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Managing financial risk in scheduling of batch plants
European Symposium on Computer Aided Process Engineering - 13 A. Kraslawski and I. Turunen (Editors) © 2003 Elsevier Science B.V. All rights reserved.
41
Managing Financial Risk in Scheduling of Batch Plants Aima Bonfiir, Jordi Canton^, Miguel Bagajewicz^, Antonio Espuna^ and Luis Puigjaner^'^ (+)DEQ-ETSEIB-UPC. Av. Diagonal 647, P. G, 2° P. (08028) Barcelona, Spain, e-mail [email protected]; (++) University of Oklahoma, School of Chemical Engineering and Materials Science. 100 E. Boyd St., T - 335, Norman, OK 73019. On sabbatical leave at ETSEIB; (#) Corresponding Author
Abstract The short-term scheduling problem of a multiproduct batch plant with uncertain product demands is addressed in this article. The problem is modelled using a two-stage stochastic approach maximising the expected profit and integrating inventory costs and penahies for production shortfalls. Additionally, risk is controlled using a recently developed methodology.
1. Introduction The scheduling problem in the chemical industry has been extensively studied and alternative methodologies and problem statements have been proposed in the literature to address the combinatorial character of this problem (Shah, 1998). However, most of the formulations presented are based on nominal parameter values without considering the uncertain requirements after the operations are planned and scheduled. The uncertainty from a real environment entails a risk that is initially traduced in a cost but may lead to an unfeasible situation. The aim of the present work is to provide a tool to support the decision making of developing a scheduling policy in an uncertain environment while controlling the variability over the possible scenarios. For this purpose, a deterministic MILP formulation is first presented and used as starting point for modelling a two-stage stochastic optimisation approach. To control the variability over the different scenarios, the concept of risk is introduced. The simultaneously reduction of risk and maximisation of profit results in a multiobjective stochastic optimisation problem. Two alternative risk definitions, one related with financial risk introduced by Barbaro and Bagajewicz (2002a,b) and another considering the downside risk defined by Eppen et al. (1989) are considered and the latter is used as an objective. The outcome is a set of parametric solutions corresponding to different levels of risk.
2. Model The scheduling problem of a multiproduct batch plant under uncertain product demand is addressed with the aim to maximise the expected profit. The scheduling policy involves the number of batches to be produced for each product, the detailed sequence and the starting and final times of each operation performed.
42 To derive the proposed models, one production line with fixed assigned equipment units and zero wait transfer policy is assumed. The scheduling is addressed within a time horizon of one month. Finally, inventory costs and penalties for production shortfalls, proportional to the amount of underproduction, are adopted for each product. 2.1. Deterministic scheduling model The deterministic scheduling model is formulated as a MILP problem based on a batch slot concept. With this formulation, the time horizon is viewed as a sequence of batches, each of which will be assigned to one particular product. The proposed mathematical model is shown in equations (1) to (13). It maximises the expected profit (sales inventory costs - production shortfalls) using X(b,p) as sequencing decision variable (see Nomenclature). A small penalty term on the sum of the initial time of all batches is added to force the schedule to finish in the smallest makespan possible. MaxP = ^[F(/?)-i^r(p) - I{p)'CI{p) - Fr{pyPe{p)]-a^Tin{o,b)
(1)
oJ>
P
subject to: H>TMo,b)
Vo,b
(2)
T(o,b) = '^X{b,p)TOP{o,p)
Vo,b
Tfn{p,b) = Tin{o,b) + T(p,b) Tin{o,b +1) > Tfn{p,b)
yo,b
Tfn{o,b) = Tin{o + \,b) ^X(b,p)
\fo,b
(4) (5)
yo,b
(6)
\/b
nip) = Y,^(b,p)
(3)
("7)
(8)
yp
b
(9)
i>^n(p)
Vip) = mm{D(p), n(p) • BSip))
^p
(10)
I(p) = n(p)-BSip)-V(p)
yp
(11)
43 F(p) = Dip)-Vip)
Vp
(12)
2.2. Stochastic model From the formulation presented in the previous subsection and to derive the two-stage stochastic program, the uncertain product demands are modelled by probability distributions and a set of scenarios is generated using sampling techniques. The number of batches to be produced and the corresponding sequence of products are considered as first-stage decisions, since it is assumed that they have to be taken at the scheduling stage before the realisation of the uncertainty. The sales, inventory and production shortfalls are then recomputed on a second-stage for each scenario generated. Therefore, equations (10) - (12) are rewritten to take into account the different scenarios s. Then the maximisation of the expected profit is defined as: EP = Y^ Prob(s) • Pis) -a'Y,
^Ho, b) =
o,b
-T
Probis) 'J^[V(p,s). Prip) - Iip,s) • CI(p) - F(p,s) • Pe(p)] -a'Y,^in(o,b) (1^) o,b
2.3. Risk management The concept of financial risk defined by Barbaro and Bagajewicz (2002a) and its alternative of downside risk (Eppen et al., 1989) have been incorporated into the stochastic model to support the decision making. Downside risk is finally used. 2.3.1. Financial risk Managing financial risk involves the maximisation of: EP = J^[Prob(s)• P(s)]-a• ^7-m(o,6)-^^^[ProfcC*)• p,. • z^sj)] o,b
s
Pis)>n(i)-U'z(s,i)
5
(14)
i
\/s,i
(15)
2.3.2. Downside risk The utilisation of the alternative downside risk concept requires the redefinition of the objective function and constraints as follows.EP = ju 'Y,[P^ob(s) • P(s)]-a • Y,THo,b) -Y^[Prob(s) • S(s)] o,b
s
S(s)>Q-P(s) ^(^)>0
\/s V^
(16)
s
(17) (18)
44
3. Results and Discussions The proposed methodology to incorporate risk management in the framework of twostage optimisation for scheduling of multiproduct batch plants under uncertainty has been applied to a case study involving four production stages and five different products adapted from Petkov and Maranas (1997). A total number of 500 independent scenarios were simulated using Monte Carlo sampling technique from the given normal demand probability distributions. To assess the importance of the stochastic formulation, the deterministic model using the nominal demand values has been first solved and compared with its stochastic counterpart. The results obtained are detailed in Table 1. The schedules are not shown for space reasons, but they are significantly different. In addition, it is noted that the makespan of the deterministic model is shorter, because the model does not generate inventory to hedge from adverse scenarios, as the stochastic one does. One important thing to notice is that the deterministic model predicts a solution that poorly represents the uncertain environment, i.e. the schedule obtained with the nominal parameters may be unrealistic when another demand is ordered. Indeed, although the profit of the deterministically generated schedule is higher than the expected value of the stochastically generated, when the first one is used to face the uncertainty the expected value of profit drops 16%. The stochastic schedule performs better as it is also reflected in the shift to the left of the risk curve (Figure 1). This is one indication of the danger of using deterministic models in the believe that stochastic models will only "refine" the solution. Table L Results of Deterministic and Stochastic Models.
Products n(p) E[Sales] E[Inv.] E[Penalt.] Profit E[Profit] Makespan
Deterministic schedule P2 PI P4 P3 P5 10 10 15 15 8 726 941 7099 3086 1884 74 59 401 214 116 56 123 25 468 168 79875 (value in nominal scenario) 67165 617
Stochastic schedule P4 PI P2 P3 P5 10 10 16 16 8 726 941 7313 3165 1884 74 59 687 355 116 25 56 254 89 123 76275 (value in nominal scenario) 68539 662
The management of risk has been next performed considering several profit targets (one at a time) along with various weight values reflecting different levels of risk exposure. At each profit target, a schedule with its correspondent risk curve is obtained. Risk values of schedules obtained by managing downside risk with a weight value \x = 0.001 are reported in Table 2. As it was expected, the expected profit value of the schedules approaches the maximum value obtained with the stochastic model (SP) as the profit target is incremented. From a comparison with the stochastic schedule it can be observed that reductions of downside risk of 30% and 16%) are attained at target profits Q. = 60,000 and 65,000, respectively. Selected risk curves are depicted in Figure 2.
45
30000
40000
50000
60000 Profit
70000
30000
40000
50000
60000
70000
80000
90000
Profit
Figure 1. Risk curves from the stochastic Figure 2. Selected downside risk curves for schedules with different profit targets. and deterministic schedules.
Table 2. Risk values of selected schedules when optimising at different profit targets fl. Stochastic solution (SP) Schedule Target Q E[Profit] DRisk FRisk (%) DRisk FRisk (%) 2 4.2 50,000 64,310 102 227 C D 55,000 64,442 275 7 516 8.2 E 60,000 67,160 770 13 1,085 15.4 F 65,000 67,160 1,787 30.6 2,140 27.6 G 70,000 67,165 4,040 60.6 4,048 49.6 H 75,000 68,539 7,170 74.6 7,170 74.6
4. Conclusions The treatment of uncertainties in batch process planning and scheduling has been addressed as a two-stage stochastic optimisation problem with the incorporation of financial risk management, thus resulting in a multiobjective stochastic system. The proposed methodology has been applied to the scheduling of a multiproduct batch plant and uncertainty in product demands has been first considered and modelled by probability distributions. A comparison between the deterministic and the stochastic formulations has been performed to assess the importance of the stochastic approach. The schedule obtained with the nominal parameter values will poorly perform in most of the probable scenarios. However, with the stochastic modelling a significant improvement is attained and schedules with a good performance over all the scenarios are obtained. The management of risk has been next introduced. From the results obtained it can be noticed that risk can only be managed at the expense of the expected profit. In addition, despite the flexibility introduced by the uncertain parameters and the inventory, different cost-dependent alternatives are required to be able to manage risk in a meaningful way. A variety of alternative schedules are obtained reflecting different risk exposure poHcies.
46
5. Acknowledgements Financial support received from the Spanish "Ministerio de Educacion Cultura y Deporte" (FPU research grant) and "GeneraHtat de Catalunya" (project GICASA-D), and from the European Community (project GlRD-CT-2000-00318 and project GRDl2000-25172) is thankfrilly appreciated. The support of the ministry of Education of Spain for the sabbatical stay of Dr. Bagajewicz at UPC is also acknowledged.
6. Nomenclature Sets: b: i: o: Ps: Parameters: BS(p): CI(p): D(p): H: L: Pe(p): Pr(p): Prob(s): TOP(o,p): U: Q(i): ^(i):
Batches Profit targets Operations Products Scenarios Batch size Inventory cost Demand Horizon time Maximum n** of batches Production shortfall cost Sales benefit Probability Operation time Big value Profit target Financial risk upper bound
P(i): 5(s): l^a: Variables: EP: F(p)/F(p,s): I(p)/I(p,s): MS: n(p): P(s): T(o,b): Tfn(o,b): Tin(o,b): V(p)/V(p,s): X(b,p): z(s,i):
Weight value Downside risk Weight value Low value Expected profit value Shortfall Inventory Makespan Number of batches Profit value Processing time Final time Initial time Sales of product/^ Batch b assigned to prod. p (Binary variable) Binary variable
7. References Barbaro, A. and Bagajewicz, M.J. submitted 2002a, Managing Financial Risk in Planning under Uncertainty - Part I: Theory. Barbaro, A. and Bagajewicz, M.J. submitted 2002b, Managing Financial Risk in Planning under Uncertainty - Part II: Applications and Computational Issues. Eppen, G.D., Martin, R.K. and Schrage, D. 1989, A Scenario Approach to Capacity Planning. Operations Research, 37, 517 - 527. Petkov, S.B. and Maranas, CD. 1997, Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty. Ind. Eng. Chem. Res. 36,4864-4881. Shah, N., 1998, Single- and Multisite Planning and Scheduling: Current Status and Future Challenges. Foundation of computer-aided process operations, AIChE symposium series, 340.
41
Managing Financial Risk in Scheduling of Batch Plants Aima Bonfiir, Jordi Canton^, Miguel Bagajewicz^, Antonio Espuna^ and Luis Puigjaner^'^ (+)DEQ-ETSEIB-UPC. Av. Diagonal 647, P. G, 2° P. (08028) Barcelona, Spain, e-mail [email protected]; (++) University of Oklahoma, School of Chemical Engineering and Materials Science. 100 E. Boyd St., T - 335, Norman, OK 73019. On sabbatical leave at ETSEIB; (#) Corresponding Author
Abstract The short-term scheduling problem of a multiproduct batch plant with uncertain product demands is addressed in this article. The problem is modelled using a two-stage stochastic approach maximising the expected profit and integrating inventory costs and penahies for production shortfalls. Additionally, risk is controlled using a recently developed methodology.
1. Introduction The scheduling problem in the chemical industry has been extensively studied and alternative methodologies and problem statements have been proposed in the literature to address the combinatorial character of this problem (Shah, 1998). However, most of the formulations presented are based on nominal parameter values without considering the uncertain requirements after the operations are planned and scheduled. The uncertainty from a real environment entails a risk that is initially traduced in a cost but may lead to an unfeasible situation. The aim of the present work is to provide a tool to support the decision making of developing a scheduling policy in an uncertain environment while controlling the variability over the possible scenarios. For this purpose, a deterministic MILP formulation is first presented and used as starting point for modelling a two-stage stochastic optimisation approach. To control the variability over the different scenarios, the concept of risk is introduced. The simultaneously reduction of risk and maximisation of profit results in a multiobjective stochastic optimisation problem. Two alternative risk definitions, one related with financial risk introduced by Barbaro and Bagajewicz (2002a,b) and another considering the downside risk defined by Eppen et al. (1989) are considered and the latter is used as an objective. The outcome is a set of parametric solutions corresponding to different levels of risk.
2. Model The scheduling problem of a multiproduct batch plant under uncertain product demand is addressed with the aim to maximise the expected profit. The scheduling policy involves the number of batches to be produced for each product, the detailed sequence and the starting and final times of each operation performed.
42 To derive the proposed models, one production line with fixed assigned equipment units and zero wait transfer policy is assumed. The scheduling is addressed within a time horizon of one month. Finally, inventory costs and penalties for production shortfalls, proportional to the amount of underproduction, are adopted for each product. 2.1. Deterministic scheduling model The deterministic scheduling model is formulated as a MILP problem based on a batch slot concept. With this formulation, the time horizon is viewed as a sequence of batches, each of which will be assigned to one particular product. The proposed mathematical model is shown in equations (1) to (13). It maximises the expected profit (sales inventory costs - production shortfalls) using X(b,p) as sequencing decision variable (see Nomenclature). A small penalty term on the sum of the initial time of all batches is added to force the schedule to finish in the smallest makespan possible. MaxP = ^[F(/?)-i^r(p) - I{p)'CI{p) - Fr{pyPe{p)]-a^Tin{o,b)
(1)
oJ>
P
subject to: H>TMo,b)
Vo,b
(2)
T(o,b) = '^X{b,p)TOP{o,p)
Vo,b
Tfn{p,b) = Tin{o,b) + T(p,b) Tin{o,b +1) > Tfn{p,b)
yo,b
Tfn{o,b) = Tin{o + \,b) ^X(b,p)
\fo,b
(4) (5)
yo,b
(6)
\/b
nip) = Y,^(b,p)
(3)
("7)
(8)
yp
b
(9)
i>^n(p)
Vip) = mm{D(p), n(p) • BSip))
^p
(10)
I(p) = n(p)-BSip)-V(p)
yp
(11)
43 F(p) = Dip)-Vip)
Vp
(12)
2.2. Stochastic model From the formulation presented in the previous subsection and to derive the two-stage stochastic program, the uncertain product demands are modelled by probability distributions and a set of scenarios is generated using sampling techniques. The number of batches to be produced and the corresponding sequence of products are considered as first-stage decisions, since it is assumed that they have to be taken at the scheduling stage before the realisation of the uncertainty. The sales, inventory and production shortfalls are then recomputed on a second-stage for each scenario generated. Therefore, equations (10) - (12) are rewritten to take into account the different scenarios s. Then the maximisation of the expected profit is defined as: EP = Y^ Prob(s) • Pis) -a'Y,
^Ho, b) =
o,b
-T
Probis) 'J^[V(p,s). Prip) - Iip,s) • CI(p) - F(p,s) • Pe(p)] -a'Y,^in(o,b) (1^) o,b
2.3. Risk management The concept of financial risk defined by Barbaro and Bagajewicz (2002a) and its alternative of downside risk (Eppen et al., 1989) have been incorporated into the stochastic model to support the decision making. Downside risk is finally used. 2.3.1. Financial risk Managing financial risk involves the maximisation of: EP = J^[Prob(s)• P(s)]-a• ^7-m(o,6)-^^^[ProfcC*)• p,. • z^sj)] o,b
s
Pis)>n(i)-U'z(s,i)
5
(14)
i
\/s,i
(15)
2.3.2. Downside risk The utilisation of the alternative downside risk concept requires the redefinition of the objective function and constraints as follows.EP = ju 'Y,[P^ob(s) • P(s)]-a • Y,THo,b) -Y^[Prob(s) • S(s)] o,b
s
S(s)>Q-P(s) ^(^)>0
\/s V^
(16)
s
(17) (18)
44
3. Results and Discussions The proposed methodology to incorporate risk management in the framework of twostage optimisation for scheduling of multiproduct batch plants under uncertainty has been applied to a case study involving four production stages and five different products adapted from Petkov and Maranas (1997). A total number of 500 independent scenarios were simulated using Monte Carlo sampling technique from the given normal demand probability distributions. To assess the importance of the stochastic formulation, the deterministic model using the nominal demand values has been first solved and compared with its stochastic counterpart. The results obtained are detailed in Table 1. The schedules are not shown for space reasons, but they are significantly different. In addition, it is noted that the makespan of the deterministic model is shorter, because the model does not generate inventory to hedge from adverse scenarios, as the stochastic one does. One important thing to notice is that the deterministic model predicts a solution that poorly represents the uncertain environment, i.e. the schedule obtained with the nominal parameters may be unrealistic when another demand is ordered. Indeed, although the profit of the deterministically generated schedule is higher than the expected value of the stochastically generated, when the first one is used to face the uncertainty the expected value of profit drops 16%. The stochastic schedule performs better as it is also reflected in the shift to the left of the risk curve (Figure 1). This is one indication of the danger of using deterministic models in the believe that stochastic models will only "refine" the solution. Table L Results of Deterministic and Stochastic Models.
Products n(p) E[Sales] E[Inv.] E[Penalt.] Profit E[Profit] Makespan
Deterministic schedule P2 PI P4 P3 P5 10 10 15 15 8 726 941 7099 3086 1884 74 59 401 214 116 56 123 25 468 168 79875 (value in nominal scenario) 67165 617
Stochastic schedule P4 PI P2 P3 P5 10 10 16 16 8 726 941 7313 3165 1884 74 59 687 355 116 25 56 254 89 123 76275 (value in nominal scenario) 68539 662
The management of risk has been next performed considering several profit targets (one at a time) along with various weight values reflecting different levels of risk exposure. At each profit target, a schedule with its correspondent risk curve is obtained. Risk values of schedules obtained by managing downside risk with a weight value \x = 0.001 are reported in Table 2. As it was expected, the expected profit value of the schedules approaches the maximum value obtained with the stochastic model (SP) as the profit target is incremented. From a comparison with the stochastic schedule it can be observed that reductions of downside risk of 30% and 16%) are attained at target profits Q. = 60,000 and 65,000, respectively. Selected risk curves are depicted in Figure 2.
45
30000
40000
50000
60000 Profit
70000
30000
40000
50000
60000
70000
80000
90000
Profit
Figure 1. Risk curves from the stochastic Figure 2. Selected downside risk curves for schedules with different profit targets. and deterministic schedules.
Table 2. Risk values of selected schedules when optimising at different profit targets fl. Stochastic solution (SP) Schedule Target Q E[Profit] DRisk FRisk (%) DRisk FRisk (%) 2 4.2 50,000 64,310 102 227 C D 55,000 64,442 275 7 516 8.2 E 60,000 67,160 770 13 1,085 15.4 F 65,000 67,160 1,787 30.6 2,140 27.6 G 70,000 67,165 4,040 60.6 4,048 49.6 H 75,000 68,539 7,170 74.6 7,170 74.6
4. Conclusions The treatment of uncertainties in batch process planning and scheduling has been addressed as a two-stage stochastic optimisation problem with the incorporation of financial risk management, thus resulting in a multiobjective stochastic system. The proposed methodology has been applied to the scheduling of a multiproduct batch plant and uncertainty in product demands has been first considered and modelled by probability distributions. A comparison between the deterministic and the stochastic formulations has been performed to assess the importance of the stochastic approach. The schedule obtained with the nominal parameter values will poorly perform in most of the probable scenarios. However, with the stochastic modelling a significant improvement is attained and schedules with a good performance over all the scenarios are obtained. The management of risk has been next introduced. From the results obtained it can be noticed that risk can only be managed at the expense of the expected profit. In addition, despite the flexibility introduced by the uncertain parameters and the inventory, different cost-dependent alternatives are required to be able to manage risk in a meaningful way. A variety of alternative schedules are obtained reflecting different risk exposure poHcies.
46
5. Acknowledgements Financial support received from the Spanish "Ministerio de Educacion Cultura y Deporte" (FPU research grant) and "GeneraHtat de Catalunya" (project GICASA-D), and from the European Community (project GlRD-CT-2000-00318 and project GRDl2000-25172) is thankfrilly appreciated. The support of the ministry of Education of Spain for the sabbatical stay of Dr. Bagajewicz at UPC is also acknowledged.
6. Nomenclature Sets: b: i: o: Ps: Parameters: BS(p): CI(p): D(p): H: L: Pe(p): Pr(p): Prob(s): TOP(o,p): U: Q(i): ^(i):
Batches Profit targets Operations Products Scenarios Batch size Inventory cost Demand Horizon time Maximum n** of batches Production shortfall cost Sales benefit Probability Operation time Big value Profit target Financial risk upper bound
P(i): 5(s): l^a: Variables: EP: F(p)/F(p,s): I(p)/I(p,s): MS: n(p): P(s): T(o,b): Tfn(o,b): Tin(o,b): V(p)/V(p,s): X(b,p): z(s,i):
Weight value Downside risk Weight value Low value Expected profit value Shortfall Inventory Makespan Number of batches Profit value Processing time Final time Initial time Sales of product/^ Batch b assigned to prod. p (Binary variable) Binary variable
7. References Barbaro, A. and Bagajewicz, M.J. submitted 2002a, Managing Financial Risk in Planning under Uncertainty - Part I: Theory. Barbaro, A. and Bagajewicz, M.J. submitted 2002b, Managing Financial Risk in Planning under Uncertainty - Part II: Applications and Computational Issues. Eppen, G.D., Martin, R.K. and Schrage, D. 1989, A Scenario Approach to Capacity Planning. Operations Research, 37, 517 - 527. Petkov, S.B. and Maranas, CD. 1997, Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty. Ind. Eng. Chem. Res. 36,4864-4881. Shah, N., 1998, Single- and Multisite Planning and Scheduling: Current Status and Future Challenges. Foundation of computer-aided process operations, AIChE symposium series, 340.