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Industrial systems, such as logistics and supply chain networks, are complex systems because they comprise a big number of interconnected actors and significant nonlinear and stochastic features. This paper analyzes a distribution network design problem for a four-echelon supply chain. The problem is represented as an inventory-location model with uncertain demand and a continuous review inventory policy. The decision variables include location at the intermediate levels and product flows between echelons. The related safety and cyclic inventory levels can be computed from these decision variables. The problem is formulated as a mixed integer nonlinear programming model to find the optimal design of the distribution network. A linearization of the nonlinear model based on a piecewise linear approximation is proposed. The objective function and nonlinear constraints are reformulated as linear formulations, transforming the original nonlinear problem into a mixed integer linear programming model. The proposed approach was tested in 50 instances to compare the nonlinear and linear formulations. The results prove that the proposed linearization outperforms the nonlinear formulation achieving convergence to a better local optimum with shorter computational time. This method provides flexibility to the decision-maker allowing the analysis of scenarios in a shorter time.

Supply chain design is a critical strategy for achieving competitiveness in current global business environments. The supply chain consists of all functions involved, directly and indirectly, in meeting customer needs. Decision-making in supply chain management is classified into three hierarchical levels: strategic (long-term), tactical (medium-term), and operational (short-term). Frequently, these decisions are analyzed and solved independently at each planning stage, which generates an overall suboptimal solution when compared to solutions from comprehensive models. Therefore, supply chain management demands the use of new strategies and technologies to meet the current challenges of economic globalization [

Facility location problems, which are typically used to design distribution networks, involve determining the sites where to install resources, as well as the assignment of potential customers to those resources. This family of problems typically assumes a linear cost function and a set of deterministic customer demand. These assumptions avoid modelling interactions between facility location and inventory management decisions. According to these facts, the inventory-location research literature is aimed to integrate the demand uncertainty and the risk-pooling effect into supply chain network design [

This paper presents a mixed integer nonlinear programming formulation aimed to optimize the distribution network design. The model includes decisions on plants and warehouses location, transportation and assignment between facilities (plants-warehouses, warehouses-retailers), supplier selection, and implicit inventory decisions. The model is named a Four-Echelon Inventory-Location Problem with Supplier Selection (4EILP-SS). The mathematical formulation models a four-echelon supply chain, with a set of potential sites for allocation of plants and warehouses, as shown in Figure

Four-echelon supply chain.

The objective is to determine the flow of material in each echelon and the location of plants and warehouses. Inventory management decisions are simultaneously modeled with the design of the distribution network. The problem of locating facilities is commonly used as a base for the design of supply chains. However, the effect of risk integration by inventory management decisions is not directly modeled by the classic facility location problem (FLP). Additionally, supplier selection is a relevant issue because it affects many important aspects of the supply chain performance. In some real cases, for example in the context of the automotive industry, it is common to consider the involvement of suppliers in the supply chain planning stage. Supplier selection is a complex process that involves identifying potential suppliers, requesting information, establishing contract terms, negotiation, and evaluation. Several criteria are considered for supplier selection: quality, delivery time, price, service, and flexibility, among others. In the problem studied, we are assuming that the available suppliers were preselected considering several of the criteria described before, and the last decision was made based on the outcome of the optimization model that considers cost, lead time, and capacity. This is a common practice in industry, especially automotive industry, which is the motivation of this work, where an exclusive group of suppliers has the appropriate certifications by Original Equipment Manufacturers to participate in bids and requests.

Additionally, in the recent years, there has been an offshore moving of automotive manufacturing plants from industrialized countries to emerging economies. This happened in China, India, Mexico, Brazil, Thailand, and Turkey, among others [

The remainder of this paper is organized as follows. Section

Inventory-location problems (ILPs) have been extensively studied over the past two decades. This section presents a review of the main contributions reported in the literature, showing trends in formulations and solution approaches. Although ILPs are extensions of Facility Location Problems (FLPs), this paper does not discuss the related FLP literature. Interested readers may consult Hamacher and Drezner [

Jayaraman [

Daskin et al. [

Snyder et al. [

ILPs with inventory capacity constraints were analyzed by Miranda and Garrido [

Notice that all previous works are focused on optimizing location and inventory costs and decisions are taken at a single stage of the supply chain (i.e., warehouses), along with transport or assignment decisions (plant-warehouses and/or warehouses-retailers). In the works described forward, some additional considerations of costs and operations management decisions are integrated into similar ILP formulations.

Kang and Kim [

Silva and Gao [

Shahabi et al. [

Shen and Daskin [

Atamtürk et al. [

Kaya and Urek [

Schuster Puga and Tancrez [

Escalona et al. [

Tapia-Ubeda et al. [

It must be noticed that most of the previous ILP formulations integrate location decisions at one or two stages (warehouse location, plant location and supplier selection, warehouse and consolidating center location, etc.). Inventory costs and decisions are modeled at one or two levels (warehouses, warehouses, and plants, warehouses and retailers, etc.), but none of these previous work simultaneously integrates location and inventory management decisions at two stages in the supply chain. Therefore, this research paper fills an important gap in the field of supply chain network design.

To the best of our knowledge, only a previous work by ourselves has considered the design of such a complex supply chain as in this research, with some differences in the model and more important, in the solution method. Perez Loaiza et al. [

The following part of the literature review discusses linearization strategies for nonlinear problems. You and Grossmann [

According to the discussed literature review, this paper contributes with an inventory-location model, under continuous review policy, to optimize a four-echelon supply chain network, encompassing multiple suppliers, plants, warehouses, and retailers. The problem is formulated as a mixed integer nonlinear programming model, considering a single item, with stochastic demands across de supply chain network. Decision variables include warehouse and plant location, plant-warehouse assignments, retailer assignment to warehouses, and shipments from suppliers to plants. This ILP integrates inventory costs and decisions at plants and warehouses. As it can be observed from the related literature, the proposed network design model of a four-echelon supply chain has a nonlinear formulation that arises when inventory aspects are integrated into facility location problems, yielding to a nonlinear mixed integer programming model.

Nonlinear mixed integer programming models are extremely hard to solve to optimality, particularly when commercial optimization solvers are employed. They combine the mixed integer programming (NP-Hard) nature with nonlinear and non-convex components. In many cases, researchers and practitioners face two major challenges: local optimum solutions and prohibitive computational times for tackling practical problems. Moreover, these challenges increase when more stages and decision variables are added to the model, such as in this research article. In order to reduce the computational effort, and as a strategy to develop a heuristic framework to tackle this complex problem, a linearization of the model is proposed based on a piecewise linear approximation of the objective function, and a linear reformulation of nonlinear constraints, yielding to a mixed integer linear programming model. The approach is developed following the approach proposed by Diabat and Theodorou [

A continuous review based inventory control policy

Continuous review inventory control policy performance.

The existing inventory level just before the order arrives at each location is known as the safety stock, while the inventory level that is observed over the safety stock is known as cyclic or working inventory.

Warehouse demand depends on customers

To calculate the working inventory cost (

In this section, the mathematical formulation of the problem studied in this paper is presented. Section

Thus, the mathematical model for the problem is formulated as

A reformulation of the constraints in (

If

This section presents a linear approximation approach for the safety stock costs

Piecewise linear function

Considering a partition of

The linear approximation of

An alternative objective function Z1’ is defined. Z1’ is equivalent to Z1 but considering the linear approximation of

In this section, five instance sizes with 10 cases each one are presented. The code of the instance size indicates the number of suppliers followed by the number of potential plants, followed by the number of potential warehouses, and the number of retailers. For example, instance size 5-3-5-10 indicates 5 suppliers, 3 potential plants, 5 potential warehouses, and 10 retailers.

A commercial license of LINGO 14 was used to solve every case, with the nonlinear and linear models. The nonlinear model corresponds to the formulation described by (

Parameters of the base case, associated with suppliers

Parameter | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

| 85 | 70 | 90 | 80 | 75 | 65 | 70 | 71 | 66 | 66 |

| 30 | 28 | 30 | 28 | 24 | 25 | 28 | 21 | 28 | 30 |

Parameters of the base case, associated with plants

Plant | | | | Warehouse | | | | | |
---|---|---|---|---|---|---|---|---|---|

| 150 | 200 | 110 | | 70 | 80 | 75 | 70 | 85 |

| 187500 | 250000 | 137500 | | 131250 | 150000 | 97500 | 131250 | 121875 |

| 167 | 134 | 164 | | 184 | 220 | 190 | 189 | 205 |

| 1 | 3 | 2 | | 2 | 1 | 2 | 1 | 2 |

| 646 | 657 | 656 | | 743 | 850 | 552 | 743 | 690 |

Parameters of the base case, demand, and variance of retailers

Retailer | | | | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Demand ( | 5 | 10 | 10 | 6 | 8 | 5 | 13 | 23 | 12 | 10 | 6 | 7 | 20 |

Variance ( | 46 | 40 | 25 | 30 | 87 | 50 | 55 | 65 | 30 | 68 | 70 | 76 | 30 |

| |||||||||||||

Retailer | | | | | | | | | | | | | |

| |||||||||||||

Demand ( | 10 | 20 | 9 | 17 | 15 | 15 | 24 | 5 | 25 | 23 | 16 | 14 | |

Variance ( | 35 | 86 | 40 | 50 | 25 | 10 | 60 | 68 | 40 | 73 | 45 | 50 |

Parameters of the base case, transportation unit cost from supplier

| | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

| 164 | 151 | 117 | 132 | 153 | 139 | 158 | 151 | 115 | 182 |

| 140 | 123 | 189 | 162 | 136 | 123 | 132 | 136 | 143 | 167 |

| 137 | 155 | 120 | 119 | 167 | 134 | 133 | 127 | 180 | 132 |

Parameters of the base case, transportation unit cost from plant

| | | | | |
---|---|---|---|---|---|

| 164 | 140 | 137 | 132 | 162 |

| 151 | 123 | 155 | 153 | 136 |

| 117 | 189 | 120 | 119 | 167 |

Parameters of the base case, transportation unit cost from warehouses

| | | | | |
---|---|---|---|---|---|

| 94 | 70 | 78 | 100 | 93 |

| 99 | 86 | 86 | 90 | 89 |

| 86 | 83 | 98 | 80 | 97 |

| 75 | 90 | 77 | 78 | 91 |

| 72 | 75 | 77 | 83 | 87 |

| 80 | 80 | 71 | 99 | 83 |

| 90 | 74 | 98 | 94 | 99 |

| 75 | 75 | 87 | 72 | 98 |

| 74 | 94 | 79 | 80 | 95 |

| 90 | 80 | 84 | 75 | 99 |

| 79 | 95 | 96 | 97 | 79 |

| 84 | 68 | 65 | 97 | 74 |

| 78 | 74 | 95 | 89 | 75 |

| 81 | 73 | 84 | 82 | 89 |

| 98 | 88 | 100 | 98 | 90 |

| 77 | 78 | 98 | 70 | 95 |

| 72 | 85 | 70 | 90 | 66 |

| 98 | 98 | 84 | 86 | 89 |

| 93 | 65 | 83 | 90 | 90 |

| 94 | 71 | 75 | 94 | 90 |

| 89 | 71 | 84 | 85 | 82 |

| 73 | 80 | 76 | 86 | 80 |

| 87 | 84 | 79 | 80 | 70 |

| 81 | 73 | 71 | 70 | 85 |

| 87 | 83 | 89 | 70 | 72 |

The uncertainty in demand is represented by the variance values shown in Table

Table _{(0.95)} = 1.64). Table

Results of applying the nonlinear model and the linear model considering a service level of 95%.

Case # | Non-Linear Model 4EILP-SS | Linear Model 4EILP-SS-L | Saving | |||
---|---|---|---|---|---|---|

Best found objective function value (Z1) | Elapsed run time (sec) | Best found objective function value (Z2) | Elapsed run time (sec) | Saved Cost | % Saving | |

| 523,690 | 5.05 | 499,861 | 2.86 | 23,829 | 4.55% |

| 451,963 | 58.68 | 451,021 | 3.92 | 942 | 0.21% |

| 589,253 | 490.76 | 589,246 | 42.80 | 7 | 0.00% |

| 734,372 | 273.47 | 733,524 | 65.91 | 848 | 0.12% |

| 514,842 | 396.13 | 514,842 | 3.64 | 0 | 0.00% |

| 506,738 | 13.35 | 505,464 | 11.79 | 1,274 | 0.25% |

| 511,549 | 116.7 | 511,549 | 8.71 | 0 | 0.00% |

| 517,406 | 98.01 | 517,406 | 10.56 | 0 | 0.00% |

| 639,303 | 213.99 | 639,303 | 83.68 | 0 | 0.00% |

| 584,845 | 48.24 | 584,845 | 15.37 | 0 | 0.00% |

AVERAGE | 557,396 | 171.44 | 554,706 | 24.92 | 0.51% | |

| ||||||

| 745,385 | 378.24 | 735,503 | 191.68 | 9,882 | 1.33% |

| 811,106 | 92.01 | 727,251 | 88.86 | 83,855 | 10.34% |

| 988,571 | 419.46 | 919,670 | 403.44 | 68,901 | 6.97% |

| 974,342 | 351.94 | 919,175 | 343.47 | 55,167 | 5.66% |

| 833,043 | 900.64 | 832,979 | 90.82 | 64 | 0.01% |

| 845,915 | 180.74 | 818,642 | 160.75 | 27,273 | 3.22% |

| 735,539 | 98.76 | 735,334 | 92.62 | 205 | 0.03% |

| 917,953 | 50.82 | 858,491 | 50.82 | 59,462 | 6.48% |

| No solution found | - | 977,396 | 191.88 | - | Only Linear Solution |

| 971,726 | 876,959.40 | 855,959 | 150.08 | 115,767 | 11.91% |

AVERAGE | 869,287 | 97,714.66 | 822,556 | 174.73 | 5.11% | |

| ||||||

| No solution found | - | 1,051,772 | 597.58 | - | Only Linear Solution |

| 993,942 | 1,587.54 | 974,498 | 461.08 | 19,444 | 1.96% |

| 1,274,019 | 212,939.00 | 1,269,256 | 1,636.17 | 4,763 | 0.37% |

| 1,285,862 | 1,636.40 | 1,268,819 | 1,454.15 | 17,043 | 1.33% |

| No solution found | - | 1,062,143 | 1,986.87 | - | Only Linear Solution |

| 1,051,350 | 16,621.00 | 1,051,299 | 859.15 | 51 | 0.00% |

| 976,769 | 3,850.00 | 974,422 | 589.93 | 2,347 | 0.24% |

| 1,112,979 | 28,959.00 | 1,112,397 | 611.73 | 582 | 0.05% |

| No solution found | - | 1,221,641 | 1,374.21 | - | Only Linear Solution |

| No solution found | - | 1,269,527 | 693.48 | - | Only Linear Solution |

AVERAGE | 1,115,820 | 44,265.49 | 1,108,449 | 935.37 | 0.66% | |

| ||||||

| 980,256 | 4,228.34 | 979,306 | 2,829.46 | 950 | 0.10% |

| 831,262 | 3,300.00 | 831,094 | 1,531.48 | 168 | 0.02% |

| 1,106,338 | 6,568.55 | 1,106,338 | 3,234.91 | 0 | 0.00% |

| No solution found | - | 1,108,531 | 2,508.82 | - | Only Linear Solution |

| No solution found | - | 1,050,292 | 1,279.51 | - | Only Linear Solution |

| 975,718 | 1,786.84 | 975,705 | 3,503.63 | 13 | 0.00% |

| 845,102 | 3,327.00 | 824,554 | 594.98 | 20,548 | 2.43% |

| No solution found | - | 976,929 | 1,151.09 | - | Only Linear Solution |

| No solution found | - | 1,109,935 | 3,214.58 | - | Only Linear Solution |

| No solution found | - | 1,109,484 | 3,532.16 | - | Only Linear Solution |

AVERAGE | 947,735 | 3,842.15 | 943,399 | 2,338.89 | 0.51% | |

| ||||||

| 1,270,846 | 923.97 | 1,270,708 | 982.18 | 138 | 0.01% |

| No solution found | - | 1,106,664 | 1381.60 | - | Only Linear Solution |

| 1,281,643 | 8,552.48 | 1,278,946 | 4,756.29 | 2,697 | 0.21% |

| 1,280,408 | 2,266.26 | 1,280,802 | 3,530.13 | -394 | -0.03% |

| 1,417,022 | 6,783.66 | 1,266,151 | 3,295.30 | 150,871 | 10.65% |

| 1,140,753 | 8,476.40 | 1,130,751 | 819.56 | 10,002 | 0.88% |

| No solution found | - | 1,127,784 | 1,273.77 | - | Only Linear Solution |

| No solution found | - | 1,433,556 | 7,271.13 | - | Only Linear Solution |

| No solution found | - | 1,277,243 | 3,500.21 | - | Only Linear Solution |

| 1,268,140 | 7,441.08 | 1,266,054 | 954.83 | 2,086 | 0.16% |

AVERAGE | 1,276,469 | 5,740.64 | 1,248,902 | 2,389.72 | 2.38% |

Table

Structure of the supply chain applying the nonlinear model and the linear model considering a service level of 95%.

Case # | Non-Linear Model 4EILP-SS | Linear Model 4EILP-SS-L | ||
---|---|---|---|---|

Number of plants selected | Number of warehouses selected | Number of plants selected | Number of warehouses selected | |

| 3 | 4 | 3 | 5 |

| 3 | 5 | 3 | 5 |

| 2 | 4 | 2 | 4 |

| 2 | 3 | 2 | 3 |

| 3 | 4 | 3 | 5 |

| 3 | 4 | 3 | 5 |

| 3 | 4 | 3 | 5 |

| 3 | 4 | 3 | 5 |

| 2 | 3 | 2 | 4 |

| 3 | 4 | 3 | 4 |

| ||||

| 3 | 4 | 3 | 4 |

| 2 | 4 | 3 | 4 |

| 2 | 3 | 2 | 3 |

| 2 | 3 | 2 | 3 |

| 2 | 4 | 2 | 4 |

| 2 | 4 | 2 | 4 |

| 3 | 4 | 3 | 4 |

| 2 | 3 | 2 | 4 |

| No solution found | - | 2 | 3 |

| 2 | 3 | 2 | 4 |

| ||||

| No solution found | - | 2 | 4 |

| 3 | 4 | 3 | 4 |

| 2 | 3 | 2 | 3 |

| 2 | 3 | 2 | 3 |

| No solution found | - | 2 | 4 |

| 2 | 4 | 2 | 4 |

| 3 | 4 | 3 | 4 |

| 2 | 3 | 2 | 4 |

| No solution found | - | 2 | 3 |

| No solution found | - | 2 | 3 |

| ||||

| 2 | 3 | 2 | 4 |

| 3 | 4 | 3 | 4 |

| 2 | 3 | 2 | 3 |

| No solution found | - | 2 | 3 |

| No solution found | - | 2 | 3 |

| 2 | 3 | 2 | 4 |

| 3 | 4 | 3 | 4 |

| No solution found | - | 2 | 4 |

| No solution found | - | 2 | 3 |

| No solution found | - | 2 | 3 |

| ||||

| 2 | 3 | 2 | 4 |

| No solution found | - | 3 | 4 |

| 2 | 3 | 2 | 4 |

| 2 | 3 | 2 | 4 |

| 2 | 3 | 2 | 4 |

| 2 | 4 | 3 | 4 |

| No solution found | - | 3 | 4 |

| No solution found | - | 2 | 3 |

| No solution found | - | 2 | 4 |

| 2 | 3 | 2 | 4 |

Figure

Average total cost, comparing the nonlinear versus the linear models.

The comparison of the number of solutions obtained between the linear and nonlinear model is presented in Figure

Percent of solutions obtained, comparing nonlinear versus linear models.

This research introduced an inventory-location model, 4EILP-SS, for the network design of a four-echelon supply chain, considering the location of warehouses and plants, transportation cost, inventory costs, and supplier selection. The inventory policy and network design decisions are tactical and strategic decisions that should be analyzed concurrently. This is attained using the aforementioned model. This model can be applied to design complex and long supply chains, like in the automotive industry because of the offshore movement of production facilities from industrialized countries to emerging economies. In many cases, there is infrastructure available in the country, and it is easy to adapt the model to that situation fixing locations for the available facilities and allowing the model to decide for the new optimum flows and locations of new facilities.

The model proposed becomes nonlinear because of the application of the continuous review inventory policy. This feature added to the computational complexity inherited from the Facility Location Problem makes the problem hard to solve. The mixed integer nonlinear model was reformulated using a piecewise linear function and a reformulation of nonlinear constraints in order to generate a mixed integer linear model. Several instances were solved using the linear and nonlinear models, comparing the results in terms of cost savings, number of solutions obtained, and computational runtime. It was observed that solving the nonlinear model with the commercial software generated local optimal solutions, and in several cases the software either did not converge or resulted in infeasible solutions. The total cost of the original objective function is lower, in most of the cases, with the solution obtained by the linear model than the solution obtained by the nonlinear model.

The main contribution of the paper is two-fold. Firstly, a network optimization model for a particular four level supply chain, where facility location and inventory management decisions are considered in two stages (plants and warehouses), and transportation or assignment decisions are modeled in three stages of the supply chain, i.e., suppliers-plants, plants-warehouses, and warehouses-customers. Secondly, a linearization approach of the proposed nonlinear model, in order to facilitate its applicability and its effective and efficient implementation, improves both solutions quality and computational times.

The proposed linear model proves to be useful for decision makers interested in analyzing and designing supply chain networks with the structure of a 4EILP-SS. The problem can be analyzed and solved in a short period of time and with significant confidence in the solution quality.

The experiments conducted allowed us to understand the complexity of the problem. For this study, LINGO was used to solve the linear and the nonlinear models, but future work may take advantage of using more efficient commercial software to solve larger instances. As observed, using commercial optimizers may not be the best alternative when it comes to solve very large instances. Therefore, immediate future research should consider the development of other heuristics aimed to tackle large instances (Lagrangian relaxation, Bender decompositions, Cutting planes, and Metaheuristics). Another direction is to extend the pricing approach proposed by Shu et al. [

In addition, a deeper study may involve the analysis of the relations between parameters like costs, capacities, lead times, and demand variability, to understand their impact in the supply chain network configuration.

Finally, it would be interesting to extend the model to analyze multiobjective situations, more common when designing supply chain networks for real situations.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research was supported by Programa para el Mejoramiento del Profesorado PROMEP (Grant ITESUR-001 Charter no. PROMEP/103.5/10/5709) and Universidad Panamericana (Grant no. UP-CI-2018-ING-GDL-06).