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Brazilian Journal of Chemical Engineering

Print version ISSN 0104-6632On-line version ISSN 1678-4383

Braz. J. Chem. Eng. vol.19 no.1 São Paulo Jan./Mar. 2002 




A.Alle and J.M.Pinto*
Department of Chemical Engineering, University of Sao Paulo
Av. Prof. Luciano Gualberto, trav. 3, 380, Phone: (55) (11) 3091-2237,
Fax: (55) (11) 3813-2380, 05508-900, São Paulo - SP, Brazil.


(Received: October 6, 2001 ; Accepted: January 4, 2002)



Abstract - The problems of scheduling and optimization of operational conditions in multistage, multiproduct continuous plants with intermediate storage are simultaneously addressed. An MINLP model, called TSPFLOW, which is based on the TSP formulation for product sequencing, is proposed to schedule the operation of such plants. TSPFLOW yields a one-order-of-magnitude CPU time reduction as well as the solution of instances larger than those formerly reported (Pinto and Grossmann, 1994). Secondly, processing rates and yields are introduced as additional optimization variables in order to state the simultaneous problem of scheduling with operational optimization. Results show that trade-offs are very complex and that the development of a straightforward (rule of thumb) method to optimally schedule the operation is less effective than the proposed approach.
: Scheduling, Operation, Mixed-Integer Optimization, Continuous Plants.




Continuous plants have been associated with single-product lines. However, with the increasing need for more flexible processing systems, continuous plants that are able to process various products become more and more frequent. The scheduling of such plants involves trade-offs between storage levels, quantities produced, cycle duration and transition costs. In a production line that has the flexibility to adjust the processing rates of one of its stages, better results could be achieved by combining scheduling with adjustment of the processing rates. A combined optimization approach would introduce synergic effects for more effective plant operation. This comprehensive simultaneous problem would provide additional degrees of freedom and thus better capture the trade-offs between cycle times, inventory levels and setup times.

Batch plants were studied under this approach by Bhatia and Biegler (1996). That work incorporated dynamic processing conditions for products in a multiproduct batch plant in the context of scheduling and equipment design. Only recently, Jain and Grossmann (1998) incorporated process models for the scheduling of continuous processes. The authors considered the case of continuous furnaces in parallel with decaying performance. The conversion yield decayed with time. However, the relation between yields and processing rates was not considered. Georgiadis et al. (2000) and Georgiadis and Papageorgiou (2000) incorporated fouling models into the problem of maintenance scheduling of heat exchanger networks. In those studies, heating rate decayed with time due to fouling, but the trade-off between flowrates and heat transfer coefficients was not included in the scheduling model.

The objectives of this paper are (1) to develop a compact and robust model for scheduling multiproduct multistage plants with intermediate storage and (2) to formulate an optimization model for such a plant by integration of scheduling with optimization of operating conditions.



A number of specified products, which are to be produced in a plant having several stages that are interconnected by intermediate inventory tanks for each product (Figure 1), is given. Each stage consists of one production line that is interconnected with a fixed topology that is similar to the one described by Pinto and Grossmann (1994). Transition times that arise between the processing of two successive products are sequence dependent. Constant demand rates in the form of lower bounds that are to be satisfied are also given. Each stage has a fixed processing capacity for each product. Moreover, the processing rates of the stages may be changed within a range.



The following are the assumptions for modeling the problem:

A1) Every product must be processed in the same sequence throughout all stages (i.e., flowshop plant).

A2) Intermediate inventory depends on the maximum level of material accumulated, as in Buzacott and Ozkarahan (1983).

A3) The inventory cost of final products depends on the average amount of material to be stored, as in Sahinidis and Grossmann (1991).

The problem then consists of determining the following for a cyclic scheduling: (a) sequence of products, (b) length of cycle time, (c) length of production times, (d) amounts of products to be produced, (e) levels of intermediate storage and final product inventories and (f) processing rates for every product in stages.

The objective is the maximization of profit, which includes income from the sales of products, inventory, transition, raw material and operational costs.



The main features of the proposed model in comparison to Pinto and Grossmann (1994) are

1) The use of binary variables zij to represent a transition from the production of product i to that of product j in an approach similar to the classic one used to model the TSP (Traveling Salesman Problem);

2) A more compact representation of intermediate storage due to the reduction in the number of binary variables;

3) The reduction in the number of continuous variables and equality constraints with respect to the aforementioned work;

4) The subtour elimination constraints associated with the TSP are not necessary, as shown in Appendix B.

As the plant is a flowshop due to assumption A1, product ordering can be viewed as a Traveling Salesman Problem (TSP). In this classic combinatorial problem, a salesman must visit a set of cities in a sequence that minimizes overall cost. Analogously, in a flowshop plant a set of products must be made in an order that minimizes transition and other costs. The classic TSP problem formulation uses binary variables to represent the transition from one city to another. In the present formulation, binary variables zij are used to represent a transition from product i to product j of the flowshop plant. The ordering of products is completely defined by assigning values to zij variables. This is the main point by which the proposed model becomes more efficient than that of Pinto and Grossmann (1994).

In the proposed model, binary variables xim indicate whether production of i in consecutive stages is simultaneous. These variables model maximum inventory levels in intermediate tanks, replacing max operators in order to maintain differentiability in the NLP subproblems of the outer approximation algorithm (Duran and Grossmann, 1986), used to solve the MINLP model (see Pinto and Grossmann, 1994 and Appendix A). An important feature of the models presented in this paper is the reduction in the number of these variables from NP2.(M-1) to NP.(M-1).

Scheduling Model: TSP Formulation

The MINLP model for the scheduling problem with fixed operating conditions is as follows:

subject to

Sequencing constraints:

Mass balance constraints:

Transition cost constraints:

Timing constraints:

Demand constraints:

Intermediate inventory constraints:

where (see Appendix A)

In (1), profit is defined as the sum of the sales revenue minus the inventory costs for intermediate products (which is a function of maximum levels), the transition costs and the average inventory costs for final products. The derivation of the average final storage level expression is found in Pinto and Grossmann (1994). The cost coefficients in the objective function are assumed to be scaled so as to yield a profit in annual terms ($/yr). Also note that the inventory cost coefficients, Cinvim and Cinvfi, have different units ($/ton and $/ton.h, respectively).

According to constraint (2), exactly one product j follows product i, and according to constraint (3), exactly one product i precedes product j .

Equation (4) relates the amount produced in each stage and the time assigned to production through the stage rate. Equation (5) is the mass balance of product i between two consecutive stages. Parameter aim is the inverse of the yield.

Constraint (6) represents transition costs incurred in the schedule. Constraint (7) states that in processing two consecutive products, the processing of the second starts exactly after the end of the processing of the first plus the transition time. Constraint (8) states that the first product in the sequence is product number one. This is an arbitrary choice since the scheduling is cyclic. Constraint (9) ensures that a product cannot be processed in any stage prior to production having started in the previous one. Similarly, constraint (10) ensures that production of any product cannot end in any stage before processing in the previous has ended. The length of the cycle time in stage m is equal to the summation of the lengths of all time slots (which include processing and transition times). Constraint (11) states that the cycle time of the plant is the longest of all the stage cycle times.

Constraint (12) states that demand must be satisfied for all products in the plant and that production may be exceeded. It should be noted that in these problems it is not always easy to specify feasible demands. A simple procedure to test the feasibility of specified demands is presented in Pinto and Grossmann (1994).

Inventory levels for intermediates are represented by the constraints in (13) that are derived in Appendix A. Variable Invim denotes the difference between the total amount produced and the maximum inventory level. Constraints (13) to (15) have this form in order to guarantee their differentiability and linearity. One could represent the inventory levels by the following form:

where I1im and I2im are respectively the first and the second discontinuity points at the intermediate inventory level (see Figure A1 in Appendix A).

However constraints (17) to (20) would be neither linear nor differentiable since they involve min and max operators.

In order to reduce the number of continuous variables and constraints, and therefore reduce the computational effort of the NLP subproblem, a few replacements can be made. Variable Wim in equation (4) can be substituted into equation (5) , yielding

As this applies to all stages, it yields

or from M to 1:

We can avoid the use of all Tpim variables except one by using equation (23). We choose every Tpim as a function of TpiM, so

Defining the parameter rim as follows:

Equation (24) yields

Therefore rim is constant since giM, g im and aim are constant. But if rates and yields are introduced into the optimization variable set, rim may not be constant.

With the aid of parameter rim, we can eliminate the Wim variables from the model. Variable Imaxim can be substituted directly into the model. Constraints (4), (5) and (13) may be removed. This yields a more compact model, referred to as TSPFLOW, which is as follows:

subject to

Sequencing constraints (2) and (3)
Transition cost constraint (6)
Timing constraints (8) to (10)

And new timing constraints:

Demand constraint (12)

Intermediate inventory constraints (15)

And new intermediate inventory constraints:


All parameters in (32) to (34) simplify the model in the sense of reducing the number of continuous variables. The TSPFLOW model saves continuous and binary variables, compared to the previous formulation of Pinto and Grossmann (1994). The TSPFLOW model reduces from an O(NP3) order of complexity to O(NP.M) with respect to continuous variables and from O(NP2.M) to O(NP2) with respect to binary variables (see Table 1).



Figure 2 and Table 2 illustrate the evolution of the number of variables in the model as a function of problem size (products and stages).





The most significant reduction is in the number of binary variables, since it reflects the number of nodes to be enumerated in the solution tree with the Branch and Bound algorithm. The algorithm will probably converge more easily and faster to the optimum in a smaller tree.

Some valid constraints may be added in order to provide better bounds to the model:

Constraint (35) requires that production of all products must begin no earlier than the end of production of product number one, which is conventionally assigned to the first position in the cycle. Constraint (36) means that Imaxim, the maximum quantity stored as stated in (13), must be nonnegative, and constraint (37) means that at the end of operation of stage m, the storage will be nonnegative even if production in stage m+1 has already begun.

The model given by (27) to (37) corresponds to an MINLP problem. Note that the only nonlinearities present in the model are in the objective function (due to cycle time Tc and TpiM). The model is linearly constrained and the feasible region is convex.

An important issue that may arise with the proposed model is with respect to the completeness of the sequencing constraints. A TSP formulation usually includes the so-called subcycle elimination constraints given as follows:

where I is the set of all products i.

Constraint (38) prevents generation of subcycles in the solution, i.e., two or more closed sequences instead of just one sequence containing all products (cities). For instance, in a five-product problem solution A-B-C-A and D-E-D is feasible if this constraint is not included. Note that yields 2NP-2 constraints. Miller et al. (1960) show that may be replaced by a more compact set of constraints as follows:

where ui and uj are arbitrary real numbers. This form gives rise to NP2-NP constraints. However, in the present formulation neither (38) nor (39) is required due to the existence of timing constraints that block all subcycles. The proof is given in Appendix B.

Introduction of Operational Variables

In a flowshop plant, it may be possible to adjust the processing rates of one of the stages within a range. The faster the unit runs, the lower its yield due to the shortening of residence times. Moreover, operational costs may increase. For instance, the heat transfer requirement could increase to assure stable temperature conditions. On the other hand, the unit would be free in a shorter time, and therefore able to process more profitable products.

The introduction of variable processing rates aims to incorporate this trade-off into the scheduling model. For the sake of simplicity, we assume that only the rate of the first stage can be changed. The relation between rate and yield is assumed to follow an exponential decaying function (40), as illustrated in Figure 3.



Raw material consumption (Fi) is incorporated into the problem since the change in the process yield of the first stage implies a variation in the demand of the plant for raw material. A negative term, ciFi/Tc, must be incorporated into the objective function where ci ($/kg) is the unit price of raw material.

The operational cost of a product i in the first stage is assumed to be directly proportional to the processing rate, gi1 (kg.h-1), and to the total amount of material that is processed in the first stage, i.e., the total amount of raw material, Fi (kg), used to produce the final product i during the cycle. Therefore, the operational cost of product i may be calculated as in (41).

where the COi coefficient ($ is a parameter that relates the processing rates and the raw material cost to the operational cost.

The objective function must be rewritten in order to reflect the new trade-offs involved. Now, besides the terms already present in (1), raw material costs and operational costs must be included. The objective function becomes as follows:

The objective function is first stated in in (42) terms of the original variables in order to make it more understandable (see eq (1)). Further variables are substituted to achieve a computationally more efficient format. After convenient replacements, the reformulated objective function yields


Essentially, the constraints of the new formulation, called TSPFLOP, are the same as those of the TSPFLOW. A few adaptations are made due to the fact that operating conditions for this stage are no longer constant parameters. Since gi1 is now a variable, related parameters di1, ri1 and Dri1 are also variables in the new optimization problem. All constraints where these new variables occur must be modified. However, ri1, Dri1 and even ai1 can be eliminated from the new formulation. The modified constraints are


Constraint (46), which states the relationship between the start times of consecutive products for stage one, and constraint (47), which is identical to (28) but valid only for the remaining stages, replace constraint (28).

Analogously, pairs (48)-(49), (50)-(51) and (52)-(53) in TSPFLOP play the roles of constraints (29),(30) and (31) in TSPLOW, respectively.

As previously stated in eqs. (54) and (55), di1 are now variables. In order to keep the model differentiable (since they involve a max operator), these new variables can no longer be defined according to equation (16). Now di1 is defined by constraints (54) and (55). Note that eq. (40) is not incorporated into the model and the inverse of yield ai1 is substituted directly by the function of processing rate gi1 expressed by eq.(40). Constraint (56) provides bounds within which gi1 is allowed to vary. Note that constraints (54) to (56) comprise a set of new constraints, introduced in order to consider the problem of operational optimization in the scheduling model. Therefore this set will be referred to as operational constraints.

Note that eq.(40) is not incorporated into the model and the inverse of yield ai1 is substituted directly by the function of processing rate gi1 expressed by eq.(40) . The introduction of parameter ri allowed elimination of variables ri1 and Dri1. Therefore, only variables gi1 and di1 were required in TSPFLOP, and this model differs from TSPFLOW by only 2.NP continuous variables.

Model TSPFLOP can be summarized as follows:

Max (43)

subject to

Sequencing constraints (2) and (3)
Transition cost constraint (6)
Timing constraints (8), (9) and (46) to (51)
Demand constraints (12)
Inventory constraints (15), (52) and (53)
And operating constraints (54) to (56)

Model TSPFLOP corresponds to an MINLP problem. Note that nonlinearities are present in the objective function (due to the cycle time Tc and the last term) and in constraints (46), (48), (50) and (52), due to variable processing rate gi1. Because of the terms di1.TpiM/gi1, di1.Tsi1 and di1.Tsi2 in (52), the feasible region becomes not only nonlinear but also nonconvex.

It is possible to linearize the feasible region by the discretization of variables gi1. This may be accomplished by introducing a new set of binary variables (yil), a new set of continuous variables (Tinvil and Tpil ) and a new set of constraints.

In order to do so, first we define

where NR is the number of discretization points. Equation (58) implies that each stage rate level, gi1l, corresponds to one yield level ai1l.

Binary variables yil are then introduced to indicate which discrete operation point must be chosen:

Equation (59) implies that just one operational point (giql, aiql) must be chosen. In order to maintain model linearity in the constraints, variables Tpi1l and Tinvi1l are introduced in the following manner:


Since gi1l, ai2 and gi2 are constant, di1l is also a constant parameter from (68).

The feeding of raw material in the first stage is also affected by the process yield due to the mass balance equation

Variables Imaxi1 and Fi may be directly substituted into the objective function that yields


The linearization presented for TSPFLOP will be called TSPFLOPLIN. This linearly constrained problem is posed as

 Max (70)

subject to

Sequencing constraints (2) and (3)
Operating point constraint (59)
Transition cost constraint (6)
Timing constraints (8), (9), (47), (51), (60), and (61)
Timing constraints for stage one (63) and (64)
Demand constraints (12)
Inventory constraints for stage one (62), (66) and (67).

Inventory constraints ( 53) and

Note that constraint (73) replaces constraint (15) that is present in the TSPFLOW and TSPFLOP models since variable Invi1 is not defined in TSPFLOPLIN. Therefore, these constraints are now defined only for stages 2 to M-1.

Unlike TSPFLOP, continuous variables gim and dim are not present in TSPFLOPLIN. On the other hand, binary variables yil and continuous variables Tinvi1l and Tpi1l are introduced into the linearized formulation. Table 3 shows a comparison in terms of the number of variables required to formulate the simultaneous problem of scheduling and operational optimization by TSPFLOP, TSPFLOPLIN and Pinto and Grossmann (1994). Only the first-stage rates and yields were allowed to vary.



TSPFLOP has the lowest number of binary and continuous variables of the models. TSPFLOPLIN has NP.L more binary variables than TSPFLOP. Depending on the number of discretization points, the number of discretization variables in TSPFLOPLIN may be even larger than that in the model of Pinto and Grossmann (1994).

Simplicity was the only reason to derive TLOP and TSPFLOPLIN, based on the assumption that stage one was the only stage with variable processing rates. If another stage, m', were chosen in the place of stage one, the number of modifications in TSPFLOW would be larger, because every parameter rim of stages prior to m' would be a function of the variable yield of this stage. Nevertheless, TSPFLOP and TSPFLOPLIN may be extended to any stage other than stage one for unfixed operating conditions, though this would demand more adaptations that are analogous to the derivation presented in this section.



The GAMS modeling system (Brooke et al., 1998) was used to implement all the models presented. The solver used in all cases was DICOPT-2 (Vecchietti and Grossmann, 2000), which uses OA/AP/ER ("outer approximation/augmented penalty/equality relaxation") (Duran and Grossmann, 1986; Viswanathan and Grossmann, 1990). The NLP subproblems were solved with the CONOPT2 solver (Drud, 1995) and the MILP subproblems with the CPLEX 6.6 solver (Ilog Co., 1999). The computational tests were performed in a Pentium III 550 MHz machine.

Fixed Operating Conditions

First, the results for fixed operating conditions are shown. The new TSPFLOW formulation is compared in terms of maximum profit and CPU time with the formulation proposed by Pinto and Grossmann (1994), from which data are taken (see Table 4).



It is worth noting that the above-mentioned solutions are not guaranteed to be global optima since the objective function is nonconvex and the OA/AP/ER method only guarantees optimality when the relaxed MINLP is convex. Strategies for achieving global optimum in these nonconvex relaxed problems is an interesting subject for future development.

Table 4 shows that TSPFLOW demands much less computational effort. For instance in the problem of a plant with five products and two stages, the effort is 120 less. Moreover, the new model is shown to be more robust in the sense that it provides better solutions and efficiently solves problems of larger sizes. The largest problem solved involved a four-stage, ten-product plant and contained 120 discrete variables and 102 continuous variables.

Variable Operating Conditions

The TSPFLOP and TSPFLOPLIN (NR = 5) models are compared (Table 5). Results show that the nonlinear constraints are more effective than the linear discrete ones from both optimality and computational effort points of view. Data for the problems are the same as those of Pinto and Grossmann (1994) except for operation rates, costs and yields in stage 1, which are shown in Appendix C.



TSPFLOP could solve up to problem number 8, while TSPFLOPLIN solved only the first three problems. In problem 3, the CPU time required by TSPFLOP is one order-of-magnitude less that required by TSPFLOPLIN. Moreover, TSPFLOP yielded a better value for the objective function in problems 1 and 3, while TSPFLOPLIN found a better solution only for problem 2. Therefore, the TSPFLOP model showed a better performance than TSPFLOPLIN in the set of problems tested. Nevertheless neither model is guaranteed to provide global optima. This is the reason that TSPFLOPLIN found a better optimum than TSPFLOP in problem 2, although the former model is just a linear approximation of the latter. Solutions for the linearized TSPFLOPLIN model are not guaranteed to be global optimal because of two factors: first, the nonconvexity of the objective function and, second, the discretization of operating variables. On the other hand, despite maintaining operating rates as continuous variables, TSPFLOP is not only nonlinear but also nonconvex and therefore a local optimum rather than a global optimum may be found by the proposed solution method. That certainly happened in the case with four products and two stages.

It must be pointed out that the number of discretization points has a strong influence on CPU time. The distribution of the discrete points along the interval of processing rates may also affect CPU time because the dependence of yield on the processing rates is exponential. In this case, yield varies strongly with the processing rate at the beginning of the range considered and slightly with that at its end, which makes the uniform distribution of points a poor approximation. However, the number of points used in the discretization process was relatively small (only five points). Despite the low level of discretization, TSPFLOPLIN was unable to solve problems larger than problem 4. This indicates that the linearized model would hardly be efficient, even with a very good selection of discrete points.

Sensitivity Analysis: Influence of Processing Rates on the Optimum Schedule

A question that should be asked is whether the impact of the simultaneous optimization of operating conditions on scheduling is large enough to justify this approach. In order to assess this, a sensitivity analysis was made to show how the choice of processing rates may affect the profit value. The plant with two stages, processing three products in problem 1 was chosen. First, the values for the processing rates of all products except the most profitable one (product 3) were fixed at the optimum value. Then, the problem of scheduling the plant was solved for several fixed values of the processing rate of product 3, which varied around its optimal value of 1.076. Results are shown in Figure 4.



As can be seen in Figure 4, the value of the processing rate of the most profitable product may have a strong impact on optimum scheduling. A bad choice of processing rate may lead to a significantly smaller profit for the plant. For instance, if a processing rate of 0.85 ton/h is chosen for product, the optimum schedule yields a value of 164.2 $/h, that is 14% less than the profit achieved with the simultaneous approach (190.7 $/h for a processing rate of 1.07 ton/h). If a processing rate above the optimum value is chosen, then the impact is even larger. If the processing rate of 1.25 ton/h is chosen, the profit would be 113.7 $/h, therefore 40% less than the optimum one.

Case Study: TSPFLOP vs. Rules of Thumb

This example aims to show the potential gains of a simultaneous scheduling and operation optimization over rules of thumb for a multiproduct continuous plant. A plant that processes five products (A-E) in three stages with variable processing rates in the first stage is scheduled in two ways: cases I and II. The typical cost distribution in this plant is illustrated in Figure 5. Plant transition times, costs and the parameters in eq. (40) were randomly generated.



In case I, the schedule was obtained heuristically. First, the products were sequenced in order to reduce overall transition time in the bottleneck (stage 2). Then, the processing rates of the first stage for all products were set equal to those in the second stage in order to reduce intermediate inventory. In case II, the TSPFLOP model was used to simultaneously schedule and optimize the plant. Table 6 and Figure 6 show the main results for both cases.





In Table 7 note that rates for case I are set according to the rule of thumb while rates for case II are optimized by TSPFLOP. Figure 5 clearly shows that intermediate inventory after the first stage is smaller for case I than it is for case II. Results show that overall transition time in the bottleneck for case I is 32 h vs. 36 h for case II and that overall inventory costs in I are 30% less than those in II. In fact, the heuristic was able to reduce intermediate inventory levels. However, case II is more profitable (5%) than case I. That occurs mainly for two reasons (1) TSPFLOP selects more compatible rates and yields, which results in more rational use of raw material (an almost 20% overall reduction in operational and raw material costs) and (2) case II shows gains due to the enlargement of the production scale (a cycle time 50% larger than that in I), thus lowering the impact of transition costs in the cycle.




The TSPLOW model for scheduling of multiproduct, cyclic continuous plants with intermediate storage and fixed operational conditions has been shown to be faster and able to solve larger problems than the model of Pinto and Grossmann (1994). Processing rates and yields were introduced as additional optimization variables in order to state the simultaneous problems of scheduling and operational optimization. This new problem was formulated in the TSPFLOP model with nonconvex constraints and a continuous interval for optimization of processing rates and in the TSPFLOPLIN model with linearized constraints and discrete values for processing rates. Results show that nonlinear restrictions in TSPFLOP are more effective than linear discrete ones in TSPLOPLIN from both the optimality and the computational effort points of view.




Products   i, j = 1,..., NP
Stages m = 1,..., M
Discretization  l = 1,…,NR

Continuous Variables

Cti total transition cost associated with the processing of product i
Imaxim maximum inventory of product i after stage m

difference between amount produced and maximum inventory level of product i between stages m and m+1

Tc cycle time
Teim end time of processing product i in stage m
Tpim processing time of product i in stage m

start time of processing product i in stage m

Wim amount of product i produced in stage m

Binary Variables

xim denotes whether the production of i in stages m and m+1 occurs simultaneously
zij  denotes whether product i is immediately preceded by j



cost coefficient for inventory of final product i


cost coefficient for inventory of product i in stage m


cost of transition between product i and product j


minimum rate demand for product i

pi price of product i
upper bound of inventory for product i in stage m

upper bound of processing time for product i


transition time from product i to product j in stage m


processing rate of product i in stage m


inverse of the yield of product i in stage m



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Since inventory in a cyclic process shows a cyclic behavior, it must not accumulate. The intermediate inventory of product i grows whenever processing of i in stage m begins and decreases when stage m+1 begins to consume i. Therefore it has a maximum level in the interval between the start and the end of production in stage m. As the level is a linear function of time for a fixed processing rate, Teim or Tsim+1 may be points of maximum inventory. Figure A1 illustrates time profiles for intermediate inventories that may occur.



The instant at which the inventory level of product i reaches a maximum depends on the relative position of final processing time in stage m (Teim) and on the start of processing time in the next stage (Tsim+1). It also depends on the relation between production rates gim and consumption rates aim+1.gim+1. If Tsim+1 is greater than or equal to Teim, then consecutive stages m and m+1 do not operate simultaneously and both points Teim and Tsim+1 have the same level that is maximum, as illustrated in Figure A1, case A. Otherwise, if Tsim+1 is less than or equal to Teim, consecutive stages operate simultaneously and the location of the maximum inventory depends on processing rates. In this case, if stage m produces i faster than stage m+1 consumes it, the maximum level is located at the end time of stage m. This is illustrated in Figure A1, case B. On the other hand, if stage m+1 consumes i faster than stage m replaces it, the maximum level is placed at the start time of stage m+1, as shown in Figure A1, case C. From the cases presented in Fig. A1, the derivation of a general expression for maximum inventory level can be done as follows:



Equation (A.3) yields

Cases B and C can be reduced to a single case. We just need to introduce parameter bim as follows:

Three cases are thereby reduced to only two.



From , (a.7),


Equation (A.8) only holds for Tsim+1£ Teim, i.e., DTim = Teim – Tsim+1 ³ 0. Otherwise (A.6) holds because this situation is the same as that of case A'. Since  gim > 0 and aim > 0, the value of dim is always positive. Therefore, equations (A.6) and (A.8)  may be written as a single equation as follows:

If  DTim is positive, product dim DTim  is also positive, thus (A.11) yields (A.8). Otherwise (A.11) yields (A.6).

Raman and Grossmann (1991) define f = max {0,f (x)} with the aid of binary variable x as follows:

Therefore, Invim =  max {0, (gim  bim ) DTim } may be written with the aid of binary variables xim as



DTim  = Teim – Tsim+1 = Tsim + TPim – Tsim+1,




Definition 1: A subcycle is a subset Ic of the complete product set I, Ic Ì I, which contains the minimum number of products such that every product in the subset is preceded and succeeded by only one product in the same subset. Thus,

Lemma 1: If equations (B.1) and (B.2) hold for one or more subsets Ic Ì I, then at least one subcycle exists in I.

Proof: If equations (B.1) and (B.2) hold for one or more subsets Ic Ì I, then there is at least one subset Ic(k) Ì Ic where the number of products satisfying (B.1) and (B.2) are minimum. Therefore, subset Ic(k) is a subcycle, according to definition 1 and at least one subcycle exists in I.

Lemma 2: Let I be the set of all products. When zij are solutions of the set of equations (2) and (3):

and equations (B.1)  and (BV.2) are satisfied for a subcycle Ic, Ic Ì I, then its complementary set Ic', I = Ic È Ic', contains at least one subcycle.

Proof: From the assumptions above, (B.1) and (B.2) hold for Ic and (2) and (3) hold for I. Thus, combining (2) and (3) with (B.1) and (B.2) yields

Substituting from (B.1) and (B.2) yields



But from

Substituting  (B.10) into  (B.8) and (B.11)into (B.9),

From Lemma 1, if (B.12) and (B.13) hold, then the subset Ic' = {i : i Î I and i Ï Ic} contains at least one subcycle. Thus, if Ic is a subcycle, then its complementary set Ic' contains at least one subcycle.

Theorem: Constraint

in conjunction with equations (2) and (3) blocks all subcycles.

Proof: If the solution of the sequencing problem has subcycles then it has more than one subcycle (Lemma 2). Operator [r] = "in position r of the subcycle" is introduced in order to facilitate the demonstration because it permits that any set be written as an ordered set in a straightforward manner irrespective of product denominations. Consider a subcycle Ic = {[1],…,[k]} of I where product 1 Ï Ic. From (7), it yields

On the other hand, start times in Ic are

As product [1] is different from 1, its start time is also given by equation (7). As [1] must start after the completion of the last product [k] in the subcycle, then

Summing up the set of equations represented by (B.16) to (B.19) yields

In the plant every r[j]m must be strictly positive (because every machine is able to process every product) and Tp[j]M and t[i][j] must be nonnegative numbers because they represent time periods of unit occupation. Therefore a necessary condition for (20) to be true is that Tp[j]M = 0 for every j. In this case, a contradiction arises because there would not be any subtour in production since none of the products in subset Ic would be produced. In the case that at least one Tp[j]M is different from zero, (B.20) cannot be satisfied. Therefore, if constraint (7) together with (2) and (3) are satisfied, no subcycle can exist in a subset not containing product 1. From Lemma 2, a subcycle containing product 1 implies the existence of at least one subcycle in its complementary set that does not contain product 1. But that may not happen if constraints (7) and (8) in conjunction with equations (2) and (3) are satisfied, as proved previously. Then no subcycle containing product one can exist. Therefore if constraints (7) and (8) in conjunction with equations (2) and (3) are satisfied, no subcycle at all can exist.




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