A branch and efficiency algorithm for the optimal design of supply chain networks
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Abstract
Supply chain operations directly affect service levels. Decision on amendment of facilities is generally decided based on overall cost, leaving out the efficiency of each unit. Decomposing the supply chain superstructure, efficiency analysis of the facilities (warehouses or distribution centers) that serve customers can be easily implemented. With the proposed algorithm, the selection of a facility is based on service level maximization and not just cost minimization as this analysis filters all the feasible solutions utilizing Data Envelopment Analysis (DEA) technique. Through multiple iterations, solutions are filtered via DEA and only the efficient ones are selected leading to cost minimization. In this work, the problem of optimal supply chain networks design is addressed based on a DEA based algorithm. A Branch and Efficiency (B&E) algorithm is deployed for the solution of this problem. Based on this DEA approach, each solution (potentially installed warehouse, plant etc) is treated as a Decision Making Unit, thus is characterized by inputs and outputs. The algorithm through additional constraints named “efficiency cuts”, selects only efficient solutions providing better objective function values. The applicability of the proposed algorithm is demonstrated through illustrative examples.
Keywords
Integer programming Branch and bound DEA Supply chain management Mixed integer linear programming (MILP)1 Introduction
Productivity measurement and efficiency in particular is a common term among the discipline of production economics. Each organization, firm, enterprise, business bank, can extract the efficiency of its branches, units if the latter are described by inputs and outputs. Efficiency is generally defined as the ratio of the amount of output produced given the unit’s inputs or resources. Thus, in order to measure the efficiency of a unit, certain inputs and outputs of a unit should be provided a priori. Deploying Data Envelopment Analysis (DEA) technique, the efficiency of each unit can be calculated (Charnes et al. 1981, 1985).
In the past decades there have been advances in Integer Programming (IP); one of which is Mixed Logical Linear Programming (MLLP) or Mixed Integer Linear (MILP) Programming models, where binary variables provide information regarding the installation of a plant or the selection of a route or a procedure, depending on the nature of the model (Hooker and Osorio 1999). This “family” of problems is generally solved with IP techniques, one of which is Branch and Bound (B&B) (Ross and Soland 1975). Based on this approach, a tree representation of the problem is provided; given a minimization direction to the problem (a transformation can be deployed if there is a maximization direction in the objective function) each node is examined separately for feasibility and for objective value improvement.
Due to the existence of binary variables B&B solution approach can be easily implemented providing solutions that are subjected to a single criterion. However, as can be seen in Fig. 1, considering each potentially installed warehouse as an entity then another conflicting objective is added to the problem. Based on this new objective facilities are not selected only based on cost minimization (or profit maximization) but also on whether these solutions are efficient. If more than one sites are selected then a possible approach is to add a single sourcing constraint which will select a single facility that reduces greatly the overall cost, but it may also cause the problem to become infeasible. In this case, other methodologies must be employed in order to provide solution to the problem.
The proposed algorithm is formulated in order to reduce the number of facilities (warehouses) in a supply network design problem. When designing the supply chain network, the Decision Maker (DM) seeks for less warehouses (facilities in general) as possible in order to reduce cost. This is obtained through economies of scale, however, reduction in facilities leads to less customers’ satisfaction. Generally the models that are used in order to design supply chain network, use single sourcing constraint in order to reduce the number of facilities by imposing the model to select one facility to accommodate a cluster of demand zones. This constraint is hard and often leads to infeasibility. In case of infeasibility, the model has to be solved without single sourcing constraint by letting the model to select as much facilities (warehouses) as needed in order to satisfy demand of the customers. Also, the cost is significantly increasing, depending on the demand pattern. On the contrary, the presented algorithm selects the warehouses based on their efficiency, leading to better results, both by reducing cost and by increasing efficiency. The algorithm filters all the optimal results derived from the initial MILP model and iteratively selects those warehouses that are fully technically efficient. Some of the methods that are employed for solving supply chain network design problems, in case of infeasibility, are Lagrangean Relaxation and Benders Decomposition, however, these methods use relaxation of “hard” constraints in order to solve the problem. The presented algorithm solves the problem based on two dimensions: cost and efficiency. Additional constraints remove inefficient solutions. In case that this new problem is now infeasible, different thresholds of efficiency are considered in the constraint, where the lower level of that percentage is left to DM to decide.
In this work, which is an extension of the work of Grigoroudis et al. (2014), a Branch and Efficiency (B&E) algorithm is proposed for the optimal design of supply chain networks. Through an iterative procedure, an initial vector of solutions is provided along with the inputs and outputs of each solution. Each solution is filtered in the following stages through constraints (efficiency cuts). The algorithm stops if the number of the nonzero solutions of the final vector is less than a certain predetermined level, or there is no change in objective function’s value. An approach that is integrating DEA technique in the selection of solutions, providing a MultiObjective Programming model, as solutions are not only subjected to constraints to the problem, but also to the “efficiency cuts” that are posed by B&E approach, has not yet been proposed in the supply chain literature.
2 Literature review
In the recent years supply chain design literature has expanded to consider the rapidly changing economic environment in which a supply chain network should be designed. There has been proposed a plethora of mathematical programming models that have been applied to supply chain network design problems of which Mixed Integer Linear Programming (MILP) and Mixed Integer Non Linear Programming (MINLP) models have been widely used, providing generic frameworks for managerial use.
Optimal supply chain network design models are divided into two categories; the steady state and the multiperiod ones. In the first case, time is absent from the analysis, and this type of formulation provides average levels of decisions, while in the multiperiod models, the decisions are made with respect to the planning horizon.
The key point in modeling supply chain networks is the demand uncertainty. A number of studies have captured the stochasticity with distribution functions that best describe demand.
Tsiakis et al. (2001) proposed a steady state model for the optimal supply chain network design, with decisions that regard the installation of facilities (distribution centers and warehouses). Demand uncertainty is modeled through different demand and capacity parameter scenarios.
The optimal design of supply chain networks has been also proposed by Petridis (2015). Demand stochasticity is proposed by Normal Distribution, while probabilistic constraint are integrated in a single framework for the optimal supply chain network providing decisions about the occurrence of stock out instances.
In their work, Rodriguez et al. (2014) employed a multiperiod model for the optimal design of supply chain taking into consideration demand uncertainty providing a production plan that integrates tactical and strategic decisions.
Supply chain management in global scale integrates decisions that take into account the factors that contribute to the sustainability of a supply chain network. A holistic approach towards the optimization of sustainability subject to the economic, ecological and social objectives has been proposed in the work of Kannegiesser and Günther (2014).
In the context of supply chain network design model, DEA formulations have been proposed in order to assess the efficiency of supply chain networks (Network DEA, Two stage DEA formulation etc). The following works demonstrate the use of DEA to supply chain area. However, most applications of DEA to supply chain systems is performed in an entirely different context to the one presented in this paper.
A non linear programming model has been proposed by Liang et al. (2006) for the evaluation of supply chain efficiency under intermediated performance evaluation.
A multiphase supply chain network design model has been proposed by Talluri and Baker (2002) utilizing DEA technique along with a game theoretic approach for pairwise evaluation of performance, designing the supply chain and providing optimal routing decisions.
Besides the evaluation of supply chain efficiency, sub operations conducted among the nodes of supply chain is also of major importance. In their work, Cheung and Hausman (2000), propose an exact measure efficiency measurement of (Q, R) policy of a twoechelon, multiple retailers system. Evaluation of supply chain performance has been proposed in the work of Forker et al. (1997) where through the combination of non linear DEA and regression analysis, Total Quality Management measures were provided.
Frota Neto et al. (2008) applied DEA and utilized DEA technique’s ability for efficiency extraction integrating into a unified framework with a multiobjective optimization model.
In their work Chen and Yan (2011) proposed a special network DEA approach in order to provide exact modeling with respect to the internal interactions of the supply chain. Similar works have been also proposed by Prieto and Zofío (2007), Huang et al. (2010) and Färe and Grosskopf (2000) proposed Network DEA models that can eventually be applied to the supply chain network evaluation framework.
Yang et al. (2011), have proposed an exact production possibility set for evaluating the performance different forms of supply chain models, while a game theoretic DEA model has been proposed by Chen et al. (2006), for analyzing the efficiency game between two supply chain parts. The model is proposed to explain bargaining supplier and manufacturer’s behavior for decision process. The internal supply chain performance, has been examined by Wong and Wong (2007) using technical and cost efficiency models. Using this DEA to measure the internal operations, inefficiencies in supply chain operations can be identified.
Generally, DEA technique has been used in order to provide efficiency of whole supply chain network system or to measure the performance of specific critical subsystems, like suppliers, plants etc. Yet, the data for the application of DEA technique are provided apriori, while the results are not taken into account during the optimization technique.
In this paper a DEA based algorithm is proposed for the optimal design of supply chain networks design. The algorithm utilizes the properties of DEA technique to provide productivity scores based on multiple inputs and outputs for each examined unit. It is assumed that in the proposed algorithm, except for the maximization of profit or revenue (minimization of cost), there is another objective based on which the selection of solutions is conducted, the maximization of selected solutions. The proposed algorithm is called Branch and Efficiency (B&E) as in each iteration the algorithm adds “efficiency cuts”, which are constraints to filter only the feasible and efficient solutions to be accepted. None of the existing papers in the literature have proposed such an algorithm until now.
3 B&E algorithm
3.1 Introduction to B&E
After efficiency calculation, Technical Efficiency (TE) is calculated as an efficiency measure, and is defined as the reciprocal of \(\varphi \) (Andersen and Petersen 1993). Additional constraints are introduced in order to allow only solutions that gather efficiency equal to 1. Correspondingly, new binary variables \(({\varvec{\upxi }})\) are introduced replacing those that concerned the selection of a solution. These variables are triggered only if the solution is efficient.
Throughout this procedure, the initially empty set of solutions N is filled with solutions that satisfy the conditions of iterative set \(J^{\gamma }\).
3.2 Notation
 i

Plant
 j

Warehouse
 k

Customer
 \(\gamma \)

Iteration
 \(P_i^U\)

Upper bound of produced quantities at plant i
 \(P_i^L\)

Lower bound of produced quantities at plant i
 \(Q_{ij}^U\)

Upper bound of transported quantities from plant i to warehouse j
 \(Q_{jk}^U\)

Upper bound of transported quantities from warehouse j customer k
 \(W_j^U\)

Upper capacity of warehouse j
 \(\beta _j\)

Coefficient relating quantity at capacity at warehouse j
 \(I_j\)

Inventory level stored warehouse j
 \(c_i^P\)

Production cost at plant i
 \(c_{ij}^V\)

Unit transportation cost of products transported from plant i to warehouse j
 \(c_{ij}^F\)

Rout transportation cost of products transported from plant i to warehouse j
 \(c_{jk}^V\)

Unit transportation cost of products transported from warehouse j to customer k
 \(c_{jk}^F\)

Route transportation cost of products transported from warehouse j to customer k
 \(c_k^{{ PEN}}\)

Penalty cost assigned to uncovered demand of customer k
 \(F_j\)

Installation cost of warehouse j
 \(P_i\)

Production quantity at plant i
 \(Q_{ij}\)

Transported quantity from plant i to warehouse j
 \(Q_{jk}\)

Transported quantity from warehouse j to customer k
 \(W_j\)

Capacity of warehouse j
 \(\lambda _j\)

Lambda (peers)
 \(g_k\)

Variable modeling deficit in demand of customer k
 \({ TC}\)

Total cost
 \(\varphi _j\)

Efficiency of each
 \(X_{ij}\)

1 if the connection between plant i and warehouse j exists, 0 otherwise
 \(X_{jk}\)

1 if the connection between warehouse j exists and customer k exists, 0 otherwise
 \(Y_j\)

1 if warehouse j will be installed, 0 otherwise
 \(\xi _j\)

1 if warehouse j will be installed under efficiency level a %, 0 otherwise
3.3 Introduction through Supply Chain Network Design (SCDN) problem
In order to demostrate the applicability of the B&E algorithm, an introduction to the proposed mathematical programming algorithm is provided, via an application of a simple SCDN problem.
In Fig. 3, the network of the supply chain is presented. In the supply chain network proposed here it is assumed that only a single product is manufactured, stored and transported throughout the channels of the network. The application of B&E algorithm is intensively demonstrated in a simple SCDN model, so that it can be analytically described, however, the algorithm can be applied in any SCDN formulation independently of whether it is theoretical or real.
In Fig. 3, the binary variables that correspond to the supply chains arcs (connections) and nodes (warehouses) are shown. The present model can be extended in order to take into account, more than one products, while there are many stages and echelons, the problem is extended to a multiproduct, multistage multiechelon SCDN problem.
 a)
The quantities produced, transported and stored
 b)
Capacity of the installed facilities and their location.
3.4 SCDN model
Besides constraints that regard to continuous variables, there are also logical constraints that are introduced in order to model the logical conditions. Constraints (9) and (10) suggest that the transported quantities from plant i to warehouse j and from warehouse j to customer k are upper bounded if—f the corresponding connection exists. Constraints (11) and (12) are introduced for the supply chain design network. In the two constraints is stated that if warehouse j is installed then the corresponding connection from warehouse j to plant i and from warehouse j to customer k exists. Constraint (13) states that warehouse’s capacity should be more than product’s quantities that will be transported to warehouse j plus the inventory level stored at warehouse j. The aforementioned quantities are multiplied by coefficient \(\beta _j\) expressing the amount of warehousing capacity required to hold a unit amount of the examined product at warehouse j. Constraint (14) is introduced to model the upper bound of warehouse capacity j.
Nonnegativity constraints are imposed to production, transportation and warehousing variables as seen in (16).
4 B&E formulation
In this section the B&E algorithm will be deployed on the SCDN model that was previously described. Based on this approach, each node acts as an unit being described by inputs and outputs. The efficiency of each unit is extracted based on their level of inputs and outputs. Thus, the incurring efficiency can be measured with DEA. The data that will be fed to DEA are acquired by solving the initial problem, while the inputs and outputs are apriori determined. In Fig. 4, the inputs and outputs of the warehouse facility is provided. If it assumed that a warehouse was a branch of an enterprise, then the most productive one would be the one that would minimize its operational cost and would provide more services.
4.1 Data
The data used for the efficiency extraction are generally provided in advance after statistical, qualitative or techno economic analysis. As mentioned in the previous section and can be seen in B&E flowchart, DEA technique is applied for the caclulation of DMUs efficiency. Here DMUs are considered to be the potentially installed warehouses. The data that are provided to DEA technique, are not externally provided, but come within the solution of the problem. The only parameter that should be predetermined is the inputs and outputs that the DM considers that capture the productivity of each warehouse.
Inputs and Outputs of the proposed supply chain
Inputs  \(C_j^{1, V} =\sum _i {c_{ij}^V} \cdot Q_{ij}^{*}\) 
\(C_j^{2, V} =\sum _k {c_{jk}^V} \cdot Q_{jk}^{*}\)  
\(C_j^{1, F} =\sum _i {c_{ij}^F} \cdot X_{ij}^{*}\)  
\(C_j^{2, F} =\sum _k {c_{jk}^F} \cdot X_{jk}^{*}\)  
\(C_j^{IN} ={\left\{ \begin{array}{ll} F_j \cdot Y_j^{*},\quad \gamma =1\\ F_j \cdot \xi _j^{*},\quad \gamma >1\\ \end{array}\right. }\)  
Output  \({ OC}_j =\sum _k {X_{jk}^{*}} \) 
\({ TQ}=\sum _k {Q_{jk}^{*}} \) 
4.2 Measuring DMUs efficiency
As in this model, the efficiency is derived through solutions of a MILP model, inputs and outputs must be selected so that the efficiency of each warehouse is captured. As it can be seen from Table 1, incoming quantities have not been taken into account because due to the mass balance constraint, incoming quantities are equal to out coming quantities of a node.
4.3 Solving problem \(P_0^\gamma \)
5 Results
5.1 Description of case study
In this section, the applicability of the model is presented through a case study. The proposed B&E algorithm can be applied in any supply chain, reducing overall cost. In the supply chain network presented in the previous section, a single homogeneous product is manufactured and transported throughout the links of the supply chain.
Data of the proposed case study
Description  Parameter  Value 

Upper bound of produced quantities of plant i  \(P_i^U\)  8000 
Lower bound of produced quantities of plant i  \(P_i^L\)  5000 
Upper bound of transported quantities from plant i to warehouse j  \(Q_{ij}^U\)  500 
Upper bound of transported quantities warehouse j to customer k  \(Q_{jk}^U\)  500 
Unit production cost at plant i  \(c_i^P\)  U[0, 200] 
Fixed route cost from plant i to warehouse j  \(c_{ij}^F\)  U[50, 100] 
Unit transportation cost from plant i to warehouse j  \(c_{ij}^V\)  U[0, 20] 
Fixed route cost from warehouse j to customer k  \(c_{jk}^F\)  U[50, 100] 
Unit transportation cost from warehouse j to customer k  \(c_{ij}^V\)  U[0, 20] 
Penalty cost assigned to uncovered demand of customer k  \(c_k^{{ PEN}}\)  \(10^{6}\) 
Fixed installation cost of warehouse j  \(F_j\)  \(U[50, 100]\times 10^{3}\) 
Coefficient relating quantity at capacity warehouse j  \(\beta _j\)  U[0.0001, 0.01] 
Inventory kept at warehouse j  \(I_j\)  U[0, 100] 
Demand of customer k  \(d_k\)  \(U[500, 1000]\times 10^{3}\) 
5.2 Numerical results
In this section the application of B&E is demonstrated through the results of the case study. The case study was model and solved in GAMS optimization software using CPLEX as LP and MIP solver on an Intel Pentium, 2.3 GHz, 2 GB RAM laptop computer. Even if the instance is medium to large with \(\left I \right =50\) plants, \(\left J \right =50\) plants and \(\left K \right =50\) customers, the problem was solved in 10 CPU seconds to optimality.
5.2.1 Initialization
Technical efficiency for all selected DMUs (warehouses)
Warehouse  Technical efficiency  Warehouse  Technical efficiency 

1  1  26  0.845 
2  0.978  27  1 
3  0.788  28  0.992 
4  0.937  29  0.914 
5  1  30  0.957 
6  1  31  1 
7  1  32  0.962 
8  0.912  33  0.923 
9  0.949  34  1 
10  0.957  35  0.999 
11  0.983  36  0.962 
12  0.967  37  0.937 
13  0.874  38  0.743 
14  0.954  39  0.898 
15  0.896  40  0.972 
16  0.965  41  0.998 
17  1  42  0.954 
18  0.999  43  1 
19  0.989  44  1 
20  0.941  45  1 
21  0.945  46  1 
22  0.932  47  0.967 
23  1  48  0.914 
24  0.965  49  0.997 
25  0.950  50  0.988 
After solving model \(P_0\) 50 warehouses are selected providing a total cost of 27,352,117.18 r.m.u. The number of warehouses (viz. the decision variables that correspond to the installation of a warehouse) is larger than the minimum DMU requirements of \(T=\max \left\{ {5\cdot 2, 3\cdot \left( {5+2} \right) } \right\} =21\) needed in order for the DEA results to have validity, so the procedure continues.
5.2.2 DEA
Results from application of B&E algorithm to larger instances
Instance  Objective (r.m.u)  No. facilities  Iterations 

\(\begin{array}{l} \left I \right =100\\ \left J \right =100 \end{array}\)  51,969,399.72  100  \(P_{0}\) 
\(\left K \right =100\)  50,682,282.31  23  \({\varvec{\gamma }}=\mathbf{1}\) 
\(\begin{array}{l} \left I \right =150\\ \left J \right =150 \end{array}\)  76,899,574.02  150  \(P_{0}\) 
\(\left K \right =150\)  73,652,608.31  19  \({\varvec{\gamma }}=\mathbf{1}\) 
\(\begin{array}{l} \left I \right =200\\ \left J \right =200 \end{array}\)  102,015,663.39  200  \(P_{0}\) 
\(\left K \right =200\)  95,210,734.52  35  \({\varvec{\gamma }}=\mathbf{1}\) 
5.2.3 Efficiency cuts
As aim of the proposed algorithm is to reduce the number of facilities in order to reduce overall cost but at the same time increase the service level of the customers, the efficiency cuts are applied to the second MILP model \(P_0^{\gamma =1}\). Model \(P_0^{\gamma =1}\) presented in Sect. 4.3 (\(\gamma =1\) denotes first iteration) receives the initial solutions from model \(P_0\). After solving \(P_0^{\gamma =1}\) the number of selected facilities has dropped from 50 to 13 which are the following: 1, 5, 6, 7, 17, 23, 27, 31, 34, 43, 44, 45, and 46. It must be noticed that from all DMUs, the model found and excluded those with Technical Efficiency equal to 1. The problem was solved to optimality and without any infeasibility occurrence. The results of model \(P_0^{\gamma =1}\) can be easily confirmed from Table 3 as all the selected facilities have a Technical Efficiency equal to 1; if any infeasibility instance would occur, based on the flowchart of B&E algorithm (Fig. 2), wider Technical Efficiency bounds would be used. The cost after the first iteration of B&E algorithm is 27,059,894.34 r.m.u. As the number of facilities that are selected from \(P_0^{\gamma =1} \) is less than the threshold set for DEA functionality (terminating criterion \(\left {N^{\gamma =1}} \right <21\)), the algorithm stops. The optimal solutions are the one derived from \(P_0^{\gamma =1}\). The number of selected facilities that are eventually selected are 13 (from 50 that have been selected after solving \(P_0)\) leading to cost reduction from 27,352,117.18 r.m.u. to 27,059,894.34 r.m.u.
5.2.4 Larger instances
In this section the application of the algorithm to larger instances is demonstrated. In Table 4 the instance characteristics (size of the problem), the results (objective function value and number of selected facilities) and the iterations for final solutions are presented.
As it can be seen from Table 4, in the proposed problems initially all the warehouses are selected. The initial cost is considered as an upper bound for the iterations of B&E algorithm and at each iteration overall cost decreases. In all instances, the algorithm terminated after the first iteration \((\gamma =1)\) either because the number of facilities is less than the threshold or because after selecting the efficient warehouses of first iteration, the MILP model \(P_0^\gamma \) was infeasible.
6 Conclusions
Efficiency measurement is applied to firms or to units when the data (inputs and outputs) are known a priori. Using DEA technique the efficiency is extracted for each DMU under examination. Considering each solution as a DMU, it is possible to evaluate the solutions efficiency using endogenous data. When modeling the supply chain network design, single sourcing constraints are used in order to provide better results in terms of minimizing the overall cost (objective function value) and economies of scale through facilities concentration. However, this type of constraint is a “hard” constraint and may eventually leads to infeasibility.
In this paper a Branch and Efficiency (B&E) algorithm has been deployed for the optimal design of supply chain networks. The algorithm integrates DEA technique in the design of supply chain network, through an iterative process and takes into advantage the strengths of DEA to provide efficiency scores for multiple inputs and outputs. Up to now, DEA technique has been applied in order to measure performance of a system based on exogenous data. The proposed B&E algorithm takes into the data provided within the optimization procedure in each iteration. Based on this approach, the problem is initially solved providing nonzero solutions for the initialization of the algorithm. The initialized values are fed to DEA to measure the efficiency of the unit under examination (in this case warehouses). Through the addition of efficiency cuts, the algorithm selects only the efficient solutions, which minimize overall cost (or maximize profit/revenue).
For illustrative purposes, a twostage supply chain model is proposed. Production units form the first link of this supply chain network while customer’s site form the last link, both of which are assumed to be already installed. The model is designed in such a way so as to provide decisions about the potential installation of warehouses. Setting apriori the characteristics that could capture the efficiency of each facility (warehouse), the data are provided to the algorithm. The proposed algorithm has two characteristics; heuristic and evaluative. The first comes from the fact that even if initial solutions are provided during the initialization process, the algorithm searches among the “efficient neighborhoods” and would accept or exclude solutions based on efficiency cuts, after the evaluation process. The algorithm is generic and can be applied in any type of supply chain, regardless the level of complexity, making it a valuable tool for long and short term managerial decisions.
Notes
Acknowledgments
The authors would like to thank Editor of Annals of Operations Research (ANOR) Prof. Endre Boros, and three anonymous reviewers for their insightful comments and suggestions. Konstantinos Petridis would like to acknowledge that part of this work was cofunded within the framework of the Action “State Scholarships Foundation’s (IKY) mobility grants programme for the short term training in recognized scientific/research centers abroad for candidate doctoral or postdoctoral researchers in Greek universities or research” from the European Social Fund (ESF) programme “Lifelong Learning Programme 2007–2013”.
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