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A mathematical programming approach including flexible recipes to batch operation rescheduling
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantehdes (Editors) © 2006 PubHshed by Elsevier B.V.
A mathematical programming approach including flexible recipes to batch operation rescheduling Sergio Ferrer-Nadal, Carlos A. Mendez, Moises Graells, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The inherent dynamic nature of industrial environments often needs not only the execution of the required rescheduling actions but also the proper adjustment of the production recipe to the current process conditions. Therefore, the concept of flexible recipe becomes an important part of the rescheduling fi-amework that allows ftiU exploitation of the batch plant intrinsic flexibility. This work introduces a rigorous mathematical approach that incorporates the concept of recipe flexibility to batch operation rescheduling. Keywords: rescheduling, flexible recipe, batch operations, MILP model. 1. Introduction Batch processes have received great attention over the last years because of their higher flexibility compared to continuous processes and the increasing demand for specialty, high added-value chemical and pharmaceutical products. Within this context, the shortterm scheduling deals with the optimal allocation of a set of scarce plant resources over time to manufacture one or more products following a batch recipe. Most of the scheduling approaches assume that batch processes are operated at nominal conditions following predefined fixed production recipes (Mendez and Cerda, 2003 a). However, in many cases a flexible recipe operation may result a more suitable way of incorporating systematic recipe adaptations depending on the actual process conditions. The flexible recipe concept was originally introduced by Rijnsdorp (1991) as a set of adaptable elements that controls the process output. Afterwards, Verwater-Lukszo (1994) presented a flexible recipe approach for the adjustment of control recipes during production which has been applied to several case studies (Sel et al., 1999; Rutten and Bertrand, 1999). One of the first attempts to extent the flexible recipe approach to a plant-wide scheduling problem was carried out by Romero et al. (2001). These authors proposed to integrate a linear flexible recipe model into a multipurpose batch process scheduling model based on an S-graph algorithm. In addition to changes in nominal process conditions, frequent unexpected events can also take place during the normal batch plant operation (equipment failures, late order arrivals, order cancellations and so on). These unforeseen changes may lead the inprogress schedule to become suboptimal or even infeasible. Although rescheduling techniques have a central role in process operations, only a few developments have focused their attention on this challenging problem (Cott and Macchietto, 1988; Mendez and Cerda, 2003b). These rescheduling approaches allow performing certain corrective actions such as partial resource re-allocation, re-sequencing and re-timing assuming a Author to whom correspondence should be addressed: [email protected].
1377
S. Ferrer-Nadal et al
1378
fixed batch production recipe. This work introduces a MILP-based framework to batch operation rescheduHng including the concept of recipe flexibihty. 2. The flexible rescheduling framework The proposed rescheduhng approach is based on the MILP model for single-stage multiproduct reactive scheduling introduced in Mendez and Cerda (2003b). The original model was adapted and extended to address the rescheduling problem of multistage multipurpose batch plants involving different storage policies, non-zero transfer times and flexible recipes. This model relies on the notion of general precedence which reduces the number of binary variables and so the computational effort, as reported in Mendez and Cerda (2003a). Flexible recipe constraints are incorporated in this model to account for the possibility of changing the processing time of some tasks tweaking the rest of the parameters of the product recipe. The cost for modifying these process variables from their optimal economic conditions is taken into account to represent how plant productivity is increased regardless of the cost of altering the nominal plant conditions. Different incidences during the processing horizon can be considered such as insertion of new orders, equipment failures, due dates changes, delay in arrivals, variations in the cost of the raw materials or products, etc. Therefore, when an unexpected event arises, the following groups of tasks can be defined: • Executed tasks (T^^^^) already processed at the rescheduling point which are not included in the rescheduling formulation since they are past events with no influence in the remaining schedule. • Non-directly affected tasks (T"^^) by the unexpected event that are being processed or still have to be processed at the rescheduling point. • Directly affected tasks (T^^) by the unexpected event. They include rejected tasks , running at the rescheduling point that have to be transferred to an alternative unit in order to be reprocessed. Successive stages of these rejected tasks in the processing sequence are also included in this group. • ^New task (T"®^) from late order arrivals to be scheduled. All these different tasks are depicted in Figure 1 which represents a production scenario where two products (dark grey and light grey) are manufactured. rnda
I
Unit U l Unit U 2 Unit U 3 Rescheduling point
Time 'Cxec rpnda rpda ,
Figure 1. Basic representation of task types: r^^^ T'^, T"^ and T"'^. Rather than re-optimizing the sequence of the remaining tasks, only local changes are allowed in order to reduce the impact over the schedule in progress. Partial rescheduling actions linked to each group of tasks are summarized in Table 1.
A Mathematical Programming Approach
1379
3. Case study The proposed rescheduling strategy under the flexible recipe framework will be illustrated by solving a modified version of the case study proposed in Romero et al. (2003). Five products are manufactured in four stages being available alternative process equipment units The flexible recipe for the production of benzyl alcohol (PI) is introduced within this production scenario. Transfer times as a 5 % of the processing time of each processing task have been also considered. Production orders comprise a single batch of one or several products with specific due dates. Batch processing times, available processing units and order due dates are reported in Table 2. Table 1. Allowable rescheduling actions (X: No ; V: Yes) ^ , Task type
.^ .... (Re)Alloc.
.^ s^ (Re)Seq.
.^ xr^. . (Re)Timing
1. Executed tasks, T""^
X
X
X
X
2. Non directly affected tasks, T"^
X
V
V
V
V V
V V
V V
V V
3.Directly affected tasks, T''* 4. New tasks, T™
Recipe adjustment ^ i / ^ ur . i N (only for flexible tasks)
The crossed-Cannizaro reaction for the batchwise production of benzylalcohol (second stage of product PI at unit U2) from the reduction of benzaldehyde has been studied by Keesman (1993). This author proposed a quadratic model to predict the yield of the reaction for a priori known disturbances in the process inputs. Equation 1 shows the linearized model required to be incorporated into the MILP formulation, assuming a small flexibility region around the current operating conditions (see Table 3). y = 4A X i + 4 JC2+95 X3+95 X4
(1)
Additional flexibility has been considered in the first stage of product PI (unit Ul). This is a preheating stage where the temperature is directly proportional to the processing duration and the temperature for the reaction in the next stage. The selected objective function to be minimized is the total cost associated to the order earliness (1 m.u./h), tardiness (5 m.u./h) as well as the corresponding cost for manipulating the process conditions (See Table 3). The proposed approach is implemented within the modelling language GAMS (Brooke et al., 1998) using CPLEX version 7.5. 4. Results This rescheduling approach is applied to the schedule in progress shown in Figure 2 which has to be updated at time 3h in order to face the breakdown of unit 7 with a repairing time of 17 hours. In addition, new batches corresponding to the arrival of late orders must be also inserted in the on-going schedule (see Table 2). Figure 3 shows the proposed reschedule plan without considering the recipe flexibility. Finally, Figure 4 depicts a flexible reschedule that, despite the recipe modification cost, results in a better solution in terms of the proposed objective function. This improvement comes not only from the recipe changes but also from the several modifications of sequencing decisions, which can be easily observed in Figures 3 and 4. It is worth nothing that the
1380
S. Ferrer-Nadal et al
computational effort for updating the current schedule remains very low, even in the case that additional rescheduling actions are considered. Short reaction times are highly important in real industrial environments. Table 4 summarizes the main features of the schedules generated without and with recipe flexibility. In order to improve the customer satisfaction in the flexible production environment, the second stage of the first, third, fifth and sixth batch of product PI (flexible reaction task) reduces their processing time at the expense of increasing the amount of formaldehyde Table 2. Process data for the case study Product PI Stage
Product P5
Unit
PT,h
Unit
PT,h
Unit
PT, h
Unit
Ul
0.5
Ul
1
U7
2
U2
1.5
U8
U2
0.75
U9
2.5
U8
2
U8
2
U3
2
U3
2
U3
1
U9
2
U4
1
U9
2.5
U6
1
U7
2
U4
2
U5
1.5
1.75
(Flexible stage)
4
Product P4
PT,h
U2
3
Product P3
Unit
(Flexible stage)
2
Product P2
U3
2
U4
1.5
U9
1.75
U7
2
U4
0.5
U6
1
U7
0.75
U5
1
PT, h
U5
1
U2
1.25
Ul
1
Due dates (h) Order 1
10
10
9
10
15
Order 2
10
20*
9
22*
15
Order 3
10
15
Order 4
15
20*
Order 5
19*
Order 6
19*
(*) New ordei
Table 3. Recipe parameters, flexibility region and cost for deviation from nominal conditions Flexible process variable
Flexibility Region Lower bound
Deviation cost
Upper bound No deviation is allowed in quality
5y
reaction yield
5xi
reaction temperature
-0.7'^C
0.5 °C
3 m.u.AC
8x2
reaction duration
-0.3 h
0.1 h
2m.u./C
5x3
amount of KOH
-2.7 g
8.5 g
5 m.u./g
6x4
amount of Formaldehyde
-30 g
7.5 g
4 m.u./g
A Mathematical Programming Approach
1381
and KOH. However, in none of the batches of PI, flexibility is exploited for the first stage. This situation arises because flexibility has always a cost and this stage does not suppose a bottleneck in the process and, consequently, no improvement can be achieved. Therefore, this example clearly reflects the high importance of recipe flexibility in the rescheduling process of critical and hard-constrained batch operations. I
Product PI ] Product P2 ] Product P3 Product P4 Product P5 111^^81 Waiting time WMMi Rejected task Figure 2. Schedule in progress at the rescheduling point. U1 U2 tJ3 LJ^
US U6 UT-
US U9
Figure 3. Optimal rescheduling considering fixed production recipe at nominal conditions.
Figure 4. Optimal rescheduling consideringflexibleproduction recipes. 5. Conclusions An efficient MILP-based reschedulingfi*ameworkthat incorporates the recipe flexibility as an additional rescheduling opportunity has been presented. The approach is based on
1382
S. Ferrer-Nadal et al
a continuous-time domain representation and the generalized notion of precedence. The rescheduling strategy considers the on-going production schedule and the current process conditions in order to simultaneously adapt the production recipe to the new scenario and re-optimize the schedule of the batches still to be processed. Different objective functions can be employed to regain feasibility or optimality at minimum cost. Efficiency and applicability of the proposed strategy is demonstrated by successfully solving a complex rescheduling problem in a multipurpose batch plant with reasonable computational effort. The results reported put clearly on evidence the significant benefits of exploiting the inherent flexibility of batch plants. Table 4. Comparison between fixed and flexible rescheduling Fixed reschedule
Flexible reschedule
Tardiness cost, m.u.
131.81
116.44
Earliness cost, m.u.
7.93
11.54
Process conditions change cost, m.u.
0.00
1.95
Objective function, m.u.
139.74
Binary vars., cont. vars., constraints
5, 555, 974
CPU time, s (AMD Athlon 2600 MHz, CPLEX 7.5 )
68
129.93
283,676,1131 230
Acknowledgements Financial support received from the European Community projects (MRTN-CT-2004512233; RFC-CR-04006; INCO-CT-2005-013359) and the Generalitat de Catalunya with the European Social Fund (FI grant) is fully appreciated. References Brooke, A., Kendrick, D., Meeraus, A. & Raman, R. (1998). GAMS - A user's guide. The Scientific Press. San Francisco. Cott B.J. & Macchietto S. (1989). Minimizing the effects of batch process variability using online schedule modification. Computers & Chemical Engineering. 13, 105 - 113. Mendez C.A. & Cerda, J. (2003a). A MINLP continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization & Engineering. 4, 7 - 22. Mendez C.A. & Cerda, J. (2003b). Dynamic scheduling in multiproduct batch plants, Computers & Chemical Engineering. 27, 1247 - 1259. Rijnsdorp, J.E. (1991). Integrated Process Control and Automation, Elsevier, Amsterdam. Romero J., Espuna A., Friedler F. & Puigjaner L. (2003). A new framework for batch process optimization using the flexible recipe. Industrial & engineering chemistry research. 42, 370 379. Rutten, W. G. M. M. & Bertrand, J. W. M. (1998). Balancing stocks, flexible recipe costs and high service level requirements in a batch process industry: A study of a small scale model. European Journal of Operational Research. 110, 626 - 642. Sel, D., Hvala, N., Strmcnik, S., Milanic, S. & Suk-Lubej, B. (1999). Experimental testing of flexible recipe control based on a hybrid model. Control Engineering Practice. 7, 1191-1208. Verwater-Lukszo, Z. (1998). A practical approach to recipe improvement and optimization in the batch processing industry. Computers in Industry. 36, 279 - 300.
A mathematical programming approach including flexible recipes to batch operation rescheduling Sergio Ferrer-Nadal, Carlos A. Mendez, Moises Graells, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The inherent dynamic nature of industrial environments often needs not only the execution of the required rescheduling actions but also the proper adjustment of the production recipe to the current process conditions. Therefore, the concept of flexible recipe becomes an important part of the rescheduling fi-amework that allows ftiU exploitation of the batch plant intrinsic flexibility. This work introduces a rigorous mathematical approach that incorporates the concept of recipe flexibility to batch operation rescheduling. Keywords: rescheduling, flexible recipe, batch operations, MILP model. 1. Introduction Batch processes have received great attention over the last years because of their higher flexibility compared to continuous processes and the increasing demand for specialty, high added-value chemical and pharmaceutical products. Within this context, the shortterm scheduling deals with the optimal allocation of a set of scarce plant resources over time to manufacture one or more products following a batch recipe. Most of the scheduling approaches assume that batch processes are operated at nominal conditions following predefined fixed production recipes (Mendez and Cerda, 2003 a). However, in many cases a flexible recipe operation may result a more suitable way of incorporating systematic recipe adaptations depending on the actual process conditions. The flexible recipe concept was originally introduced by Rijnsdorp (1991) as a set of adaptable elements that controls the process output. Afterwards, Verwater-Lukszo (1994) presented a flexible recipe approach for the adjustment of control recipes during production which has been applied to several case studies (Sel et al., 1999; Rutten and Bertrand, 1999). One of the first attempts to extent the flexible recipe approach to a plant-wide scheduling problem was carried out by Romero et al. (2001). These authors proposed to integrate a linear flexible recipe model into a multipurpose batch process scheduling model based on an S-graph algorithm. In addition to changes in nominal process conditions, frequent unexpected events can also take place during the normal batch plant operation (equipment failures, late order arrivals, order cancellations and so on). These unforeseen changes may lead the inprogress schedule to become suboptimal or even infeasible. Although rescheduling techniques have a central role in process operations, only a few developments have focused their attention on this challenging problem (Cott and Macchietto, 1988; Mendez and Cerda, 2003b). These rescheduling approaches allow performing certain corrective actions such as partial resource re-allocation, re-sequencing and re-timing assuming a Author to whom correspondence should be addressed: [email protected].
1377
S. Ferrer-Nadal et al
1378
fixed batch production recipe. This work introduces a MILP-based framework to batch operation rescheduHng including the concept of recipe flexibihty. 2. The flexible rescheduling framework The proposed rescheduhng approach is based on the MILP model for single-stage multiproduct reactive scheduling introduced in Mendez and Cerda (2003b). The original model was adapted and extended to address the rescheduling problem of multistage multipurpose batch plants involving different storage policies, non-zero transfer times and flexible recipes. This model relies on the notion of general precedence which reduces the number of binary variables and so the computational effort, as reported in Mendez and Cerda (2003a). Flexible recipe constraints are incorporated in this model to account for the possibility of changing the processing time of some tasks tweaking the rest of the parameters of the product recipe. The cost for modifying these process variables from their optimal economic conditions is taken into account to represent how plant productivity is increased regardless of the cost of altering the nominal plant conditions. Different incidences during the processing horizon can be considered such as insertion of new orders, equipment failures, due dates changes, delay in arrivals, variations in the cost of the raw materials or products, etc. Therefore, when an unexpected event arises, the following groups of tasks can be defined: • Executed tasks (T^^^^) already processed at the rescheduling point which are not included in the rescheduling formulation since they are past events with no influence in the remaining schedule. • Non-directly affected tasks (T"^^) by the unexpected event that are being processed or still have to be processed at the rescheduling point. • Directly affected tasks (T^^) by the unexpected event. They include rejected tasks , running at the rescheduling point that have to be transferred to an alternative unit in order to be reprocessed. Successive stages of these rejected tasks in the processing sequence are also included in this group. • ^New task (T"®^) from late order arrivals to be scheduled. All these different tasks are depicted in Figure 1 which represents a production scenario where two products (dark grey and light grey) are manufactured. rnda
I
Unit U l Unit U 2 Unit U 3 Rescheduling point
Time 'Cxec rpnda rpda ,
Figure 1. Basic representation of task types: r^^^ T'^, T"^ and T"'^. Rather than re-optimizing the sequence of the remaining tasks, only local changes are allowed in order to reduce the impact over the schedule in progress. Partial rescheduling actions linked to each group of tasks are summarized in Table 1.
A Mathematical Programming Approach
1379
3. Case study The proposed rescheduling strategy under the flexible recipe framework will be illustrated by solving a modified version of the case study proposed in Romero et al. (2003). Five products are manufactured in four stages being available alternative process equipment units The flexible recipe for the production of benzyl alcohol (PI) is introduced within this production scenario. Transfer times as a 5 % of the processing time of each processing task have been also considered. Production orders comprise a single batch of one or several products with specific due dates. Batch processing times, available processing units and order due dates are reported in Table 2. Table 1. Allowable rescheduling actions (X: No ; V: Yes) ^ , Task type
.^ .... (Re)Alloc.
.^ s^ (Re)Seq.
.^ xr^. . (Re)Timing
1. Executed tasks, T""^
X
X
X
X
2. Non directly affected tasks, T"^
X
V
V
V
V V
V V
V V
V V
3.Directly affected tasks, T''* 4. New tasks, T™
Recipe adjustment ^ i / ^ ur . i N (only for flexible tasks)
The crossed-Cannizaro reaction for the batchwise production of benzylalcohol (second stage of product PI at unit U2) from the reduction of benzaldehyde has been studied by Keesman (1993). This author proposed a quadratic model to predict the yield of the reaction for a priori known disturbances in the process inputs. Equation 1 shows the linearized model required to be incorporated into the MILP formulation, assuming a small flexibility region around the current operating conditions (see Table 3). y = 4A X i + 4 JC2+95 X3+95 X4
(1)
Additional flexibility has been considered in the first stage of product PI (unit Ul). This is a preheating stage where the temperature is directly proportional to the processing duration and the temperature for the reaction in the next stage. The selected objective function to be minimized is the total cost associated to the order earliness (1 m.u./h), tardiness (5 m.u./h) as well as the corresponding cost for manipulating the process conditions (See Table 3). The proposed approach is implemented within the modelling language GAMS (Brooke et al., 1998) using CPLEX version 7.5. 4. Results This rescheduling approach is applied to the schedule in progress shown in Figure 2 which has to be updated at time 3h in order to face the breakdown of unit 7 with a repairing time of 17 hours. In addition, new batches corresponding to the arrival of late orders must be also inserted in the on-going schedule (see Table 2). Figure 3 shows the proposed reschedule plan without considering the recipe flexibility. Finally, Figure 4 depicts a flexible reschedule that, despite the recipe modification cost, results in a better solution in terms of the proposed objective function. This improvement comes not only from the recipe changes but also from the several modifications of sequencing decisions, which can be easily observed in Figures 3 and 4. It is worth nothing that the
1380
S. Ferrer-Nadal et al
computational effort for updating the current schedule remains very low, even in the case that additional rescheduling actions are considered. Short reaction times are highly important in real industrial environments. Table 4 summarizes the main features of the schedules generated without and with recipe flexibility. In order to improve the customer satisfaction in the flexible production environment, the second stage of the first, third, fifth and sixth batch of product PI (flexible reaction task) reduces their processing time at the expense of increasing the amount of formaldehyde Table 2. Process data for the case study Product PI Stage
Product P5
Unit
PT,h
Unit
PT,h
Unit
PT, h
Unit
Ul
0.5
Ul
1
U7
2
U2
1.5
U8
U2
0.75
U9
2.5
U8
2
U8
2
U3
2
U3
2
U3
1
U9
2
U4
1
U9
2.5
U6
1
U7
2
U4
2
U5
1.5
1.75
(Flexible stage)
4
Product P4
PT,h
U2
3
Product P3
Unit
(Flexible stage)
2
Product P2
U3
2
U4
1.5
U9
1.75
U7
2
U4
0.5
U6
1
U7
0.75
U5
1
PT, h
U5
1
U2
1.25
Ul
1
Due dates (h) Order 1
10
10
9
10
15
Order 2
10
20*
9
22*
15
Order 3
10
15
Order 4
15
20*
Order 5
19*
Order 6
19*
(*) New ordei
Table 3. Recipe parameters, flexibility region and cost for deviation from nominal conditions Flexible process variable
Flexibility Region Lower bound
Deviation cost
Upper bound No deviation is allowed in quality
5y
reaction yield
5xi
reaction temperature
-0.7'^C
0.5 °C
3 m.u.AC
8x2
reaction duration
-0.3 h
0.1 h
2m.u./C
5x3
amount of KOH
-2.7 g
8.5 g
5 m.u./g
6x4
amount of Formaldehyde
-30 g
7.5 g
4 m.u./g
A Mathematical Programming Approach
1381
and KOH. However, in none of the batches of PI, flexibility is exploited for the first stage. This situation arises because flexibility has always a cost and this stage does not suppose a bottleneck in the process and, consequently, no improvement can be achieved. Therefore, this example clearly reflects the high importance of recipe flexibility in the rescheduling process of critical and hard-constrained batch operations. I
Product PI ] Product P2 ] Product P3 Product P4 Product P5 111^^81 Waiting time WMMi Rejected task Figure 2. Schedule in progress at the rescheduling point. U1 U2 tJ3 LJ^
US U6 UT-
US U9
Figure 3. Optimal rescheduling considering fixed production recipe at nominal conditions.
Figure 4. Optimal rescheduling consideringflexibleproduction recipes. 5. Conclusions An efficient MILP-based reschedulingfi*ameworkthat incorporates the recipe flexibility as an additional rescheduling opportunity has been presented. The approach is based on
1382
S. Ferrer-Nadal et al
a continuous-time domain representation and the generalized notion of precedence. The rescheduling strategy considers the on-going production schedule and the current process conditions in order to simultaneously adapt the production recipe to the new scenario and re-optimize the schedule of the batches still to be processed. Different objective functions can be employed to regain feasibility or optimality at minimum cost. Efficiency and applicability of the proposed strategy is demonstrated by successfully solving a complex rescheduling problem in a multipurpose batch plant with reasonable computational effort. The results reported put clearly on evidence the significant benefits of exploiting the inherent flexibility of batch plants. Table 4. Comparison between fixed and flexible rescheduling Fixed reschedule
Flexible reschedule
Tardiness cost, m.u.
131.81
116.44
Earliness cost, m.u.
7.93
11.54
Process conditions change cost, m.u.
0.00
1.95
Objective function, m.u.
139.74
Binary vars., cont. vars., constraints
5, 555, 974
CPU time, s (AMD Athlon 2600 MHz, CPLEX 7.5 )
68
129.93
283,676,1131 230
Acknowledgements Financial support received from the European Community projects (MRTN-CT-2004512233; RFC-CR-04006; INCO-CT-2005-013359) and the Generalitat de Catalunya with the European Social Fund (FI grant) is fully appreciated. References Brooke, A., Kendrick, D., Meeraus, A. & Raman, R. (1998). GAMS - A user's guide. The Scientific Press. San Francisco. Cott B.J. & Macchietto S. (1989). Minimizing the effects of batch process variability using online schedule modification. Computers & Chemical Engineering. 13, 105 - 113. Mendez C.A. & Cerda, J. (2003a). A MINLP continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization & Engineering. 4, 7 - 22. Mendez C.A. & Cerda, J. (2003b). Dynamic scheduling in multiproduct batch plants, Computers & Chemical Engineering. 27, 1247 - 1259. Rijnsdorp, J.E. (1991). Integrated Process Control and Automation, Elsevier, Amsterdam. Romero J., Espuna A., Friedler F. & Puigjaner L. (2003). A new framework for batch process optimization using the flexible recipe. Industrial & engineering chemistry research. 42, 370 379. Rutten, W. G. M. M. & Bertrand, J. W. M. (1998). Balancing stocks, flexible recipe costs and high service level requirements in a batch process industry: A study of a small scale model. European Journal of Operational Research. 110, 626 - 642. Sel, D., Hvala, N., Strmcnik, S., Milanic, S. & Suk-Lubej, B. (1999). Experimental testing of flexible recipe control based on a hybrid model. Control Engineering Practice. 7, 1191-1208. Verwater-Lukszo, Z. (1998). A practical approach to recipe improvement and optimization in the batch processing industry. Computers in Industry. 36, 279 - 300.