Arabian Journal for Science and Engineering

, Volume 44, Issue 8, pp 7219–7233 | Cite as

A Maximal Covering Problem in Supply Chain Considering Variable Radius and Gradual Coverage with the Choice of Transportation Mode

  • Alireza EydiEmail author
  • Hossein Torabi
Research Article - Systems Engineering


Today, supply chain management is a fundamental principle of business infrastructure implementation in the world. One of the most important pillars of the supply chain management is distribution network design. A distribution network with appropriate structure reduces costs and enhances customer-perceived service level, thereby improving competitive advantages of the company. In supply chain analysis, it is essential to focus on customers, as they greatly affect the formation of chain. This requires a balance between associated costs and the level of service provided. Variable radius and gradual radius maximal covering problem focus on the service levels in relation to the costs incurred to the system. In this study, a bi-objective mixed-integer nonlinear programming model was presented to determine optimal number, location and capacity of factories, distribution centers, and retailers, select transportation modes between the chain elements, and evaluate coverage radius of retailers, in such a way to minimize total transportation costs, maximize the covered demand, and achieve gradual coverage of facilities in a variable radius scheme. The proposed model was validated on a number of sample problems produced and solved by GAMS optimization software. Being NP-Hard, the problem was proposed to be solved via NSGA-II algorithm when its dimensions go large. Analysis of the computational results and respective comparisons indicated good performance of the presented algorithm.


Supply chain design Multi-objective Variable radius and gradual covering Transportation mode selection NSGA-II algorithm 

Set of Indexes


Set of customers


Set of retailers


Distribution centers




Mode of transportation


Set of capacity levels available for factories


Set of levels of capacity available for distribution centers


Set of levels of capacity for retailers

Model Parameters

\(\hbox {DE}_{{{i}}}\)

The demand of customer i


The distance of customer i from the potential location of retailer j


The potential location of distribution center k from the potential location of retailer j


The potential location of factory p from the potential location of retailer k

\(\upvarphi _{j} (r)\)

The variable establishment cost function of retailer j with radius of r


The budget allocated to the establishment of the facility


Covering function determined based on the distance (decreasing)


The cost of establishing a distribution center of the capacity h at potential location k


Fixed cost of establishing a retailer facility of the capacity u at location j


The cost of establishing the factory of the capacity m at potential location of p


Shipping cost of a single product from a potential distribution center k to the potential retailer j via the transportation mode i


Shipping cost of a single product from the potential factory at location p to the potential distribution center k via the transportation mode i


Shipping cost of a single product from the potential retailer j to customer i

\(\hbox {Cap}_{kh}\)

Capacity of the distribution center k at level h

\(\hbox {Cap}_{pv}\)

Capacity of the factory p at level m

\(\hbox {Cap}_{ju}\)

Capacity of retailer j at level u

Model Variables


The amount of product transported from factory p to distribution center k via the transportation mode l


The amount of product transported from distribution center k to retailer j via the transportation mode l


A binary variable, 1 if a distribution center of the capacity h is established at location k, and 0 otherwise


A binary variable, 1 if a factory of the capacity v is constructed at location p, and 0 otherwise


A binary variable, 1 if a retailer is constructed at location j, and 0 otherwise


A binary variable, 1 if customer i is assigned to retailer j, and 0 otherwise


A binary variable, 1 if transportation mode i is selected to serve the path between distribution center k and factory p, and 0 otherwise


A binary variable, 1 if transportation mode i is selected to serve the path between distribution center k and retailer j, and 0 otherwise


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Industrial EngineeringUniversity of KurdistanSanandajIran
  2. 2.Master of Industrial Engineering, Faculty of EngineeringUniversity of KurdistanSanandajIran

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