | Type III: DCI, variance dunction estimation, prediction interval approach, etc. |

IV | Uncertainty created in process chain | The bullwhip effect | Type IV: satisficing, ESS, etc. |

Types of robust design | Types of uncertainty | Example in SC | Quantification |
---|---|---|---|

I | Uncertainty in parameter | Uncertainty in the demand side, such as unpredictable order | Type I, II: EMI, Monte Carlo simulation, Latin hypercube, first/second moment method, etc. |

II | Variable uncertainty | Uncertainty in the supply side, such as variation in productivity | |

III | Uncertainty in model structure | Supply chain broken due to unexpected disasters (e.g., the COVID-19 pandemic) | Type III: DCI, variance dunction estimation, prediction interval approach, etc. |

IV | Uncertainty created in process chain | The bullwhip effect | Type IV: satisficing, ESS, etc. |

Taguchi develops the “parameter design concept” to use a two-part orthogonal array for experimental designs using the “signal-to-noise-ratio” as an optimization criterion. This is later defined by Chen et al. [11]. With the identification of one limitation of Taguchi’s method—no accurate solution can be yielded for highly nonlinear problems, and given the method is criticized by the statistical community [66], Chen et al. proposed a variation to Taguchi’s method by integrating the response surface methodology with the cDSP to manage both Type I and Type II uncertainties. Meanwhile, Simpson et al. [67] and Chen et al. [68,69] proposed the robust concept exploration method (RCEM) and used the principles to determine a range set of top-level design specifications to realize the robust design. Later, Choi et al. [70] used RCEM with error margin index (RCEM-EMI) to realize the robust design of materials by employing the EMI to indicate the mean and spread of system performance considering the variability in design variables and models. The authors of Ref. [70] also proposed RCEM with design capability index (RCEM-DCI) to determine whether a ranged design specification is capable of satisfying a ranged set of design requirements. Choi et al. [14,71] proposed an inductive design exploration method (IDEM) for designing materials and products concurrently and systematically to manage Type IV uncertainty by sequentially identifying a ranged set of feasible specifications and searching the feasible spaces regarding the top-level design requirements.

Given the foundation of four types of robust design—the RCEM and IDEM with indices, the satisficing community later expands the method, derives RCEM methods with new indices, and applies them in different fields of engineering design. Wang et al. [72] proposed a design exploration method for adaptive design systems that include using local regression and inverse IDEM using an example problem in the design of a photonic crystal coupler and waveguide. Sinha et al. [73–75] proposed to use the inductive discrete constraints evaluation, which is to sequentially identify feasible regions in design space using a metric called hyper dimensional-error margin index (HD-EMI) indicating the degree of reliability of the model. Kulkarni et al. [76] and Gautham et al. [77] later applied HD-EMI in the multistage, inverse design of the heat treatment operation. Samadiani et al. [78] and Panchal et al. [79] integrated the proper orthogonal decomposition-based (POD-based), multiscale model with cDSP to realize the robust design of datacenter cell in thermal design. Messer et al. [80] and Rippel et al. [81] applied and enriched the RCEM using the early stage of design of a pressure vessel by presenting three alternative methods, namely, second derivative, multiple derivatives, and multiple point method, to measure the robustness through estimating the variance. Goh et al. [82–84] proposed the integrated multiscale robust design for traversing the integrated computational materials engineering (ICME) in multiscale heterogeneous internal structure evolution.

In summary, the four types of robust design using satisficing strategy are using the cDSP and ALP as the basic modeling construct and solution algorithm, applying indices to measure and control variations from multiple sources, and sometimes integrating other modeling techniques to identify and work on a range of feasible solutions to ensure the design is insensitive to uncertainties.

### 4.2 Multiscale and Microstructure Design in Material Design.

Material design is a field that often encounters nonlinear equations, variations from multiple sources, multiscale or hierarchical structures, etc., so it requires satisficing as a design strategy. For this topic, the contributions are mainly from the applications of the robust design method in material design.

Seepersad et al. [85] presented the Type I, II, and III robust design methods for designing materials in mesoscopic scales by topologically and parametric tailoring them to achieve desired properties. Allen et al. [12] used the first three types of the robust design methods on multidisciplinary, multiscale design with multiple sources of uncertainty. Thompson et al. [86] considered the material properties as uncertain variables and applied the robust design method to provide ranges of acceptable material properties for framing subsequent material design or selection for designing blast-resistant panels. McDowell et al. [87] gave an example for designing a plasticity-related microstructure—a four-phase reactive power metal–metal oxide mixture for initiation of exothermic reactions under shock-wave loading. Later, McDowell et al. [88] integrated the design of multiscale material, multifunctional material, and product design, and the proposed integrated method also works for other systems-based design of materials. Seepersad et al. [89] proposed a two-stage topology design approach that requires customized multifunctional properties for designing cellular materials. Samadiani et al. [78] used a test problem of datacenter cell internal design to demonstrate the efficacy of the proposed method with the integration of POD-based, multiscale modeling with the cDSP. The method can be used for other simulation-based, multiscale designs. Shukla et al. [17,90] dealt with a ladle refining and continuous casting problem by using the satisficing strategy to facilitate producing new grades of steel to meet stringent sets of property specifications by exploring the solution space. Goh et al. [84] selected the satisficing strategy for it allows finding feasible ranged sets of solutions and avails the ICME horizontal process-structure-property-performance in multiscale heterogeneous internal structure evolution. Beemaraj et al. [91] proposed a method for designing a robust composite structure subjected to different loading conditions.

Why do designers choose the satisficing strategy in material design, especially when designing multiscale, microstructures, or needing to cope with the material properties with other requirements in a process or product? Typical reasons include (i) the product is expected to be robust to the variation in the material property, (ii) the material attributes can be designed to be robust against uncertainties due to random variation of microstructure, (iii) designers only need to determine the value of decision variables that satisfy the rigid requirements (constraints) and achieve the desired properties (goals) as closely as possible, which means they do not have to squeeze to the optimal properties, (iv) material design for a product is often a concurrent design that comprises material design or selection and the dimension or geometry design of the product, (v) robust solutions can be founded to reduce the variability in response for simulation-based design or metamodel-based design, and (vi) the solution space can be explored relatively sufficiently not only to output a solution but to observe and predict model behaviors and in the wide design space, which can be useful information for new material experiment and innovation.

### 4.3 Multistage Design and Concurrent Design.

Multistage design and concurrent design are also applications of robust design methods, specifically RCEM. The applications include engine design, aircraft design, gear transmission, multifunctional material design, process chain design, product and product family design, and steel production.

#### 4.3.1 Concurrent Design.

Rangarajan et al. [92] adopted RCEM as a framework to address tribological considerations in the concurrent design of automobile engine lubricated components. Chen et al. [93] used RCEM to evaluate design alternatives and develop top-level specifications for high-speed civil transport aircraft. Simpson et al. [94] proposed a product variety tradeoff evaluation method for assessing alternative product platform concepts with varying levels of commonality—the challenge lies in balancing the commonality and performance of products in a product family. The authors verified the method using gear transmission and general aviation aircraft design. McDowell et al. [88] performed concurrent design by integrating multiscale, multifunctional, and product design at the same time.

#### 4.3.2 Inverse Design.

Nellippallil and coauthors dealt with a series of multistage, inverse design problems in steel manufacturing. Nellippallil et al. [95] presented an inverse design method based on empirical models and response surface models to support the integration that facilitates information flow between stages of the hot rod process chain. The stages are designed by passing the information obtained after exercising the end-stage cDSP to the earlier-stage cDSPs. Then, the authors improved the inverse design method [23] to achieve the integrated design exploration of materials, products, and manufacturing processes through the vertical and horizontal integration of models. They demonstrated the efficacy of the method using a hot rod rolling and cooling process chain problem by exploring the processing paths and microstructure in an inverse manner. Nellippallil and the coauthors further improved the method into a goal-oriented inverse design method [96] to empower the capability to carry out a microstructure-mediated design satisficing specific processing performance of a product. They applied the method to design the thermos-mechanical processing of a steel rod. The authors then summarized their work as an approach [97] that facilitates co-design across the materials, products, and manufacturing processes. With this goal-oriented, inverse co-design approach, Fonville et al. [21] designed two components of an American football helmet, the composite shell and foam liner. The method can be used in other fields when the multistage, concurrent design of multiple elements (material, product, manufacturing, etc.) is needed and when only the end-stage performance or requirements are available.

Designers choose the robust design method under the satisficing strategy for concurrent, multistage, and inverse design projects because they need to manage the uncertainties brought by connecting the stages, concurrent decision-making, and the lack of knowledge on the association between the early-stage decision variable and the end performance of a product or a system.

### 4.4 Coupled Decision-Making.

Coupled decision-making problems often have common features with the multistage, concurrent design problems—designers need to determine two things while interdependent on each other in a system, for example, the material and the product specification. In this subsection, we focus on the problems that require hierarchical ways of coupling the selection decisions and compromise decisions. Another type of coupled decision-making is making decisions by different teams with multiple backgrounds or fulfilling multiple disciplines and managing the interactions. We categorize those problems in multidisciplinary design and illustrate them in Sec. 3.4.

Besides the works on steel manufacturing described in Sec. 3.3, which require the design of materials and products at the same time, some applications and derived methods in steel casting and gear design also require material selection and geometric design simultaneously. Kumar et al. [16] developed an integrated design framework, platform for the realization of engineered materials and products (PREMΛP), based on metamodels and cDSP. The authors utilized the framework to assist in making decisions when an existing configuration for continuous casting is unable to meet the requirements. The approach can be adopted for integrating the host of operations for materials development with specific properties and the coupled design of products and materials. Gautham et al. [77] envisaged PREMΛP as a platform for the purpose of integrating models, knowledge, and data for designing both the material and the product. Kulkarni et al. [98] used PREMΛP as a platform for the realization of engineered materials and products for gear design by exploiting the synergy between component design, material design, and manufacturing. Later, Kulkarni et al. [99] proposed a method, the concept exploration method, based on the cDSP, which is demonstrated for gear design which requires the simultaneous exploration of geometry, material, and manufacturing spaces to exploit synergies. Sharma et al. [100] presented a method for robust design using coupled decisions to identify design scenarios that are relatively insensitive to uncertainties. Those design scenarios are modeled as coupled decisions to account for influence among decisions. The method is tested using a gearbox design problem—selecting gear material and determining gearbox geometry. Based on this work, Sharma et al. [101] later proposed a decision classification scheme, the multilevel decision scenario matrix, to develop design processes involving decision interactions.

As DSIDES can couple the cDSP and sDSP in different levels, hierarchies, and sequences, when the selection and compromise decisions need to be made at the same time, designers tend to adopt cDSP-ALP-DSIDES and derived design frameworks or methods in coupled decision-making.

### 4.5 Multigoal (Multi-Objective) or Multidisciplinary Design Through Exploring the Solution Space.

Engineering-design problems often have multiple goals (or objectives). Most test problems referred to in Secs. 3.1–3.4 are multigoal problems. However, in this subsection, we only introduce the publications that focus on the methods of the exploration of the design preferences, tradeoffs among goals, and interrelationship among goals, especially when there are three or more goals and the design encompasses multiple disciplines. Since multigoal problems are usually managed through exploring the solution space (consists of vectors of decision variables) and design space (consists of scenarios of formulating and compromising multiple goals), we combine the reviewing of the work in two categories—“(v) exploration of the solution space and design space” and “(vi) multigoal (multi-objective) or multidisciplinary design”—here in one section.

Xiao et al. [102] proposed the collaborative multidisciplinary decision-making methodology (CMDM) and used it to design a robot arm with three goals—deformation toward 0.5 mm, von Mises stress around 6 MPa, and weight around 3.5 g under working loads around. The CMDM enables designers to integrate the cDSP with game theory and apply DCI to obtain robust solutions. The cDSP is used to implement and exchange design scenarios while making coupling decisions. Game theory is adopted for managing interactions between teams’ decisions. DCI is used for maintaining design freedom to accommodate downstream changes. Wang et al. [72] proposed the design exploration method for adaptive design systems to design a photonic crystal coupler and waveguide using the mean response function from simulations as the objective and setting the EMIs between the objective and its upper and lower bounds as goals. The method is a derived method of the IDEM in the robust design. Ahmed et al. [103] designed a multigoal gear blank without prior knowledge of preferences among goals. They also explore different scenarios in the two ways (Archimedean and Preemptive) of goal formulation and gain knowledge on design scenarios associated with the design preferences. Smith et al. [104] designed a thermal plant based on the Rankine cycle with six goals and demonstrated the exploration of the solution space to compromise the goas using both the Archimedean (weighted combination) and Preemptive (Lexicographic) methods. Gautham et al. [105] used the IDEM to design a heat treatment process. The five material properties become goals in the decision model. They studied the tradeoffs among the properties and found ranged sets of specifications by exploring the design preferences. Anapagaddi et al. [106] and Sabeghi et al. [107,108] explored the solution space of multigoal design problems in continuous casting and ladle refining by evolving the weights of the goals so as to produce new grades of steels to meet stringent sets of property specifications. Goh et al. [83], Sabeghi et al. [108], Nellippallil et al. [22,23,95–97], and Guo et al. [65,109] applied the Ternary plot to visualize the desired weight range for three-goal problems using applications in steel manufacturing, material design, and supply chain design. Guo et al. [9] proposed a domain-independent algorithm, the adaptive leveling-weighting-clustering algorithm (ALWC), to explore the design scenarios for multigoal and many-goal (when there are more than three goals) problems. Using ALWC, designers may explore combinations and priorities of the goals based on their interrelationships.

Engineering-design problems in nature have multiple goals, and often those goals are formulated using knowledge and requirements from various disciplines. In the early stage of design, designers rely on their domain expertise or intuition to compromise the goals with certain design scenarios and meet different design preferences. However, a robust method should output desired performance without too much domain expertise or assumptions. The methods of the exploration of the solution space and design space can reduce designers’ dependence on prior domain knowledge and attain knowledge along with the exploration. The design methods in satisficing strategy well support solution space exploration as sets of satisficing solutions under multiple design scenarios can be identified, and model behaviors associated with those solutions and design scenarios can be recorded and analyzed.

### 4.6 Knowledge-Based Design and Platformation.

Theoretically, the awareness, discovery, reconstruction, storage, and reuse of knowledge can take place in any design problem. Not only the satisficing design strategy supports the knowledge-based design, but all kinds of methods should do. However, because the engineering designs using satisficing strategy, especially the RCEM, enforce rich post-solution analyses, designers can obtain relatively more knowledge about the robust design, solution space and design space performance, and model behaviors. The beauty of knowledge recognized in design is that it can be reused and guide the next design project, and it can be generalized into rules and packaged into templates or platforms that ease future designs in various fields.

#### 4.6.1 Knowledge and Templates/Tools About Design Problems.

Chen et al. [93] proposed the RCEM not only as a method for facilitating robust design but also as a method for enhancing design productivity by increasing design knowledge in the early stages of designs. “Design knowledge is increased by implementing sophisticated integrated systems analysis in the early stages of design and making decisions based on better information.” Zha et al. [110] integrated the cDSP with the fuzzy synthetic decision model as a knowledge-intensive collaboration paradigm to support hybrid decisions. Given that collaborative design decisions have objective and subjective features, the proposed method hybrid the selection and compromise decisions and archive knowledge on collaborations. Given the demand for a reusable design process and tool to support interactions in designs, Panchal et al. [111] derived a method, the modular decision-centric approach. The authors develop generic computational templates to instantiate four foundations to support concurrent design. The four foundations include hierarchical systems, separate information, decision-centric activities, and model interactions. Pederson et al. [112] proposed a domain-independent approach for realizing hierarchical product platforms. The authors focused on how to synthesize numerical taxonomy and technology diffusion in a systematic multi-objective decision-making method.

#### 4.6.2 Knowledge and Platforms About Design Methods.

Wang et al. proposed an ontology-based uncertainty management approach [113] in designing robust decision workflows and a knowledge representation approach [114] for designing complex engineered systems. In their two publications [113,114], they proposed a decision-centric design process representation scheme, the phase-event-information X diagram, for designing workflows. Later, Wang et al. [115] proposed a knowledge-based design guidance system (KBDGS) to support cyber-physical-product-service systems design. Ming et al. [116,117] used an ontology to represent knowledge of decision interactions and realize it as a cloud-based platform [118]. The authors architected the cloud-based platform to support a human-cyber-physical view of systems realization ecosystem. The decision support platforms, KBDGS and PDSIDES (platform for decision support in the design of engineered systems), can be extended as cloud-based platforms, which lead to possibilities in artificial intelligence (AI)-based design and self-organizing systems design in the age of Industry 5.0.

## 5 Way Forward

Industry 5.0, a transformative industrial revolution, with new technologies and applications like cloud-based design and manufacturing, cyber-physical-social systems, and the networked intelligent systems, make it possible for all participants in the whole process chain to be networked, self-organized, and collaborate on decentralized decision-making [119]. The question is, why satisficing strategy? Especially, why whole process satisficing?

The answer is: A networked, self-organized system that requires decision-making may have all the features that engineering-design problems have—nonlinear, nonconvex mathematical relations among factors, a number of conflicting goals, variations in controllable and uncontrollable factors, coupled decisions required, multistage and inverse decisions required, knowledge is created and can be generalized and reused anywhere all the time, etc. Above all, robust design is still the urged demand in Industry 5.0. We have used the previous three sections to demonstrate that satisficing strategy allows decision-makers to manage the features above relatively well. Some instructive concepts and ideas have already been proposed or foreseen by the satisficing community.

Nellippallil et al. [119] presented the architecture and functionalities of a cloud-based computational platform to support mass collaboration and open innovation. By reviewing the evolution of engineering design in connected products, end-to-end digital integration, mass customization and personalization, data-driven design, digital twins, etc., Jiao et al. [120] envisioned a human-cyber-physical view of the systems realization ecosystems. Milisavljevic-Syed et al. [121] proposed to leverage the advantages of digitization and AI to address smart manufacturing in a networked manufacturing system. Ming et al. [122] identified the requirements of the decision support platform for design Engineering 5.0 and proposed an architecture to resolve the challenges. Later, they proposed a framework for a cloud-based platform for decision support in the design of engineered systems [123].

For the whole process satisficing strategy, there are two research directions that may help the community expand its research power—dealing with large-scale engineering design and cyber security. cDSP-ALP-DSIDES and PDSIDES have been proven to be efficient when managing engineering problems with regular size. However, in the new age, as the size of a networked system and system of systems may grow exponentially, the traditional tools may not provide sufficient computing power. Problem partition and parallel computing will be needed.

As CB-PDSIDES can provide cloud-based design support, aware and store knowledge for reuse, learn, and record users’ preferences, cyber security may become a critical issue. Designing secure protocols and mechanisms for data transmission and storage in the cloud to protect engineering designs from unauthorized access, interception, or tampering will become a necessity. This includes encryption, access control, and secure key management. Investigating strategies to enhance the resilience of cloud-based engineering design systems against cyber-attacks, natural disasters, or system failures is another urgent task.

In summary, future research on engineering design is mainly about (i) how to graft new technologies and ideas into the satisficing methodology, (ii) how to utilize new technologies and ideas to better derive and innovate design methods under the satisficing strategy, and (iii) make the knowledge recognition and knowledge-based design available for all participants in the cyber-physical-social system.

## 6 Closing Remarks

In this paper, we first discuss the motivation for developing and employing the satisficing strategy in engineering designs. Optimization, as a popular method and strategy, is based on the utility theory. However, decision-makers, just like all other human beings, cannot be completely rational and knowledgeable to develop a utility function to map a problem in the physical world as a model in the mathematical world; therefore, the optimal solution to an optimization problem may not exist or be accessible. Even if the optimal solution is obtained, it may not be useful when being implemented in the physical world. In this sense, we need another modeling strategy that facilitates finding useful, “good enough” (but not necessarily optimal) solutions, which we define as satisficing solutions.

In addition, for engineering-design problems, besides solving the problem and obtaining a solution, there are other tasks, such as acquiring knowledge and insight about the output performance in certain ranges of solution space, improving the robustness of the design, coupling selection decisions with compromise decisions, integrating multiple models with different types or levels of fidelity, from multiple stages, etc. Therefore, a design strategy that enables designers to avoid the drawbacks of the optimizing strategy and work on the tasks above is needed. And that is satisficing strategy. There are different methods that give satisficing solutions and attain knowledge through post-solution analysis. In this paper, we focus on the methods based on the cDSP-ALP-DSIDES due to the consistency, productivity, and richness of their relevant work.

Then, we analyze the specialty of the formulation construct (cDSP), the solution algorithm (ALP), and the solver (DSIDES) that supports yielding satisficing solutions and realizing the features that support the design. Those features include the usefulness of the solutions, insensitivity of the solutions to uncertainties from various sources, and relatively easy and cheap to obtain.

By reviewing the publications in the cDSP-ALP-DSIDES community in the past 30 years, we classify the contributions in seven subjects: (i) robust design methods, (ii) multiscale and microstructure design, (iii) multistage design and concurrent design, (iv) coupled decision-making, (v) exploration of the solution space and design space, vi) multigoal (multi-objective) or multidisciplinary design, and (vii) knowledge-based design and platformation. We review the representative work in each subject, summarize their contributions and applied field, and discuss why the authors chose satisficing methods as the foundation to derive their methods.

Finally, we envision the value and development of satisficing methodologies in the age of Industry 5.0. Given the complexity of design in the digital era, satisficing strategy has advantages in producing robust design. We hope this paper can provide useful information on when and why designers may choose the satisficing strategy, concepts, methods, and tools to make their designs robust, smart, and valuable.

## Footnote

For complex problems that apply an approximation algorithm to solve, the optimal solution may not be found within the available computing power. Near-optimal solutions are found instead. There is not rigid definition of “how close” a near-optimal solution should be to the optimal solution. It depends on the problem and algorithm.

## Acknowledgment

Lin Guo acknowledges the financial support from the Pietz Professorship and Start-Up Fund from the Office of the Vice President for Research and Partnerships and the Office of the Provost at the South Dakota School of Mines and Technology.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

No data, models, or code were generated or used for this paper.

## Appendix: Detailed Information About the Milestone Publications—The Complexities Being Managed, Proposed Methods, Foundational Theories, Design Stages, Limitations, and Application Fields

Authors | Problem features and complexities managed | Proposed method | Method description | Foundational theory or Baseline method | Stage (F, A, S, E) | Limitations | Application field |
---|---|---|---|---|---|---|---|

Kiyota et al. [25] | Multi-objective, interdependent, complex systems, and system of systems | Satisficing squared S-2 method | Fuzzy satisficing method using generic algorithms; set aspiration level of each objective, minimizing unsatisfaction rating | Principles of bounded rationality, generic algorithm | F, A, S | May lead to unmature convergence | Multi-objective problems with an aspiration for each objective can be identified but not reached |

Toyoda and Kogiso [40] | Multi-objective, variation in variables and parameters | Satisficing tradeoff method | Add mean and variance of the variation as objectives, apply tradeoff analysis, and investigate the effect of uncertainty in Pareto surface | Taguchi’s robust design | F | Only investigate the desired space; keen in Pareto solutions | Multi-objective problems with uncertainties, and when designers want to explore how variation may affect Pareto set |

Binazadeh and Shafiei [54] | Nonlinear, time-vary systems with slowly varying parameters | An approach to design a stabilizing control law | Use a parametric version of satisficing control strategy, develop a time-frozen parameter to make a stabilizing control law, and evaluate the maximum admissible rate of change of system dynamics | Satisficing control strategy | F, E | The theory suggesting the stabilizing control law may vary with problems | Slowly varying control systems |

Li et al. [37] | Conflict resolution among subsystems | A satisficing algorithm to mitigate the conflict resolution problem | Formulate a conflict resolution problem as a game model, use a satisficing game theory to mitigate the model, and develop a judgment parameter to ensure a socially acceptable compromise | Game theory | F, E | Rely on utility theory assuming subsystems’ rationality and satisficing level are the same | Systems with multiple subsystems with conflict resolutions, such as unmanned aerial vehicles |

Nguyen et al. [33] | Uncertainty that can be formulated as chance constraints or stochastic model | Satisficing measurement approach (SMA) | Use SMA to mitigate customers’ dissatisfaction, and integrate the SMA into Tabu search heuristics to solve a set of Solomon instances | Stochastic programming | S, E | Assuming uncertainties can be quantified and represented as stochasticity | Stochastic vehicle routing problems with time windows |

Stirling and Nokleby [39] | Robotic systems capable of sophisticated social behavior | Multi-agent satisficing | Use conditional utility considering interest of others, use multi-agent utility aggregation to avoid group interests dominating individual interests, and apply Bayesian network to artificial societies to gain obtain satisficing social welfare | Game theory and Bayesian network theory | F, A, E | Rely on conditional utility assuming individual’s rationality and level of consideration of others are the same among individuals | Systems have participants or designers with interaction and gaming nature |

Okamoto et al. [49] | Multi-objective, nonlinear | A solution search method with satisficing tradeoff method | Develop a search method to find a preferred solution from Pareto optimal solutions, visualize the Pareto set using a self-organizing map, and perform satisficing tradeoff analysis | Pareto optimum | S, E | Assume Pareto front is easy to acquire | Material design |

Ming et al. [42] | Multitype decision-making—selection, compromise, and coupled | Template-based method configuration and execution of decision workflows | Use template to provide typical decision workflows for designers to execute, identify satisficing solutions, and explore solution space | Bounded rationality realized by cDSP and ALP | F, A, S, E | Did not explain and prove why using cDSP and ALP can obtain satisficing solutions | Product design, system design where coupled decisions are required |

Nellippallil et al. [22] | Multi-objective, nonlinear, multistage, multiple ways to integrate models | A goal-oriented inverse design method | Introduce robust design goals and indices, EMIs, and DCIs to obtain satisficing robust design specifications | Bounded rationality realized by cDSP and ALP; robust design realized by EMI, DCI | F, A, S, E | The solution space exploration can be varied from case to case | Multi-objective, nonlinear, and multistage design problems, such as the hot rod rolling process chain design |

Meng [31] | Fuzzy variables | A satisficing data envelopment analysis (DEA) model | For general fuzzy input and output variables, design a hybrid particle swarm optimization (PSO) algorithm by integrating approximation method, neural network (NN), and PSO to solve the proposed DEA model | PSO, NN, DEA | F, A, S | Based on seeking optimality and utility theory which assume decision-makers have perfect rationality | Problem with fuzzy input and output variables |

Lewis and Mistree [44] | Complex problems that finding a global optimum is difficult, discrete and continuous variables | Foraging-directed ALP | Hybrid foraging-directed algorithm with ALP—incorporate interior-point searching algorithm with sequential linearization and dual simplex | Bounded rationality realized by cDSP and ALP | F, A, S, E | Only efficient for problems with multiple local optima and easy to stuck in local optima | Complex problems that finding a global optimum is difficult |

Kanno et al. [38] | Inherent uncertainty, exogenous uncertainty, vulnerable to certain events | An info-gap robust-satisficing approach | Reverse performance between competing designs is quantified by intersection between the info-gap robustness curves | Info-gap robust-satisficing | F, E | The min–max objective is based on utility theory which assumes the model is complete and accurate | Structure design |

Wang and Wang [35] | Reliability-based problem | A maximum confidence enhancement-based sequential sampling approach | Employ surrogate model and define a cumulative confidence level (CCL) to measure reliability estimation accuracy, and select sampling points with the largest CCL improvement | Surrogate modeling, reliability-based programming | F, A, E | Only apply to reliability-based design | Reliability-based design, such as vehicle side crash |

Authors | Problem features and complexities managed | Proposed method | Method description | Foundational theory or Baseline method | Stage (F, A, S, E) | Limitations | Application field |
---|---|---|---|---|---|---|---|

Kiyota et al. [25] | Multi-objective, interdependent, complex systems, and system of systems | Satisficing squared S-2 method | Fuzzy satisficing method using generic algorithms; set aspiration level of each objective, minimizing unsatisfaction rating | Principles of bounded rationality, generic algorithm | F, A, S | May lead to unmature convergence | Multi-objective problems with an aspiration for each objective can be identified but not reached |

Toyoda and Kogiso [40] | Multi-objective, variation in variables and parameters | Satisficing tradeoff method | Add mean and variance of the variation as objectives, apply tradeoff analysis, and investigate the effect of uncertainty in Pareto surface | Taguchi’s robust design | F | Only investigate the desired space; keen in Pareto solutions | Multi-objective problems with uncertainties, and when designers want to explore how variation may affect Pareto set |

Binazadeh and Shafiei [54] | Nonlinear, time-vary systems with slowly varying parameters | An approach to design a stabilizing control law | Use a parametric version of satisficing control strategy, develop a time-frozen parameter to make a stabilizing control law, and evaluate the maximum admissible rate of change of system dynamics | Satisficing control strategy | F, E | The theory suggesting the stabilizing control law may vary with problems | Slowly varying control systems |

Li et al. [37] | Conflict resolution among subsystems | A satisficing algorithm to mitigate the conflict resolution problem | Formulate a conflict resolution problem as a game model, use a satisficing game theory to mitigate the model, and develop a judgment parameter to ensure a socially acceptable compromise | Game theory | F, E | Rely on utility theory assuming subsystems’ rationality and satisficing level are the same | Systems with multiple subsystems with conflict resolutions, such as unmanned aerial vehicles |

Nguyen et al. [33] | Uncertainty that can be formulated as chance constraints or stochastic model | Satisficing measurement approach (SMA) | Use SMA to mitigate customers’ dissatisfaction, and integrate the SMA into Tabu search heuristics to solve a set of Solomon instances | Stochastic programming | S, E | Assuming uncertainties can be quantified and represented as stochasticity | Stochastic vehicle routing problems with time windows |

Stirling and Nokleby [39] | Robotic systems capable of sophisticated social behavior | Multi-agent satisficing | Use conditional utility considering interest of others, use multi-agent utility aggregation to avoid group interests dominating individual interests, and apply Bayesian network to artificial societies to gain obtain satisficing social welfare | Game theory and Bayesian network theory | F, A, E | Rely on conditional utility assuming individual’s rationality and level of consideration of others are the same among individuals | Systems have participants or designers with interaction and gaming nature |

Okamoto et al. [49] | Multi-objective, nonlinear | A solution search method with satisficing tradeoff method | Develop a search method to find a preferred solution from Pareto optimal solutions, visualize the Pareto set using a self-organizing map, and perform satisficing tradeoff analysis | Pareto optimum | S, E | Assume Pareto front is easy to acquire | Material design |

Ming et al. [42] | Multitype decision-making—selection, compromise, and coupled | Template-based method configuration and execution of decision workflows | Use template to provide typical decision workflows for designers to execute, identify satisficing solutions, and explore solution space | Bounded rationality realized by cDSP and ALP | F, A, S, E | Did not explain and prove why using cDSP and ALP can obtain satisficing solutions | Product design, system design where coupled decisions are required |

Nellippallil et al. [22] | Multi-objective, nonlinear, multistage, multiple ways to integrate models | A goal-oriented inverse design method | Introduce robust design goals and indices, EMIs, and DCIs to obtain satisficing robust design specifications | Bounded rationality realized by cDSP and ALP; robust design realized by EMI, DCI | F, A, S, E | The solution space exploration can be varied from case to case | Multi-objective, nonlinear, and multistage design problems, such as the hot rod rolling process chain design |

Meng [31] | Fuzzy variables | A satisficing data envelopment analysis (DEA) model | For general fuzzy input and output variables, design a hybrid particle swarm optimization (PSO) algorithm by integrating approximation method, neural network (NN), and PSO to solve the proposed DEA model | PSO, NN, DEA | F, A, S | Based on seeking optimality and utility theory which assume decision-makers have perfect rationality | Problem with fuzzy input and output variables |

Lewis and Mistree [44] | Complex problems that finding a global optimum is difficult, discrete and continuous variables | Foraging-directed ALP | Hybrid foraging-directed algorithm with ALP—incorporate interior-point searching algorithm with sequential linearization and dual simplex | Bounded rationality realized by cDSP and ALP | F, A, S, E | Only efficient for problems with multiple local optima and easy to stuck in local optima | Complex problems that finding a global optimum is difficult |

Kanno et al. [38] | Inherent uncertainty, exogenous uncertainty, vulnerable to certain events | An info-gap robust-satisficing approach | Reverse performance between competing designs is quantified by intersection between the info-gap robustness curves | Info-gap robust-satisficing | F, E | The min–max objective is based on utility theory which assumes the model is complete and accurate | Structure design |

Wang and Wang [35] | Reliability-based problem | A maximum confidence enhancement-based sequential sampling approach | Employ surrogate model and define a cumulative confidence level (CCL) to measure reliability estimation accuracy, and select sampling points with the largest CCL improvement | Surrogate modeling, reliability-based programming | F, A, E | Only apply to reliability-based design | Reliability-based design, such as vehicle side crash |