Ind. Eng. Chem. Res. 1994,33, 1930-1942
1930
Novel Optimization Approach of Stochastic Planning Models Marianthi G. Ierapetritou and Efstratios N. Pistikopoulos' Centre for Process Systems Engineering, Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, London, SW7 ZBY, U.K. In this paper the problem of planning under uncertainty is addressed. Short term production planning with a time horizon of a few weeks or months and long-range planning including capacity expansion options are considered. Based on the postulation of general probability distribution functions describing process uncertainty, a two-stage stochastic programming formulation is developed where the objective is to determine an optimal plan (i.e., process utilization levels, purchases and sales of materials) and/or an optimal capacity expansion policy that maximize an expected profit. A decomposition-based optimization approach is proposed, where planning decisions are taken by coupling economic optimality and plan feasibility without requiring an "a priori" discretization of the uncertainty. The proposed algorithmic procedure features a highly parallel solution structure which can be exploited for computational efficiency. Three example problems are presented to illustrate the steps of the novel planning under uncertainty optimization algorithm. 1.
Introduction
In any production system, many key parameters such as product demands, prices, and availabilities are only partially known; typically there is significant uncertainty regarding their future values. Consequently, in making planning decisions a company must not only consider a "short-term" economiccriterion but also identify and assess the impact on ita business of important uncertainties so as to develop strategies for dealing with them by providing contingency plans to be put into effect as the uncertainties are revealed. However, the proper treatment of uncertainty in planning decision making transforms the representation of the problem from a pure deterministic model to a stochastic programming model whose solution typically requires the application of specialized decomposition techniques and advanced computer technologies (for example, parallel processors). Solution approachesdealing with uncertainties in model parameters have broadly proceeded along two directions: deterministic methods and probabilistic programming techniques. The former includes (a) the scenario analysis approach, which is characterized by discretization over the parameter space (Brauers and Weber, 1988) and (b) the use of multiperiod models, which is characterized by discretization over the time horizon (Beale et al., 1980; Grossmann et al., 1983; Sahinidis et al., 1989). Probabilistic programming analysis has been mainly based on stochastic programming models with recourse (Birge, 1982, 1985;Wallace, 1987;Birge and Wets, 1989). Bloom (1983) and Bienstock and Shapiro (1988) described resource acquisition problems faced by an electric utility company; a Benders decomposition method was applied to solve the proposed two-stage programming with recourse model. For electric utility planning problems, Borison et al. (1984) presented a stochastic dynamic programming model for the determination of optimal purchase policies of generating technologies in the face of uncertainty; Modiano (1987) developed a stochastic programming with recourse model to analyze the impact of demand uncertainties on capacity expansion policies. More recently, Dantzig (1989) proposed a Benders decomposition algorithm to solve the twostage stochastic programming formulation of the resource planning problem for large-scale electric power systems under uncertainty. A common feature of the above ~~
~
* Author to whom correspondence should be addressed.
approaches is that they are based on an "a priori" and somewhat arbitrary discretization ("scenarios") of the uncertainty involved; consequently, even feasibility of the consecutive planning decisions cannot in general be ensured. Friedman and Reklaitis (1975) and Shimizu (1989)suggested an approach for incorporating flexibility in a system where, by allowing for possible future additive corrections on the current decisions, the system can be optimized by applying an appropriate cost-for-correction in the objective function. For short-/medium-termlinear planning models where capacity expansion policies are not taken into account, Ierapetritou et al. (1994) proposed an algorithm for two-stage stochastic linear planning models based on the solution of multiparametric linear programs; this however may become a nontrivial task especially as the number of first-stage decisions and uncertain parameters increases. The problem of capacity expansionsin planning models has also received a lot of attention in the literature (Himmelblau and Bickel, 1980; Luss, 1982; Shimizu and Takamatau, 1985;Jimenez and Rudd, 1987). Sahinidis et al. (1989) presented a multiperiod planning model for the optimal selection of the capacity expansion and/or shutdown policies, where forecasts for prices and demands of chemicals are taken into account through the examination of different scenarios. A different approach has been followed by Davis et al. (1986), who formulated the mathematical model of the capacity expansion problem considering the demand as a random point process and studied the optimization of capacity expansion by means of stochastic control theory. The objective of this paper is to propose a new planning methodology to deal with the production and capacity planning problems involving stochasticparameters. Based on a two-stage programming formulation, the basic idea is to perform the plan's expected economic evaluation while ensuring future feasibility via a suitable decomposition schemewithout requiring the multiparametric solution of linear problems. The proposed approach is based on a modified generalized Benders decomposition algorithm. The primal step involves the evaluation of the multiple integral of the expected profit within the feasible region of the plan through a Gaussian quadrature formula with the corresponding quadrature points simultaneously obtained as a result of the optimization procedure. In the master problem a valid overestimator function is constructed by properly utilizing the dual information of the
0888-5885/94/2633-1930$04.50/0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1931
Figure 1. Chemical complex.
primal step; its solution results in a new potentially optimal plan to be examined in the next iteration. The paper is organized as follows. In section 2, the twostage programming planning model is introduced; the main difficulties regarding its solution are underlined motivating the basic ideas of the proposed approach. The key building blocks of the proposed algorithmic procedure are discussed in detail in section 3. Finally, three example problems are presented in section 4 to illustrate the basic steps of the proposed approach and to provide more insight on the main features of the algorithm. 2. Two-State Stochastic Formulation
The planning problem to be considered in this paper can be stated as follows: A network of M continuous processes and N chemicals is given (see Figure 1). Also given are the prices, demands, and availabilities of chemicals as well as investment and operating cost data over a time period (first stage), whereas their exact values are not known for the future (second stage); instead they are considered uncertain, described by continuous distribution functions. The problem then consists of determining (i) production profiles, (ii) sales and purchases of chemicals, and/or (iii) capacity expansionsfor the existing processes over each stage which will maximize an expected profit over the two stagesby also ensuring future feasibility. A typical aggregate linear planning model consists of the following set of constraints: (i) production capacity constraints:
xht = J C ~ , , -+ ~ CE,.,
j = 1, ...,M
CEkgj,t ICEj,tICEFaj,t t = 1, ...,T where Xi,, denotes the production capacity of process j 0' = 1, ..., M) during time period t; CEj,t represents the (potential) capacity expansion of process j in t; the occurrence or not of an expansion is denoted by a set of (0,l)variables yj,t; CEkt and CEit are constant lower and upper bounds of the capacity expansion variables CEj,t. (ii) demand constraints: d i e ISi,t Idyt
i = 1, ...,N , t = 1, ..., T
where Si,t denotes the amount of chemical i (i = 1, ...,N) sold in time period t; dft and d; are the lower and upper bounds on the demands of chemical i during period t, respectively.
(iii) availability constraints: L ui,tIPi,, Iuyt i = 1, ...,N , t = 1, ..., T where Pi,t denotes the amount of chemical i purchased in time period t; akt and ug are the lower and upper bounds on the availability of chemical i during period t, respectively. (iv) inventory requirements:
4yI EPL = 323.3, set k = k + 1 and return to step 2. The proposed approachrequires eight iterations to reach the optimal solution within a tolerance of 0.1 for the lower and upper bounds. The optimal solution is the plan that corresponds to the following values of the operating variables (R11, R12, S13,s14,91, P12) = (93.3,36.67,40,20, 37.3,112.67). This plan has an expected profit of 504.2 units within a corresponding feasible region defined by the following bounds: for 81, 40 Idl I60, and for each quadrature point by bounds for 82 as shown in Table 3. The progress of the convergence of the algorithm is shown in Figure 5. Using GAMS/MINOS for the solution of the linear subproblems and the master problem, it requires a total of approximately 254 CPU s on a SPARC 2 workstation (32 CPU s for each iteration with approximately 30.5 CPU s spent on the solution of the subproblems and 1.5 CPU s on the solution of the master problem). Although the parallelization of the solution by itself may not fully resolve the problem of the increased computational requirements (especially when the number of uncertain parameters and the quadrature points increased), one can easily envisage that for the specific example if the independent optimization taskswere solved on parallel processors (with an almost perfect speed-up factor) then the CPU time needed for each iteration could be reduced by 80%; this issue will become even more apparent in the next example. Example 2. Figure 6 shows a chemical complex involving two processes where a single material is transformed into two products. The notation used in example 1 is also followed here. Then the long-range two-stage
s,3
=0
where = level at which process j is run at stage s I$, ti = level of the inventory of material i at the beginning and at the end of stage s, respectively (with = 0,Gi = S a i = amount of material i sold during stage s Pi1 = amount of material 1 purchases during stage s (ii) limitations on the utilization levels over the two stages: Raj
e
e)
0 IR,, I 2 0 0 IR,, I20
+ CE, I 2 0 + CE,
0 IR,, I 2 0 0 IR,,
0 ICE, I20y, 0 ICE2 I20y,
where CEj = capacity expansion of unit j yj = 0, 1 variables to represent the decision of capacity expansion of unit j, 1 if capacity expansion of unit j takes place, 0 otherwise (iii) limitations on sales due to product demands: 15 IS,,I 3 0 15 ISi3 I25
e,
Is,
s 30
e,
Is ,
s 25
(iv) limitations on purchases due to the availability of raw material: 22 IP,,I 4 0 22 IP,, Ie, (v) maximum and minimum inventory requirements at the end of each stage: O I I 3 2
1938 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 4. Data for Uncertain Parameters for Example 2 mean positive negative uncertain distrib dev dev value Pafunction 81 N(20,2.5) 20 10 10 e2 N(15,2.5) 15 10 10 83 N(40,2.5) 40 10 10
05 EPL= 998.06, the iterative procedure proceeds. Convergence within a tolerance of 0.1 for the lower and upper bounds is achieved after five iterations. The optimal solution is the plan that corresponds to the following values of the operating variables (R11, RIZ,SI,, s 1 3 , PII)= (20,20,15,15,40) and a capacity expansion of process 1 by 6.6 units. This plan has an expected profit of 1189.5 units. The progress of the convergence of the algorithm is shown in Figure 7. Using GAMWMINOS for the solution of the linear subproblems and SCICONIC for the solution of the mixed-integer linear-programming (MILP) master problem, it requires a total of 205 CPU s on a SPARC 2 workstation per iteration with approximately 199 CPU s spent on the solution of the subproblems and 3 CPU s on the solution of the master problem. The increase in the required computation time compared to example 1 is mainly due to the large number of subproblems that have to be solved per iteration, namely 156 optimization subproblems for the case of 5 quadrature points (as compared to 31 for the case of 2 uncertain parameters). It is interesting to note that if only two uncertain parameters are considered, the time spent per iteration is reduced to 20 CPU s. Again however, most of the optimization subproblems are independent and can be solved in parallel to enhance computational performance. For example, after the bounds of el, (e,: e?) are evaluated
M
1-
5oD
Sm
*OD
kdkm
Figure 7. Progress of bounds (example 2). cotta
Figure 8. Refinery input and output schematic. Table 5. Convergence of the Algorithm for Different Starting Points initial plan operational plan capacity expansions (yi, yz, CEi, CEz) (Rii, Riz, Siz,Sia,Pii) (1,0,10,0) (20,20, 15,20,40) (1,1,10,10) (20,20,15,15,40) (0,L 0 , m (20,20, 15,20,40) (20,20, 15,15,40) (O,O,0,O)
comput time CPU s per iter 205 205 205 205
no. of iter 5 6 5 5
Table 6. Data for the Refinery Crudes and Products
gasoline kerosene fuel oil residual processing cost (per thousand bbl)
yield crude crude oil 1 oil 2 0.8 0.44 0.5 0.4 0.2 0.36 0.05 0.1 0.5 1
max capacity 24 15 1
and the quadrature pointsef' are placed within [OF, e:], the five subproblems for the determination of the bounds of 82 can be solved simultaneously and independently. The effect of the consideration of different starting points on the performance of the algorithm is shown in Table 5. Note that since the size of the problem remains unchanged, the CPU time needed per iteration is the same regardless of the initial point used; note also that the number of iterations required for the algorithm to converge from different starting points to the optimal solution does not change significantly. Example 3. This example is a variation of the refinery planning problem considered in Edgar and Himmelblau (1988). Figure 8 is a simplified schematic of feedstocks and products for the refinery (where costs and prices are also given). Two crude oils are available for purchase subject to supply limitations. Four products are produced according to a yield matrix with limits on the production capacities (see Table 6). Both crudes and products can be stored subject to tank inventory limits. Products are available for sales according to market demands which are considered uncertain for gasoline and fuel oil during the second stage following normal distribution functions N(15,1.25) and N(5,1), respectively. The objective is the
Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 1939 Table 7. Bounds for 68 at 91
v &Jqi 2
1 13.31 4.69 7.01
2 14.55 3.45 7.05
for the Initial Plan 3 16.35 3.00 7.12
4 18.15 3.29 7.13
a1
5 19.39 4.04 6.61
wY I
*I.)
DI
P
Table 8. Optimal Profit at the Quadrature Points
-
Y-
101
u91
92
1 2 3 4 5
1 420.31 418.95 467.13 499.05 515.75
2 432.98 444.76 491.24 513.89 525.68
Table 9. Bounds for 68 at a1
e;ll
e eYq1
1 13.31 4.69 7.01
2 14.55 3.45 7.05
3 446.13 477.99 514.51 535.63 540.21
101
4 459.27 498.34 532.57 547.70 554.75
for the Initial Plan 3 4 16.35 18.15 3.04 2.50 7.13 7.12
5 468.25 512.23 540.82 555.40 560.13
5 19.39 4.04 6.61
maximization of the expected value of profit function, defined as the difference between income (from product sales) and cost (operating cost and cost of purchasing), over the two stages. The two-stage planning model consists of 28 inequalities and 36 equalities involving 40 variables. The application of the proposed approach leads to the following results. S t e p 1: An initial plan is selected to purchase 20 and 15 thousand bbl/day of crude oil 1 and 2, respectively, and to utilize 18 and 15 thousand bbl/day of crude oil 1 and 2, respectively, for the first stage; EPL = -03, k = 1, and t = 0.1. S t e p 2 The solution of the feasibility problem (Bl) leads to the following bounds of 81: 13 I81 I 19.7. Considering five quadrature points for 81and solving ( B 2 9 at each of these points, the results shown in Table 7 are obtained. Step 3 Considering five quadrature points for 82 and solving problem (PQ) at each of these points, the results shown in Table 8 are obtained. S t e p 4: The expected profit is evaluated through a Gaussian quadrature formula yielding a value of $939.98 thousand/day; the lower bound is updated, EPL = 939.98. Steps 5a, 5b: After the evaluation of the corrected multipliers the master problem is formulated and solved. A plan is then obtained that corresponds to purchasing and utilizing 20 and 12.5 thousand bbl/day of crude 1 and 2, respectively; an upper bound is found of EP; = $950.93 thousand/day. S t e p 6 Since the stopping criterion is not satisfied, EP; = 950.93 > EPL = 939.98, set k = k + 1 and return to step 2. The proposed approach requires only two iterations to reach the optimal solution within a tolerance of 0.1 for the lower and upper bounds. The optimal solution is the plan that corresponds to purchasing and utilizing 20 and 12.5 thousand bbl/day of crude 1 and 2, respectively. This plan has an expected profit of $948.5thousandlday within a corresponding feasible region defined by the following bounds: for el, 13 I81 I19.7, and for each quadrature point by bounds for 8 2 as shown in Table 9. The progress of the convergence of the algorithm is shown in Figure 9. Using GAMWMINOS for the solution of the linear subproblems and for the linear master problem, it requires a total of approximately 33 CPU s on a SPARC 2 workstation (16.5 CPU s for each iteration
-
--
9omY 1
.--
.--
. e '
I
i
u
U
n Figure 9. Progress of bounds (example 3).
with approximately 15.5 CPU s spent at the solution of the subproblems and 1 CPU s at the solution of the master problem). Note that the overall computational time remains small despite the relative increase of the model size.
5. Summary The problem of selecting an optimal plan by also identifying opportunities for optimal capacity expansion policies of stochastic planning models has been addressed. A two-stage stochastic optimization formulation has been presented with the objective to maximize an expected profit while simultaneously ensuring future plan feasibility. Uncertainty in second-stage demands, prices, and availabilities is considered to be described by a general distribution function. A decomposition-basedalgorithm has been proposed which employs a Gaussian quadrature scheme for the expectedprofit evaluation with quadrature points obtained as part of the optimization procedure. Thus, the need for an "a priori" discretization of the uncertain parameters can be effectively avoided; this also results in obtaining an accurate approximation of the feasible region while all quadrature points are properly placed within it. The proposed algorithm constitutes a theoretical advancement over previously published work, since it enables plan optimization and feasibility to be performed simultaneously. An obvious limitation of the current implementation of the algorithm is ita high computational requirements especially when the number of uncertain parameters is large; however, its highly distributed solution structure offers enough hope for future exploitation.
Acknowledgment The authors gratefully acknowledge financial support from the Commission of European Communities under Grants ERB CHBI CT93 0484 and BRE2 CT92 0355.
Nomenclature AI, Az, As, B2 = constant matrices Bl(8) = matrix that may also involve uncertainty bl, b2(@= current and future demands and availabilities, respectively CE1, CE2 = vectors of first- and second-stage capacity expansion variables, respectively cl, c2(@= current and future costa of sales and purchases, respectively E&) = expectation operator of over 0 EP = expected profit over the two stages Epu = predicted upper bound of expected profit at k iteration (a)
1940 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994
EPL = lower bound of expected profit e, ci = level of the inventory of material i at the beginning and at the end of stage s, respectively J(S) = probability distribution function of the uncertain parameters 8 R = feasible region (xl, yl, y2, CE1, CE2),R = (OlV 0 E R3 x2: second-stage constraints are satisfied) PBi= amount of material i purchased during stage s R8j = level at which process j is run at stage s S8i= amount of material i sold during stage s x1, x2 = vectors of first- and second-stageoperational decision variables, respectively y1,y2 = vectors of binary variablesdenoting first- and secondstage capacity expansion alternatives, respectively wf', w p = weights corresponding to each quadrature point
Greek Symbols q ,a2 = current and future variable-size cost coefficients for the investment cost of capacity expaneions fll, f l 2 = current and future fixed-cost charges for the investment cost of capacity expansions qq1qz = corrected multipliers for the constraints of problems
The KT conditions corresponding to L1are shown below:
(PQ) e = vector of uncertain parameters fly, 8: = upper and lower bounds of
el, respectively
the uncertain parameter
u:l))
= 0 (Al)
Oql = quadrature points in O1-space
e? = upper and lower bounds of the uncertain parameter 02, at eil, respectively e q l q Z = quadrature points in &space b :A = corrected multipliers for the Constraints of problem A,, (B1) A?, A? = corrected multipliers for the constraints of problems (B291) p~ = scalar variable in the master problem
Q,
Appendix. Derivation of Correction Factors For the derivation of correction factors the KuhnTucker (KT) conditions for problem (P2) for fixed plan and capacity expansion policy (XI, y1, y2, CE1, CE2) will (B299, and (PQ). be compared with those of problems (Bl), The Lagrangian of problem (P2) for fixed plan is given by the following equation: L' = EP(q,y,,y2,CE,,CE,)61 Qz
aepl aql 92'1
eki 1- " q z 2
(
2
11)
Similarly, the Lagrangians of problems (Bl),B2q1), and (PQ) are given by the following equations:
ae? aql
1
- pp' = (A3)
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1941
Similarly, by comparing (A5) with (A101 and (A6) with (All), the correction factors C p and Cl@ can be determined such that
Ap1=C F P P ,
= C$4l*hl 2
%
Similarly, for L2, L*l, and L*lqz the following KT conditions are required
ec
891 ff1Q) 2( 1 ) 2
- iq1qzBI( #" ff1@) = 0 1 3 2
(A13
By comparing (Al) with (A81 and (A2) with (A9), the correction factors C c and C e are determined such that A; = CFYR?,
A: = CeLf
Literature Cited Beale, E. M.; Forrest, J. J. H.; Taylor, C. J. Multi-time-period Stochastic Programming. Stochastic Programming; Academic Press: New York, 1980. Bienstock, D.; Shapiro, J. F. Optimizing Resource Acquisition Decisions by Stochastic Programming. Manage. Sci. 1988, 34, Q,
These equations involve the multipliers pyl of the equations defining the quadrature points e? that are not available from the solution of (Bl). However, they can be determined from the solution of (A3) for p y :
215.
Birge,J. R. The Value of the Stochastic Solution in StochasticLinear Programs with Fixed Recourse. Math. Program. 1982,24, 314. Birge, J. R. Aggregation Bounds in Stochastic Linear Programming. Math. Program. 1986,25,31. Birge,J. R.; Wets,R. Sublinear UpperBoundsfor StochasticPrograms with Recourse. Math. Program. 1989,43, 131. Bloom, J. A. Solving an Electricity Generating Capacity Expansion PlanningProblemby GeneralizedBenders Decomposition. Oper. Res. 1983, 31, 84.
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Eng.Chem. Res., Vol. 33, No. 8, 1994
Borison, A. B.; Morris, P. A.; Oren, S. S. A State-of-the World Decomposition Approach to Dynamics and Uncertainty in Electric Utility Generation Expansion Planning. Oper,Res. 1984,32,1062. Brauers, J.; Weber, M. A New Method of Scenario Analysis for Strategic Planning. J. Forecast. 1988, 7, 31-47. Edgar, T. F.; Himmelblau,D. M. Optimization of Chemical Processes; McGraw Hill: New York, 1988. Dantzig, G. B. Decomposition Techniques for Large-Scale Electric Power Systems Planning Under Uncertainty. Ann. Oper. Res. 1989.
Davis, M. H. A.; Dempster, M. H. A.; Sethi, S. P.; Vermes, D. Optimal Capacity Expansion Under Uncertainty. Adv. Appl. B o b . 1986. Friedman, Y.; Reklaitis, G. V. Flexible Solutions to Linear Programs under Uncertainty: Inequality Constraints. AIChE J . 1975,21, 77-83.
Gassmann, H. I. MSLiP A Computer Code for the Multistage Stochastic Linear Programming Problem. Math. Program. 1990, 47, 407.
Geoffrion, A. M. Generalized Benders Decomposition. J . Optim. Theor. Appl. 1972,10, 237. Grossmann, I. E.; Halemane, K. P.; Swaney, R. E. Optimization Strategies for Flexible Chemical Processes. Comput. Chem.Eng. 1983, ?, 439-462.
Himmelblau, D. M.; Bickel, T. C. Optimal Expansion of Hydrodesulfurization Process. Comput. Chem. En#. 1980, 4 , 101. Ierapetritou, M. G.; Pistikopoulos, E. N.; Floudi, C. A. Operational Planning Under Uncertainty. Comput. Chem. Eng. 1994, 18S,
Modiano, E. M. Derived Demand and Capacity Planning Under Uncertainty. Oper. Res. 1987, 35, 185-197. Pistikopoulos, E. N.; Grossmann, 1. E. Optimal Retrofit Design for Improving Process Flexibility in Linear Systems. Comput. Chem. Eng. 1988,12, 719-731. Pistikopoulos, E. N.; Ierapetritou, M. G. A Novel Approach for Optimal Process Design Under Uncertainty. Submitted for publication 1993. Sahinidis, N. V.; Grossmann, I. E.; Fornari, R. E.; Chathrathi, M. Optimization Model for Long-Range Planning in Chemical Industry. Comput. Chem. Eng. 1989,9, 1049. Shmizu, Y, Application of flexibilityanalysisfor compromisesolution in large-scale linear systems. J . Chem. Eng. Jpn. 1989,22,189193.
Shimizu, Y.; Takamatsu, T.Application of mixed-integer linear programming in multiterm expansion planning under multiobjectives. Comput. Chem. Eng. 1985,9, 367. Straub, D. A.; Grossmann, I. E. Evaluation and optimization of stochasticflexibilityin muhiproductbatch plants. Comput.Chem. Eng. 1992,16, 69. Straub, D. A.; Grossmann, I. E. Design Optimization of Stochastic Flexibility. Comput. Chem. Eng. 1993, 17, 339. Wallace,S. W. A piecewise linear upper bound on the network recourse function. Math. Program. 1987, 38, 133. Wellons, H. S.; Reklaitis, G. V. The design of multiproduct batch plants under uncertainty with staged expansion. Comput. Chem. Eng. 1989,13, 115-126.
S259.
Jimenez, A. G.; Rudd, D. F. Use of a recursive mixed-integer programming model to detect an optimal integration sequence for the Mexican petrochemical industry. Comput. Chem. Eng. 1987, 11, 291.
Luss, H. Operations Research and Capacity Expansion Problems: A Survey. Oper. Res. 1982,30,907.
Received for review October 12, 1993 Revised manuscript received May 10, 1994 Accepted May 24, 1994O
Abstract published in Advance ACS Abstracts, July 1,1994.
