Abstract
The forest supply chain encompasses different closely related operations. Harvesting and transportation decisions are interdependent, where a modification in the former has a considerable impact on the latter. In the literature, these decisions are usually approached in a decoupled way, leading to suboptimal solutions. In this work, a mixed integer linear programming model that integrates both problems for a weekly planning horizon is presented. In addition to decisions about bucking patterns selection in each harvest area and the trucks routing, the composition of the load and the scheduling of the harvesting crews are considered. In this way, the different involved tradeoffs are simultaneously addressed and solved. Through the obtained results, the capabilities of the proposed model are analyzed.
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Abbreviations
- \(B\) :
-
Set of bucking patterns, b = b1, b2,…, bmax
- \(BF_{b,f}\) :
-
Set of bucking patterns b that can be used in harvest area f
- \(C\) :
-
Set of trucks, c = c1, c2,…, cmax
- \(C_{p}\) :
-
Set of trucks belonging to regional base p
- \(D\) :
-
Set of log diameters, d = d1, d2,…, dmax
- \(F\) :
-
Set of harvest areas, f = f1, f2,…, fmax
- \(I\) :
-
Set of plants, i = i1, i2,…, imax
- \(L\) :
-
Set of log lengths, l = l1, l2,…, lmax
- \(P\) :
-
Set of regional bases, p = p1, p2,…, pmax
- \(T\) :
-
Set of time periods, t = t1, t2,…, tmax
- \(V\) :
-
Set of possible truck trips, v = v1, v2,…, vmax
- \(cd_{p,f}^{{}}\) :
-
Cost per travelled kilometer between p and f ($/km)
- \(cl_{f,i}^{{}}\) :
-
Cost per travelled kilometer between f and i ($/km)
- \(cu_{i,f}^{{}}\) :
-
Cost per travelled kilometer between i and f ($/km)
- \(cr_{i,p}^{{}}\) :
-
Cost per travelled kilometer between i and p ($/km)
- \(cap_{i,t}\) :
-
Plant processing capacity during period t (logs)
- \(capmax_{c}\) :
-
Maximum truck capacity (ton)
- \(capmin_{c}\) :
-
Minimum truck capacity (ton)
- \(cfix_{c,p}\) :
-
Fixed cost per use of truck ($/truck)
- \(closs_{b}\) :
-
Cost for loss of wood when applying pattern b ($/ton)
- \(cqrl_{l,d,f}\) :
-
Inventory cost of log of length l and diameter d in f ($/log)
- \(cstock_{l,d,i}\) :
-
Inventory cost of log of length l and diameter d in i ($/log)
- \(dpf_{p,f}^{{}}\) :
-
Distance between p and f (km)
- \(dfi_{f,i}^{{}}\) :
-
Distance between f and i (km)
- \(dif_{i,f}^{{}}\) :
-
Distance between i and f (km)
- \(dip_{i,p}^{{}}\) :
-
Distance between i and p (km)
- \(dmin_{l,d,i,t}\) :
-
Minimum committed demand (logs)
- \(dtot_{l,d,i}\) :
-
Total (weekly) demand (logs)
- \(fconv_{l,d,b}\) :
-
Log conversion (stem-to-log) by using bucking pattern b (log)
- \(loss_{b}\) :
-
Loss of raw material after applying bucking pattern b (%)
- \(maxstock_{l,d,i}\) :
-
Storage capacity, by type of log, in plant (logs)
- \(qcsmax_{f}\) :
-
Maximum amount of stems to cut in harvest area f (stems)
- \(qcsmin_{f}\) :
-
Minimum amount of stems to cut in harvest area f (stems)
- \(qend_{f}\) :
-
Desired amount of standing stems at the end of the planning horizon (stems)
- \(qini_{f}\) :
-
Initial stock of standing stems (stems)
- \(qminb_{b}\) :
-
Minimum number of times to apply bucking pattern b if used (stems)
- \(qttup_{l,d,c,v,f,i,t}\) :
-
Maximum amount of logs that a truck can load on a trip (logs)
- \(stockini_{l,d,i}\) :
-
Initial stock of logs, by type, in plant i (logs)
- \(maxt_{c,t}^{{}}\) :
-
Maximum route duration time (h)
- \(vd_{p,f}^{{}}\) :
-
Average travel speed between p and f (km/h)
- \(vl_{f,i}^{{}}\) :
-
Average travel speed between f and i (km/h)
- \(vu_{i,f}^{{}}\) :
-
Average travel speed between i and f (km/h)
- \(vr_{i,p}^{{}}\) :
-
Average travel speed between i and p (km/h)
- \(weight_{l,d}\) :
-
Weight of log of length l and diameter d (ton)
- \(CUT_{f,t}\) :
-
Binary variable that indicates if harvest area f is cut in period t or not
- \(Q_{b,f,t}^{BP}\) :
-
Continuous variables that indicates the number of times bucking pattern b is used in f in period t
- \(Q_{l,d,f,t}^{CL}\) :
-
Continuous variable that indicates the amount of logs of each type that are generated in harvest area f in period t
- \(Q_{f,t}^{CS}\) :
-
Continuous variable that indicates the amount of stems that are cut in harvest area f in period t
- \(Q_{l,d,i,t}^{PL}\) :
-
Continuous variable that indicates the amount of logs of each type that are processed above the minimum demand in plant i in period t
- \(Q_{l,d,f,t}^{RL}\) :
-
Continuous variable that indicates the amount of logs of each type that remain on the roadside in harvest area f at the end of period t
- \(Q_{l,d,i,t}^{SL}\) :
-
Continuous variable that indicates the amount of logs of each type that are kept in inventory at plant i at the end of period t
- \(Q_{f,t}^{SS}\) :
-
Continuous variable that indicates the amount of stems that are left standing in harvest area f at the end of period t
- \(Q_{l,d,f,t}^{TL}\) :
-
Continuous variable that indicates the amount of logs of each type that are transported from harvest area f during period t
- \(Q_{l,d,c,v,f,i,t}^{TT}\) :
-
Continuous variable that indicates the amount of logs of each type that are transported from harvest area f to plant i on trip v of truck c in period t
- \(X_{c,p,f,f}^{D}\) :
-
Binary variable that indicates whether truck c during trip v travels from p to f in period t or not
- \(X_{c,v,f,i,t}^{L}\) :
-
Binary variable that indicates whether truck c during trip v travels from f to i in period t or not
- \(X_{c,v,i,f,t}^{U}\) :
-
Binary variable that indicates whether truck c during trip v travels from i to f in period t or not
- \(X_{c,v,i,p,t}^{R}\) :
-
Binary variable that indicates whether truck c during trip v travels from i to p in period t or not
- \(Y_{b,f,t}^{BP}\) :
-
Binary variable that indicates whether bucking pattern b is used in harvest area f in period t or not
- \(Y_{c,p,t}^{T}\) :
-
Binary variable that indicates whether truck c belonging to regional base p is used in period t or not
- \(Y_{l,c,v,f,i,t}^{TT}\) :
-
Binary variable that indicates whether logs of length l are loaded into truck c during trip v that travels from f to i in period t or not
- \(Z^{{}}\) :
-
Objective function ($)
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Acknowledgements
This work was supported by the CONICET, FONCyT and UTN through their projects PIP 1401, PICT 2017-4004 and PID SIUTIFE0005246TC, respectively.
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Bordón, M.R., Montagna, J.M. & Corsano, G. Operational transportation planning in the forest industry integrating bucking decisions. Ann Oper Res 322, 385–411 (2023). https://doi.org/10.1007/s10479-022-04688-2
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DOI: https://doi.org/10.1007/s10479-022-04688-2