A robust multiobjective global supplier selection model under currency fluctuation and price discount
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Abstract
Robust supplier selection problem, in a scenariobased approach has been proposed, when the demand and exchange rates are subject to uncertainties. First, a deterministic multiobjective mixed integer linear programming is developed; then, the robust counterpart of the proposed mixed integer linear programming is presented using the recent extension in robust optimization theory. We discuss decision variables, respectively, by a twostage stochastic planning model, a robust stochastic optimization planning model which integrates worst case scenario in modeling approach and finally by equivalent deterministic planning model. The experimental study is carried out to compare the performances of the three models. Robust model resulted in remarkable cost saving and it illustrated that to cope with such uncertainties, we should consider them in advance in our planning. In our case study different supplier were selected due to this uncertainties and since supplier selection is a strategic decision, it is crucial to consider these uncertainties in planning approach.
Keywords
Supplier selection Robust programming Discount Currency exchange rate uncertaintyAbbreviations
 DEA
Data envelopment analyses
 DMU
Decision making unit
 CNY
China, Yuan Renminbi
 Toman
Iranian Toman
Introduction
Supplier selection is a strategic decision that determines the long viability of a company, particularly when purchasing costs represents a significant portion of the operating costs. The supplier selection is considered as a crucial part in achieving the objectives of an effective supply chain (Ng 2008). The decisions related to the supplier selection problem in the relevant literature are mainly related to which supplier to select and how much to order from each supplier in each period and over the planning horizon. Some researchers outlined and classified the significant risks associated with global outsourcing (Christopher et al. 2011; Sawik 2011).
One of the significant risks of global purchasing is the risk of uncertain changes in the currency exchange rates that affects the total supply chain cost. If the buyer does not consider these changes, selection and purchasing decisions will be suboptimal. It seems necessary that global supplier selection models consider exchange rates fluctuations to cope with challenges that purchasing managers are facing in global environments. The literature seems to be sparse on analytical approaches involving the currency fluctuation uncertainty in the supplier selection process. It should be noted that it is one of our underlying principles in this research paper. We model the currency fluctuation through a scenariobased approach. The motivation of this modeling will be presented in detail in “Literature review” and “Overall comment and problem statement”.
The remainder of this paper is organized as follows. “Literature review” presents a literature review on existing suppliers’ selection models, and variety of risks associated with the selection of suppliers. In particular, emphasis was placed on currency fluctuation risk and discounts. “Overall comment and problem statement” is dedicated to problem statement. “Problem formulation” addresses the mathematical formulation for the problem under study. In “Solution method” a solution method has been provided. In “Numerical study”, the computational experiments are presented. “Conclusions” argues the concluding remarks and future research direction.
Literature review
Many analytical techniques have been used to address the supplier’s selection problem. Ghodsypour and O’Brien (1998) integrated an analytical hierarchy process with a linear programing model to cope with both tangible and intangible factors in supplier selection problem. Thus, their objective function was total value of purchasing.
Ghodsypour and O’Brien (2001) developed a mixed integer nonliner programing model to solve a multiple sourcing problem. It takes into account the total cost of logistics, including net price, storage, and transportation and ordering costs in which the buyer has limitation on budget, quality, and service. For this purpose, they proposed an algorithm to solve the problem.
An innovative literature review has been done on supplier selection problem by De Boer et al. (2001). They classified this problem according to the supplier’s selection stage.
Wadhwa and Ravindran (2007) broadly clustered the related literature by the method used in modeling the program into three clusters including: single or multiobjective mathematical programming methods, and game theoretic methods. For a detailed review of these three clusters, we kindly refer the reader to (Wadhwa and Ravindran 2007).
Other researchers have addressed risk uncertainty or sourcing risk through quantitative models. A twostage stochastic model has been introduced by Xu and Nozick (2009) in which they considered different disruption scenarios. One of the scenarios specified the severity of the production capability lost at each supplier’s site in each time period. In this work, the researchers integrated the transportation model selection with the supplier selection. However, their model ignored price discounts, inventory costs, exchange rate fluctuations. Their objective function was to minimize the expected cost over the planning horizon.
Hammami et al. (2012) focused on low cost supplier issue with the uncertainty of delivery lead times from supplier sites to buyer’s sites and emphasized the impact of the safety stock at the buyer’s site while ignoring the price discount and exchange rate uncertainty of their model.
Amid et al. (2009) presented a fuzzy multiobjective model to deal with the supplier selection problem. In their research suppliers offered price breaks as a function of order quantity. Their three objective functions were (a) minimizing the net cost, (b) minimizing the net rejected items, and (c) minimizing the net late deliveries.
Hammami et al. (2014), proposed a stochastic model for supplier selection problem which considers price discount and inventory decisions for different scenarios.
Mazdeh et al. (2015) investigated the problem of supplier selection and lotsizing in two cases with and without discount. Given the npharded origin of the problem, the researchers developed a heuristic algorithm to solve the problem. Azadnia et al. (2015) integrated sustainable supplier selection with lotsizing decision in multiperiod multiproduct situation.
Overall comment and problem statement
Problem statement
The supplier selection problem is typically considered for long term, ranging from one to three years. The long lasting nature of strategic decisions requires to have a robust planning for every possible scenario. In this research, we developed a multiobjective supplier selection model that integrates the exchange rate fluctuation uncertainties with price discount while explicitly considering transportation and inventory costs. Then, the robust optimization is used in our modeling approach which is regarded as the best method in overtaking the uncertainties.
Our company is concerned with purchasing a product from heterogeneous suppliers over the planning horizon. The total amounts to be allocated to each supplier must be decided. Similarly, agreements must also be established. The set of suppliers are denoted by i. The total quantity bought from supplier i over the planning horizon is denoted by Q _{ i } based on which discounts are offered. A given supplier (i) offers discounts in N _{ i } intervals and a set of threshold quantities \(A_{i}^{n}\). The quantity \(A_{i}^{n}\) is the lower limit on the total purchased quantity bought from supplier i over the planning horizon. If \(A_{i}^{n} \le Q_{i} < A_{i}^{n + 1}\), then the unit price offered by supplier i on all purchased units is \(Pr_{i}^{n}\). Clearly, the larger n is, the smaller \(Pr_{i}^{n}\) will be, for each supplier i, the quantity \(A_{i}^{1}\) of the first discount interval is equal to the null value.
Since payment currencies of different suppliers differ, all prices are expressed in the currency of the parent company. Changes in the currency exchange rates are very difficult to predict in the longterm period. The difficulty economists have faced in finding an empirically successful exchange rate theory is well documented (Taylor 1995).
In this research to predict currency exchange rate fluctuations, different exchange rate scenarios on a quarterly basis are elaborated, some of which are published by financial organizations. Therefore, they are used in the current modeling approach.
We formulated the problem under study as a scenariobased stochastic model to address currency fluctuation. On the other hand, to cope with these uncertainties, we used a recent extension in robust optimization theory.
Supplier’s data
Supplier  Management cost (USD)  Unit base price (USD)  Transportation cost (USD) 

China  5.25  100  3000 
China  5  100  3000 
France  6  100  7000 
India  3.25  100  2000 
Problem formulation
We developed a mixed integer scenariobased stochastic programing model for our supplier selection problem. First stage decisions (must be made immediately) included which suppliers to select and the total quantity to purchase from each supplier over the planning horizon. The second stage variables are the quantity ordered each period and inventory level. These variables are associated to each scenario and depend on firststage variables.
Notations

\(Y_{i}\): Binary integer variables, \(Y_{i} = 1\) if supplier i was selected, 0 otherwise.

\(Q_{i}\): Total amount ordered from supplier i over the planning horizon.

\(e_{i}^{n}\): Binary integer variables, \(e_{i}^{n} = 1\) if the quantity purchased from supplier.

\(q_{i}^{{t{\text{s}}}}\): Quantity ordered from supplier i in period t under scenarios \({\text{s}}\).

\({\text{Inventory}}^{{t{\text{s}}}}\): Inventory level at the beginning of period t under scenario \({\text{s}}\).
Cost factors

\(Pr_{i}^{n}\): Supplier i’s Unit purchasing price in the currency of the supplier associated with its discount interval.

\({\text{TC}}_{i}\): Unit transportation cost from supplier i.

\(h^{t}\): Unit inventory holding cost over period t.

\({\text{MC}}_{i}\): Supplier i’s management cost.

\(D^{t}\): Demand in period t.

\(A_{i }^{n}\): Lower limit on the business volume of supplier i that corresponds to the discount interval n \(\left( {0 = A_{i}^{1} ,\forall i \in I} \right)\).

\(L_{i}^{t}\): Capacity of supplier i in period t.

\(\alpha_{i}^{{t{\text{s}}}}\): Exchange rate from the currency of supplier i to the standard currency.

\(p_{\text{s}}\): Probability of scenario \({\text{s}}\).
Objective functions
The objective function (1) minimized total cost z expressed in currency of reference. This cost was the sum of the supplier management cost, purchasing cost, and the inventory cost. All cost factors are expressed in the currency of reference except purchasing price.
Once the supplier is selected \((Y_{i} = 1)\), we have management cost. The quantity \(\mathop \sum \nolimits_{i \in I} {\text{MC}}_{i} Y_{i}\) is the total management cost for all selected suppliers. The unit purchasing price based on supplier i’s discount interval is \(\mathop \sum \nolimits_{n = 1}^{n = N} e_{i}^{n} Pr_{i}^{n}\). Under scenario \({\text{s}}\), the purchasing price is associated with the quantity \(q_{i}^{{t{\text{s}}}}\) in the currency of supplier i is \(\left( {\sum\nolimits_{n = 1}^{n = N} {e_{i}^{n} } Pr_{i}^{n} } \right)q_{i}^{t}\). It becomes \(\alpha_{i}^{t} \left( {\sum\nolimits_{n = 1}^{n = N} {e_{i}^{n} } Pr_{i}^{n} } \right)q_{i}^{t}\) in the currency of reference. Hence, the total purchasing price becomes \(\sum\nolimits_{i \in I} {\sum\nolimits_{t = 1}^{t = T} {\alpha_{i}^{t} } } \left( {\sum\nolimits_{n = 1}^{n = N} {e_{i}^{n} } Pr_{i}^{n} } \right)q_{i}^{t}\).
Constraints
The secondstage decisions might vary from one scenario to another but have to be in line with the firststage decisions. According to constraint (7), the total quantity purchased from supplier i \(\sum\nolimits_{t = 1}^{t = T} {q_{i}^{t} }\) must be equal to the total quantity allocated to supplier (i) \((Q_{i} )\).
Solution method
The proposed model is actually a multiobjective stochastic mixed integer linear programing. To solve this model, a twophased approach was proposed. In the first phase, we solved the multiobjective problem using lexicographic method. However, in the second phase, to cope with uncertainty, we adopted (Aghezzaf et al. 2010) method for the developed robust counterpart.
Coping with multiple objective
Here, \(i\) represents a function’s position in the preferred sequence, and \(F_{j} (x_{j}^{*} )\) represents the optimum of the \(j{\text{th}}\) objective function, found in the \(j{\text{th}}\) iteration (Stadler 1988).
To solve the multiobjective problem in our model, lexicographic method was applied which optimized second objective function while maintaining optimality of the first objective. Meanwhile, DM determined the weighting factor (B). We avoided using analytical methods to determine the weighting factor. However, multicriteria decisionmaking techniques such as analytic hierarchy process (AHP) could be used to precisely determine the weighting factor.
St.
St.
We called the above formulation here after model (B), where the objective function is (2) under (3)–(5) and (7)–(20) constraints.
Coping with uncertainty
\(Z_{\text{s}}^{*}\) is the optimal objective function value for each scenario. The corresponding value was considered as a given input parameter in model (R). As a result, by applying (21), robust counterpart would be:
Model (R):
Min (22)
St.
Numerical study
Discount schedule for different supplier
n  \(A_{i}^{n}\)  Quantity  Discount (%) 

1  0  0 to under 2000  0 
2  2000  2000 to under 3000  1 
3  3000  3000 to under 4000  2 
4  4000  4000 and over  3 
Exchange rates for different suppliers in periods
Exchange rate/period  1Q  2Q  3Q  4Q 

China  700  542  480  500 
China  700  542  480  500 
France  3700  3200  3500  4000 
India  660  500  460  520 
Value of stochastic programing
The value of the stochastic solution (VSS) was used in order to illustrate the importance of applying exchange rate uncertainty in supplier selection problem.VSS measures the expected cost saving from stochastic model rather than its deterministic counterpart.
In our case, VSS equals the value of optimal objective function in deterministic solution subtracted by the value of optimal objective function in stochastic solution.
We considered 3 forecasts for exchange rates for each supplier over the planning horizon.
DMC: is the model in which there is only one scenario (the average value) in each period.
DMF: the model considered fluctuation in three different scenarios in each period with their prospective probabilities. This model is multiperiod and multidynamic.
Example of exchange rate calculation for supplier France
Q1  Q2  Q3  Q4  Prob.  

Forecast 1  3550  3600  3490  4000  0.4 
Forecast 2  3500  3489  3700  3870  0.3 
Forecast 3  3700  3590  3750  3690  0.3 
Forecast DMC  3560  3580  3600  3990  – 
On the other hand, the exchange rate for supplier France in quarter 1 in DMF model would be 3550 × 0.4 + 3500 × 0.3 + 3700 × 0.3 = 3580.
Value of stochastic solution
Prob.  DMC model objective function  DMF model objective function  VSS 

0.5–0.25–0.25  5860456951  5616482220  243974731 
0.4–0.3–0.3  5895423564  5605464848  289958716 
0.35–0.35–0.3  5886426594  5647981354  238445240 
0.4–0.2–0.4  5845824689  5645816664  200008025 
We suggest different scenarios according to the probability associated with the different forecasts.
Stochastic and deterministic solutions
The French supplier was selected in the deterministic and stochastic models. However, it was not selected in robust model. Since supplier selection is a strategic decision which might be impossible to change in short terms, it is important to be prepared for even worst case possible scenario. Therefore, the robust optimization method is needed in this content.
Conclusions
In the present study, a multiobjective mixed integer programing model and its robust counterpart was developed. The buyer sources a product from an international network of diverse suppliers. Prices were offered based on suppliers’ local currencies which were subject to uncertainty over time. In addition, suppliers offered discounts.
An industrial case study was used to demonstrate the performance and applicability of the proposed model. Numerical studies suggested that industry practitioner or leaders could apply this model to their problem to save money and have robust contracts with the best possible suppliers. It was indicated that in the presence of exchange rate fluctuations decision variables were different. It illustrated that to cope with such uncertainties, we should consider them in advance in our planning. In our case study, different suppliers were selected due to these uncertainties and since supplier selection is a strategic decision, it is crucial to consider these uncertainties in planning approach. This would help purchasing manager to save money and time by selecting the best suppliers. With robust optimization method we prepared ourselves for even worst case scenario in both demand and exchange rates, thereby, the decisions were made according to these circumstances. Therefore, the purchasing manager could be ascertain that they would not lose demands even in worst case scenario. At the same time, the supplier manager will be assured to save remarkable amount of money. For future research, it would be important to incorporate other uncertainty parameters and develop efficient heuristic approach that explores and uses the specific characteristics of our model.
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