Abstract
A mixed integer linear programming (MILP) for the detailed production planning of multiproduct batch plants is presented in this work. New timing decisions are incorporated to the model taking into account that an operation mode based in campaigns is adopted. This operation mode assures a more efficient production management adjusted to the specific context conditions of the considered time horizon. In addition, special considerations as sequence-dependent changeover times and different unit sizes for parallel units in each stage are taken into account. The problem consists of determining the amount of each product to be produced, stored and sold over the given time horizon, the composition of the production campaign (number of batches and their sizes), the assignment, sequencing and timing of batches, and the number of repetitions of the campaign, for a given plant with known product recipes. The objective is to maximize the net profit fulfilling the minimum and maximum product demands. The proposed model provides a useful tool for solving the optimal campaign planning of installed facilities in reasonable computation time, taking different decisions about the operations management.
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Abbreviations
- \(b, b'\) :
-
Batch
- \(i, i'\) :
-
Product
- j :
-
Stage
- k :
-
Unit
- \(l, l'\) :
-
Slot
- m :
-
Index utilized for the representation in base 2 for the number of repetition of the campaign
- r :
-
Raw material
- I :
-
Products
- \({ IB}_{i}\) :
-
Admitted batches of product i in the campaign composition
- \(K_{j}\) :
-
Nonidentical parallel batch unit that operate out-of-phase in stage j
- \(B_i^{{ max} } \) :
-
Maximum feasible batch size for product i
- \(B_i^{{ min} } \) :
-
Minimum feasible batch size for product i
- \(c_{ii^{\prime }k}\) :
-
Sequence-dependent changeover time between products i and \(i'\) at unit k
- \({ co}_{i}\) :
-
Operating cost coefficient of product i
- \({ DE}_i^L\) :
-
Minimum demand of product i
- \({ DE}_i^U \) :
-
Maximum demand of product i
- \(F_{ri}\) :
-
Raw material conversion factor
- H :
-
Planning horizon
- \({ IMinit}_{r}\) :
-
Inventory of raw material r at the beginning of planning horizon
- \({ IPinit}_{i}\) :
-
Inventory of product i at the beginning of planning horizon
- L :
-
Number of slots postulated for all units of each stage
- \(M_{b}\) :
-
Big-M constant parameters for b = 1, 2, 3, 4
- \({ np}_{i}\) :
-
Selling price of product i
- \(NBC_i^{{ UP}} \) :
-
Maximum number of batches of product i in the campaign
- \({ NC}^{{ UP}} \) :
-
Maximum number of times that the campaign can be repeated over the planning horizon
- \(Q_i^L \) :
-
Lower bound for production level of product i
- \(Q_i^U \) :
-
Upper bound for production level of product i
- \({ SF}_{ij}\) :
-
Size factor of product i in stage j
- \(t_{ik}\) :
-
Processing time of product i in batch unit k
- \(V_{k}\) :
-
Size of unit k
- \(\alpha _{ik}\) :
-
Equipment utilization minimum rate for product i at unit k
- \(\beta _{r}\) :
-
Inventory cost coefficient for raw material r
- \(\delta _{i}\) :
-
Inventory cost coefficient for product i
- \(\kappa _{r}\) :
-
Price of raw material r
- \(\lambda \) :
-
Weighting factor for the variable CT in the objective function
- \(x_{m}\) :
-
Variable utilized for the representation in base 2 for the number of campaign repetition
- \(X_{kl}\) :
-
Indicates if slot l of unit k is employed
- \(Y_{bkl}\) :
-
Indicates if batch b is assigned to slot l of unit k
- \(z_{ib}\) :
-
Indicates if batch b of product i is selected
- \(B_{ib}\) :
-
Size of batch b of product i
- \({ CR}_{r}\) :
-
Amount of raw material r purchased during the planning horizon
- CT :
-
Campaign cycle time
- \({ IM}_{r}\) :
-
Inventory of raw material r at the end of the time horizon
- \({ IP}_{i}\) :
-
Inventory of final product i at the end of the time horizon
- \({ NB}_{it}\) :
-
Total number of batches of product i processed in the time horizon
- \({ NBC}_{i}\) :
-
Number of batches of product i included in the campaign
- NC :
-
Number of times that the campaign is cyclically repeated over the time horizon
- NP :
-
Net profit
- \(Q_{i}\) :
-
Amount of product i to be produced in the planning horizon
- \({ QS}_{i}\) :
-
Amount of product i sold at the end of the planning horizon
- \({ RM}_{r}\) :
-
Amount of raw material r used for producing of all products during the time horizon
- \({ RM}_{ri}\) :
-
Amount of raw material r used to make product i
- \({ TF}_{kl}\) :
-
Final processing time of slot l in unit k for the first campaign cycle
- \({ TI}_{kl}\) :
-
Initial processing time of slot l in unit k for the first campaign cycle
- \(w_{ibm}\) :
-
Variable that represents the product \(B_{ib} \quad x_{m}\)
- \({ww}_{m}\) :
-
Variable that represents the product \({ CT x}_{m}\)
- \({YY}_{blb^{\prime }l^{\prime }k}\) :
-
Continuous variable on interval [0, 1] that indicates if batch b is assigned to slot l of unit k and batch \(b'\) to slot \(l'\) of the same unit
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Acknowledgments
The authors appreciate the financial support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT) from Argentina.
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Fumero, Y., Corsano, G. & Montagna, J.M. An MILP model for planning of batch plants operating in a campaign-mode. Ann Oper Res 258, 415–435 (2017). https://doi.org/10.1007/s10479-016-2301-6
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DOI: https://doi.org/10.1007/s10479-016-2301-6