A Combined Inventory and Lateral Re-Supply Model for Repairable Items—Part I: Modeling
4.4 Summary, Conclusions and Recommendations
Let us give an example of Eqs. (4.15) and (4.16):
(4.21)
(4.22) Here big-M is set at 50, i = 0, 2, 3 and j = 1. This suggests that in our four-depot ex- ample network, we are focusing on depot 1, with the other depots being the home depot 0, depot 2 and depot 3. The local resources available at depot 1 is seven units as shown in Eq. (4.22)—corresponding to the sum of the inventory at hand (β1 = 1 unit) and the repair capability ( U1( u) = 6 units). If y1 is zero, or there is no demand for spare parts placed on depot 1, then Eqs. (i) and (iii) will hold, allowing depot 1 to receive spare parts from other depots. Meanwhile, Eqs. (ii) and (iv) will reflect no delivery from depot 1 to other depots. If y1 equals unity, or there is demand for spare parts placed on depot 1, then Eqs. (ii) and (iv) would hold. This allows the delivery of spare parts from depot 1 to the other three depots. Meanwhile, Eqs. (i) and (iii) translate to no deliveries from the other depots to depot 1.
Equation (4.22) suggests that if y1 is equal to zero, or no spare-part demand is placed on depot 1, all available resources are to be “delivered” locally–a total of seven units in this case. If y1 is unity, or there is spare-part demand placed on depot 1, then Eq. (4.22) is relaxed, or that the resource may not be used locally.
Equation (4.17) relates to the repair capacity at the depot. This repair capacity Ui( u) includes potentially what it repairs for itself and for the other depots. Equa- tion (4.18) is the constraint related to available inventory at each depot. This con- straint is similar to the repair capacity. It includes what it supplies to itself and others out of its depot inventory. We modified the VRP constraints by adding Eq. (4.19), which suggests that we operate within available crew-duty hours. Here we also add a constant loading and unloading time. Specifically, thi is the given amount of time vehicle h spends at demand point i. Uh is the time that a vehicle h spends “on the road” (or the crew duty hours). Equation (4.20) is a natural statement to accompany the “placing order” variable yij when there are multiple sources of supply.
and thus can re-generate a failed item. Items can be transported from one depot to another to satisfy the demand, using a fleet of vehicles stationed at the main de- pot. Each vehicle makes a tour to visit one or more depots before returning to the main depot. Transportation cost is charged against the total operating cost. Inven- tory holding and shortage costs for unsatisfied demands are charged at the depots, contributing toward the total operating cost. The objective is to decide the amount of items to repair at the depots and a distribution plan of the available items among the depots so as to minimize the total transportation and expected holding and short- age costs. We formulate the problem as a nonlinear MIP, consisting of a generally nonlinear objective-function and nonlinear constraints. Generalized Benders’ de- composition is proposed to solve the resulting RLS problem.
Acknowledgements The authors like to acknowledge the valuable inputs from M. Arostegui, M.
Niklas, J. Stewart, and J. Weir, for their contribution during the conduct of this study. They also benefited from the insightful comments of reviewers. The study was conducted under the spon- sorship of the Air Force Materiel Command at Wright-Patterson AFB while the first author was stationed at the Base and while the second author was a faculty member of the Air Force Institute of Technology. Obviously, the authors alone are responsible for the document as presented. The views expressed here do not represent those of the organizations the authors are/were affiliated with.
References
Axater, S.: A new decision rule for lateral transshipments in inventory systems. Manage. Sci. 49, 1168–1179 (2003)
Baita, F., Ukovich, W., Pesenti, R., Favaretto, D.: Dynamic routing-and–inventory problems: A review. Transp. Res-A. 32, 585–598 (1998)
Berman, O., Larson, R.C.: Deliveries in an inventory/routing problem using stochastic dynamic programming. Transp. Sci. 35, 192–213 (2001)
Bertazzi, L., Paletta, G., Speranza, M.G.: Deterministic order-up-to-level policies in an inventory routing problem. Transp. Sci. 36, 119–132 (2002)
Bowersox, D.J., Closs, D.J.: Logistical Management: The Integrated Supply Chain Processes.
McGraw-Hill, New York (1996)
Chan, Y.: Location, Transport and Land-Use: Modelling spatial-temporal information. Springer, Heidelberg (2005)
Christofides, N., Beasley, E.J.: The period routing problem. Networks. 14, 237–256 (1984) Coelho, L.C., Cordeau, J.-F., Laporte, G.: (n.d.): “Thirty years of inventory-routing.” Working
paper, Faculte des sciences de l’administration, Laval University, Quebec City
DeCroix, G., Song, J.-S., Zipkin, P.H.: A series system with returns: Stationary analysis. Op. Res.
53, 350–362 (2005)
Dror, M., Ball, M.: Inventory/routing: Reduction from an annual to a short-period problem. Nav.
Res. Logist. 34, 891–905 (1987)
Federgruen, A., Zipkin, P.: A combined vehicle routing and inventory/allocation problem. Op. Res.
32, 1019–1037 (1984)
Federgruen, A., Prastacos, G., Zipkin, P.H.: An allocation and distribution model for perishable products. Op Res. 31, 75–82 (1986)
Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)
Haughton, M.A., Stenger, A.J.: Comparing strategies for addressing delivery shortages in stochas- tic demand settings. Transp. Res-E. 35, 25–41 (1999)
Herer, Y.T, Levy, R.: The metered inventory problem, an integrative heuristic algorithm. Prod.
Econ. 51, 69–81 (1997)
Herer, Y.T., Rashit, A.: Lateral stock transshipments in a two-location inventory system with fixed and joint replenishment costs. Nav. Res. Logist. 46, 525–547 (1999)
Hooker, J. N.: Planning and scheduling by logic-based benders decomposition. Oper. Res. 55(3), 588–602 (2007)
Jung, B.-R., Sun, B.-G., Kim, J.-S., Ahn, S.-E.: Modeling lateral transshipments in multiechelon repairable-item inventory systems with finite repair channels. Comput. Op. Res. 30, 1401–
1417 (2003)
Kaihara, T.: Multi-agent based supply chain modelling with dynamic environments. Int. J. Prod.
Econ. 85, 263–269 (2003)
Khouja, M.: The single-period (news-vendor) problem: Literature review and suggestions for fu- ture research. Omega—Int. J. Manage. Sci. 27, 537–553 (1999)
McGrath, R.N.: Maintenance: The missing link in supply chain management. Ind. Manage. 41, 29–33 (1999)
Meloa, M.T., Nickel, S., Saldanha da Gama, F.: Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning. Comput.
Op. Res. 33, 181–208 (2005)
Shen, Z-.J. M., Coullard, C., Daskin, M.S.: A joint location-inventory model. Transp. Sci. 37, 40–55 (2003)
Sherbrook, C.C.: Optimal Inventory Modeling of Systems: Multi-echelon Techniques. Wiley, New York (1992)
Slikker, M., Fransoo, J., Wouters, M.: Cooperation between multiple new-vendors with transship- ments. Eur. J. Oper. Res. 167, 370–380 (2005)
Stadler, H.: Supply chain management and advanced planning––basics, overview and challenges.
Eur. J. Oper. Res. 163, 575–588 (1995)
Tersine, R.J.: Principles of Inventory and Materials Management. PTR Prentice-Hall, Saddle River (1994)
Webb, I.R., Larson, R.C.: Period and phase of customer replenishment: A new approach to the strategic inventory/routing problem. Eur. J. Oper. Res. 85, 132–148 (1995)
Yang, W-W; Mathur, K.; Ballou, R.H.: Stochastic vehicle routing problem with restocking. Transp.
Sci. 34, 99–112 (2000)
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