Robust vehicle routing and cross-dock scheduling ...

Robust vehicle routing and cross-dock scheduling with uncertain loading and unloading time Ali Rahbari Department of Industrial Engineering, Alborz Campus University of Tehran Tehran, Iran [email protected]

Mohammad Mahdi Nasiri * School of Industrial Engineering, College of Engineering University of Tehran Tehran, Iran [email protected]

Abstract—Cross-docking is a recently interesting logistics technique that consolidates incoming freights from suppliers and rapidly loads them on the outgoing vehicles to realize the economy of scale in the delivery process. Simultaneously considering the outgoing vehicles routing problem and the cross-dock scheduling is known as Vehicle Routing Problem with Cross-Docking (VRPCD) that is increasingly gaining ground for 3PL companies from the viewpoint of cost efficiency. This paper, minimizing the total cost (including transporting, cross-docking, inventory holding costs and the penalty cost of early and tardy deliveries), presents a rigorous MIP model for the VRPCD in which doors of the cross-dock are different in loading/unloading time, and a heterogeneous fleet of vehicles is used for delivery process. Considering the uncertainty of the loading/unloading times in dockdoors and unloading times in delivery nodes to retailers, a robust reformulation of the model is presented by applying intervalpolyhedral (Bertsimas and Sim, 2004) approach. The robustness of the solutions and the effect of conservatism and variability levels on the solutions have been evaluated using simulation. The results of numerical analysis confirm the performance of the applied robust optimization approach and validate the robustness of the model. Keywords: Cross-docking; Robust optimization; Vehicle routing problem; Uncertain loading and unloading time

1. Introduction In today's competitive environment of business, many companies are trying to develop cost efficient strategies to control the physical flow of the supply chain. As stated by Apte and Viswanathan (2000), the distribution process constitutes 30% of the product prices. Cross-docking is one of the appropriate distribution techniques in which products are transshipped from incoming to outgoing vehicles and then delivered to the customers without being held in a warehouse. The cross-dock eliminates the storing and picking activities from the traditional warehouse, which are more expensive activities (Schaffer, 1998); Indeed, it can decrease warehousing costs up to 70% (Vahdani and Zandieh, 2010). Moreover, cross-docking center enables consolidation of orders and deliver them in full-truck-load (FTL) rather than less-than-truck-load (LTL). The literature of cross-docking, as a logistics technique in supply chain, can be classified into two categories; strategic and operational level. At the strategic level, Jayaraman and Ross (2003) designed the PLOT (Production, Logistics, Outbound, Transportation) systems. Sung and Song (2003) designed an integrated service network in that they located the cross-docks and allocated inbound and outbound trucks. Ross and Jayaraman (2008) described and evaluated new heuristic solution procedures for the cross-dock location planning problem. Kreng and Chen (2008) studied the production and distribution planning in a supply chain and compared two strategies, namely, traditional warehousing and cross-docking. Ma et al. (2011) considered a cross-docking distribution network to explore the tradeoffs between transportation cost, inventory and scheduling requirements. At the operational level, most of the researches are concerned with vehicle routing and scheduling. Yu and Egbelu (2008), Chen and Lee (2009), Chen and Song (2009) and Boysen et al. (2010) scheduled inbound and outbound trucks by considering crossdocks as a two-machine flow shop problem and minimized cycle time in the cross-dock. Li et al. (2004) proposed a machine scheduling problem to minimize the earliness and tardiness penalty cost for the containers processing at the cross-dock. Lee et al. (2006) considered time windows and presented a pickup and delivery model in a distribution network with a single cross-dock. Wen et al. (2009) proposed a model for determining the pickup and delivery routing of vehicles in a cross-dock. Moreover, Santos et al. (2011) and Santos et al. (2013) have proposed branch-and-price algorithms for somewhat similar problems. Dondo et al. (2011) have formulated VRPCD as a mixed integer linear programming model in which hybrid multi-echelon distribution networks transport products from manufacturers to customers, directly and/or using warehousing and cross-docking strategies. Agustina et al. (2014) presented a model that focuses specifically on joint optimization of scheduling and vehicle routing at a cross-docking center for outgoing vehicles, considering customer delivery time windows. Dondo and Cerda (2015) have formulated the VRPCD with heterogeneous vehicle fleet. Mokhtarinejad et al. (2015) presented a mathematical model for location-routing and scheduling problem with cross-docking. They *

Corresponding author

solved the model by using a machine-learning-based heuristic method in which the customers, manufacturers and locations of the crossdocking centers are grouped through a bi-clustering approach. Ahmadizar et al. (2015) have proposed a model to assign products to suppliers and cross-docks, to optimize the routes and schedules of inbound and outbound vehicles. They presented a hybrid genetic algorithm to solve this two-level VRPCD problem. Chen et al. (2016) presented a variant of Particle Swarm Optimization with a SelfLearning strategy to solve VRPCD with multiple products. Moghadam et al. (2014) formulated a VRPCD problem in which both suppliers and customers must be visited within their time windows, also a customer can be visited more than once by different vehicles. Vincent F. Yu et al. (2016) have introduced an Open VRP with cross-docking problem. Recently, Alinaghian et al. (2016) have proposed a model in which split service is allowed for both pickup and delivery processes and vehicles in pickup process are rental. Yin et al. (2016) addressed the problem of the Integrated Green Vehicle Routing and Scheduling for Cross-docking with CO2 emission constraint. Maknoon and Laporte (2017) presented a two-level planning model for vehicle routing with cross-dock selection problem. Robust optimization is increasingly gaining ground as a preeminent methodology to cope with the uncertainty. Compared to stochastic programming techniques, the robust optimization has two main advantages: (i) the robust counterpart remains computationally tractable, (ii) instead of the probability distribution function of the uncertain parameters, only the range of uncertain parameters is required in the robust optimization. In contrast with the deterministic approaches, there is less attention to the cross-docking under uncertainty. Mousavi et al. (2013) presented a novel fuzzy mathematical programming-based possibilistic approach for the VRPCD. In another study, Mousavi et al. (2014) proposed a fuzzy possibilistic–stochastic programming model for location of cross-docking facilities and routing-scheduling of vehicles. Seyedhoseini et al. (2014) considered two M/M/c queues to describe operations of inbound and outbound trucks for cross-docking under uncertain environment. Further, Shi et al. (2013) designed a robust configuration for a cross-docking distribution center such that the disturbances of supply uncertainty cannot considerably affect the system. In a recent study, Ladier and Alpan (2016) proposed robust models for the truck scheduling with time windows. They concluded that an effective approach to ensure robustness in the schedule is to minimize the average number of trucks assigned to a given door, but it also increases storage. Up to our knowledge, none of the previous studies implemented the interval-polyhedral (Bertsimas and Sim, 2004) robust optimization approach to the VRPCD. So in this paper, we consider cross-dock scheduling with multiple products, and vehicle routing of outgoing delivery to satisfy customer’s demand. We encounter the uncertainty of loading/unloading times. The remainder of the paper is organized as follows. Section 2 contains problem description, assumptions, and mathematical model of vehicle routing and scheduling with cross-docking problem. Then the robust reformulation of the model are presented in Section 3. In Section 4, computational results are provided and finally Section 5 summarizes the results and proposes some future research directions.

2. The deterministic problem description and model formulation Problem description Let consider a three echelon supply chain, including suppliers, a distributor and retailers in which the distributor as a 3PL company uses a cross-docking center to provide logistics service. The retailers place the orders to the suppliers, specifying the product type and quantity as well as the delivery time window. The ordered freights are transported from suppliers to the cross-dock using the vehicles owned by the suppliers, so-called incoming vehicles. Each of the suppliers is responsible for sharing the information about orders loaded on its incoming vehicles with the distributor. The distributor, as the decision maker of the problem, plans the cross-docking and delivery operations in order to minimize the total cost, including transporting, cross-docking, inventory holding and earliness and tardiness penalty costs. According to the schedule specified by the plan, the incoming vehicles transport the freights from suppliers to the cross-docking center. When an incoming vehicle arrives at the cross-dock, unloads its freight on a specified receiving dock-door according to the sequence and schedule determined by the plan. The unloaded products may be sent directly to the shipping dock-doors in order to be immediately loaded onto outgoing vehicles and delivered to their destinations, or may be stored in a temporary storage area and wait to be consolidated with other incoming products. On the other side of the cross-dock, loading of consolidated products onto outgoing vehicles are carried out according to the specified sequence and schedule on each shipping dock-door. After the complete loading of the products of an outgoing vehicle, it is dispatched to the retailers, according to the route and schedule specified by the plan. Outgoing vehicles deliver the orders to the specified retailers and then come back to the cross-docking center. The main assumptions are as follows:           



The temporary storage area of the cross-dock is unlimited. The demand of a retailer on a product type will be transported by only one incoming vehicle. Each of the outgoing vehicles may have a different capacity as well as different variable and fixed costs. Each of the outgoing vehicles can be used at most one time during the planning horizon. All of the vehicles are available at the beginning of the planning horizon at the cross-dock. The loading/unloading of a vehicle at the cross-dock cannot be interrupted (Pre-emption is not allowed). Each dock-door is either exclusively dedicated to inbound or outgoing vehicles (Exclusive service mode). The time required for loading/unloading a same product type may differ between different dock-doors. Transshipment of products loaded on an incoming vehicle can be started only after full unloading of the vehicle. The delivery routes start and end at the cross-dock (Closed-loop). The demand of a retailer should be delivered by only one outgoing vehicle (Split delivery is not allowed). The whole process is completed within the planning time horizon.

Notations and the model formulation 𝑟 𝑝 𝑘 𝑗

The indices, parameters and variables used for the formulation of the model are as follows: Indices: Index of retailers {𝑟 = 1,2, … , 𝑅} Index of product types {𝑝 = 1,2, … , 𝑃} Index of incoming vehicles {𝑘 = 1,2, … , 𝐾} where k = 0 and K+1 are considered as dummy vehicles Index of outgoing vehicles {𝑗 = 1,2, … , 𝐽} where j = 0 and J+1 are considered as dummy vehicles

𝑑 𝑑′

Index of receiving doors {𝑑 = 1,2, … , 𝐷} Index of shipping doors {𝑑′ = 1,2, … , 𝐷′ }

Parameters: 1, if product type p ordered by retailer r is loaded on incoming vehicle k; 0, otherwise 𝐺𝑟𝑝𝑘 1, if some/all of the demands of retailer r is loaded on incoming vehicle k 𝐺 ′ 𝑟𝑘 Unit volume of product type p 𝑉𝑝 Capacity of outgoing vehicle j 𝐶𝐴𝑃𝑗 Fixed cost of using outgoing vehicle j 𝐹𝐶𝑗 Unit variable cost of outgoing vehicle j 𝑉𝐶𝑗 Unit variable cost of using the cross-dock [$ per hour] 𝑁 Unit inventory holding cost of product type p at the cross-dock 𝐻𝑝 Transfer time at the cross-dock; from receiving doors to shipping doors [min] 𝑇𝑇 Truck changeover time [min] 𝑇𝐶𝑇 Planning horizon 𝑇 Travel time between the cross-dock and retailer r [min] 𝑂𝑟 Travel time from retailer r to r' [min] 𝑂𝑟𝑟′ Time required to unload a unit of product type p at receiving door d of the cross-dock [min] 𝑈𝑑𝑝 Time required to load a unit of product type p at shipping door d' of the cross-dock [min] 𝐿𝑑′𝑝 Time required to unload a unit of product type p at retailer r [min] 𝑈𝑟𝑝 Demand of retailer r for product type p [pallet] 𝐷𝑟𝑝 Unit earliness penalty cost of retailer r [$ per pallet per min] 𝐸𝑃𝑟 Unit tardiness penalty cost of retailer r [$ per pallet per min] 𝑇𝑃𝑟 Lower bound of delivery time for retailer r 𝐿𝐵𝑟 Upper bound of delivery time for retailer r 𝑈𝐵𝑟 A big constant number 𝑀 Decision variables: Binary variables 1, if receiving door d of the cross-dock is used 𝑒1𝑑 1, if incoming vehicle k is processed at receiving door d 𝑝1𝑑𝑘 1, if incoming vehicle k & k' are processed on the same receiving door and k immediately precedes 𝑞1𝑘𝑘 ′ 1, if shipping door d' of the cross-dock is used 𝑒2𝑑 ′ 1, if outgoing vehicle j is processed at shipping door d' 𝑝2𝑑 ′𝑗 1, if outgoing vehicle j & j' are processed on the same shipping door and j immediately precedes 𝑞2𝑗𝑗 ′ 1, if the demand of retailer r is delivered by outgoing vehicle j through shipping door d' 𝑧𝑑 ′𝑟𝑗 1, if retailer r is the first retailer visited by vehicle j 𝑦𝑟𝑗 1, if retailer r' is immediately visited after retailer r by vehicle j 𝑦𝑦𝑟𝑟 ′𝑗 1, if retailer r is the last retailer visited by vehicle j 𝑦𝑦𝑦𝑟𝑗 Continuous positive variables arrival time of incoming vehicle k at the cross-dock 𝑎𝑡1𝑘 arrival time of outgoing vehicle j at the cross-dock 𝑎𝑡2𝑗 departure time of outgoing vehicle j from the cross-dock 𝑑𝑡2𝑗 arrival time of outgoing vehicle j at retailer r 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 return time of outgoing vehicle j to the cross-dock 𝑟𝑡𝑗 departure time of order of retailer r from the cross-dock 𝑑𝑡_ℎ𝑟 arrival time of product type p ordered by retailer r to the cross-dock 𝑎𝑡_ℎ𝑟𝑝 start of working time of the cross-dock 𝑠𝑡 finish of working time of the cross-dock 𝑓𝑡 earliness at retailer r 𝑒𝑒𝑒𝑟 tardiness at retailer r 𝑡𝑡𝑡𝑟 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑂𝐹 = ∑𝑟,𝑗 𝑦𝑟𝑗 𝐹𝐶𝑗 + ∑𝑗 (𝑟𝑡𝑗 − 𝑎𝑡2𝑗 )𝑉𝐶𝑗 + (𝑓𝑡 − 𝑠𝑡)𝑁 + ∑𝑟,𝑝(𝑑𝑡_ℎ𝑟 − 𝑎𝑡_ℎ𝑟𝑝 )𝐻𝑝 𝐷𝑟𝑝 + ∑𝑟,𝑝(𝑡𝑡𝑡𝑟 𝑇𝑃𝑟 + 𝑒𝑒𝑒𝑟 𝐸𝑃𝑟 )𝐷𝑟𝑝 The first term of OF computes the fixed and variable costs of using outgoing vehicles, the second term states the variable costs of cross-docking and the inventory holding costs at the temporary storage area, and the third term determines the earliness and tardiness penalty costs of products delivery to the retailers. Subject to: ∑𝑑 𝑝1𝑑𝑘 = 1

∀𝑘

(1)

∑𝑘 𝑝1𝑑𝑘 ≤ 𝑀 × 𝑒1𝑑

∀𝑑

(2)

𝑒1𝑑 ≤ ∑𝑘 𝑝1𝑑𝑘

∀𝑖, 𝑑(𝑑 ∈ 𝐷𝑖 )

(3)

𝑞1𝑘𝑘 ′ − 1 ≤ 𝑝1𝑑𝑘 − 𝑝1𝑑𝑘 ′ ≤ 1 − 𝑞1𝑘𝑘 ′

∀𝑑, 𝑘, 𝑘 ′ (𝑘 ≠ 𝑘 ′ )

(4)

∑𝑘 ′=0,…,𝐾,(𝑘 ′≠𝑘) 𝑞1𝑘 ′𝑘 = 1

∀𝑘

(5)

∑𝑘 ′=1,…,𝐾+1,(𝑘 ′≠𝑘) 𝑞1𝑘𝑘 ′ = 1

∀𝑘

(6)

∑𝑘 𝑞10𝑘 = ∑𝑑 𝑒1𝑑

(7)

∑𝑘 𝑞1𝑘(𝐾+1) = ∑𝑑 𝑒1𝑑 𝑞10𝑘 + 𝑞10𝑘 ′ + 𝑝1𝑑𝑘 + 𝑝1𝑑𝑘 ′ ≤ 3

(8) ∀𝑑, 𝑘, 𝑘′(𝑘, 𝑘 ′

= 1, … , 𝐾 + 1; 𝑘 ≠ 𝑘′) ∀𝑘, 𝑘 ′ (𝑘

𝑎𝑡1𝑘 ′ ≥ 𝑎𝑡1𝑘 + ∑𝑑,𝑟,𝑝 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 𝑈𝑑𝑝 + 𝑇𝐶𝑇 − 𝑀(1 − 𝑞1𝑘𝑘 ′ )

≠

𝑘′)

(9) (10)

Equation (1) implies that an incoming vehicle must be processed on only one receiving door. Constraints (2) and (3) determine if a receiving door is used. Constraint (4) ensures that if an incoming vehicle precedes another incoming vehicle, they must be on the same receiving door. Equations (5) and (6) imply that each non-dummy incoming vehicle has exactly one preceding and exactly one succeeding incoming vehicle (possibly a dummy incoming vehicle). Constraints (7), (8) and (9) restrict dummy incoming vehicles ‘0’ and ‘K+1’ to be the first and the last incoming vehicles on each used receiving door, respectively. Constraint (10) states that if an incoming vehicle precedes another incoming vehicle, then the arrival time of the latter must ensure that there is enough time for the former to complete its unloading. ∑𝑑 ′ 𝑝2𝑑 ′𝑗 ≤ 1

∀𝑗

(11)

∑𝑟 𝑧𝑑 ′𝑟𝑗 ≤ 𝑀 × 𝑝2𝑑 ′𝑗

∀𝑑′ , 𝑗

(12)

𝑝2𝑑 ′𝑗 ≤ ∑𝑟 𝑧𝑑 ′𝑟𝑗

∀𝑑′ , 𝑗

(13)

∑𝑗 𝑝2𝑑 ′𝑗 ≤ 𝑀 × 𝑒2𝑑 ′

∀𝑑′

(14)

𝑒2𝑑 ′ ≤ ∑𝑗 𝑝2𝑑 ′𝑗

∀𝑑′

𝑞2𝑗𝑗 ′ − 1 ≤ 𝑝2𝑑 ′𝑗 − 𝑝2𝑑 ′𝑗 ′ ≤ 1 − 𝑞2𝑗𝑗 ′

∀𝑑′ , 𝑗, 𝑗 ′ (𝑗

∑𝑗 ′=0,…,𝐽(𝑗 ′≠𝑗) 𝑞2𝑗 ′𝑗 = ∑𝑑 ′ 𝑝2𝑑 ′𝑗

∀𝑗

(17)

∑𝑗 ′=1,…,𝐽+1(𝑗 ′≠𝑗) 𝑞2𝑗𝑗 ′ = ∑𝑑 ′ 𝑝2𝑑 ′𝑗

∀𝑗

(18)

(15) ≠

𝑗′)

(16)

∑𝑗 𝑞20𝑗 = ∑𝑑 ′ 𝑒2𝑑 ′

(19)

∑𝑗 𝑞2𝑗(𝐽+1) = ∑𝑑 ′ 𝑒2𝑑 ′

(20)

𝑞20𝑗 + 𝑞20𝑗 ′ + 𝑝2𝑑 ′𝑗 + 𝑝2𝑑′𝑗 ′ ≤ 3

∀𝑑′ , 𝑗, 𝑗 ′ (𝑗, 𝑗 ′

𝑎𝑡2𝑗 + 𝑑𝑡2𝑗 ≤ 𝑀 ∑𝑑 ′ 𝑝2𝑑 ′𝑗

∀𝑗

(22)

𝑎𝑡2𝑗 ′ ≥ 𝑑𝑡2𝑗 + 𝑇𝐶𝑇 − 𝑀(1 − 𝑞2𝑗𝑗 ′ )

∀𝑗, 𝑗 ′ (𝑗 ≠ 𝑗 ′ )

(23)

= 1, … , 𝐽 + 1; 𝑗 ≠

𝑗′)

(21)

∀𝑑′ , 𝑗

(24) 𝑑𝑡2𝑗 ≥ 𝑎𝑡2𝑗 + ∑𝑟 (𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝐷𝑟𝑝 𝐿𝑑 ′𝑝 ) (25) 𝑑𝑡2𝑗 ≥ 𝑎𝑡1𝑘 + ∑𝑑,𝑝 𝑝1𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 𝑈𝑑𝑝 + ∑𝑑 ′,𝑝 𝑧𝑑 ′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 𝐿𝑑 ′𝑝 + 𝑇𝑇 − ∀𝑟, 𝑗, 𝑘 𝑀(2 − 𝐺 ′ 𝑟𝑘 − ∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 ) Equation (11) implies that an outgoing vehicle can be processed on only one shipping door. Constraints (12) and (13) enforce the integration of decisions in vehicle assignment to the retailers and shipping doors. Constraints (14) and (15) determine if a shipping door is used and Constraint (16) ensures that if an outgoing vehicle precedes another outgoing vehicle, they must be on the same shipping door. Equations (17) and (18) imply that each non-dummy outgoing vehicle has exactly one preceding and exactly one succeeding outgoing vehicle (possibly a dummy outgoing vehicle). Constraints (19), (20) and (21) restrict dummy outgoing vehicles ‘0’ and ‘J+1’ to be the first and the last outgoing vehicles on each used shipping door, respectively. Constraint (22) restricts the arrival and departure time of outgoing vehicles to the cross-dock. Constraints (23) and (24) schedule the arrival, loading and departure time of outgoing vehicles at shipping doors and Constraint (23) states that if an outgoing vehicle precedes another outgoing vehicle, then the arrival time of the latter must ensure that there is enough time for the former to complete its loading. Constraint (25) enforces the dependency of outgoing vehicles on the incoming vehicles and connects the departure time of an outgoing vehicle to the arrival time of an incoming vehicle if any of the products is transferred between the vehicles. 𝑠𝑡 ≤ 𝑎𝑡1𝑘

∀𝑘

(26)

𝑓𝑡 ≥ 𝑑𝑡2𝑗

∀𝑗

(27)

𝑎𝑡_ℎ𝑟𝑝 ≤ 𝑎𝑡1𝑘 + ∑𝑑 𝑝1𝑑𝑘 𝐷𝑟𝑝 𝑈𝑑𝑝

∀𝑟, 𝑝, 𝑘 𝑖𝑓 𝐺𝑟𝑝𝑘 = 1

(28)

𝑑𝑡_ℎ𝑟 ≥ 𝑑𝑡2𝑗 − 𝑀(1 − ∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 ) ∑𝑑 ′,𝑗 𝑧𝑑 ′𝑟𝑗 = 1

∀𝑟, 𝑗

(29)

∀𝑟

(30)

∑𝑟 𝑦𝑟𝑗 =∑𝑑 ′ 𝑝2𝑑 ′𝑗

∀𝑗

(31)

∑𝑟 (∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝑉𝑝 𝐷𝑟𝑝 ) ≤ 𝐶𝐴𝑃𝑗 ∑𝑟 𝑦𝑟𝑗

∀𝑗

(32)

∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 ≤ 𝑀(𝑦𝑟𝑗 + ∑𝑟 ′(𝑟 ′≠𝑟) 𝑦𝑦𝑟 ′𝑟𝑗 ) 𝑦𝑟𝑗 + ∑𝑟 ′(𝑟 ′≠𝑟) 𝑦𝑦𝑟 ′𝑟𝑗 ≤ 𝑀 ∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗

∀𝑟, 𝑗

(33)

∀𝑟, 𝑗

(34)

∑𝑟,𝑟 ′(𝑟≠𝑟 ′) 𝑦𝑦𝑟𝑟 ′𝑗 ≤ 𝑀 ∑𝑟 𝑦𝑟𝑗

∀𝑗

(35)

𝑦𝑟𝑗 + ∑𝑟′(𝑟 ′≠𝑟) 𝑦𝑦𝑟′𝑟𝑗 = ∑𝑟′(𝑟 ′≠𝑟) 𝑦𝑦𝑟𝑟′𝑗 + 𝑦𝑦𝑦𝑟𝑗

∀𝑟, 𝑗

(36)

𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 ≤ 𝑀(𝑦𝑟𝑗 + ∑𝑟 ′(𝑟 ′≠𝑟) 𝑦𝑦𝑟 ′𝑟𝑗 )

∀𝑟, 𝑗

(37)

𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 ≥ 𝑑𝑡2𝑗 + 𝑂𝑟 − 𝑀(1 − 𝑦𝑟𝑗 )

∀𝑟, 𝑗

(38)

𝑎𝑡_𝑑𝑒𝑙𝑟 ′𝑗 ≥ 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 + ∑𝑑 ′(𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝐷𝑟𝑝 𝑈𝑟𝑝 ) + 𝑂𝑟𝑟 ′ − 𝑀(1 − 𝑦𝑦𝑟𝑟 ′𝑗 )

∀𝑟, 𝑟 ′ (𝑟 ≠ 𝑟 ′ ), 𝑗

(39)

𝑡𝑡𝑡𝑟 ≥ 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 − 𝑈𝐵𝑟

∀𝑟, 𝑗

(40)

𝑒𝑒𝑒𝑟 ≥ 𝐿𝐵𝑟 − 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 − 𝑀(1 − ∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 )

∀𝑟, 𝑗

(41)

𝑟𝑡𝑗 ≥ 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 + ∑𝑝 𝐷𝑟𝑝 𝑈𝑟𝑝 + 𝑂𝑟 − 𝑀(1 − 𝑦𝑦𝑦𝑟𝑗 )

∀𝑟, 𝑗

(42)

𝑟𝑡𝑗 ≤ 𝑇

∀𝑗

(43)

Constraints (26) and (27) compute the start and finish of working time of the cross-dock that are respectively the minimum of incoming vehicles’ arrival time and the maximum of outgoing vehicles’ departure time. Constraints (28) and (29) determine the arrival and departure time for each type of products ordered by each retailer in order to compute the holding time in the temporary storage area. Equation (30) implies that the demand of each retailer must be delivered by only one outgoing vehicle from one shipping door (prevents split delivery). Constraint (31) implies that each used outgoing vehicle visits at most one retailer at the beginning of its delivery tour, and Constraint (32) restricts the load of outgoing vehicles. Constraints (33) and (34) ensure that an outgoing vehicle can deliver products to a retailer only if the vehicle visits that retailer. Constraints (35) and (36) enforce route continuity of outgoing vehicles. Constraint (37) restricts the arrival time of outgoing vehicles at retailers. Constraints (38) and (39) schedule the arrival, unloading and departure of outgoing vehicles at retailers (Constraint (39) prevents sub-tours in delivery process). Constraints (40) and (41) compute the earliness and tardiness of delivery for each retailer and Constraint (42) determines the return time of outgoing vehicles to the cross-dock. Constraint (43) implies that the whole operations of the outgoing vehicles (including cross-docking, delivery and coming back to the cross-dock) must be completed within the planning time horizon.

3. Robust reformulation of the model Three sources of uncertainty are considered: (i) the unit unloading time of products at the receiving doors of the cross-dock (𝑈𝑑𝑝 ), (ii) the unit loading time of products at the shipping doors of the cross-dock (𝐿𝑑 ′𝑝 ), and (iii) the unit unloading time of products at retailers (𝑈𝑟𝑝 ). The robust optimization approach proposed by Bertsimas and Sim (2004) is applied to formulate the robust model. This approach allows to control the level of conservatism in the solution, by considering the degree of conservatism for every constraint subject to uncertainty. Due to the uncertainty of 𝑈𝑑𝑝 , 𝐿𝑑 ′𝑝 , and 𝑈𝑟𝑝 , Constraints (10), (24), (25), (28), (39), and (42) are subject to uncertainty. We discuss the procedure of obtaining the robust reformulation for only one of the constraints in detail, i.e., Constraint (25). The rest of (45) constraints which include uncertain parameters can be reformulated using the same reasoning. Let 𝒥𝑟𝑗𝑘 be the set of coefficients 𝑈𝑑𝑝 ̃𝑑𝑝 and 𝐿̃𝑑 ′𝑝 take values in [𝑈 ̅𝑑𝑝 − 𝑈 ̂𝑑𝑝 , 𝑈 ̅𝑑𝑝 + 𝑈 ̂𝑑𝑝 ] and and 𝐿𝑑 ′𝑝 that are subject to uncertainty in Constraint (25); i.e., 𝑈 ̅ ̂ ̅ ̂ ̅ ̂ [𝐿𝑑 ′𝑝 − 𝐿𝑑 ′𝑝 , 𝐿𝑑 ′𝑝 + 𝐿𝑑 ′𝑝 ], respectively, without assuming a specific distribution, e.g., 𝑈𝑑𝑝 and 𝑈𝑑𝑝 represent estimates for the nominal value and range of the uncertain unit unloading time of product type p at receiving door d, respectively. The total scaled deviation of ̃𝑑𝑝 from 𝑈 ̅𝑑𝑝 and 𝐿̃𝑑 ′𝑝 from 𝐿̅𝑑 ′𝑝 cannot exceed from a specified value Γ (45) , called the uncertainty budget of Constraint (25). variable 𝑈 𝑟𝑗𝑘 So, the uncertainty set for the coefficients of Constraint (25) is given by: (45)

(45)

̃𝑑𝑝 , 𝐿̃𝑑 ′𝑝 ), (𝑑, 𝑑′ , 𝑝) ∈ 𝒥 ∑ 𝑈𝑟𝑗𝑘 = {((𝑈 𝑟𝑗𝑘 ) ∶ (𝑑,𝑝)∈𝒥

̃𝑑𝑝 −𝑈 ̅𝑑𝑝 | |𝑈 ̂𝑑𝑝 𝑈

+ ∑(𝑑 ′,𝑝)∈𝒥

|𝐿̃𝑑′𝑝 −𝐿̅𝑑′ 𝑝| 𝐿̂𝑑′𝑝

(45)

≤ 𝛤𝑟𝑗𝑘 ,

̃𝑑𝑝 ∈ [𝑈 ̅𝑑𝑝 − 𝑈 ̂𝑑𝑝 , 𝑈 ̅𝑑𝑝 + 𝑈 ̂𝑑𝑝 ] , 𝐿̃𝑑 ′𝑝 ∈ [𝐿̅𝑑 ′𝑝 − 𝐿̂𝑑 ′𝑝 , 𝐿̅𝑑 ′𝑝 + 𝐿̂𝑑 ′𝑝 ] ∀(𝑑, 𝑑′ , 𝑝) ∈ 𝑈

(44) (45) 𝒥𝑟𝑗𝑘 }

̃𝑑𝑝 and 𝐿̃𝑑 ′𝑝 , respectively, ensuring that the constraint applies for all occurrences of coefficients 𝑈 ̃𝑑𝑝 and Replacing 𝑈𝑑𝑝 and 𝐿𝑑 ′𝑝 by 𝑈 ̃𝐿𝑑 ′𝑝 such as defined in the uncertainty set 𝑈 (45) , the robust (still nonlinear) counterpart for Constraint (25) can be formulated as follows: 𝑟𝑗𝑘 ̅𝑑𝑝 (𝑝1𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + ∑𝑑 ′,𝑝 𝐿̅𝑑 ′𝑝 (𝑧𝑑′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + 𝑇𝑇 − 𝑀(2 − 𝐺 ′ 𝑟𝑘 − ∑𝑑 ′ 𝑧𝑑 ′𝑟𝑗 ) 𝑑𝑡2𝑗 ≥ 𝑎𝑡1𝑘 + ∑𝑑,𝑝 𝑈 +

(45)

𝑚𝑎𝑥 (45)

(45)

{𝒮∪{𝜏}|𝒮⊆𝒥𝑟𝑗𝑘 ,|𝒮|=⌊𝛤𝑟𝑗𝑘 ⌋,𝜏∈𝒥𝑟𝑗𝑘 \𝒮}

̂𝑑𝑝 (|𝑝1𝑑𝑘 |𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + ∑(𝑑 ′,𝑝)∈𝒮 𝐿̂𝑑 ′𝑝 (|𝑧𝑑 ′𝑟𝑗 |𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + {∑(𝑑,𝑝)∈𝒮 𝑈 (45)

(45)

(45)

̂𝑑𝑝 (|𝑝1𝑑𝑘 |𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + 𝐿̂𝑑 ′𝑝 (|𝑧𝑑′𝑟𝑗 |𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ))} (𝛤𝑟𝑗𝑘 − ⌊𝛤𝑟𝑗𝑘 ⌋) (𝑈

It is unlikely that all the unit unloading/loading times of all product types for all receiving/shipping doors take their worst-case values, (25)

and the goal of this approach is to guarantee that the robust solutions are feasible concerning Constraint (25) if up to ⌊Γ𝑟𝑗𝑘 ⌋ of the unloading/loading times 𝑈𝑑𝑝 and 𝐿𝑑 ′𝑝 take their worst values and one of the rest of the unloading/loading times 𝑈𝑑𝑝 or 𝐿𝑑 ′𝑝 changes to ̅𝑑𝑝 + (Γ (25) − ⌊Γ (25) ⌋) 𝑈 ̂𝑑𝑝 or 𝐿̅𝑑 ′𝑝 + (Γ (25) − ⌊Γ (25) ⌋) 𝐿̂𝑑 ′𝑝 . Since the number of coefficients 𝑈𝑑𝑝 and 𝐿𝑑 ′𝑝 subject to uncertainty in 𝑈 𝑟𝑗𝑘

𝑟𝑗𝑘

𝑟𝑗𝑘

𝑟𝑗𝑘

(45)

(25)

Constraint (25) is equal to 𝑃(𝐷 + 𝐷′ ), and according to the definition of 𝑈𝑟𝑗𝑘 , parameter Γ𝑟𝑗𝑘 can take real values in the interval (25) (25) [0, 𝑃(𝐷 + 𝐷′ )]. Parameter Γ𝑟𝑗𝑘 is used to control the level of conservatism of the solution: Γ𝑟𝑗𝑘 = 0 is equivalent to the deterministic (25)

model; and Γ𝑟𝑗𝑘 = 𝑃(𝐷 + 𝐷′ ) corresponds to the worst-case scenario. Therefore, we gain the flexibility of adjusting the robustness of (25) the solution against the level of conservatism by varying Γ𝑟𝑗𝑘 ∈ [0, 𝑃(𝐷 + 𝐷′ )]. In order to linearize Constraint (45), we use the following procedure of Bertsimas and Sim (2004), knowing that 𝑝1𝑑𝑘 , 𝑧𝑑 ′𝑟𝑗 ≥ 0 and thus |𝑝1𝑑𝑘 | = 𝑝1𝑑𝑘 and |𝑧𝑑 ′𝑟𝑗 | = 𝑧𝑑 ′𝑟𝑗 . Given a vector 𝑝1∗𝑑𝑘 and 𝑧𝑑∗ ′𝑟𝑗 the protection function of Constraint (25):

(45)

𝛽 (25) (𝑝1∗𝑑𝑘 , 𝑧𝑑∗ ′𝑟𝑗 , 𝛤𝑟𝑗𝑘 ) =

(45)

𝑚𝑎𝑥 (45)

(45)

{𝒮∪{𝜏}|𝒮⊆𝒥𝑟𝑗𝑘 ,|𝒮|=⌊𝛤𝑟𝑗𝑘 ⌋,𝜏∈𝒥𝑟𝑗𝑘 \𝒮}

̂𝑑𝑝 (𝑝1∗𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + ∑(𝑑 ′,𝑝)∈𝒮 𝐿̂𝑑 ′𝑝 (𝑧𝑑∗ ′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + {∑(𝑑,𝑝)∈𝒮 𝑈 (45)

(45)

̂𝑑𝑝 (𝑝1∗𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + 𝐿̂𝑑 ′ 𝑝 (𝑧𝑑∗ ′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ))} (𝛤𝑟𝑗𝑘 − ⌊𝛤𝑟𝑗𝑘 ⌋) (𝑈 equals the objective function of the following linear optimization problem: (45)

̂𝑑𝑝 (𝑝1∗𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 )𝜓𝑑𝑝𝑟𝑗𝑘 + ∑𝑑 ′,𝑝 𝐿̂𝑑 ′𝑝 (𝑧𝑑∗ ′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 )𝜒 𝑑′𝑝𝑟𝑗𝑘 ) 𝛽 (25) (𝑝1∗𝑑𝑘 , 𝑧𝑑∗ ′𝑟𝑗 , 𝛤𝑟𝑗𝑘 ) = 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 (∑𝑑,𝑝 𝑈 (45) ∑𝑑,𝑝 𝜓𝑑𝑝𝑟𝑗𝑘 + ∑𝑑 ′,𝑝 𝜒 𝑑 ′𝑝𝑟𝑗𝑘 ≤ 𝛤𝑟𝑗𝑘 0 ≤ 𝜓𝑑𝑝𝑟𝑗𝑘 , 𝜒 𝑑′𝑝𝑟𝑗𝑘 ≤ 1

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:

(46) ∀𝑑, 𝑑′ , 𝑝

The dual of problem (46) is given by: (45) (45)

(45)

(45)

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑥𝑟𝑗𝑘 𝛤𝑟𝑗𝑘 + ∑𝑑,𝑝 𝜇𝑑𝑝𝑟𝑗𝑘 + ∑𝑑 ′,𝑝 𝛿𝑑 ′𝑝𝑟𝑗𝑘 (45)

(45)

̂𝑑𝑝 (𝑝1∗𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) 𝑥𝑟𝑗𝑘 + 𝜇𝑑𝑝𝑟𝑗𝑘 ≥ 𝑈 (45) (45) 𝑥 +𝛿 ′ ≥ 𝐿̂𝑑 ′ 𝑝 (𝑧𝑑∗ ′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 )

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:

𝑟𝑗𝑘 𝑑 𝑝𝑟𝑗𝑘 (45) (45) (45) 𝑥𝑟𝑗𝑘 , 𝜇𝑑𝑝𝑟𝑗𝑘 , 𝛿𝑑 ′𝑝𝑟𝑗𝑘

∀𝑑, 𝑝

(47)

∀𝑑′ , 𝑝

≥0

∀𝑑, 𝑑′ , 𝑝

So, the linear reformulation of Constraint (45) as the robust linear counterpart for Constraint (25) is as follows: (45) (45)

(45)

̅𝑑𝑝 (𝑝1𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + ∑𝑑 ′,𝑝 𝐿̅𝑑 ′𝑝 (𝑧𝑑′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + 𝑥 ∑ 𝑑𝑡2𝑗 ≥ 𝑎𝑡1𝑘 + ∑𝑑,𝑝 𝑈 𝑟𝑗𝑘 𝛤𝑟𝑗𝑘 + 𝑑,𝑝 𝜇𝑑𝑝𝑟𝑗𝑘 +

∀𝑟, 𝑗, 𝑘

′ ∑𝑑 ′,𝑝 𝛿𝑑(45) ′ 𝑝𝑟𝑗𝑘 + 𝑇𝑇 − 𝑀(2 − 𝐺 𝑟𝑘 − ∑𝑑 ′ 𝑧𝑑 ′ 𝑟𝑗 ) (45)

(45)

̂𝑑𝑝 (𝑝1𝑑𝑘 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) 𝑥𝑟𝑗𝑘 + 𝜇𝑑𝑝𝑟𝑗𝑘 ≥ 𝑈

∀𝑑, 𝑝, 𝑟, 𝑗, 𝑘

(45) (45) 𝑥𝑟𝑗𝑘 + 𝛿𝑑 ′𝑝𝑟𝑗𝑘 ≥ 𝐿̂𝑑 ′𝑝 (𝑧𝑑′𝑟𝑗 𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) (45) (45) (45) 𝑥𝑟𝑗𝑘 , 𝜇𝑑𝑝𝑟𝑗𝑘 , 𝛿𝑑 ′𝑝𝑟𝑗𝑘 ≥ 0

∀𝑑′ , 𝑝, 𝑟, 𝑗, 𝑘

(48)

∀𝑑, 𝑑′ , 𝑝, 𝑟, 𝑗, 𝑘

Using the same reasoning for considering the uncertainty in Constraint (25), the robust counterpart for Constraints (10), (24), (28), (39), and (42) can be formulated as Constraints (49)-(53), respectively: (10) (10)

(10)

̅𝑑𝑝 ∑𝑟(𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) + 𝑥 ′ 𝛤 ′ + ∑𝑑,𝑝 𝜇 𝑎𝑡1𝑘 ′ ≥ 𝑎𝑡1𝑘 + ∑𝑑,𝑝 𝑈 + 𝑇𝐶𝑇 − 𝑀(1 − 𝑞1𝑘𝑘 ′ ) 𝑘𝑘 𝑘𝑘 𝑑𝑝𝑘𝑘 ′ (10)

(10)

̂𝑑𝑝 ∑𝑟(𝐺𝑟𝑝𝑘 𝐷𝑟𝑝 ) 𝑥𝑘𝑘 ′ + 𝜇𝑑𝑝𝑘𝑘 ′ ≥ 𝑈 (10)

∀𝑑, 𝑝, 𝑘, 𝑘 ′ (𝑘

(10)

𝑥𝑘𝑘 ′ , 𝜇𝑑𝑝𝑘𝑘 ′ ≥ 0

∀𝑑′ , 𝑝, 𝑗

(24)

𝑥𝑑 ′𝑗 + 𝜇𝑑 ′𝑝𝑗 ≥ 𝐿̂𝑑 ′𝑝 ∑𝑟 𝑧𝑑 ′𝑟𝑗 𝐷𝑟𝑝 (24) (24) 𝑥𝑑 ′𝑗 , 𝜇𝑑 ′𝑝𝑗

≠

(50)

∀𝑑′ , 𝑝, 𝑗

≥0

(28) (28) (28) ̅𝑑𝑝 𝑎𝑡_ℎ𝑟𝑝 + 𝑥𝑟𝑝𝑘 𝛤𝑟𝑝𝑘 + ∑𝑑 𝜇𝑟𝑝𝑑𝑘 ≤ 𝑎𝑡1𝑘 + ∑𝑑 𝑝1𝑑𝑘 𝐷𝑟𝑝 𝑈

∀𝑟, 𝑝, 𝑘 𝑖𝑓 𝐺𝑟𝑝𝑘 = 1

(28) (28) ̂𝑑𝑝 𝑝1𝑑𝑘 𝐷𝑟𝑝 𝑥𝑟𝑝𝑘 + 𝜇𝑟𝑝𝑑𝑘 ≥ 𝑈

∀𝑟, 𝑝, 𝑑, 𝑘 𝑖𝑓 𝐺𝑟𝑝𝑘 = 1

(28) (28) 𝑥𝑟𝑝𝑘 , 𝜇𝑟𝑝𝑑𝑘

∀𝑟, 𝑝, 𝑑, 𝑘 𝑖𝑓 𝐺𝑟𝑝𝑘 = 1

≥0 (39) (39)

(39)

̅𝑟𝑝 ) + 𝑥 ′ 𝛤 ′ + ∑𝑝 𝜇 ′ + 𝑂𝑟𝑟 ′ − 𝑀(1 − 𝑦𝑦𝑟𝑟 ′𝑗 ) 𝑎𝑡_𝑑𝑒𝑙𝑟 ′𝑗 ≥ 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 + ∑𝑑 ′(𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝐷𝑟𝑝 𝑈 𝑟𝑟 𝑗 𝑟𝑟 𝑗 𝑟𝑟 𝑗𝑝

∀𝑟, 𝑟 ′ (𝑟 ≠ 𝑟 ′ ), 𝑗, 𝑝

(39) (39) 𝑥𝑟𝑟 ′𝑗 , 𝜇𝑟𝑟 ′𝑗𝑝

∀𝑟, 𝑟 ′ (𝑟

≥0 (42) (42)

(42)

𝑥𝑟𝑗

(42)

(42)

̂𝑟𝑝 𝐷𝑟𝑝 + 𝜇𝑟𝑗𝑝 ≥ 𝑈 (42)

𝑥𝑟𝑗 , 𝜇𝑟𝑗𝑝 ≥ 0

(42)

+ ∑𝑝 𝜇𝑟𝑗𝑝 + 𝑂𝑟 − 𝑀(1 − 𝑦𝑦𝑦𝑟𝑗 )

(51)

∀𝑟, 𝑟 ′ (𝑟 ≠ 𝑟 ′ ), 𝑗

(39) (39) ̂𝑟𝑝 ) 𝑥𝑟𝑟 ′𝑗 + 𝜇𝑟𝑟 ′𝑗𝑝 ≥ ∑𝑑 ′(𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝐷𝑟𝑝 𝑈

̅𝑟𝑝 + 𝑥 𝑟𝑡𝑗 ≥ 𝑎𝑡_𝑑𝑒𝑙𝑟𝑗 + ∑𝑝 𝐷𝑟𝑝 𝑈 𝑟𝑗 𝛤𝑟𝑗

(49)

𝑘′ )

∀𝑑′ , 𝑗

(24) (24) (24) 𝑑𝑡2𝑗 ≥ 𝑎𝑡2𝑗 + ∑𝑟 (𝑧𝑑 ′𝑟𝑗 × ∑𝑝 𝐷𝑟𝑝 𝐿̅𝑑 ′𝑝 ) + 𝑥𝑑 ′𝑗 𝛤𝑑 ′𝑗 + ∑𝑝 𝜇𝑑 ′𝑝𝑗 (24)

∀𝑘, 𝑘 ′ (𝑘 ≠ 𝑘 ′ ) ∀𝑑, 𝑝, 𝑘, 𝑘 ′ (𝑘 ≠ 𝑘 ′ )

≠

(52)

𝑟 ′ ), 𝑗, 𝑝

∀𝑟, 𝑗 ∀𝑟, 𝑗, 𝑝 ∀𝑟, 𝑗, 𝑝

Therefore, the robust reformulation of the model with uncertain loading/unloading times is given by: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑂𝐹 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 Error! Reference source not found. −(9), (11) − (23), (26), (27), (29) − (38), (40), (41), (43), 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝑠𝑒𝑡 (48) − (53)

(53)

4. Numerical analysis

The model is coded in GAMS software and solved using the CPLEX solver on a PC with Intel® Core™ i7 3520M 2.90 GHz CPU and 8.0 GB of RAM. A small-size example is considered in which 𝑅 = 6, 𝑃 = 2, 𝐾 = 4, 𝐽 = 4, 𝐷 = 2, 𝐷′ = 2 (The parameters of the example are available at https://doi.org/10.6084/m9.figshare.5234797). ̂𝑑𝑝 = 𝜑𝑈 ̅𝑑𝑝 , 𝐿̂𝑑 ′𝑝 = 𝜑 is defined as the level of variability of the uncertain parameters with regard to their nominal values, i.e., 𝑈 (45) (45) (45) ̂𝑟𝑝 = 𝜑𝑈 ̅𝑟𝑝 . Also, we define 𝜆 as the level of conservatism, e.g., Γ 𝜑𝐿̅𝑑 ′𝑝 and 𝑈 = 𝜆Γ , where Γ is the maximum value rjk

(45)

rjk max

rjk max

of Γrjk , and so on for the uncertainty budget of the other constraints. To analyze the effect of robustness on the optimal solutions, three levels of variability 𝜑 ∈ {10%, 20%, 40% }, and 10 levels of conservatism 𝜆 ∈ {0.1, 0.2, … ,1} were considered. To evaluate the robustness of the solutions for each level of conservatism (only for 𝜑 = 20%) , the uncertainty of the parameters is simulated by randomly generated uncertain parameters, and the feasibility of the optimal solution obtained from robust models is checked. The ratio of the number of infeasible simulation runs to the total number of runs is considered as the probability of constraint violation. The percentage increase in the optimal value, i.e.,

𝑂𝐹𝑅∗ −𝑂𝐹𝐷∗ 𝑂𝐹𝐷∗

where 𝑂𝐹𝐷∗ and 𝑂𝐹𝑅∗ are the optimal value of deterministic and (10)

robust models, respectively, and the probability of constraint violation are displayed in Fig. 1 (𝛤𝑘𝑘 ′ = 8,

(24) 𝛤𝑑′𝑗 𝑚𝑎𝑥

=

(39) 𝛤𝑟𝑟 ′𝑗 𝑚𝑎𝑥

=

= 𝑃 = 2,

(28) 𝛤𝑟𝑝𝑘 𝑚𝑎𝑥

𝑚𝑎𝑥

=

= 𝐷 = 2)

𝜑=0.1 𝜑=0.2 𝜑=0.4 The probability of violating constraints on variability level of 𝜑=0.2

70

Objective Function (𝑂𝐹∗) Increase (%)

(42) 𝛤𝑟𝑗 𝑚𝑎𝑥

= 𝐷𝑃 = 4, 𝛤𝑟𝑗𝑘

100 60

90 80

50

70 40

60

50

30

40

20

30 20

10

10

0

0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 𝜆 (level of conservatism)

0.8

0.9

Probability Of Constraint Violation (%)

𝑃(𝐷 +

𝐷′ )

(45)

𝑚𝑎𝑥

1

Fig. 1. The percentage increase in the optimal value and the probability of constraint violation for variability level of 𝜑 = 0.2

As expected, the optimal value of objective function deteriorates as the robustness is implied. As shown in Fig. 1, the optimal value of objective function deteriorates between 3% (when variability and conservatism levels are both 10%) and 63% (when variability and conservatism levels are 40% and 100%, respectively). For both variability levels of 10% and 20%, the worst-case scenario is reached before the theoretical situation, i.e., 𝜆 = 0.7, while for variability level of 40%, the it is reached at the highest level of conservatism. In order to guarantee less than 10% of the probability of constraints violation, it is needed to choose 𝜆 = 0.6, which increases the objective value by 27% (when 𝜑 = 0.2). The probability of constraint violation (for variability level of 𝜑 = 0.2) tends to be high as the conservatism level is less than 30%, and then, the probability decreases dramatically to reach 10% in conservatism level of 60%. Table 1. The percentage increase in the partial costs of the optimal value of objective function (%) outgoing vehicles cost cross-docking and inventory holding costs 10% 20% 40% 10% 20% 40% 𝜆 𝜑 0.1 3.17 1.28 2.56 5.13 6.35 12.82 0.2 6.23 2.56 5.13 22.17 12.57 25.43 0.3 8.15 3.85 21.18 27.82 15.39 32.64 0.4 9.76 4.83 23.22 31.90 18.31 39.26 0.5 10.20 5.67 24.87 35.20 19.20 41.04 0.6 10.45 6.41 26.17 37.81 19.86 42.04 0.7 10.68 7.02 27.24 39.96 20.63 42.96 0.8 10.91 7.37 28.80 41.17 22.03 43.88 0.9 10.91 7.72 29.44 42.38 22.03 45.02 1.0 10.91 8.07 30.08 46.91 22.10 45.61

earliness and tardiness penalty costs 10% 20% 40% 74.09 131.87 184.45 232.59 241.96 251.34 259.50 266.51 266.51 266.51

132.64 278.92 116.33 181.87 199.62 220.21 244.69 227.67 227.67 227.72

282.69 365.36 520.39 697.24 720.86 744.48 771.76 801.43 844.34 787.82

According to Table 1, the effect of uncertainty in the loading/unloading times of products on the deterioration of the optimal value of tardiness penalty cost is higher than the effect of uncertainty in the other partial costs of the objective, i.e., around 75% for 𝜆 = 0.1 and 𝜑 = 0.1, and about eight times larger than the associated nominal value for 𝜆 = 1 and 𝜑 = 0.4.

5. Conclusions In this paper, the vehicle routing and scheduling with cross-docking problem is considered under the uncertainty of (i) the unit unloading time of products at the receiving doors of the cross-dock, (ii) the unit loading time of products at the shipping doors of the cross-dock, and (iii) the unit unloading time of products at retailers. The robust optimization approach proposed by Bertsimas and Sim (2004) is employed to properly protect the solutions against uncertainty in parameters. It is provided insights into the performance of the robust optimization model in a small-size example. It is also shown that in order to guarantee a low value for the probability of constraints violation, i.e., 10%, it is sufficient to protect about 60% of the resource of uncertainty, which increases the optimal value of objective function by 27% (when 𝜑 = 0.2). Numerical experiments show that the effect of uncertainty in the loading/unloading times of products on the deterioration of the optimal value of tardiness penalty cost is higher than the effect of uncertainty in the other partial costs of the objective, i.e., the fixed and variable costs of vehicles, the cross-docking costs, and the inventory holding costs at the temporary storage area. In this paper, it is assumed that the cross-dock operates in pre-distribution mode and split delivery is not allowed. Therefore, as a future research suggestion, researchers can consider the effects of post-distribution mode and split delivery on the robustness of the solutions against the uncertainty of loading/unloading time of products. Moreover, another sources of uncertainty in the cross-dock, e.g., truck changeover time and transfer time at the cross-dock, can be considered.

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