Developing a crossdocking network design model under uncertain environment
 2k Downloads
 2 Citations
Abstract
Crossdocking is a logistic concept, which plays an important role in supply chain management by decreasing inventory holding, order packing, transportation costs and delivery time. Paying attention to these concerns, and importance of the congestion in cross docks, we present a mixedinteger model to optimize the location and design of cross docks at the same time to minimize the total transportation and operating costs. The model combines queuing theory for design aspects, for that matter, we consider a network of cross docks and customers where two M/M/c queues have been represented to describe operations of indoor trucks and outdoor trucks in each cross dock. To prepare a perfect illustration for performance of the model, a real case also has been examined that indicated effectiveness of the proposed model.
Keywords
Crossdocking Network design Truck allocation Response time Queuing theoryList of symbols
 fl_{ij}
Rate of flow goes to customer j from customer i
 \( {\text{C}}_{\text{ijtl }} \)
Transportation cost between customer i and j where flow goes through cross docks of t and l
 C_{1mt}
Costs of allocating m outdoor trucks to cross dock t for transportation between cross docks
 C_{2mt}
Costs of allocating m indoor trucks or outdoor trucks to cross dock t for transporting for customers
 F_{t}
Establishing cost of cross dock t
 M_{I}
Highest possible Number of indoor trucks
 M_{1O}
Highest possible Number of outdoor trucks for transportation between cross docks
 M_{2O}
Highest possible Number of outdoor trucks for a cross dock to transport goods for customers
 N
Number of candidate nodes for cross docks
 L
Number of customers
Variables
 x_{ijt1}
\( 1 {\text{ if there is any way from customer i to customer j which uses cross dock t and }}\,0\,{\text{otherwise}} \)
 y_{mt}
1 if m indoor truck be allocated to cross dock t \( {\text{and}}\,0\,{\text{otherwise}} \)
 z_{mt}
1 if m outdoor truck be allocated to cross dock t to transport goods for customers \( {\text{and}}\,0\,{\text{otherwise}} \)
 l_{mt}
1 if m outdoor truck be allocated to cross dock t for transportation to other cross docks \( {\text{and}}\,0\,{\text{otherwise}} \)
 N_{t}
\( 1 {\text{ if cross dock j establish and}}\,0\,{\text{otherwise}} \)
 λ_{It}
Rate of indoor demand for cross dock t
 λ_{O1t}
Rate of outdoor flows that need to be transported from cross dock t to another cross dock
 λ_{O2t}
Outdoor flows go through cross dock t and customers
Introduction
In the competitive environment, companies must satisfy more complicated demands with less response time. Crossdocking is a relatively new warehousing strategy in logistic that involves moving products directly from the receiving dock to the shipping dock (Bellanger et al. 2013). It can be defined as a transshipment platform which receives flows from various suppliers and consolidates them with other flows for a common final delivery to a destination (Kinnear 1997). Also the efficiency of crossdocking will influence the lead time, inventory level and response time to the costumer (Kuo 2013).
In the literature, there are some researchers who considered crossdocking problem. For instance, Bellanger et al. (2013) dealt with the problem of finding optimal schedule in crossdocking, where the main goal was to minimize the completion time of the latest order. In their paper, crossdocking was modeled as a threestage hybrid flow shop, and for obtaining good feasible solutions, they have developed several heuristic schemes. They also proposed a branchandbound algorithm to evaluate the heuristics. Chen and Song (2009) considered twostage hybrid crossdocking scheduling problem, where the objective was minimizing the make span. To do so, a mixedinteger programming and four heuristics were presented.
Kuo (2013), considered another interesting aspect in optimization of crossdocking; he presented a model for optimizing both inbound and outbound truck sequencing and both inbound and outbound truck dock assignment. Jayaraman and Ross (2003) considered supply chain design problem which incorporates crossdocking into a supply chain environment. Agustina et al. (2010) provide a literature review of mathematical models in crossdocking, where the models were classified into three levels of operational, tactical and strategic.
In recent years, Santos et al. (2013) dealt with pickup and delivery in cross docks, and proposed an integer programming model and a Branchandprice for the problem. Liao et al. (2012) considered problem of inbound and outbound truck sequencing for cross docks, and proposed two new hybrid differential evolution algorithms for the problem. Agustina et al. (2014) considered perishable food products, and proposed a mixedinteger model to minimize earliness, tardiness, inventory holding, and transportation cost.
Location analysis and network design are two major research areas in supply chain optimization, location problems deal with the decisions of where to optimally locate facilities whereas network design involves activating optimal links (Contreras and Fernández 2012). In this area, Ross and Jayaraman (2008) studied location planning for the cross docks and distribution centers in supply chains. Later, Babazadeh et al. (2012) proposed a new network design mathematical model for an agile supply chain.
LüerVillagra and Marianov (2013) considered price and location; they proposed a competitive hub location and pricing problem for the air passenger industry. Mousavi and TavakoliMoghaddam (2013) considered location and routing scheduling problems with crossdocking, and present a twostage mixedinteger programming model.
Tavakkolimoghaddam et al. (2013) considered a network design problem for a three level supply chain, and proposed a new mathematical model, where their aims were to determine the number of located distribution centers, their locations, capacity level, and allocating customers to distribution centers.
One of the most important factors in supply chain management is response time, which consists of production, handling and waiting times (Vis and Roodbergen 2011). Some researchers modeled response time in stochastic environment, and some of them used queuing theory to represent a mathematical model. Some researchers believe that it makes the problem hard to solve, and some suggest cutting planes for obtaining optimal solutions in small and mediumsized problem instances (KarimiNasab and Seyedhoseini 2013).
In this area, Ha (1997) considered Poisson demand and exponential production times for a single item maketostock production system. He proposed an M/M/1/S queuing system for modeling the system. Later, KarimiNasab and Fatemi Ghomi (2012) argued that it is not a practical assumption that production times are fixed input data of the problem. They proposed that in many cases the production time of the item may be considered as either a decision variable or an uncertain input data other than a fixed value. Nonetheless, many operational managers believe that making decisions with minimal total costs is of crucial importance (KarimiNasab and SabriLaghaie 2014).
Kerbache and Smith (2004) proposed context of supply chain and application of queue approach. They modeled supply chain network as a queuing system and analyzed it, in particular, they used queuing network methods to evaluate the performance measures of a supply chain. In this research, queuing theory has been used to describe stock control system of retailers and response of suppliers, where each retailer has been assumed to be an M/M/1 queue with balk arrival, and each supplier has been assumed to be an M/M/1 queue.

Where cross docks should be located?

What is the optimal number of indoor trucks and outdoor trucks?

How customers should be allocated to cross docks?

Location problem of cross docks has been considered in an uncertainty environment

A mathematical model has been developed which simultaneously optimizes location of cross docks, number of indoor trucks and outdoor trucks.

A congestion constraint has been considered in the model which restricts waiting time of customers to not be more than t_{I}, and another constraint has been considered that restricts waiting time of shipments in cross docks to not be more than t_{I}.
The reminder of this paper is organized as follows: section two provides a brief definition of the problem, and also represents a mathematical model and a linearization approach to solve the problem. Section three is for computational results which consist of an example and a real case. Section four concludes the paper.
Problem definition
 1.
Demand size for each customer is less than size of a truck, and each truck can give service to n customers.
 2.
Each truck gives service to exactly n customers, and if number of customers be lesser than n, then truck waits to service with full capacity.
 3.
Location of customers is fixed.
 4.
Between each pair of nodes—customers and cross docks—at most one link can be constructed.
 5.
Candidate nodes for cross docks are fixed.
 6.
Demand of customer k has been assumed Poisson distribution with rate of \( \lambda_{\text{k}} \).
 7.
Service time for each truck has been assumed exponentially distribution with mean value of \( \frac{1}{\mu } \).
 8.
Waiting time for each shipment in cross dock must not be more than t_{I}.
 9.
Waiting time for customers that are ready to get service must not be more than t_{I}.
In this queue each customer has some waiting time but as \( \lambda_{\text{j}} \) is a big amount for cross docks, waiting time in this queue could be ignorable. For that matter only waiting time for trucks are considered in this paper.
Model development
Objective
Objective function has composed of five sections. First section is for transportation costs, second section relates to costs of establishing cross docks, and the other three sections are for operating indoor and outdoor trucks.
Constraint five calculates indoor flows for cross dock t, constraint six calculates outdoor flows that need to be transported from cross dock t to another cross dock, and constraint seven calculates outdoor flows go through cross dock t and customers. Constraints eight and nine ensure that between each pair of customers, only one path be constructed. Constraint 10 states that a path goes through a cross dock only if it is established. Constraints 11, 12 and 13 states that no truck can be allocated to a cross dock unless it has been established. Constraints 14, 15 and 16 ensure that waiting time of customer for indoor trucks or outdoor trucks not to be more than t_{It}, where constraint 14 is for indoor trucks, constraint 16 is for outdoor trucks which give service to shipments between two cross docks and constraint 15 is for outdoor trucks which goes through cross docks and customers.
Solving approach
Computational result
In this paper, using a numerical example, performance of the proposed model has been evaluated, and then efficiency of the model has been examined by a real case.
Numerical result
Case study
In this section, we represent a case study, which is for Dried fruit products in Iran. In this case, we only considered two major products of raisin and pistachios. Based on information of 2012, Iran is capable to produce 154,000 tons of raisin and 192,000 tons of pistachios per year. Where it exports 138,000 tons of raisin and 150,000 tons of pistachios. These two products need to be gathered from farmers or factories that pack the products, then transfer them to wholesalers.
There are five wholesalers in countries of Qatar, Iraq, Russia, Azerbaijan and Turkey which consist of major exports of these products, and also more than 15 wholesalers are existing in different cities of Iran. Also there are 14 cities which produce major raisins of Iran and 30 cities which produce major pistachios of Iran. In these cities, Ghazvin produces with 50,000 tons of raisin is the most capable producer of raisin, and Rafsanjan with 21,000 tons of pistachios is the most capable producer of pistachios.
Production rate of raisin and pistachios for different cities in year of 2012, and also products flow rate between each pair of cities for both of the products have been presented in “Appendix” B. In this case, it has been assumed that all the cross docks must be established in the Iran. Consequently, transportation of the exports out of the Iran can be ignored, and only it be considered in Iran. Another reason for this is for different transportation types which could be used out of the Iran. For instance, if a shipment wants to go Qatar, it must go to Bandarabas dock, and then it goes to Qatar.
Optimal solution for the problem (costs are in 1,000,000 Tomans)
Cross docks  Indoor trucks  Outdoor trucks (cross dock to customers)  Outdoor trucks (cross dock to cross dock  Allocated nodes  Optimal costs  

Scenario 1  Raisin  Ghazvin  6  3  4  Tehran, Esfahan, Shiraz, Sari, Yazd, Khoram Shahr, Ghazvin, Malayer, Hamadan  462 
Mashad  6  2  3  Mashad, Ghochan, Gorgan, Dargaz, Bardaskan, Kashmar, Shirvan, Farouj, Esfrayen, Bojnourd  
Bandarabas  0  1  0  Bandarabas, Kerman  
Oromiye  2  4  0  Oromiye, Tabriz, Bazargan, Astara, Bandar anzaly, Rasht, Malekan  
Pistachios  Rafsanjan  7  4  4  Rafsanjan, Kerman, Sirjan, Zarand, Anar, Shahr babak, Ravar, Mehriz, Ardakan, Yazd, Khatam, Sarvestan, Shiraz, Neyriz, Khash, Esfahan, Bandarabas, Khoram Shahr  511  
Mashad  5  3  2  Tabas, Ghayen, Bardaskan, Bajestan, Kashmar, Ferdos, Torbat heydarie, sabzevar, Mashad, Boujnord, Gorgan, Damghan  
Bandaranzaly  2  6  0  Bandar anzaly, Bazargan, Astara, Rasht, Sari, Tabriz, Tehran, Zarandiye, Save, Ghom, Ghazvin  
Scenario 2  Raisin and pistachios  Oromiye  4  8  0  Oromiye, Tabriz, Bazargan, Astara, Bandar anzaly, Rasht, Malekan  720 
Mashad  9  4  4  Mashad, Ghochan, Gorgan, Dargaz, Bardaskan, Kashmar, Shirvan, Farouj, Esfrayen, Bojnourd, Tabas, Ghayen,, Bajestan, Ferdos, Torbat heydarie, sabzevar, Damghan  
Ghazvin  6  2  4  Tehran, Esfahan, Sari, Ghazvin, Malayer, Save, Ghom, Zarandiye  
Rafsanjan  7  4  4  Rafsanjan, Kerman, Sirjan, Zarand, Anar, Shahr babak, Ravar, Mehriz, Ardakan, Yazd, Khatam, Sarvestan, Shiraz, Neyriz, Khash, Bandarabas, Khoram Shahr 
As it is demonstrated in Table 1, if raisin and pistachios could be transported together, optimal costs would plunge. For this scenario, costs are 720 million Tomans, which is significantly lesser than when these products need to be transported separately. Because of consolidation of raisin and pistachios, numbers of trucks and cross docks have been decreased. In this case, 64 trucks and seven cross docks are needed for the first scenario and 56 trucks and four cross docks are needed for the second scenario.
Conclusion
In this paper, we considered the problem of crossdocking, and proposed a mathematical model to simultaneously optimize location of cross docks and number of trucks. Assigning optimal trucks, and sequencing of trucks in cross docks are some historical problems, but considering location of cross docks with variable number of trucks is a new problem.
In this model, we used one of the most important concepts of cross docks, which state that no shipment in cross docks can wait more than a specific time. Some of the researchers state that this waiting time is equal to 12 h and some defined 24 h. For this reason we proposed two stochastic constraints. We also improved the model by considering another waiting time constraint for shipments in customer nodes. Consequently, to satisfy these constraints, numbers of indoor trucks and outdoor trucks in each cross dock have been considered variable.
In this paper, the model has been examined due to different parameters of the model, and differences among our model and previous models have been analyzed. We also prepared a real case to illustrate performance of the model, which is for products of raisin and pistachios in Iran. For this case, two scenarios have been considered and proposed model has been analyzed.
For future studies this research can be extended by considering scheduling of trucks, this may increase complexity of the problem but the model would become more realistic. Another extension of this research is possible by considering rate of corruption for perishable inventories. Another aspect which is of further interest is to consider waiting time costs in objective, although it may be expected that this will lead to intractable formulations.
References
 Agustina D, Lee CKM, Piplani R (2010) A review : mathematical models for cross docking planning. Int J Eng Bus Manag 2(2):47–54Google Scholar
 Agustina D, Lee CKM, Piplani R (2014) Vehicle scheduling and routing at a cross docking center for food supply chains. Int J Prod Econ 152:29–41CrossRefGoogle Scholar
 Babazadeh R, Razmi J, Ghodsi R (2012) Supply chain network design problem for a new market opportunity in an agile manufacturing system. J Ind Eng Int. doi: 10.1186/2251712X819
 Bellanger A, Hanafi S, Wilbaut C (2013) Threestage hybrid flow shop model for crossdocking. Comput Oper Res 40:1109–1121CrossRefMathSciNetGoogle Scholar
 Chen F, Song K (2009) Minimizing make span in twostage hybrid cross docking scheduling problem. Comput Oper Res 36:2066–2073CrossRefzbMATHGoogle Scholar
 Contreras I, Fernández E (2012) General network design: a unified view of combined location and network design problems. Eur J Oper Res 219(3):680–697CrossRefzbMATHGoogle Scholar
 Ha AY (1997) Stockrationing policy for a maketostock production system with two priority classes and backordering. Nav Res Logist (NRL) 44(5):457–472CrossRefzbMATHGoogle Scholar
 Jayaraman V, Ross A (2003) A simulated annealing methodology to distribution network design and management. Eur J Oper Res 144:629–645CrossRefzbMATHMathSciNetGoogle Scholar
 KarimiNasab M, Fatemi Ghomi SMT (2012) Multiobjective production scheduling with controllable processing times and sequencedependent setups for deteriorating items. Int J Prod Res 50(24):7378–7400CrossRefGoogle Scholar
 KarimiNasab M, SabriLaghaie K (2014) Developing approximate algorithms for EPQ problem with process compressibility and random error in production/inspection. Int J Prod Res 52(8):2388–2421CrossRefGoogle Scholar
 KarimiNasab M, Seyedhoseini SM (2013) Multilevel lot sizing and job shop scheduling with compressible process times: a cutting plane approach. Eur J Oper Res 231(3):598–616CrossRefMathSciNetGoogle Scholar
 Kerbache L, Smith JM (2004) Queueing networks and the topological design of supply chain systems. Int J Prod Econ 91(3):251–272CrossRefGoogle Scholar
 Kinnear E (1997) Is there any magic in crossdocking? Supply Chain Manag Int J 2:49–52CrossRefGoogle Scholar
 Kuo Y (2013) Optimizing truck sequencing and truck dock assignment in a cross docking system. Expert Syst Appl 40:5532–5541CrossRefGoogle Scholar
 Liao TW, Egbelu PJ, Chang PC (2012) Two hybrid differential evolution algorithms for optimal inbound and outbound truck sequencing in cross docking operations. Appl Soft Comput 12:3683–3697CrossRefGoogle Scholar
 LüerVillagra A, Marianov V (2013) A competitive hub location and pricing problem. Eur J Oper Res 231:734–744CrossRefGoogle Scholar
 Mousavi SM, TavakoliMoghaddam R (2013) A hybrid simulated annealing algorithm for location and routing scheduling problems with crossdocking in the supply chain. J Manuf Syst 32:335–347CrossRefGoogle Scholar
 Ross A, Jayaraman V (2008) An evaluation of new heuristics for the location of crossdocks distribution centers in supply chain network design. Comput Ind Eng 55:64–79CrossRefGoogle Scholar
 Santos FA, Mateus GR, da Cunha AS (2013) The pickup and Delivery problem with crossdocking. Comput Oper Res 40:1085–1093CrossRefMathSciNetGoogle Scholar
 Vis IFA, Roodbergen KJ (2011) Layout and control policies for cross docking operations. Comput Ind Eng 61:911–919CrossRefGoogle Scholar
 Tavakkolimoghaddam R, Forouzanfar F, Ebrahimnejad S (2013) Incorporating location, routing, and inventory decisions in a biobjective supply chain design problem with riskpooling. J Ind Eng Int 9:19Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.