A robust fuzzy-stochastic optimization model for managing open innovation uncertainty in the ambidextrous supply chain planning problem

This paper studies the effect of the open innovation concept in the product design process and supply chain master planning. Complex uncertainties caused by using outbound resources within co-design processes, financial challenges between the collaborating parties, and integrating outbound innovative designs with supply chain tactical planning problem are taken into account. To this end, a robust fuzzy-stochastic optimization model is proposed which can integrate the product design with the main activities of a multi-product supply chain. The proposed model is able to cope with different type of uncertainties including random, epistemic and deep uncertainties. Integrating the financial and physical flows, using a novel pricing mechanism, and considering the outside-in innovations in the product design process are the outstanding contributions of the proposed model. Furthermore, to cover both the short-term and long-term success criteria, ambidexterity of the studied supply chain is taken into account via two conflicting explorative and exploitive objectives. Results indicate the superiority of the presented model and its ability in supporting managerial decisions in the mid-term planning process.


Introduction
Open innovation concept firstly entered the literature domain by Chesbrough (2003) as benefiting the outbound capabilities for innovation activities interacting with other firms and people. Achieving innovative ideas and creative solutions tend pioneer companies including IBM, Samsung, Google, and Apple to consider open innovation concept as a complementary section on traditional research & development (R&D) issues (Carroll 2010;Cheng et al. 2015;Samsung Electronics Co., Ltd., et al. v. Apple Inc. 35 U.S.C. 2017). Alongside many attentions have been presented in the last studies on open innovation concept, there is not convenient support on how open innovation is integrated with traditional supply chain problems. In more detail, how a large number of designs are selected, verified, and implemented in a current production system. Therefore, we are faced with a new supply chain system in which the designs are not determined before the planning process and must be considered as an uncertain event during the supplying, manufacturing, and distributing activities. However, some key challenges should be addressed in the managers' decisions as follows:

1) How individuals that collaborate within an open
innovation process can be benefited from its financial advantages? 2) How input designs are integrated with the production process within the supply chain environment? 3) How the uncertainties caused by outbound resources within the open innovation process are handled? 4) How the proposed designs are verified and validated to fulfill new product development requirements of collaborating parties? Therefore, there is an uncovered gap in how the individuals can be engaged in the designing processes in a supply chain planning system. As a better description, many pioneer companies involve a large group of individuals in the designing process applying a predefined award and the top-rated patterns are collected, modified, and selected for the manufacturing process. In this way, the key research questions that are going to be addressed in this research are how the public collaboration in the product design process is handled in the supply chain tactical planning problems.
The main idea of the current study is developing a supply chain master planning model that appropriately considers open innovation characteristics including the innovation award, customers rating systems, demand forecasting procedures, and price determination techniques integrated with the decisions in the production process. Alongside the supply chain mid-term objectives, some strategic decisions may affect future insights in the open innovation concept. Product portfolio management and brand development issues are the strategic issues that need to be modified besides the tactical ones. To cover the longterm and mid-term intentions simultaneously, the model benefits from ambidexterity objectives. However, tactical decisions are optimized according to maximization of the total net profit considering the supply chain revenue and related costs. Furthermore, the model proposes the number of selected designs (i.e., the variety in product portfolio) and total paid awards as the supply chain reputation in the form of long-term objectives. Additionally, the developed SCMP model can integrate the financial and physical flows through different supply chain stages.
On the other hand, the problem faces some inherent uncertainties in supplying, manufacturing, and marketing processes concluded from designing characteristics that are not determined before the planning process, unlike the R&D outputs. Moreover, deficiency of historical data in open innovation activities and the effect of environmental changes on the designing process lead to an ambiguous situation for future trends. However, to deal with different sources of uncertainty, including randomness, epistemic and deep uncertainties in such problems, this paper proposes a robust fuzzy-stochastic optimization model. In this regard, this study applies a possibilistic chance-constrained programming with fuzzy parameters to cover the uncertainties caused by outbound resources within the open innovation process. The applicability of the proposed model is evaluated through a case study conducted within an ecosystem formed in the home appliance industry in Iran.
The remainder of this paper is modified as follows. In Sect. 2, the relevant studies in SCMP and open innovation context are reviewed and analyzed. The description and formulation of the model are presented in Sect. 3. In Sect. 4, a hybrid robust optimization model is elaborated to cope with hybrid uncertainty in model parameters. Additionally, Sect. 5 investigates the numerical results and sensitive analysis and finally, research conclusions and future directions are devoted in Sect. 6.

Literature review
In the last decade, a large number of studies emphasized managing supply chain main decisions in a mid-term planning horizon (e.g., Alemany et al. 2010;Peidro et al. 2010;Söderberg and Bengtsson 2010;Bashiri et al. 2012;Vafa Arani and Torabi 2018). Considering products' financial and physical flows in supplying, manufacturing, distributing, and marketing processes are the core contribution of the previous studies in supply chain planning problems (see Torabi and Hassini 2009;Peidro et al. 2010;Vafa Arani and Torabi 2018;Yousefi and Pishvaee 2018;Ghafarimoghadam et al. 2019). Below the literature related to the addressed research issue is reviewed and analyzed.

Supply chain master planning
Improving the efficiency and responsiveness throughout the supply chain and also delivering improved value from the upstream to downstream in a mid-term planning horizon is the main goal of SCMP (Vafa Arani and Torabi 2018). SCMP deals with a three-echelon supply chain architecture that consists of suppliers, manufacturers, and distribution centers, and try to find the best managerial decisions by integrating the supply chain main processes (Torabi and Hassini 2008). In this section, we review the SCMP literature according to both physical and financial flow considerations. Additionally, the uncertainty in the SCMP and the proposed solving approaches are discussed in more detail.

Physical and financial flow
Generally, the main scope of previous researches in SCMP is focused on the integration of procurement, production, and distribution processes (e.g., Kim 2013;Vafaeenezhad and Tavakkoli-Moghaddam 2016;Heydari et al. 2018;Li and Chen 2020). However, limited number of research works focused on integrated physical-financial planning models (see Vafa Arani and Torabi 2018; Silvestro and Lustrato 2014;Bian 2017). Furthermore, to develop a supply chain tactical planning according to financial and physical purposes, Bian (2017) served an interface between operations and financial activities. This study developed a classical Economic Order Quantity (EOQ) model to maximize supply chain profit value by considering the financing cost of working capital requirement. Moreover, a multi-level uncapacitated model with lot sizing-based discount is developed based on the financial flow assumptions and finally, the research proposed sequential approach as solution contribution which determines the production program level by level based on the material requirements planning logic. Furthermore, Vafa Arani and Torabi (2018) reviewed the supply chain master planning literature and presented a tactical planning model that considers financial and physical activities simultaneously. Additionally, in this study, a mixture of fuzzy and random fuzzy variables is investigated dealing with different sources of uncertainty.
As the literature shows, the optimization of physical flow has received more attention than the financial flow between the different supply chain partners. However, to enhance the cash flow predictability, decrease the risk-related costs, and enhance the working capital and cash flows, supply chain managers are willing to engage novel approaches for managing financial flows throughout the supply chain. In practice, both academics and practitioners agree that the integration of the physical and financial flows can improve the efficiency and practicality of supply chain planning decisions.

Uncertainty in SCMP
Besides identifying various sources of uncertainty, many research studies have attended to different approaches dealing with uncertainty (see Stefansdottir and Grunow 2018;Gholamian et al. 2015a, b;Naderi et al. 2016). Liu (2007) introduced uncertainty theory as a branch of axiomatic mathematics for modeling belief degrees. Analyzing uncertain phenomena such as randomness and fussiness is the main goal of uncertainty theory. Moreover, this concept was widely applied in different fields including graph theory, network science, price optimization, and portfolio selection.
In another research, Bairamzadeh et al. (2018) investigated uncertainty of input parameters in three main categories including (1) randomness, (2) epistemic, and (3) deep uncertainty according to the availability of related information. Randomness uncertainty demonstrates the random nature of input parameters that are derived from a given system. Furthermore, accessing adequate and sufficient historical data as well as confirming that the achieved data are valid for future considerations are the main decision criteria for supposing randomness uncertainty of input data (Bairamzadeh et al. 2018). Moreover, epistemic uncertainty is derived from the non-deterministic pattern of limited data inspired by the system or the environment (Naderi et al. 2016). On the other hand, lack of information or ambiguous nature of the input data leads to a deep uncertainty on how the behaviors are investigated. Therefore, Table 1 represents a general classification, including sources and types of uncertainty and the methods implemented to handle these uncertainties in SCMP literature.
Therefore, in real-world optimization problems including open innovation operations, according to availability of data it would not be acceptable to consider all of imprecise numbers in a single uncertainty form. Hereby, by reviewing the previous literatures that considered various modeling approaches, it would be more welcomed to apply a mixed optimization technique that considers all types of uncertainties based on the different input sources.

Gap analysis
There is enriched literature on SCMP under traditional SCM considerations. Within the scenarios considered in these studies, the products are designed and validated using internal R&D activities. The proposed SCMP models focused on finding the best decision variables to determine how many predefined products are produced within the planning horizons. As a better expression, the product's features including the shape, quality, size, color, etc., are determined and modified before the planning process. Moreover, by introducing the open innovation concept into the SCM domain, the basic characteristics of products are faced with different sources of uncertainty, and this leads to considerable changes in the presented planning models. In this regard, Rahmanzadeh et al. (2019) presented some developments that address changing ahead to cover the open innovation assumptions in the designing process. However, the developed model improves Rahmanzadeh et al. (2019) study in four outstanding scopes.
First, this paper considers three different types of financial methods including (1) cash investment, (2) bank loans, and (3) debt of suppliers in the supply chain master planning problem. Secondly, the uncertainty input parameters are categorized in randomness, epistemic, and deep uncertainties according to the availability of historical data. Additionally, the paper considers a robust approach coping with the uncertainty condition in a hybrid robust optimization problem. Third, despite Rahmanzadeh et al. (2019) study, this paper assumed that the production of selected design is not limited to a single period and can be planned on the other next periods. At the final scope, this paper focuses on long-term objectives alongside mid-term purposes to enhance the supply chain reputation issues. Therefore, we can depict outstanding gaps between the previous studies, and to cover the addressed gaps, conceptual developments are considered in this paper. Therefore, the prominent contribution of this paper can be presented as follows.
• The open innovation process is engaged in supply chain master planning problems.
• Different sources of uncertainty including randomness, epistemic, and deep uncertainties are handled in the planning process. Moreover, a hybrid robust  Therefore, this paper presents a novel SCMP model that considers the open innovation assumptions as well as managing physical-financial flows. On the other hand, managing the different sources of uncertainty using a hybrid robust optimization model is another outstanding contribution of this paper. Generally, a comparative view of the current study and the recent SCMP literatures is provided in Table 2.

Problem description and formulation
This paper proposes a bi-objective mathematical model that considers supply chain mid-term planning decisions as well as long-term concerns. As shown in Fig. 1, the studied problem is a three-echelon supply chain consisting of multiple suppliers, one manufacturer, and multiple distribution centers regarding multi-period and multi-product considerations.
To apply the open innovation concept in the concerned problem, first, the company presents complete information about the requested designs including main characteristics, materials, and operational constraints. Then, the outbound individuals propose innovative designs and at the next step, the received designs are rated by a large group of customers. To simplify the selecting process, only a small set of top-rated designs are candidate for the production process. The developed tactical supply chain planning model can integrate the product design process with the supplying, manufacturing, and distributing activities throughout the supply chain.
Without loss of generality, it is considered that each product is produced and delivered to customers based on the received orders and pull production strategy. On the other hand, the model justifies the co-design process assumptions in the form of open innovation competition and users' rating system. Besides implementing open innovation concept in the design process, synchronizing the physical and financial flows along the entire supply chain as well as supporting efficient use of supply, production, and distribution capacities are the main purposes of supply chain planning problems (Rohde and Wagner 2002). However, to deal with co-design in supply chain processes, some inherent uncertainties such as selected design characteristics and users' rating values must be included in the physical and financial planning that are never presented in previous studies. Generally, the model assumptions can be summarized as follows: • Product designs achieved from the open innovation competition and the top-rated ones are selected as a candidate for the production process. • A negative linear relationship is assumed between the product's order quantity (OQ) and its price (P).
(i:e:; OQ ¼ a À bP 8b [ 0). • Raw material supplying process accomplished based on pricing discount strategy. In this regard, by increasing the ordered quantity, raw material price is decreased generally. • Input designs are classified into predefined categories.
However, the products with the same category are produced based on similar production steps. • The model considers three different types of financing methods including (1) cash investment, (2) bank loans, and (3) debt of suppliers with an interest rate for repayment. • The award accrued to the individuals that their designs are selected to produce. • The amount of award is considered as a variable and is determined after the planning process. • There are several product groups that designers should submit their designs to these groups. • The production resources and storage spaces for inventories are restricted and not allowed more than a predefined value. • The debts received from the suppliers must be paid in the next time period.
The proposed mathematical model developed based on Rahmanzadeh et al. (2019) study tries to determine production and order quantities, inventory levels, products' price, estimated order quantity, awards, and the payments according to the cash flow and the amount of loan received from the bank. Based on the ambidexterity considerations, the model objective function consists of two separated expressions with long-term and mid-term perspectives. The mathematical notations applied in the proposed model are given as follows.   Production cost per unit in designs in class con machine p DI jrmt Lower level of m th discount interval for purchasing raw material r from supplier j g UR rc Quantity of raw material r is required per unit of input designs in class c g CM pc Capacity of machine p is required to produce one unit of designs in class c PRR jrmt Price of raw material r determined by supplier j based on discount interval m in period t

TC pt
Total capacity is available on machine p in period t

IC t
Total storage capacity is available in period t R I ct User rate of designs in the set I ct a ct Fixed coefficient represents the relationship between order quantity and user rating value for designs in class c in period t b ct Variable coefficient represents the relationship between order quantity and price for designs in class c in period t c ct Fixed coefficient represents the relationship between award paid and the order quantity in class c in period t

HF c
Holding cost of products in class c stored in the manufacturing center HR r Holding cost of raw material r stored in the manufacturing center TRR rjt Transportation cost of raw material r between supplier j and manufacturing center in period t TRF ilt 0 Transportation cost of final product i from set I ct between manufacturing site and distribution center l in period t' LN Large number g InLoan Interest rate of loans per period g InDebt Interest rate of supplier debt MaxCash t Maximum cash available in period t MaxAW ct Maximum award is allowed to pay for the selected products in category c at period t

Decision variables
FXP it 0 Quantity of product i from set I ct is produced in period t' SV i Binary variable indicating whether design i from set I ct is selected to produce or not RXO jrt Order quantity of raw material r from supplier j in period t FXT ilt 0 Quantity of product i from set I ct transferred to distribution center l in period t' Pr it 0 Price of final product i from set I ct in period t' Order quantity of product i from set I ct that must be fulfilled in distribution center l in period t' IR rt Quantity of raw material r stored at the manufacturing center in period t IF it 0 Quantity of product i from set I ct stored at the manufacturing center in period t' M rjmt Binary variable indicating whether raw material r purchasing from supplier j in period t placed in discount interval m or not Aw kt Award value is paid to each designer for selected products from the set I ct Cash t Amount of money must be paid in cash by the manufacturer in period t RecLoan t Amount of bank loan gotten by the manufacturer in period t RecDebt jt Amount debt from supplier j that could be used by the manufacturer in period t

Objective function
To achieve ambidexterity, this paper considers two separated objective functions according to the mid-term and long-term planning perspectives. At the first expression, the model aims to maximize the total net profit considering the supply chain revenue and related costs. In the second part, the model considers the number of selected designs (variety in product portfolio) and total paid awards as criteria for supply chain reputation on a long-term horizon. The verbal description and mathematical formulation of the mid-term and long-term objective functions are described below.

Objective function 1
Max (net profit) = sales revenue -raw material purchasing cost -raw material transportation cost -raw material inventory holding cost -costs production cost -final product holding cost -final product transportation costcompetition award cost -total additional repayment for the received loan and supplier debt.

Objective function 2
Max (supply chain reputation) = number of products are selected to produce ? total number of awards paid to designers (a and b considered as weighting factors).

Constraints
Generally, the model consists of three sets of constraints, namely (1) material flow, (2) financial flow, and (3) open innovation constraints.

Material flow
Constraint (3) represents the capacity limitation on the production machines in each period. Constraint (4) confirms that the inventory levels don't exceed the storage space limitation. Moreover, raw materials and final products' inventory balancing are guaranteed in Constraints (5) and (6), respectively. Moreover, Constraints (7) and (8) demonstrate pricing discount strategy, and Constraint (9) represents the number of products transferred to each distribution center must less than the product's order quantity. Additionally, this paper uses a weighted sum method as the most commonly approach in solving multi-objectives problems. To deal this purpose, the e-constraint method is applied generating Pareto optimal solutions. Furthermore, the results are applied to determine the weight factors by the decision makers to the objectives.

Financial flow
DebtRep Cash t MaxCash t 8t 2 T ð14Þ Constraints (10) and (11) calculate the additional fees that must be paid to the bank and suppliers according to the predetermined interest rates. Moreover, Constraint (12) guarantees that total repayments must be equal to the received loan and profit refund at the end of the planning horizon. Constraint (13) determines the suppliers' debt repayment in each period, and Constraint (14) presents the maximum allowable cash that can be invested in each period. Constraint (15) shows the financial balancing must be positive in each period. According to Constraint (15), the receiving money must be greater than the repayment and costs in each period.

Open innovation constraints
Aw ct MaxAw ct 8t 2 T; 8c 2 C ð18Þ Constraint (16) shows inequality between product order quantity and the linear combination of the product price and user rates. Constraint (17) emphasizes that awards value can affect the order quantity of products. Moreover, Constraint (18) presents an upper-level constraint on the payable awards, and Constraint (19) guarantees that the decision variable SV i assigned correctly according to the production plan.

Hybrid robust optimization method
Describing the hybrid robust optimization method, first, it is needed to clarify the different sources of uncertainty in SCMP input parameters. In the proposed model, six sets of input parameters, namely production capacity, production cost, bill of materials, user rates, loan, and debt interest rates, are tainted by uncertainty.
In this paper, to deal with six uncertain sets of parameters, a hybrid robust optimization model is presented. In this regard, according to available historical data about the loan and debt interest rates, possible values of interest rates are shown in a scenario-based framework with randomness probabilities. The expected technology-related parameters like production capacity and cost are not applicable based on historical data and will vary with the different design characteristics. Therefore, these parameters are formulated as triangular fuzzy numbers considering the expert's knowledge in extracting possibility distributions. Furthermore, users' rate of the input designs is modified as deep uncertainty according to the lack of information achieving to the probability or possibility distributions. Additionally, considering the ambiguous nature of the input designs and its effect on the number of raw materials that are required, a deep uncertainty is assumed for the bill of material parameters.

The proposed hybrid robust programming model
As described before, randomness, epistemic, and deep uncertainties are considered as different sources of uncertainty that are presented in this study. To handle the uncertainty of loan and debt interest rates, the robust scenario-based stochastic programming technique is implemented based on Yu and Li (2000) research. Additionally, a possibilistic robust optimization approach and Bertsimas and Sims's (2004) methods are applied, respectively, to deal with epistemic and deep uncertainty parameters, respectively. Therefore, the randomness parameters (loan and debt interest rates) included in Eqs. (1) and (10(-)13) can be reformulated as follows: LoanRep s þ DebtRep s f g À P s 0 p s 0 LoanRep s 0 þ ½ È À DebtRep s 0 g þ 2u s Þ is included in the objective function. Additionally, k considers the weight of average value against the optimality robustness. e þ ns ; e À ns n 2 1; 2; 3 f g and e þ 4tis ; e À 4tis represent the violations of Constraints 27-30, respectively. Additionally, w þ ns ; w À ns 8n 2 1; 2; 3 f g and w þ 4tjs ; w À 4tjs are the penalty of the presented violations. As mentioned before, in this research, the technological coefficients including the production capacity and costs considered as epistemic parameters formulated as possibilistic chance-constrained programming. In this regard, Pishvaee et al. (2012) presented a possibilistic chanceconstraint programming approach in more detail. However, assume PC 1 ð Þ cp ; PC 2 ð Þ cp ; PC 3 ð Þ cp and Cm 1 ð Þ cp ; Cm 2 ð Þ cp ; Cm 3 ð Þ cp as smallest likely value, most probable value, and largest possible value for the production cost and the production capacity parameters (triangular fuzzy numbers) respectively. Then Eqs. (1), (3), and (15) can be presented as possibilistic programming as below: where d is the penalty cost of constraint violation and a is the confidence level of chance-constrained and is reasonable to be greater than 0.5 (Pishvaee et al. 2012). In Eq. (30), the production cost is substituted by the expected value according to the triangular fuzzy number considerations. Moreover, the necessary condition is applied in Eqs. (32) and (33) to achieve the possibilistic chanceconstrained requirements. Dealing with the deep uncertainty parameters, we applied Bertsimas and Sim (2004) method. As presented in the proposed model, Constraint (5) formulated as equality constraints that must be converted into the inequality constraint forms. In this regard, we applied the Bertsimas and Thiele technique (Bertsimas and Thiele 2006) to deal with equality constraints containing the uncertain parameters in the inventory holding problem. Therefore, Constraints (5) can be reformulated as inequality constraint forms. 8r 2 R where C and C 0 are the budget uncertainty and varies in 0; C rt j j ½ and 0; C 0 ict Â Ã , respectively. In better expression, C and C 0 present the number of parameters in worst-case value and controls the trade-off between the optimality of the solution and its robustness to parameter perturbation.

Dealing with nonlinearity
As described before, this paper proposed an MINLP model to optimize supply chain decisions considering binary and continuous variables. In the proposed model, the objective function and constraints consist of two different types of nonlinear expressions including multiplication of two continuous variables (  (McCormick 1976) are applied which have attended as a basis for many global optimization solvers like Couenne and BARON. Generally, a linear programming relaxation is achieved by replacing each bilinear term involving variables x, y with new variable w ¼ x:y and setting the additional constraints. In this regard, suppose x and y are two continues variables in the interval x L ; x U ½ and y L ; y U ½ , respectively. Therefore, the linear programing relaxation can be constructed as below: w ! x:y L þ y:x L À x L :y L w ! x:y U þ y:x U À x U :y U w x:y L þ y:x U À x U :y L w x:y U þ y: To linearize the nonlinear expressions caused by the multiplication of continuous and binary variables, we applied the Williams (2013) method. Hence, suppose x 1 and x 2 be binary and continues variables, respectively. However, a continuous variable y can be defined and replaced with nonlinear expression x 1 :x 2 as y ¼ x 1 :x 2 . Therefore, the additional constraints can be presented as below:

Case study
To evaluate the applicability of the proposed model, two separated case studies within a value network formed by Iranian companies in the home appliance industry namely Pars Khazar and Snowa Companies are conducted. The importance of the home appliances design characteristics on the market share and company's sales revenue, the possibility of using individual capabilities to design different home appliance devices, and accessing historical data during the planning process are the main reasons for selecting Pars Khazar and Snowa as the case studies. Furthermore, these companies have a direct relationship with their end-users, and therefore, they can appropriately employ the crowd power within an open innovation process. To investigate the case study in more detail, first, the used parameters and assumptions are explained, and then the achieved results are analyzed according to the different experiments.

Parameters and assumptions
In this paper, we considered two sets of products namely stand fan and microwave oven in Pars Khazar company and vacuum machine in Snowa. As presented in Fig. 2, individuals must present innovative designs including six, eleven, and seven components regarding the main parts of stand fan, microwave oven, and vacuum machine, respectively. Pars Khazar as the main Iranian manufacturing company is located in Rasht, and the parts are supplied from seven internal suppliers located in Qazvin, Tehran, Arak, Isfahan, Tabriz, Bandar-e Abbas, and Mashhad. Additionally, the manufactured products are transferred to ten distribution centers located in Tabriz, Isfahan Tehran, Kerman, Babol, Mashhad, Shiraz, Rasht, Ahvaz, and Hamedan (cities are located in Iran). Moreover, Snowa is a well-known Iranian home appliance brand that located in Isfahan. In this study, we considered that the company manufactures the vacuum machine product through the open innovation process. The main Snowa suppliers are located in Tehran, Bandar-e Abbas, Mashhad, and Yazd, and the manufactured products are transferred to Tehran, Tabriz, Ahvaz, Shiraz, Mashhad, and Sari. In this study, to model the randomness parameters, three distinct scenarios according to pessimistic, most likely and, optimistic condition is considered. Moreover, triangular fuzzy parameters are presented dealing with epistemic uncertainties including production costs and capacity constraints. To handle the deep uncertain parameters, it is assumed that 50% of the input parameters are assigned to the worst cases.
On the other hand, the study considers some assumptions are presented below: 1) Interests rate of loans and debts is assumed [12%, 15%, 18%] and [10%, 14%, 17%] according to optimistic, most likely and, pessimistic conditions. 2) Suppliers suggest three discount intervals according to the number of parts are purchased. 3) All parts and products are transferred by trucks. 4) To deal with ambidexterity objectives, a weighted linear combination achieved is applied using experts' opinions in an AHP mechanism. 5) It should be noted that the model is solved on a personal computer with Core i5/2.6 GHz processor and 4 GB of RAM by using of Cplex 12.6 library in Visual Studio 2010 software. 6) It is assumed that the maximum award to the selected designs must be less than 40,000 €.
To evaluate the performance of the presented robust programming model, this study applies the proposed deterministic and robust models and analyzes the comparative results. In this regard, the deterministic model is presented considering a set of Eqs. (1)-(23) in which the nominal data are used for uncertain parameters. Moreover, the detailed formulation of the robust model has been modified in Sect. 4. It should be noted that to deal with the randomness uncertainty, Constraints (10)-(13) are converted to sets of Constraints (26)-(29). Furthermore, Constraints (3) and (15) are substituted by Constraints (32) and (33), respectively, according to epistemic uncertainties. Finally, Constraints (35) -(39) are presented as the robust counterpart of Constraints (5) and (16) to handle the deep uncertainties.

Achieved results
Achieved results of solving the deterministic and robust model using the nominal data are summarized in Table 3. On the other hand, to create a comparison report between the deterministic and robust models, only the first term of the robust objective function p s :OF s ð Þis considered in the computational results. As Table 3 shows, in Pars Khazar study, the robust model suggests 75,858 € less profit than the deterministic model that is considered as the cost of robustness to overcome the model's uncertainty. By focusing on the achieved results, it should be noted that the total number of manufactured products recommended by the robust model is 447 units less than the deterministic one. Additionally, in Snowa study, the achieved results showed the deterministic model obtains 42,623 € more profit according to the robust one by suggesting 267 more manufactured units. Additionally, Fig. 3 shows the detailed information of parts purchased from the suppliers according to the deterministic and robust model.
On the other hand, to validate the proposed robust optimization model, two different models namely R deterministic and R robust based on the solution of the above deterministic and robust models are presented. Moreover, 30 sets of uncertain parameters generated by uniform distribution considered to compare the robustness of achieved solutions. However, the output of using the deterministic and robust model in the form of average and standard deviation value is presented in Table 4. As the results show, the robust model presents a better solution in 80 percent of cases. Moreover, in Pars Khazar study, the robust model increases the objective function value 11,90,017-10,55,061 = 134,956 € in average based on the random realizations of uncertain parameters. Therefore, as a result, the deterministic model presents an efficient solution based on the nominal data, i.e., 12,08,340-11,32,482 = 75,858 € that is considered as the cost of robustness in the robust model. On the other hand, supposing 30 sets of random input uncertain parameters for realizing the future changes leads to an additional profit of 134,956 €. However, according to the cost of robustness that is paid at present to obtain a robust solution, 59,098 € (134,956 -75,858) was modified as 'value of robustness'. Additionally, as we expect, the scattering of the achieved solutions in the robust model is much lower than the To show the financial flow in the investigated supply chain, we reported total investment accomplished by three different types of financing methods including (1) cash investment, (2) bank loans, and (3) debt of suppliers with a specific interest rate for repayment. However, the model was run with different maximum cash value (MaxCash t ) and the result showed using the cash investment is the priority for financing supply chain operations. Additionally, by limiting the MaxCash t value, the model suggests more financing investment by bank loans and supplier debts. Additionally, as shown in Fig. 4, bank loan investment plays a more important role than the debt investment in large MaxCash t value (300,000 €).
In another examination, we studied the effect of the award that was paid to designers on the supply chain total net profit. In this regard, the maximum award that is allowed to pay for the selected products is changed from 10 to 120% of nominal values. As the result in Fig. 5 shows, the value of objective function 1 is increased by raising the maximum payable award (MaxAw kt ) from 10 to 60%. Then, the objective function 1 remains constant by changing MaxAw kt from 60 to 120%. Therefore, it seems Constraint (18) is considered as an active constraint before 60% and as a passive constraint after 60%. On the other hand, considering the effect of the payable award ðAw kt Þ value on the objective function 2 expression, it is enhanced linearly by increasing in the maximum payable award.

Conclusions and future research
This paper presents a mathematical model to integrate the open innovation assumptions with the supply chain tactical planning process. This study assumes that the designs of the products are created by a large group of outbound designers and verified and selected to implement in the production process. For this purpose, the proposed model considers three sets of constraints that are concerned, respectively, with material flow, financial flow, and open innovation processes. Moreover, the ambidextrous objectives that are focused on the explorative and exploitive criteria are considered as the model's objective function. Additionally, the proposed model takes into consideration three different sources of uncertainty including randomness, epistemic and deep uncertainties. To handle these uncertainties, a hybrid robust optimization model is developed. To modify the mathematical model in more detail and also to investigate its applicability, a real-world case study is conducted within a value network formed in a home appliance industry in Iran. The performance of the deterministic and presented robust models is indicated under nominal data and 30 sets of random parameters. The results showed that the proposed robust model presents efficient solutions based on the random uncertain parameters for realizing the future changes. Therefore, this paper presented an integrated innovative product design and supply chain tactical planning model considering different sources of uncertainty, financial investments, and ambidextrous objectives.
Future researches must go beyond studies that focus on engaging individuals in different supply chain areas like supplying, manufacturing, distributing, and marketing processes. Design of crowdsourcing platforms, development of new smart manufacturing technologies, and enhancement of public-based delivery systems are examples of future researches direction.
Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Data availability The datasets generated during and/or analysed during the current study are not publicly available due to company information confidentiality but are available from the corresponding author on reasonable request.