Abstract
A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.
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Abbreviations
- i,j=0,1,…,N :
-
Index of customer or depot, where i,j=1,…,N are customer indexes, and 0 is the depot index,
- t=1,…,T :
-
Period index,
- C :
-
Vehicle capacity in volume,
- c ij :
-
Variable shipping cost per unit of product along arc (i,j) where c ij =c ji and triangle inequality holds (c ij +c jk ≥c ik ),
- \(c_{i0}^{b}\) :
-
Traveling cost of an empty vehicle from customer i back directly to the depot,
- f t :
-
Fixed vehicle cost per tour in period t,
- h it :
-
Holding cost per unit product for customer i in period t,
- I i0 :
-
Initial inventory level at the beginning of period 1,
- I it :
-
Inventory level of customer i at the end of period t,
- \(I_{it}^{+}=\max\mathrm{max}\,(0,I_{it})\) :
-
On-hand inventory of customer i at the end of period t, which excludes the stockout of I it <0,
- V i :
-
The inventory capacity for customer i’s warehouse,
- α it :
-
Service level for customer i’s demand in period t (probability that customer i’s demand is satisfied in period t),
- β it :
-
The service level of customer i’s warehouse in period t (probability that customer i’s warehouse is not overfilled in period t),
- ζ it :
-
Stochastic demand of customer i in period t,
- ζ i,(1,t) :
-
\(=\sum_{s=1}^{t}\zeta_{is}\), cumulative stochastic demand from period 1 to t,
- F i,(1,t)(.):
-
Accumulative probability distribution function of stochastic demand ζ i,(1,t),
- d it :
-
Delivery volume to customer i in period t,
- q ijt :
-
Demand quantity transported on directed arc (i,j) in period t,
- x ijt :
-
Number of the times that customer j is visited directly after customer i in period t.
References
Adelman, D. (2004). A price-directed approach to stochastic inventory/routing. Operations Research, 52(4), 499–514.
Archetti, C., Savelsbergh, M. W. P., & Speranza, M. G. (2006). Worst-case analysis for split delivery vehicle routing problems. Transportation Science, 40(2), 226–234.
Archetti, C., Speranza, M. G., & Savelsbergh, M. W. P. (2008). An optimization-based heuristic for the split delivery vehicle routing problem. Transportation Science, 42(1), 22–31.
Belenguer, J. M., Martinez, M. C., & Mota, E. (2000). A lower bound for the split delivery vehicle routing problem. Operations Research, 48(5), 801–810.
Bell, W. J., Dalberto, L. M., Fisher, M. L., Greenfield, A. J., Jaikumar, R., Kedia, P., Mack, R. G., & Prutzman, P. J. (1983). Improving the distribution of industrial gases with an on-line computerized routing and scheduling optimizer. Interfaces, 13(6), 4–23.
Chen, H. (2007). A Lagrangian relaxation approach for production planning with demand uncertainty. European Journal of Industrial Engineering, 1(4), 370–390.
Chen, S., Golden, B., & Wasil, E. (2007). The split delivery vehicle routing problem: applications, algorithms, test problems, and computational results. Networks, 49(4), 318–329.
Ciupala, L. (2005). A scaling out-of-kilter algorithm for minimum cost flow. Control and Cybernetics, 34(4), 1169–1174.
Dror, M., & Ball, M. (1987). Inventory/routing: reduction from an annual to a short period problem. Naval Research Logistics Quarterly, 34(6), 891–905.
Dror, M., & Trudeau, P. (1990). Split delivery routing. Naval Research Logistics, 37, 383–402.
Dror, M., & Trudeau, P. (1996). Cash flow optimization in delivery scheduling. European Journal of Operational Research, 88(3), 504–515.
Federgruen, A., & Zipkin, P. (1984). A combined vehicle routing and inventory allocation problem. Operation Research, 32, 1019–1037.
Fumero, F., & Vercellis, C. (1999). Synchronized development of production, inventory, and distribution schedules. Transportation Science, 33(3), 330–340.
Hvattum, L. M., Lokketangen, A., & Laporte, G. (2009). Scenario tree-based heuristics for stochastic inventory-routing problems. Informs Journal on Computing, 21(2), 268–285.
Jaillet, P., Bard, J. F., Huang, L., & Dror, M. (2002). Delivery cost approximations for inventory routing problems in a rolling horizon framework. Transportation Science, 36(3), 292–300.
Jin, M. Z., Liu, K., & Bowden, R. O. (2007). A two-stage algorithm with valid inequalities for the split delivery vehicle routing problem. International Journal of Production Economics, 105(1), 228–242.
Kleywegt, A. J., Nori, V. S., & Savelsbergh, M. W. P. (2002). The stochastic inventory routing problem with direct deliveries. Transportation Science, 36(1), 94–118.
Kleywegt, A. J., Nori, V. S., & Savelsbergh, M. W. P. (2004). Dynamic programming approximations for a stochastic inventory routing problem. Transportation Science, 38(1), 42–70.
Koksalan, M., & Tuncer, C. (2009). A DEA-based approach to ranking multi-criteria alternatives. International Journal of Information Technology & Decision Making, 8(1), 29–54.
Lejeune, MA, & Ruszczynski, A. (2007). An efficient trajectory method for probabilistic production-inventory-distribution problems. Operations Research, 55(2), 378–394.
Li, H. L., & Ma, L. C. (2008). Ranking decision alternatives by integrated DEA, AHP and Gower plot techniques. International Journal of Information Technology & Decision Making, 7(2), 241–258.
Patriksson, M. (1993). Partial linearization methods in nonlinear-programming. Journal of Optimization Theory and Applications, 78(2), 227–246.
Peng, Y., Kou, G., Shi, Y., & Chen, Z. X. (2008). A descriptive framework for the field of data mining and knowledge discovery. International Journal of Information Technology & Decision Making, 7(4), 639–682.
Schwarz, L., Ward, J., & Zhai, X. (2006). On the interactions between routing and inventory-management policies in a one-warehouse n-retailer distribution system. Manufacturing & Service Operations Management, 8(3), 253–272.
Tchangani, A. P. (2009). Evaluation model for multiattributes-multiagents decision making: satisfying game approach. International Journal of Information Technology & Decision Making, 8(1), 73–91.
Trudeau, P., & Dror, M. (1992). Stochastic inventory routing: route design with stockouts and route failures. Transportation Science, 26(3), 171–184.
Wolfram. Mathematica 6.0: Wolfram Research, Inc. 2007.
Wolsey, L. A. (1998). Integer programming (pp. 38–40). New York: Wiley.
Yu, Y., Chen, H., & Chu, F. (2005). Large scale inventory routing problem with split delivery: a new model and Lagrangian relaxation approach. In: 2005 IEEE international conference on service operations and logistics, and informatics, Beijing, China.
Yu, Y., Chu, F., & Chen, H. (2007). A note on coordination of production and distribution planning. European Journal of Operational Research, 177(1), 626–629.
Yu, Y., Chen, H., & Chu, F. (2008). A new model and hybrid approach for large scale inventory routing problems. European Journal of Operational Research, 189(3), 1022–1040.
Yu, Y., Chu, F., & Chen, H. (2009a). A Stackelberg game and its improvement in a VMI system with a manufacturing vendor. European Journal of Operational Research, 192(3), 929–948.
Yu, Y., Huang, G. Q., & Liang, L. (2009b). Stackelberg game theory model for optimizing advertising, pricing and inventory policies in vendor managed inventory (VMI) supply chains. Computers & Industrial. Engineering, 57(1), 368–382.
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Yu, Y., Chu, C., Chen, H. et al. Large scale stochastic inventory routing problems with split delivery and service level constraints. Ann Oper Res 197, 135–158 (2012). https://doi.org/10.1007/s10479-010-0772-4
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DOI: https://doi.org/10.1007/s10479-010-0772-4