Chinese Journal of Chemical Engineering, 16(2) 241 246 (2008)

A Hybrid Programming Model for Optimal Production Planning under Demand Uncertainty in Refinery*

LI Chufu ( )1, HE Xiaorong ( )1,**, CHEN Bingzhen ( )1, XU Qiang ( )2

and LIU Chaowei ( )21 Department of Chemical Engineering, Tsinghua University, Beijing 100084, China 2 Department of Chemical Engineering, Lamar University, Beaumont 77710, USA

Abstract Production planning under uncertainty is considered as one of the most important problems in plant-wide optimization. In this article, first, a stochastic programming model with uniform distribution assumption is developed for refinery production planning under demand uncertainty, and then a hybrid programming model in-corporating the linear programming model with the stochastic programming one by a weight factor is proposed. Subsequently, piecewise linear approximation functions are derived and applied to solve the hybrid programming model under uniform distribution assumption. Case studies show that the linear approximation algorithm is effec-tive to solve the hybrid programming model, along with an error 0.5% when the deviation/mean 20%. The simulation results indicate that the hybrid programming model with an appropriate weight factor (0.1 0.2) can ef-fectively improve the optimal operational strategies under demand uncertainty, achieving higher profit than the lin-ear programming model and the stochastic programming one with about 1.5% and 0.4% enhancement, respectively. Keywords production planning, demand uncertainty, stochastic programming, linear programming, hybrid pro-gramming

1 INTRODUCTION

The deterministic programming (Linear Pro-gramming, LP; Mixed Integer Linear Programming, MILP; Nonlinear Programming, NLP) models are adopted in most of the current refinery production planning [1 5]. However, because of volatile raw ma-terial prices, fluctuating products demands, and other changing market conditions, several parameters in a production planning model are uncertain. Most com-monly, the dominant uncertain parameters in the pro-duction planning problems are the product demands and prices [6 10]. Failure to incorporate a stochastic description of the uncertain parameters in the produc-tion planning model can lead to a non-optimal solu-tion in reality. Thus, production planning under un-certainty is considered as one of the most important problems in plant-wide optimization [11, 12].

However, to date, only few researchers have studied refinery production planning under uncertainty. Clay and Grossmann [13] used refinery planning as the case studies to illustrate the solution algorithm. Li et al. [14] applied fuzzy programming to optimize refin-ery production planning under demand uncertainty. Liao et al.[15] adopted multi-objective programming to make the refinery production planning considering the price uncertainty. Neiro and Pinto [16, 17] consid-ered uncertain product demands and prices as a set of discrete scenarios in the multi-period model for refin-ery production planning, and then applied the La-grangean decomposition algorithm to reduce the solu-tion time. Li et al. [18] presented a stochastic pro-gramming model with loss functions for refinery planning under product demand uncertainty, and pro-posed a piecewise linear approximation algorithm to obtain good results with improved solution speed.

Commonly, the optimal results obtained by the deterministic programming model are aggressive un-der demand uncertainty, while the optimal results ob-tained by the stochastic programming model are con-servative. This article presents a hybrid programming (HP) model for refinery production planning under uncertain demand, which incorporates the LP model with the stochastic programming (SP) one by using a weight factor. Subsequently, a piecewise linear ap-proximation algorithm is developed to solve the hy-brid programming model with uniform distribution assumption under demand uncertainty.

2 AN SP MODEL FOR REFINERY PRODUC-TION PLANNING UNDER DEMAND UNCER-TAINTY

An SP model for refinery production planning under demand uncertainty is developed as follows.

rw op other1 1

0

L U

0

max

s.t. min , d

, , 0, 1,2, ,

J J

j j j jj j

j j j j j

j j j

j j j

j j j

j j j

P p s h I c c c

s y

I y s

I I I

y x I

s I y j J

0Ax bx

(1)

Received 2007-05-08, accepted 2007-09-20. * Supported by the Specialized Research Fund for Doctoral Program of Higher Education of China (20060003087).

** To whom correspondence should be addressed. E-mail: hexr@tsinghua.edu.cn

Chin. J. Chem. Eng., Vol. 16, No. 2, April 2008 242

where x is a vector of production variables, A is a co-efficient matrix, and b is a requirement vector. In the model, the objective is to maximize plant profit under demand uncertainty, while the constraints mainly in-volve mass balance, capacity, inventory, raw material supply, production yield, and quality requirement.

The probability distribution assumption for un-certain demand j has an important impact on the optimal results obtained by the SP model, but this im-pact can be cut down well by the adjustment of pa-rameters of the probability distribution functions. Li et al. [18] presented an SP model with normal distribu-tion assumption; here, a detailed SP model with uni-form distribution assumption is derived as follows.

Assuming that the uncertain demand j con-forms to uniform distribution with mean j and de-viation j , written as ~ ( , )j j j j jU , the probability density function is

1 , 2

0, else

j j j j jjj (2)

Using Eq. (2), js in Eq. (1) is written as

, 0

1min , d ,2

,

j j

j j

j j j j

j j jjj

j j j j j

j j j j

y y

ys

y

y

(3)

where

22

1 min , d2

1 1d d2 2

1 24

j j

j j

j j j

j j j

j j jj

yj j j jy

j j

j j j j j jj

y

y

y y

Thus, Eq. (3) is represented as

22

,0

1 2 ,4

,

j j j j

j j j j j jjj

j j j j j

j j j j

y y

y ys

y

y

(4)

Eq. (4) is a piecewise nonlinear function. Using Eq. (4) to replace the js in Eq. (1), the SP model is translated into one equivalent to a mixed-integer nonlinear pro-gramming (MINLP) model. Replacing the js in Eq. (1)

using the second item 2 2j j j jy y

2 / 4j j j of Eq. (4), the SP model is translated

into an approximate NLP one.

3 AN HP MODEL FOR REFINERY PRODUC-TION PLANNING UNDER DEMAND UNCER-TAINTY

As already known, for an LP model, the sale of product j is calculated as

,

,j j j

jj j j

y ys

y (5)

In the case if the weighted average of Eq. (4) and Eq. (5) is used to calculate the sale of product j, then the HP model is developed. The sale of product j in the HP model is written as

2

2

2

2

,0

(1 ) 24

,

(1 ) 24

,

,

j j j j

j j j j jj

j j j j j j

j

j j j j jj

j j j j j j

j j j j

y y

y y y

ys

y y

y

y

(6)

where 0 1 is the weight factor. If 0 , the HP model becomes the LP one; if 1 , the HP model returns to the SP one. In the same way, using Eq. (6) to replace the js in Eq. (1), the HP model is translated into an equivalent MINLP one. Using the second and third items of Eq. (6) to replace the js in Eq. (1), the HP model is translated into an approxi-mate NLP one.

Using piecewise linear functions to approximate js when jy is in the interval ,j j j j , the

HP model can be translated into an approximate LP one to solve. Eight piecewise linear functions are proposed to approximate the sale js in the HP model. The eight linear functions are obtained by dividing the interval ,j j j j into eight equal subinter-vals and then finding the secant in each subinterval (Fig. 1), written as

1

2

3

4

5

4 4 316 ,16 64 4

12 10 316 3 ,16 64 4 2

20 1416 5 ,16 64 2 4

28 1616 7 ,16 64 4

28 167 ,16 64

j j jj j j j j j

j j j jj j j j j

j j j jj j j j j

j j jj j j j j

j jj j j j

L y y y

L y y y

L y y y

L y y y

L y y

6

7

8

420 1415 ,

16 64 4 212 10 33 ,

16 64 2 44 4 3

,16 64 4

jj j

j j j jj j j j j j

j j j jj j j j j j

j j jj j j j j j j

y

L y y y

L y y y

L y y y

(7)

Chin. J. Chem. Eng., Vol. 16, No. 2, April 2008 243

For the profit maximizing problem, since the product sale curve under demand uncertainty is a convex func-tion, the value of js in Eq. (1) can thus be formulated as

, 1,2, ,8j j

j k j

j j

s y

s L y k

s

(8)

The HP model is changed to an approximate LP one using

Eq. (8) to replace the 0

min , dj j j j js y

in Eq. (1).

4 CASE STUDIES

Three examples with different problem sizes and complexities illustrate the capabilities and effective-ness of the new approaches and formulations proposed in this article.

4.1 Case 1: Blending case

A simple example in Fig. 2, initially proposed by Li et al. [18], is used to illustrate the new approach. Fig. 2 consists of one gasoline blending (GB) unit, which is very common in refineries. Methyl tert-butyl ether (MTBE) and gasoline enter the GB unit to pro-duce two products: 90# and 93# gasoline. The blending requirement is that the octane number of each product should be equal to or greater than the required octane number of that product. The demands for 90# and 93#

gasoline are uncertain. A detailed description of the example can be found in Li et al. [18].

Figure 2 Configuration of the blending example

In this case, no initial inventory and inventory cost are considered, and the demands of 90# and 93#

gasoline are assumed to conform to U (40, 60) and U(60, 80) tons, respectively. For comparison, the LP, SP and HP models are adopted for this case study, respec-tively. Note that the HP model becomes the LP one when 0 and the HP model turns into the SP one when 1 ; thus, only the HP model is repeatedly used by changing the value of . Firstly, three HP models ( 0 , 0.1 and 1) are solved respectively to obtain their optimal profit. Subsequently, a simulation procedure (Fig. 3) [19] is employed to calculate the simulated profit obtained using these models under demand uncertainty. The simulation procedure is car-ried out for 1000 periods for obtaining an average profit over the randomly generated demands.

Table 1 lists the optimal profit and simulated profit obtained by solving the HP models with differ-ent algorithms. The optimization results show that the optimal profits obtained by solving the HP models using linear approximation, nonlinear approximation and scenarios [20] are very close, with the maximum error less than 0.1%, which indicates that the linear approximation algorithm is effective. Moreover, using the outer approximation [21] algorithm to solve the equivalent MINLP model can only acquire a poor lo-cal optimal solution. The simulation results indicate that there is a gap between the optimal profit obtained by the LP/SP model and the simulated profit, and the optimal profit obtained by the HP model with 0.1is close to the simulated profit. Thus, the HP model can effectively calculate the simulated profit obtained under demand uncertainty.

4.2 Case 2: Monthly production planning for a small refinery

Case 2 is a real industrial example provided by a small refinery. The refinery processes 2.5 million tons crude oil per year, and its primary products are 90#

gasoline, 93# gasoline, diesel oil and liquefied petro-

Figure 1 Eight piecewise linear approximation functionsof the sale curve ( 50, 10, 50)

Figure 3 Simulation procedure adopted for calculating simulated profit

Chin. J. Chem. Eng., Vol. 16, No. 2, April 2008 244

leum gas (LPG). It involves one crude distillation unit (CDU), one fluid catalytic cracking (FCC) unit, one fluid catalytic reforming (FCR) unit, one diesel hy-drotreating unit (DHU), one desulfurization unit (DU), one gasoline blending (GB) unit, and one diesel blending (DB) unit (Fig. 4). The proportion rule is used for gasoline and diesel blending. In this case, there are totally 4 uncertain demands considered for the primary products (90# gasoline, 93# gasoline, die-sel oil and LPG), and their expected means are 30000, 18000, 60000 and 21000 tons, respectively. All the 4 uncertain demands are assumed to conform to the uni-form distribution with the deviation/mean 20%. Based on the HP model, the simulation procedure is carried

out for 500 periods to obtain an average profit over the randomly generated demands.

Table 2 lists the optimization and simulation re-sults obtained by the HP models, which indicate the same conclusions as Case 1. In this case, the maxi-mum error of the optimal profits obtained using linear approximation, nonlinear approximation and scenarios is less than 0.3%.

4.3 Case 3: Monthly production planning for a complex refinery

Case 3 is also a real industrial example provided

Table 1 Comparison of models and algorithms in Case 1

HP model Optimal profit/RMB Yuan Algorithm Simulated profit/RMB Yuan Profit error

0 (LP model) 53033.55 simplex 51314.97 3.35% 0.1 51383.05 linear approximation 51314.97 0.131% 51383.05 nonlinear approximation 44142.58 outer approximation 51389.66 scenarios ( 500 )

1 (SP model) 45333.76 linear approximation 51098.84 11.28% 45344.42 nonlinear approximation 44142.58 outer approximation 45362.28 scenarios ( 500 )

Figure 4 Flowsheet of the small refinery

Table 2 Comparison of models and algorithms in Case 2

HP model Optimal profit×10 4/RMB Yuan Algorithm Simulated profit×10 4/RMB Yuan Profit error

0 (LP model) 3847.54 simplex 3678.77 4.59%

0.17 3676.18 linear approximation 3676.89 0.02%

3678.01 nonlinear approximation

2732.36 outer approximation

3678.79 scenarios (= 500)

1 (SP model) 3209.73 linear approximation 3563.98 9.94%

3216.04 nonlinear approximation

2665.43 outer approximation

3218.29 scenarios (= 500)

Chin. J. Chem. Eng., Vol. 16, No. 2, April 2008 245

by a complex refinery, which involves three crude distillation units, three fluid catalytic cracking units, one fluid catalytic reforming unit, three gas separators, and several other processing units. The refinery can process 8 million tons crude oil per year. It mainly produces gasoline, diesel oil, kerosene, fuel oil, naph-tha, asphalt, lube, and LPG, and the naphtha is pro-vided to petrochemical plant. In this case, the common linear blending rule is used for gasoline blending and diesel blending, and 8 uncertain demands for the pri-mary products (90# gasoline, 93# gasoline, 0# diesel oil, kerosene, fuel oil, naphtha, asphalt, and LPG) are considered, the expected means of which are 86000, 55000, 286000, 33000, 65000, 60000, 26000, and 24000 tons, respectively. These are all assumed to conform to uniform distribution with the deviation/ mean=20%.

Figure 5 depicts the accumulative mean profit curve along with the period number during the simu-lation procedure. It indicates that the accumulative mean profit obtained using the HP model with

0.17 is always higher than the LP model (HP model with 0 ) and the SP model (HP model with

1 ) in each period. At the 200th period, the accu-mulative mean profits obtained by using the HP, LP and SP models are 14779.55×104, 14723.14×104 and 14569.45×104 RMB Yuan, respectively. Thus, the profit obtained using the HP model is enhanced by about 1.5% and 0.4% as compare with that obtained using the LP model and the SP model, respectively.

Figure 5 Accumulative mean profit obtained using the HP models

LP model (HP model with 0); HP model with 0.17;SP model (HP model with 1)

5 CONCLUSIONS

This article at first developed an SP model with uniform assumption for refinery production planning under demand uncertainty, and then proposed an HP model, which incorporates the LP model with the SP model using a weight factor. The HP model becomes the LP one in the case if the weight factor is equal to zero and returns to the SP model when the weight factor is equal to one. Subsequently, piecewise linear approximation functions are derived and applied to solve the HP model under uniform distribution assump-tion. Three case studies show that the linear approxi-mation algorithm is effective to solve the HP model with an error 0.5% when the deviation/mean 20%.

The simulation results indicate that the HP model with an appropriate weight factor (0.1 0.2) can effectively improve the optimal operational strategies under de-mand uncertainty, and thus, the profit obtained using the HP model is enhanced by about 1.5% and 0.4% as compare with that obtained using the LP model and the SP model, respectively. The HP model provides an appropriate approach for refinery production planning under demand uncertainty.

NOMENCLATURE

cop total operating cost of units cother other cost crw total cost of raw materials hj inventory holding cost of product jIj inventory of product j

LjI lower bound of inventory of product j

UjI upper bound of inventory of product j

0jI initial inventory of product j

J number of products pj price of product jsj sale amount of product jxj production amount of product jyj available amount of product j

weight factor j deviation of demand of product jj expected mean of demand of production j(·) probability density function j random demand of product j

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