
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39237558
71527
10.1038/s41598-024-71527-8
Article
An extended goal programming approach with piecewise penalty functions for uncertain supplier-material selection problem in cardboard box manufacturing systems
Najibzadeh Zahra 1
Maleki Hamid Reza maleki@sutech.ac.ir

1
Niroomand Sadegh 2
1 https://ror.org/04bxa3v83 grid.444860.a 0000 0004 0600 0546 Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
2 https://ror.org/04bxa3v83 grid.444860.a 0000 0004 0600 0546 Department of Industrial Engineering, Firouzabad Higher Education Center, Shiraz University of Technology, Shiraz, Iran
5 9 2024
5 9 2024
2024
14 2071421 11 2023
28 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In this study a real case multi-objective material and supplier selection problem in cardboard box production industries is studied. This problem for the first time optimizes the objective functions such as total wastage amounts remained from all raw sheets, total costs of the system including purchasing cost and transportation cost (including fixed and variable costs) of the raw sheets, and total overplus of produced cardboard boxes. To be closer to the real situations, as a novelty, the problem is formulated in belief-degree-based uncertain environment with normal distribution where this type of uncertainty applies the ideas of experts. A solution approach including two steps is proposed to solve the problem. In the first step, the proposed uncertain formulation is converted to a crisp form using a typical chance constrained programming scheme. In the second step, a new goal programming approach containing a piecewise penalty function is developed in order to solve the obtained multi-objective crisp formulation. In this approach, based on the ideas of experts, multiple goals are considered with different penalty values. A case study from cardboard box industries is considered to evaluate the proposed formulations and solution approach. According to the obtained results, the proposed solution approach is compared to similar approaches of the literature and its efficiency is studied.

Keywords

Supplier selection
Cardboard box manufacturing
Belief-degree-based uncertainty
Multi-objective optimization
Goal programming
Subject terms

Engineering
Applied mathematics
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pmcIntroduction

Regarding the importance of selling/purchasing performance issues, the manufacturing companies are more trying to increase their ability to attract customers, communicate suppliers, and compete with their competitors and surpass them. A challenging problem in this field is choosing the right suppliers (known as supplier selection problem (SSP)) (see Ayhan1, Ayhan and Kilic2, Tan and Alp3, Asgari et al.4, Hajiaghaei-Keshteli and Fathollahi-Fard5, Khalili Goudarzi et al.6, Heydari et al.7, Aghaei Fishani et al.8, Ma and Zhou9, Peukert et al.10, Hajiaghaei-Keshteli et al.11, Heraud et al.12). In this problem, the choice between suppliers that provide raw materials with better quality, lower prices, and shorter delivery times is significant. However, among the SSP criteria, the most effective factor on the company’s performance is the price of raw materials. For example, this cost may change from 70% of total revenue in automotive industry to 80% in high-tech industries13. So that the price factor can be considered as a priority criterion for selecting the suppliers of a factory. In the SSP, there are other important factors such as waste of time and raw materials, transportation cost, etc. Therefore, this problem can be considered as a multi-objective problem based on some constraints such as meeting the customers’ demands under the supplier’s capacities, etc. (see Mosallaeipour et al.14).

The literature of supplier material selection topic contains many interesting studies and here we will briefly explain some of them. Awasthi et al.15 considered a supplier selection problem with a single producer to find a low-cost set of suppliers that can satisfy stochastic demand. Aghai et al.16 considered the performance of potential suppliers against multiple criteria in the supplier selection problem with fuzzy type uncertainty. Also, they proposed a discount method to determine the best suppliers. Nekooie et al.17 considered supplier selection problem with qualitative and quantitative factors. Strategic and operational risks were taken into account. Arikan18 considered multiple sourcing supplier selection problem under fuzzy uncertainty with three objective functions minimizing total monetary costs, maximizing total quality, and maximizing service level of purchased items, considered SSP under stochastic demand with two extensions: one restricts the number of suppliers and the other allows multi-period sourcing. Also, Abdollahi et al.19 proposed the method of selecting an appropriate supplier based on product-related and organization-related factors. Niromand et al.20,21 considered multi-criteria supplier material selection in carton box manufacturing under robust uncertainty. Also, in continue they studied the simultaneous selection of suppliers and materials in the cardboard box manufacturing industry and proposed some new hybrid form of the fuzzy programming approach to solve the problem in fuzzy environment20–22. Proposed a fuzzy interactive approach for a multi-objective SSP model including three minimization functions such as cost delivery, time, and rejected items.

In this study, we discuss supplier selection programming with a multi-objective optimization model in a real cardboard box manufacturing company. The purpose of this issue is to minimize the wastage of cutting raw materials, and at the same time minimizing the purchasing cost of raw materials and its transportation cost, and also minimizing the surplus of production. So, it is modeled as a multi-objective mixed integer linear programming model. This problem is an extended version of the problem considered by Niroomand et al.20,21 where a different discount logic for purchasing the raw materials is applied and transportation costs with fixed and variable charges (see Ref.23) are added. Since the problem is investigated in real situation some parameters cannot be precisely determined, so uncertainty is an important issue in this problem. But in some situations, decision maker cannot obtain obvious historical data of parameters, and this sometimes happens due to starting the project for the first time in a new location, marketing fluctuations, etc. Due to these conditions, the exact value of the parameters cannot be easily predicted and requires expert opinion, so the uncertain values are modeled by the uncertainty theory based on the degree of belief24, Niroomand et al.25, Niroomand et al.20,21). According to the conditions some parameters of the problem such as demand, price and transportation cost coefficients cannot be clearly determined, and are considered under belief-degree-based uncertainty. This type of uncertainty arises from the lack of precise historical data or the influence of unpredictable market conditions, which makes it challenging to assign exact numerical values to these parameters. In real-world scenarios, especially in new projects or volatile markets, this uncertainty needs to be carefully managed to ensure robust decision-making. By incorporating belief-degree-based uncertainty, the model accounts for the expert opinions and subjective judgments that are often necessary when precise data is unavailable. So, it is the first time that these types of problems are investigated under this type of uncertainty. This novel approach allows for a more realistic representation of the decision-making environment, enhancing the model's applicability and reliability. In the process of problem solving, there is a need to eliminate uncertainty and develop crisp equations, so the chance constraint method is used to convert the uncertain model to a multi-objective crisp form. The chance constraint method provides a systematic way to handle uncertainty by transforming the probabilistic constraints into deterministic equivalents, thereby simplifying the problem without losing the essence of uncertainty. This transformation is crucial for applying standard optimization techniques and obtaining feasible solutions that are practical and implementable. Moreover, this approach ensures that the solutions are robust and reliable, offering a higher degree of confidence in the decision-making process. As another novelty, the method proposed by Jadidi et al.26 is extended as a new goal programming method where several aspiration levels are determined for each objective function and apply piece-wise penalty function in order to achieve a better result. This extension allows for a more flexible and nuanced approach to goal programming, accommodating varying levels of aspirations and priorities for each objective. By implementing piece-wise penalty functions, the model can better handle deviations from the desired goals, leading to more balanced and effective solutions. This innovative method not only enhances the precision and adaptability of goal programming but also provides a more comprehensive framework for decision-makers to navigate complex multi-objective scenarios. Finally, a numerical instance from the case study is used to evaluate the performance of the proposed new goal programming approach and compare it to the methods of the literature. The results of this evaluation demonstrate the superiority of the extended method in achieving optimal trade-offs among multiple objectives, showcasing its potential as a powerful tool in multi-objective optimization. Detailed comparisons with existing methods highlight the advantages and improvements brought by the new approach, underscoring its practical relevance and applicability in real-world situations.

The paper is organized in several sections. Section “Literature review” represents literature on the subject in different fields of this article. Section “Problem description and formulation” describes and models the multi-objective supplier and material selection of the cardboard box manufacturing company in uncertain and crisp forms. Section “Solution approach: goal programming with piecewise penalty function” presents the new goal programming approach. Section “Computational experiment of the case study” reports the computational experiments. Section “Conclusion” represents the concluding remarks.

Literature review

In this part, we will discuss the subject literature in the fields of goal programming, cardboard box manufacturing and the belief-degree-based uncertainty, and we will briefly mention several articles that have researched these issues.

Goal programming

Since SSP can be formulated in the category of multi-objective problems, so they can be optimized by using appropriate multi-objective methods such as goal programming (GP), fuzzy multi-objective programming, weighted maximum optimization, etc. as have been proposed in the literatures and some of them are described here. Chang27 presented a new technique called multi-choice goal programming (MCGP) for supply chain modeling. Afterwards he improved the method and provided aspiration levels for each objective for obtaining better result (see Chang28). Afterwards, Chang et al.29 extended the proposed method of Chang28 and developed fuzzy multi-choice goal programming for SSP and defined membership function in the process of solving. Tabrizi et al.30, extended the model of Chang28 and applied ambiguous aspiration level to introduce fuzzy multi-choice goal programming. Extended the previous methods and proposed lower and upper bounds for the aspiration level of each objectives. Rabieh et al.31 presented an integrated robust-fuzzy method to select suppliers by considering risk and it was applied to a real case of supplier selection in automobile industry. Mirzaee et al.32 discussed the problem of supplier selection and order allocation with multi-period, multi-product, multi-supplier, multi-objective cases. The problem was formulated by a mixed integer linear programming model and for solving they applied, preemptive fuzzy goal programming approach. Jokar et al.33 suggested weighted fuzzy AHP method for the SSP model based on two-stage stochastic programming and robust optimization approaches. Kilic and Yalcin34 considered the environmental factors that are the main decision point in the supply chain in the form of supplier selection problem and to find the best suppliers applied two-phase modified fuzzy objective programming.

Cardboard box manufacturing

Cardboard box manufacturing industry is one of the most important manufacturing industries in which cardboard boxes are obtained by cutting the raw materials, which are in the form of rectangular raw sheets, into smaller parts. The raw materials required by these factories are provided to the manufacturer so choosing a supplier who provides the raw materials to the manufacturer in a shorter time and at a lower cost is very important, and therefore these types of issues are among the issues of supplier selection problem. In this section, we will briefly describe the articles that have been researched on this issue.

Mosallaeipour35 has addressed the issue of selecting raw material suppliers in the cardboard box production industry, which is obtained by cutting rectangular raw materials into smaller pieces, and in this issue, uncertainty of the fuzzy type has been considered. Also, Ref.20,21 investigated the problem of selecting suppliers and raw materials simultaneously in carton manufacturing industries and developed a multi-criteria mixed integer formula to select the size, quantity and supplier of raw sheets used and with the aim of minimizing objectives such as cost, the wastage of sheets and the excess of cardboard boxes at the same time. Later Gavrilescu et al.36, had a challenging work program through the reuse of production waste in the form of cardboard strips, edges and other waste to create a new engineered product that is environmentally friendly and sustainable that the market needs, by extending the life cycle of cardboard waste. We created and applied the basis of an innovative approach in the environment, which follows practical environmental design. Lo-Iacono-Ferreira et al.37, scrutinized the carbon footprint of cardboard box manufacturing and the impact of using cardboard box containers to store and transport 1000 tons of fruit and vegetable products by road from their origin in Almeria, Spain, to their destination market. Afterwards Betul and Akpinar38 investigated the weight and thickness of E flute and BC flute corrugated cardboards, which are prepared using different types of paper, and for this purpose, they were subjected to ECT crushing test, and the obtained results show that the strength of corrugated cardboard increases with the increase in paper weight. Buendía et al.39, investigated the effects of active packaging (a cardboard box containing a complex containing bicyclodextrin (βCD) with a mixture of EOs) on the quality of bell peppers (green, red and yellow) at a certain temperature. Also Tannady and Purwanto40, studied the corrugated cardboard box and the advantages of using it and solving the causes of defects in their production, which included human factors, poor training of machine operators, motor-operated machines, etc. Considering that in the production of cartons, features such as diversity in the type and processing method for each carton, special operating machines that are set up by a technician are important, therefore, in this study, Recently, You and Hsieh41 created a mathematical model and a safety-based algorithm ((IBA) and Genetic Based Algorithm (GBA) to develop decision making and production schedule and determine the best decisions to minimize total cost as well as early product delivery.

Belief degree-based uncertainty

When examining the problem in real conditions, some parameters of the problem cannot be determined definitively, so these parameters are affected by the uncertainties such as fuzzy theory, robust probability theory, uncertainty of degree of belief, etc. But sometimes, due to the lack of historical data or their invalidity or factors such as weather conditions, sudden accidents and crowding, the probability distribution of these parameters is not available. In some cases, it is not even possible to conduct experiments and collect data. Like checking the resistance of a built bridge against the force applied to it or the number of volcanoes recorded in a certain area. In these cases, one of the solutions is to use the opinion of experts in the relevant field to express their opinion based on the available evidence and documentation and their experiences. Experts and specialists are used to evaluate the degree of belief in the occurrence of each event to describe these parameters. In this case, the uncertainty theory proposed by Liu24 can be used to deal with the degree of belief. In this section, we will briefly describe articles in this field.

Chen et al.42 proposed a semi-variance method for choosing diversified portfolios, where security returns are considered as uncertain variables and are presented based on experts' estimates. Based on the concept of semi-variance and presenting three features for this method, he has proposed two different types of mean-semi variance diversified models for selecting the portfolio of uncertain variables, and to solve the models due to their complexity, a combined intelligent algorithm based on 99 methods and genetic algorithm has designed. Mahmoodirad et al.43, examined a linear fractional transportation problem in an uncertain environment with uncertain parameters of belief-degree-based uncertainty, which is transformed into an explicit form using three approaches: expected value model and chance constraint model and the combination of two methods. We can also mention Mahmoodirad and Niroomand44, who for the first time investigated the problem of designing a supply chain network with environmental effects and direct transportation in an uncertain environment based on the degree of belief. They used the three approaches of expected value model, chance constraint method and their combination to transform the proposed uncertain problem into its explicit form. Also, to solve the obvious problems obtained from two multi-objective optimization approaches, Goal Programming (GP) and Global Criterion Method (GCM) used. Also, for the first time, Mahmoodirad and Niroomand45, studied the problem of designing an uncertain dual-purpose supply chain network with cost and environmental effects under belief-degree-based uncertainty. He used the two approaches of expected value and chance constraint method to transform the uncertain problem to transform the uncertain model into its explicit form. Shen and Zhu46, investigated a two-echelon fixed charge transportation problem under uncertainty and variables like the demands, supplies, availabilities, fixed charges, and transported quantities in this problem are assumed as belief-degree-based uncertainty variables. they applied the expected value, a chance-constrained method to find the deterministic equivalent, and then proposed a Genetic algorithm and particle swarm optimization to solve the model. Also, Song et al.47 investigated uncertain mixed-integer programming for uncertain product configuration model with consideration uncertain lead-time and time-sensitive demand. Niroomand et al.48 takes into account the efficiency of production boxes, the waste value of raw sheets, and the inventory cost of excess boxes produced in the carton industry. This problem is considered under the belief-degree-based uncertainty, and the expected value method was used to achieve a crisp model.

Summary of the contributions of this study

In this article, as previously explained in the previous chapter, the problem of making a cardboard box from rectangular raw sheets with the three objectives of minimizing the cost of supplying and transporting raw materials, minimizing waste caused by cutting raw materials, and reducing surplus production. It has been noticed that the variables of the problem are of belief-degree-based uncertainty, which according to the fluctuations affecting the market have been used to estimate them from the point of view of experts, therefore, the uncertainty of belief has been applied to the variables and then using the chance constraint method of equations. After obtaining the crisp model, we solved the problem by using the modified method of goal programming. The contributions of this study can be summarized as below.Comparing to the literature of the cardboard box manufacturing planning problem, a new discount policy is considered in the proposed formulation.

Such problem, for the first time is formulated with belief-degree-based uncertain parameters.

According to the preferences of the managers, a new goal programming approach with piece-wise penalty function is introduced.

Problem description and formulation

In this section, the multi-objective problem of material and supplier selection (MSMSP) in the cardboard box production environment is described and formulated. The problem consists of different aspects, so, is defined as a multi-objective programming problem. Since the problem is considered in real conditions, some parameters are affected by uncertainty, which according to the conditions of the problem and the lack of historical data, the belief-degree-based uncertainty is appropriate for that. More details are given in the following sub-sections.

Description of the MSMSP

As mentioned earlier, the problem of this study is related to the suppliers and the materials in a cardboard box manufacturing industry which involves suppliers, producer, and consumer together. Cardboard boxes are produced in different sizes and usually used to package and transport goods. In this problem, raw paper sheets are provided by suppliers in different sizes. Each size of the raw sheets can be cut to produce a certain quantity of a certain size of cardboard box. After cutting, an unused area of the raw sheet is remained. This unused area is called wastage. On the other hand, as all raw sheets should be converted to cardboard boxes in the production system, some quantity of each cardboard box may be produced extra than the considered demands. This extra quantity is called product overplus. In this production system, the costs play critical role. The costs include purchasing costs and transportation costs. The summation of these two costs is called net cost. Notably, the purchasing cost follow a discount policy where for each size of raw sheet for orders more than a certain value (break point) the purchasing cost of the extra units is decreased. Also, transportation cost includes fixed transportation cost and variable cost which is depended to the number of raw sheets transported by each transportation device. Notably, the variable transportation cost follows a discount policy where for each raw sheet size for orders more than a certain value (break point) the variable transportation cost of the extra units is decreased. The considered problem is more complete than the study of Niroomand et al.20,21. Other than the concepts of the mentioned study, in this study we consider different discount policy and also fixed and variable transportations costs.

In addition, some parameters of this problem such as discount break point, raw material price, raw sheet transportation cost, etc. may not be possible to be determined exactly. Since we consider a real company and due to the market fluctuations, the historical data of the parameters may not be available certainly, so, the uncertainty theory explained by the Appendix is considered to represent the problem. Therefore, in the rest of this section, first the problem is formulated as a deterministic model, then its belief-degree-based uncertain form is represented, and finally the uncertain form of the problem is converted to a crisp form to be solved.

Deterministic mathematical formulation

In order to formulate the problem described by Section “Description of the MSMSP”, the following notations are defined in advance.Indexes:	
i	index used for types of cardboard box,	
j	index used for types (sizes) of raw sheet,	
k	index used for suppliers	
Parameters:	
I	number of cardboard box types available in the planning horizon,	
J	number of different types of row sheet to be supplied by the suppliers,	
K	number of suppliers,	
M	a large positive value,	
di	demand of cardboard box type i displayed by unit amount,	
aij	number of cardboard box type i that can be cut from raw sheet type j,	
wij	wastage amount residuary after cutting cardboard box type i from raw sheet type j,	
gjk	discount break point for purchasing raw sheet j from supplier k,	
g′jk	discount break point for transportation of raw sheet j from supplier k,	
pcjk1	normal price of row sheet j provided by supplier k,	
pcjk2	reduced price of row sheet j provided by supplier k (pcjk1≥pcjk2),	
fcijk	fixed cost for transportation of raw sheets type j which is purchased from supplier k and used to produced box type i,	
tcjk1	normal cost for transportation of raw sheet j provided by supplier k,	
tcjk2	reduced cost for transportation of raw sheet j provided by supplier k (tcjk1≥tcjk2)	
Variables:	
Tjk1	binary variable demonstrates if normal price is applied for row sheet j provided by supplier k,	
Tjk2	binary variable demonstrates if reduced price is applied for row sheet j provided by supplier k,	
Tjk′1	binary variable demonstrates if normal price is applied for transporting raw sheet j provided by supplier k,	
Tjk′2	binary variable demonstrates if reduced price is applied for transporting raw sheet j provided by supplier k,	
Yijk	quantity of raw sheet j purchased from supplier k used for producing box type i,	

According to the above-mentioned notations, the following multi-objective non-linear model is proposed for the MSMSP described by Section “Description of the MSMSP”.1 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

2 minOF2=∑j=1J∑k=1KTjk1pcjk1∑i=1IYijk+Tjk2pcjk1gjk+Tjk2pcjk2∑i=1IYijk-gjk+∑i=1I∑j=1J∑k=1KfcijkYijk+∑j=1J∑k=1KT′jk1tcjk1∑i=1IYijk+T′jk2tcjk1g′jk+Tjk2tcjk2∑i=1IYijk-g′jk

3 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subject to4 ∑j=1J∑k=1KaijYijk≥di∀i

5 Tjk1+Tjk2≤1∀j,k

6 ∑i=1IYijk≥Tjk1∀j,k

7 ∑i=1IYijk≤Tjk1gjk+MTjk2∀j,k

8 ∑i=1IYijk≥gjk+1Tjk2∀j,k

9 T′jk1+Tjk′2≤1∀j,k

10 ∑i=1IYijk≥Tjk′1∀j,k

11 ∑i=1IYijk≤Tjk′1g′jk+MT′jk2∀j,k

12 ∑i=1IYijk≥g′jk+1T′jk2∀j,k

13 Tjk1,Tjk2,T′jk1,Tjk′2∈0,1∀j,k

14 Yijk≥0and integer∀j,k

Objective function (1) minimizes total wastage amounts remained from all raw sheets used for producing all cardboard boxes. Objective function (2) minimizes total costs of the system where its first term calculates total purchasing cost, the second term calculates total fixed transportation cost, and the third term calculates total variable transportation cost. Objective function (3) minimizes total number of produced cardboard boxes. Constraints set (4) satisfies the demand values. It is notable to mention that Objective function (3) and Constraint (4) together minimized total overplus of the cardboard boxes. Constraints sets (5)–(8) all together respect to the price discount policy of raw sheet. Constraints sets (9)–(12) together respect to the discount policy of the variable transportation cost of the raw sheets. Constraints sets (13) and (14) are sign constraints of the model.

In order to linearize the above-mentioned model, the non-linear terms of objective function (2) such as Tjk1∑i=1IYijk, Tjk2∑i=1IYijk, T′jk1∑i=1IYijk, and Tjk′2∑i=1IYijk should be linearized. For this aim, the continuous variables Zjk1=Tjk1∑i=1IYijk, Zjk2=Tjk2∑i=1IYijk, Zjk′1=Tjk′1∑i=1IYijk, and Zjk′2=T′jk2∑i=1IYijk are introduced and instead of each of the non-linear terms a set of constraints is defined. For example, for non-linear term Tjk1∑i=1IYijk and its associated continuous variable Zjk1 the below set of constraints are defined and the same is done for other non-linear terms.15 Zjk1≤MTjk1∀j,k

16 Zjk1≤∑i=1IYijk∀j,k

17 Zjk1≥∑i=1IYijk-M1-Tjk1∀j,k

18 Zjk1≥0∀j,k

Therefore, the following multi-objective mixed integer linear model is obtained instead of the non-linear model (1)–(14).19 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

20 minOF2=∑j=1J∑k=1KZjk1pcjk1+Zjk2pcjk2+Tjk2gjkpcjk1-pcjk2+∑i=1I∑j=1J∑k=1KfcijkYijk+∑j=1J∑k=1KZ′jk1tcjk1+Zjk′2tcjk2+Tjk′2g′jktcjk1-tcjk2

21 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subject to22 Zjk1≤MTjk1∀j,k

23 Zjk1≤∑i=1IYijk∀j,k

24 Zjk1≥∑i=1IYijk-M1-Tjk1∀j,k

25 Zjk2≤MTjk2∀j,k

26 Zjk2≤∑i=1IYijk∀j,k

27 Zjk2≥∑i=1IYijk-M1-Tjk2∀j,k

28 Z′jk1≤MTjk′1∀j,k

29 Z′jk1≤∑i=1IYijk∀j,k

30 Z′jk1≥∑i=1IYijk-M1-T′jk1∀j,k

31 Z′jk2≤MT′jk2∀j,k

32 Zjk′2≤∑i=1IYijk∀j,k

33 Zjk′2≥∑i=1IYijk-M1-Tjk′2∀j,k

34 Zjk1,Zjk2,Z′jk1,Zjk′2≥0∀j,k

35 Constraints4-14.

Uncertain formulation

As mentioned earlier, in lack of historical data the belief-degree-based uncertainty theory proposed by Liu24 can be a suitable tool to deal with uncertainty in optimization problems. In the case of the proposed MSMSP, according to the reasons like starting a new project and market fluctuations, the belief-degree-based uncertainty can be used to formulate the problem. All required definitions, theorems and lemmas of the belief-degree-based uncertainty are explained in the Appendix. Therefore, the demand, cost, and some other parameters of the MSMSP are expressed by uncertain variables and the uncertain variables ζdi, ζpcjk1, ζpcjk2, ζtcjk1, ζtcjk2, ζfcijk, ζgjk, and ζg′jk with normal type distribution are defined. The normal distribution is considered as it needs just estimation of the mean and standard deviation parameters. Therefore, it can be more practical when asking from experts to estimate it. The following uncertain form of the deterministic model (19)–(35) is obtained.36 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

37 minOF2=∑j=1J∑k=1KZjk1ζpcjk1+Zjk2ζpcjk2+Tjk2ζgjkζpcjk1-ζpcjk2+∑i=1I∑j=1J∑k=1KζfcijkYijk+∑j=1J∑k=1KZ′jk1ζtcjk1+Zjk′2ζtcjk2+Tjk′2ζg′jkζtcjk1-ζtcjk2

38 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subject to39 ∑j=1J∑k=1KaijYijk≥ζdi∀i

40 ∑i=1IYijk≤Tjk1ζgjk+MTjk2∀j,k

41 ∑i=1IYijk≥ζgjk+1Tjk2∀j,k

42 ∑i=1IYijk≤Tjk′1ζg′jk+MT′jk2∀j,k

43 ∑i=1IYijk≥ζg′jk+1T′jk2∀j,k

44 Constraints5,6,9,10,13,14,22-34.

As the uncertain model (36)–(44) cannot be solved by the available exact solution methods of optimization software, first it should be converted to a crisp form and then be solved exactly. The next sub-section represents a crisp form of this model.

Equivalent crisp model

According to the literature, three popular methods have been proposed to convert a belief-degree-based uncertain model to its equivalent crisp form24. These methods are (1) expected value model (EVM) which considers the expected value of uncertain parameters in the crisp form, (2) chance-constrained model (CCM) where instead of each uncertain objective function or uncertain constraint its chance-constrained form is considered, and (3) mix of the expected value and chance-constrained models (EVCCM) which considers the expected value of uncertain objective functions and the chance-constrained form of uncertain constraints. In this section we consider the chance-constrained model in order to obtain the equivalent crisp form of uncertain formulation (36)–(44) where the uncertain parameters have normal distribution. In order to apply this method, each uncertain objective function is converted to a constraint that is less than or equal to a variable and then the introduced variable is minimized instead. Then for each uncertain constraint its chance-constrained form is defined easily. Therefore, the following model is introduced instead of uncertain formulation (36)–(44).45 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

46 minOF2=f2~

47 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subject to48 M∑j=1J∑k=1KZjk1ζpcjk1+Zjk2ζpcjk2+Tjk2ζgjkζpcjk1-ζpcjk2+∑i=1I∑j=1J∑k=1KζfcijkYijk+∑j=1J∑k=1KZ′jk1ζtcjk1+Zjk′2ζtcjk2+Tjk′2ζg′jkζtcjk1-ζtcjk2≤f2~≥λ1

49 M∑j=1J∑k=1KaijYijk≥ζdi≥αi∀i

50 M∑i=1IYijk≤Tjk1ζgjk+MTjk2≥β1jk∀j,k

51 M∑i=1IYijk≥ζgjk+1Tjk2≥β2jk∀j,k

52 M∑i=1IYijk≤Tjk′1ζg′jk+MT′jk2≥γ1jk∀j,k

53 M∑i=1IYijk≥ζg′jk+1T′jk2≥γ2jk∀j,k

54 Constraints23,24,27,28,31-45.

In continue, the following theorems and corollaries are used to obtain the CCM crisp form of uncertain model (45)–(54).

Theorem 8

Suppose that independent uncertain variables ζdi, ζpcjk1, ζpcjk2, ζtcjk1, ζtcjk2, ζfcijk, ζgjk, and ζg′jk have uncertain distributions ϕdi, ϕpcjk1, ϕpcjk2, ϕtcjk1, ϕtcjk2, ϕfcijk, ϕgjk and ϕg′jk respectively . Then, uncertain model (45)–(55) is converted to the below model.55 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

56 minOF2=f2~

57 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subjectto

58 ∑j=1J∑k=1KZjk1ϕ-1pcjk1(λ1)+Zjk2ϕ-1pcjk2(λ1)+Tjk2ϕ-1gjkϕ-1pcjk1(λ1)-ϕ-1pcjk2(1-λ1)+∑i=1I∑j=1J∑k=1Kϕ-1fcijk(λ1)Yijk+∑j=1J∑k=1KZ′jk1ϕ-1tcjk1(λ1)+Zjk′2ϕ-1tcjk2(λ1)+Tjk′2ϕ-1g′jk(λ1)ϕ-1tcjk1(λ1)-ϕ-1tcjk2(1-λ1)≤f2~

59 ∑j=1J∑k=1KaijYijk-ϕ-1di(αi)≥0∀i

60 ∑i=1IYijk-Tjk1ϕ-1gjk(1-β1jk)-MTjk2≤0∀j,k

61 ∑i=1IYijk-ϕ-1gjk(β2jk)+1Tjk2≥0∀j,k

62 ∑i=1IYijk-Tjk′1ϕ-1g′jk(1-γ1jk)-MT′jk2≤0∀j,k

63 ∑i=1IYijk-ϕ-1g′jk(γ2jk)+1T′jk2≥0∀j,k

64 Constraints5,6,9,10,13,14,22-34.

Proof.

By applying Theorem 2 and Lemma 1 (see the Appendix), constraint (48) is converted to the below constraints respectively.65 ∑j=1J∑k=1KZjk1ϕ-1pcjk1(λ1)+Zjk2ϕ-1pcjk2(λ1)+Tjk2ϕ-1gjkϕ-1pcjk1(λ1)-ϕ-1pcjk2(1-λ1)+∑i=1I∑j=1J∑k=1Kϕ-1fcijk(λ1)Yijk+∑j=1J∑k=1KZ′jk1ϕ-1tcjk1(λ1)+Zjk′2ϕ-1tcjk2(λ1)+Tjk′2ϕ-1g′jk(λ1)ϕ-1tcjk1(λ1)-ϕ-1tcjk2(1-λ1)≤f2∼

On the other hand, according to the theorems and definitions of the Appendix, the following conversion is done for constraint (49).66 ∑j=1J∑k=1KaijYijk≥ζdi≥αi⇔Mζdi-∑j=1J∑k=1KaijYijk≤0≥αi⇔∑j=1J∑k=1KaijYijk-ϕ-1di(αi)≥0

Other constraints of uncertain model (45)–(54) are converted like conversion (66). ∎

Corollary 1

Assume normal distribution of N(μ,σ) for the independent uncertain variables ζdi, ζCjk1, ζCjk2, ζCjk′1, ζCjk′2, ζCijk, ζgjk, and ζg′jk. The CCM formulation (55)–(64) is written as the below model.67 minOF1=∑i=1I∑j=1J∑k=1KwijYijk

68 minOF2=∑j=1J∑k=1KZjk1μpcjk1+ρ3πlnλ11-λ1+Zjk2μpcjk2+ρ3πlnλ11-λ1+Tjk2μgjk+ρ3πlnλ11-λ1μpcjk1+ρ3πlnλ11-λ1-μpcjk2-ρ3πlnλ11-λ1+∑i=1I∑j=1J∑k=1Kμfcijk+ρ3πlnλ11-λ1Yijk+∑j=1J∑k=1KZ′jk1μtcjk1+ρ3πlnλ11-λ1+Zjk′2μtcjk2+ρ3πlnλ11-λ1+Tjk′2μg′jk+ρ3πlnλ11-λ1μtcjk1+ρ3πlnλ11-λ1-μtcjk2-ρ3πlnλ11-λ1

69 minOF3=∑i=1I∑j=1J∑k=1KaijYijk

subjectto

70 ∑j=1J∑k=1KaijYijk-μdi-ρ3πlnαi1-αi≥0∀i

71 ∑i=1IYijk-Tjk1μgjk-ρ3πlnβ1jk1-β1jk-MTjk2≤0∀j,k

72 ∑i=1IYijk-μgjk+ρ3πlnβ2jk1-β2jk+1Tjk2≥0∀j,k

73 ∑i=1IYijk-Tjk′1μg′jk-ρ3πlnγ1jk1-γ1jk-MT′jk2≤0∀j,k

74 ∑i=1IYijk-μg′jk+ρ3πlnγ2jk1-γ2jk+1T′jk2≥0∀j,k

75 Constraints5,6,9,10,13,14,22-34.

The formulation (67)–(75) is the final crisp model (CCM) to be used instead of each of the uncertain formulations (36)–(44) and (45)–(54) where the uncertain variables have normal distribution. This is a multi-objective model and in the next section a typical goal programming approach is introduced to solve it.

Solution approach: goal programming with piecewise penalty function

As the crisp model (67)–(75) is a multi-objective model, in this section a typical goal programming approach is introduced to solve it. The proposed approach is based on the preferences of the decision maker (DM). In the rest of this section, first, classical goal programming approaches are reviewed and then the proposed goal programming approach of this study is presented.

Goal programming approach

Goal programming is of the useful optimization approaches to deal with multi-objective optimization problems. It provides Pareto-optimal solutions for multi-objective problems. Considering a certain multi-objective problem, the main idea of goal programming is to decrease the sum of violations of the objective functions from their pre-determined goals. The classical goal programming approach has been many applications in multi-objective optimization domain.

As a newer version of the goal programming, a weighted goal programming (WGP) approach was introduced. In this approach the positive and negative deviations from the goals are weighted by decision maker (DM). The original version of multi-choice goal programming (MCGP) approach was introduced by Chang27. In this approach among several goals (aspiration levels) considered for each objective function by DM, only one of them is selected by the optimization procedure (not DM). For this aim some binary variables are used in the formulation of the MCGP. Later, the original multi-choice goal programming approach was revised by Chang28. In their proposed version (R-MCGP) the binary variables are removed and are replaced by some continuous variables and constraints. So, the complexity of the model is decreased comparing to the original MCGP. Later, Chang49 presented the MCGP with utility function (MCGP-U). The aim of this version of the MCGP is to apply preferences of DM by using a utility function. Another version of the MCGP was introduced by Jadidi et al.50 considering an interval goal for each objective function. They were inspired by the fuzzy model of Tiwari et al.51 and considered two pivot points for optimizing each objective function by enabling DM in order to have more control on the interval of each goal.

A goal programming with piecewise penalty function

In order to be closer to real situations specially the preferences of the managers of the case study related to the MSMSP described by Section "Problem description and formulation", a typical multi-choice goal programming approach is introduced in this section. As one of the main preferences of the managers, there could be multiple goals for each objective function. These goals for objective function i start from the positive ideal solution (OFi+) of the objective function and are increased gradually where the goals are determined by DM. Considering this issue, for minimization type objective function i, a set of m aspiration levels (m goals) as OFi1,⋯,OFij,⋯,OFim is determined where OFi1=OFi+ and OFi1<⋯<OFij<⋯<OFim. According to this set of aspiration levels there are m-1 intervals and for each objective function in each interval a penalty value is considered by DM. Therefore, a set of penalty values for each objective function is determined as pi1,⋯,pij,⋯,pi,m-1 where pij is the penalty value and pi1<⋯<pij<⋯<pi,m-1. These penalty functions for each objective function can be normalized easily. Now, two main issues are considered as below while modeling the proposed goal programming approach.Optimizing each objective function (minimization type) as much close to its smallest aspiration

Level OFi1.

Optimizing each objective function (minimization type) in order to get a value from the interval with lowest penalty value.

Now the following steps are introduced to construct the proposed goal programming approach for the crisp version of the MSMSP represented by the crisp formulation (67)–(75).

Step 1. Determine all parameters of the crisp formulation (67)–(75).

Step 2. Solve each of the objective functions (67), (68), and (69) separately in their minimization form subject to constraints (70)–(75) in order to obtain the positive ideal solution of each objective function (OF1+, OF2+, and OF3+).

Step 3. Determine the importance weight values of the objective functions as ω1, ω2, and ω3 by the DM.

Step 4. Determine the membership function of each objective function in each of its aspiration level’s interval as αij=OFij+1-OFiOFij+1-OFij for all i and j.

Step 5. Construct the below non-linear goal programming formulation to solve the formulation (67)–(75) where variable λij∈0,1 shows whether or not the value of objective function i is in the j-th interval.76 min∑i=13ωi∑j=1m-1λij1-αij+∑i=13ωi∑j=1m-1pijλij

subject to77 αij=OFij+1-OFiOFij+1-OFij∀i,j

78 λij+αijλij≤2∀i,j

79 αijλij≥0∀i,j

80 ∑j=1m-1λij=1∀i,j

81 λij∈0,1∀i,j

82 Constraints70-75.

In this model, the weighted objective function (76) minimizes total penalty values and total difference of the objective functions from the aspiration level of their selected intervals. This objective function satisfies the above-mentioned two issues expected from the proposed approach. Equality (77) calculates the membership function of each objective function according to each interval aspiration level. As can be seen, if an objective function be in one of the intervals, the membership function of that interval is between 0 and 1, some membership functions are greater than 1 and some other membership functions are negative. Constraints (78)–(80) simultaneously guarantee that only if 0<αij≤1 (which means OFij≤OFi<OFij+1) then λij=1.

As the above model is in a non-linear form, it should be linearized before solving it. For this aim, the continuous variable rij=αijλij is defined and the model (76)–(82) is rewritten as the below linearized model where M is a pre-determined large positive value.83 min∑i=13ωi∑j=1m-1λij-rij+∑i=13ωi∑j=1m-1pijλij

subject to84 αij=OFij+1-OFiOFij+1-OFij∀i,j

85 λij+rij≤2∀i,j

86 ∑j=1m-1λij=1∀i

87 rij≤Mλij

88 rij≤αij+M1-λij

89 rij≥αij-M1-λij

90 rij≥0∀i,j

91 λij∈0,1∀i,j

92 Constraints70-75.

Finally, the linearized model (83)–(92) is used as the proposed goal programming with piecewise penalty function to solve the multi-objective crisp formulation (67)–(75) of the MSMSP of this study.

Computational experiment of the case study

In this section a case study from cardboard box production industries of Iran is considered to solve the MSMSP of this study by the goal programming approach proposed by Section "Experiments, results, and comparative study". The mathematical models of the proposed approach are coded in GAMS and are solved on a computer with AMD E2-1800 APU processor and 8.00 GB RAM. The data of the case study, the obtained results, and the related interpretations are described by the rest of this section.

Data of the case study

The data of the considered case study are represented by this sub-section. For this aim, the following points should be considered.The data is obtained from a company of the cardboard box production industries of Iran. These data are gathered according to the characteristics of the problem under study.

There are 10 types of cardboard box, 5 types (sizes) of raw sheets, and 3 suppliers where all suppliers can provide all raw sheets. Therefore, I=10, J=5, and K=3.

The mean (μ) of the normal uncertain values of the parameters are determined by the experts of the company of the case study. According to these experts 10% of each of these normal uncertain values is considered as its standard deviation (σ).

The uncertain demand values for 10 types of boxes provided by the experts is as follows:μd1=19975, μd2=21930, μd3=35814, μd4=55257, μd5=17707, μd6=17313, μd7=51508, μd8=46582, μd9=20762 and μd10=16161.

For the transportation costs the values of μtcjk1=3, μtcjk2=2, and μfcijk=10 are considered by the experts.

Also, for other parameters, the values of Table 1 and Table 2 are determined by the experts. This is notable to mention that due to the available cutting patterns of the company, the values of wij, and aij are deterministic.

It is notable to mention that the data are original, and no update or modification is done on the data.

Table 1 Data of the case study such as wij, and aij values.

Cardboard box (i)	wij(mm2)	aij	
j=1	j=2	j=3	j=4	j=5	j=1	j=2	j=3	j=4	j=5	
1	804	939	1074	1209	1330	65	78	91	104	126	
2	2945	1001	3251	1307	4335	12	18	18	24	24	
3	3348	1314	3564	1530	4511	10	15	15	20	21	
4	2290	1390	490	2740	476	15	20	25	25	35	
5	5096	7346	2630	4880	3507	3	3	6	6	8	
6	3805	6055	8305	10,555	9832	2	2	2	2	3	
7	1593	513	2763	1683	3684	9	12	12	15	16	
8	1593	513	2763	1683	3684	9	12	12	15	16	
9	4450	6700	1300	3550	600	2	2	4	4	6	
10	1250	3500	5750	8000	6000	2	2	2	2	3	

Table 2 Data of the case study such as μgjk, μg′jk, μpcjk1, and μpcjk2 values.

Raw sheet (j)	μgjk, μg′jk	μpcjk1	μpcjk2	
k=1	k=2	k=3	k=1	k=2	k=3	k=1	k=2	k=3	
1	2650	1459	1738	1092	1105	1053	928	973	955	
2	1746	2080	1137	1250	1337	1227	1092	1129	1155	
3	1407	1877	1309	1543	1559	1417	1382	1323	1285	
4	851	1031	884	1769	1713	1708	1578	1524	1533	
5	933	750	1279	2018	1911	2074	1883	1715	1707	

Experiments, results, and comparative study

In order to perform the computational experiments of the case study, the equivalent crisp formulations of the proposed MSMSP represented by Section "Equivalent crisp model" are solved by the proposed goal programming approach of Section "Experiments, results, and comparative study". For this aim the following issues are considered in advance.Five confidence levels are considered as 0.5, 0.6, 0.7, 0.8, and 0.9.

Six goals are considered for each objective function. The first goal is the positive ideal solution and other goals are set by the experts. Therefor for each objective function (say i), the set of goals OFi1=OFi+<OFi2<OFi3<OFi4<OFi5<OFi6 is considered. This means that for each objective function five interval goals are considered. The penalty values of pi1=0, pi2=OFi2OFi6, pi2=OFi3OFi6, pi2=OFi4OFi6,

and pi2=OFi5OFi6 are considered for the interval goals 1 to 5 respectively.

Three combinations of objective function weight values for the integrated objective function (86) are considered as shown by Table 3.

Fifteen scenarios are applied to the hyperparameters of the objective function (83), including three weight combinations for five confidence levels as shown in Tables 3 and 4. The weight values are differed from 0.1 to 0.6 for each objective function. Therefore, the importance of each objective function is considered here.

For comparing the proposed goal programming approach to the approaches of the literature, the goal programming approach of Jadidi et al.50 is considered. The multi-choice goal programming (MCGP) approach of this study is the most close method to the approach presented in Section "Computational experiment of the case study". The MCGP introduced by Jadidi et al.50 considers an interval goal for each objective function. They were inspired by the fuzzy model of Tiwari et al.51 and considered two pivot points for optimizing each objective function by enabling DM to have more control on the interval of each goal. In our study, the common point of these intervals is considered as OFi1+OFi62 to implement the method of Jadidi et al.50.

Table 3 Weight combinations of the objective functions.

Weight combination	ω1	ω2	ω3	
1	0.1	0.3	0.6	
2	0.3	0.6	0.1	
3	0.6	0.1	0.3	

Table 4 Different scenarios considered for the numerical experiments.

Scenario	Weight combination	Confidence level	
1	1	0.5	
2	1	0.6	
3	1	0.7	
4	1	0.8	
5	1	0.9	
6	2	0.5	
7	2	0.6	
8	2	0.7	
9	2	0.8	
10	2	0.9	
11	3	0.5	
12	3	0.6	
13	3	0.7	
14	3	0.8	
15	3	0.9	

First in each confidence level the positive ideal solution of model (67)–(75) is obtained. For this aim the data of Sect. “Data of the case study” is used to solve model (67)–(75). This model is solved three times and, in each time, one of the objective functions is optimized individually according to all of the constraints in order to obtain the positive ideal solution of the objective function. Then according to the experts of the company of the case study, in addition to the positive ideal solution of each objective function, five more goals are determined. Therefore, totally six goals for each objective function in each confidence level are considered and represented by Table 4.

Using the data of Sect. “Data of the case study”, the weight values of Table 3, and the goal values of Table 5, the required final experiments by applying the integrated model (83)–(92) are done, and the obtained results (the obtained Pareto-optimal solutions) are represented in terms of objective function values by Table 6. In order to be more clear, deviation of each objective function value of Table 6 from its positive ideal solution (represented by Table 5) is calculated and reported by Table 7. In addition, the results obtained by the method of Jadidi et al.50 are represented by these tables.Table 5 The obtained positive ideal value and the goal values proposed by the experts for all confidence levels.

Confidence level	Objective function	OFi1=OFi+	OFi2	OFi3	OFi4	OFi5	OFi6	
0.5	1	61,970,510	62,500,000	63,500,000	64,500,000	65,500,000	66,500,000	
2	43,097,940	44,000,000	45,000,000	46,000,000	47,000,000	48,000,000	
3	303,081	320,000	340,000	360,000	380,000	400,000	
0.6	1	63,358,550	64,000,000	65,000,000	66,000,000	67,000,000	68,000,000	
2	43,883,510	44,500,000	45,500,000	46,500,000	47,500,000	48,500,000	
3	296,313	310,000	330,000	350,000	370,000	390,000	
0.7	1	64,863,610	65,500,000	66,500,000	67,500,000	68,500,000	69,500,000	
2	44,688,450	45,000,000	46,000,000	47,000,000	48,000,000	49,000,000	
3	288,894	300,000	320,000	340,000	360,000	380,000	
0.8	1	66,709,240	67,500,000	68,500,000	69,500,000	70,500,000	71,500,000	
2	43,945,110	45,000,000	47,000,000	49,000,000	51,000,000	53,000,000	
3	279,927	290,000	310,000	330,000	350,000	370,000	
0.9	1	69,480,800	70,000,000	71,000,000	72,000,000	73,000,000	74,000,000	
2	44,276,850	47,000,000	50,000,000	53,000,000	55,000,000	58,000,000	
3	266,332	280,000	300,000	320,000	340,000	360,000	

Table 6 The results obtained by the proposed goal programming approach and the goal programming approach of Jadidi et al.50.

Scenario	The proposed goal programming approach	The goal programming approach of Jadidi et al.50	Improvement*	
OF1	OF2	OF3	OF1	OF2	OF3	OF1	OF2	OF3	
1	61,971,570	46,000,010	303,099	62,395,840	43,640,420	303,083	− 424,270	2,359,590	16	
2	63,360,460	47,500,110	309,799	63,795,370	45,308,580	310,001	− 434,910	2,191,530	− 202	
3	64,894,200	48,330,980	317,178	65,308,220	47,778,000	320,000	− 414,020	552,980	− 2822	
4	67,205,040	51,013,510	326,185	69,480,770	51,044,410	330,000	− 2,275,730	− 30,900	− 3815	
5	69,511,470	56,334,370	339,744	69,955,390	55,734,570	340,000	− 443,920	599,800	− 256	
6	61,971,610	44,390,990	303,101	62,395,560	43,098,490	303,100	− 423,950	1,292,500	1	
7	63,360,280	47,500,010	309,815	63,791,340	45,309,220	310,009	− 431,060	2,190,790	− 194	
8	64,894,200	48,733,490	320,000	64,864,390	48,289,880	317,197	29,810	443,610	2803	
9	66,750,450	51,832,470	326,183	69,480,800	51,045,520	330,015	− 2,730,350	786,950	− 3832	
10	69,511,540	56,334,150	339,744	69,481,530	56,540,180	340,000	30,010	− 206,030	− 256	
11	61,970,900	46,000,000	303,113	61,977,080	45,255,540	303,094	− 6180	744,460	19	
12	63,360,300	47,500,110	309,903	63,386,550	46,028,370	310,000	− 26,250	1,471,740	− 97	
13	64,894,110	48,372,570	317,188	65,307,030	47,778,130	320,014	− 412,920	594,440	− 2826	
14	66,749,140	53,677,560	326,205	67,163,570	50,886,420	330,004	− 414,430	2,791,140	− 3799	
15	69,482,940	56,545,510	339,753	69,509,170	56,329,460	340,005	− 26,230	216,050	− 252	
*The negative values are the improvements made by the proposed approach and the positive values are the improvements made by the approach of Jadidi et al.50.

Table 7 Deviations from the positive ideal solution for the results of the proposed goal programming approach and the goal programming approach of Jadidi et al.50.

Scenario	The proposed goal programming approach	The goal programming approach of Jadidi et al.50	
OF1(%)	OF2(%)	OF3(%)	OF1(%)	OF2(%)	OF3(%)	
1	0.002	6.734	0.006	0.686	1.259	0.001	
2	0.003	8.241	4.551	0.689	3.247	4.619	
3	0.047	8.151	9.790	0.685	6.914	10.767	
4	0.743	16.085	16.525	4.155	16.155	17.888	
5	0.044	27.232	27.564	0.683	25.877	27.660	
6	0.002	3.000	0.007	0.686	0.001	0.006	
7	0.003	8.241	4.557	0.683	3.249	4.622	
8	0.047	9.052	10.767	0.001	8.059	9.797	
9	0.062	17.948	16.524	4.155	16.157	17.893	
10	0.044	27.232	27.564	0.001	27.697	27.660	
11	0.001	6.734	0.011	0.011	5.006	0.004	
12	0.003	8.241	4.586	0.044	4.888	4.619	
13	0.047	8.244	9.794	0.684	6.914	10.772	
14	0.060	22.147	16.532	0.681	15.795	17.889	
15	0.003	27.709	27.567	0.041	27.221	27.662	
Minimum value of each objective function in each scenario is in bold.

The results shown by Table 7 can be used for comparing the applied methods and also for further interpretations. Based on the obtained results (bolded values of Table 7 show better performances), in 13 out of 15 experiments the proposed goal programming provides better value for objective function 1. In the case of objective function 2, the approach of Jadidi et al.50 performs better than the proposed goal programming approach. Based on the obtained results, in 13 out of 15 experiments the goal programming approach of Jadidi et al.50 provides better value for objective function 2 and in two experiments, the better value is obtained by the proposed goal programming approach of this study. In the case of objective function 3, based on the obtained results, in 11 out of 15 experiments the proposed goal programming approach of this study provides better value than the approach of Jadidi et al.50.

Behavior of objective functions 1, 2, and 3 over changing confidence level value are also depicted by Figs. 1, 2, and 3 respectively. In addition, the improvements of Table 6 are depicted by Fig. 4 where the negative values are the improvements made by the proposed approach and the positive values are the improvements made by the approach of Jadidi et al.50.Fig. 1 Behavior of objective function 1 over changing confidence level value.

Fig. 2 Behavior of objective function 2 over changing confidence level value.

Fig. 3 Behavior of objective function 3 over changing confidence level value.

Fig. 4 The improvements in the objective functions obtained by the proposed approach and the approach of Jadidi et al.50.

Managerial insights

As the managerial implications of the proposed problem and solution methodology of this study, the followings can be mentioned,For cardboard box industries the model and solution approaches can be used to optimize their supplier and material selection decisions.

The proposed mathematical model can be used for cardboard box industries, in order to reduce purchasing and transportation costs.

Using the proposed formulation, in cardboard box industries, the amount of wastage of the raw sheets can be decreased.

The proposed solution methodology and the uncertainty applied to define the problem, can be easily applied to supply chain related problems.

The belief-degree-based uncertainty can be a useful method for employing the idea of experts.

The belief-degree-based uncertainty can be a good tool for tackling a new problem in any organization when there is no historical data.

Conclusion

In this study, we discussed a supplier and material selection programming with a multi-objective optimization model in a real cardboard box manufacturing company. The purpose of the problem and its model was to minimize the wastage of cutting raw materials, and at the same time minimizing the purchasing cost of raw materials and its transportation cost, and also minimizing the surplus of production. So, it was modeled as a multi-objective mixed integer linear programming model. This problem is an extended version of the problems of literature where a different discount logic for purchasing the raw materials was applied and transportation costs with fixed and variable charges were added. According to the conditions some parameters of the problem such as demand, price and transportation cost coefficients could not be clearly determined, therefore, were considered under belief-degree-based uncertainty. So, for the first time these types of problems were investigated under this type of uncertainty in this study. As solution approach first, the chance constraint method was used to convert the uncertain model to a multi-objective crisp form. Then, as another novelty, the method proposed by Jadidi et al.26 was extended as a new goal programming method where several aspiration levels are determined for each objective function and apply piece-wise penalty function in order to achieve a better result. Finally, a numerical instance from the case study was used to evaluate the performance of the proposed new goal programming approach and compare it to the methods of the literature.

For further study, the problem can be examined in different and larger sizes, and due to the fact that it becomes larger and may take time to solve by GAMS software, the meta-heuristic algorithms can be used to solve it. On the other hand, other assumptions like multi-mode demand satisfaction strategy, selling prices consideration, etc. can be considered while modeling the problem.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-71527-8.

Author contributions

Z.N. contributed to basic concepts, formulation development, methodology design, and implementation. H.M contributed to basic concepts, methodology design, and results interpretation. S.N. contributed to formulation development, methodology design, and writing the paper.

Data availability

Data is provided within the manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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