International Journal of Academic Management Science Research (IJAMSR) ISSN: 2000-001X Vol. 2 Issue 9, September – 2018, Pages: 1-7 www.ijeais.org/ijamsr 1 Vehicle Routing Problems with Simultaneous Pickup and Delivery Soft Time Window Manar Mustafa 1 , Hesham Abdel Salam 2 , Sally Kassem 3 1 Master Student, Decision Support and Operation Research Faculty of Computers and Information, University Cairo Egypt manar.mustafa59@yahoo.com 2,3 Professors, Decision Support and Operation Research Faculty of Computers and Information, University Cairo Egypt Abstract: A mixed integer linear programing (MILP) model is presented to formulate the vehicle routing problems with simultaneous pickup and delivery soft time window where products’ delivery o r collecting of EOL products are performed simultaneously and early and late time intervals have to be violated (Soft) with adding some penalty costs. The presented a model is more generic that obtain a large solution space where more feasible solutions could be found for larger number of problems depending on its characteristics. The model is solved and tested with exact approach using standard benchmark problems. 85% of tested problems with large travel time got a feasible solution with the proposed VRPSPD-STW model while it didn’t have a feasible solution with the HTW model. Keywords: Vehicle Routing Problem, simultaneous pick and delivery, soft time window, hard time window 1. INTRODUCTION The vehicle routing problem (VRP) is an important problem in the fields of transportation that was first proposed by Dantzig and Ramser in 1960 [1]. It is a combinatorial optimization problem [2]seeking to serve a number of customers with a fleet of vehicles. VRP has a wide range of applications with the objective of determining the minimum cost for all routs resulting from serving those customers and minimum number of vehicles used. Each of these optimal routs is performed by a single vehicle that start and end at the central node, each customer is visited exactly once by one vehicle. A reverse logistics networks where delivering of a demanded goods and collecting end of life products (EOL) [3] [4]within an illustrated time frame are called Vehicle Routing Problems with Simultaneous Pickup and Delivery with time window (VRPSPD-TW) its proposed by Kassem, S. and Chen, M. (2012) [5], this a hard time window constraint couldn’t be violated. In a real life environment vehicles are operated in a network with different traffic levels of congestion base on the day and time. Being restricted with the predefined time frame is not valid for some applications of VRPSPD such as food industries where delivery of products is a must even if violating the time interval. In this paper our routing problem has a simultaneous pickup and delivery with a soft time window where violating a time window constraint is allowable with adding a Penalty cost in case of arriving at the customer before the early or after the late allowable arrival time. This is called a Vehicle Routing Problems with Simultaneous Pick and Delivery Soft Time Window (VRPSPDSTW). VRPSPDSTW is first proposed by Bhusri, N., Qureshi, A.G., Taniguchi, E. (2013) [6]. They developed a mathematical model and a genetic algorithm metaheuristic-based procedure to solve this kind of problems. In their model they present a distribution and routing system for Convenience Store chains for a food retail business in Japan. One of their objective is to attaining satisfaction of their customers so they allow violating time window [a, b], (a) represent an early allowable arrival time and (b) represent the late allowable arrival time for vehicles at each franchise convenience store, with adding a penalty cost if violation happened. A tradeoff is considered between if adding an early or late penalty cost or using new vehicle with its vehicle cost. This violation is also limited by a new early and late time window denoted by [a’, b’] where vehicle is not allowed to arrive beyond this new time window, arriving after the allowed b time and before b’ a tradeoff will considered between if will serving this customer and adding a late arrival penalty cost or using a new. vehicle with its fixed cost. In case of vehicle arrived before the allowed early time a, and after the new limit a’, vehicle will be wait and start the service at the original allowed early time and at same time an early penalty cost will be added. In our paper the soft time window is obtained by allowing a time violation if vehicle is not arrived within