강건한 최적화 기법을 이용한 화학 공정 네트워크

단기 생산 계획 모델의 개발

이 희만, 박 선원

한국과학기술원 화학공학과

Development of a short-term production planning model for the chemical processing networks using the robust optimization technique

Heeman Lee and Sunwon Park

Dept. of Chem. Eng., KAIST

Introduction

While the multiperiod optimization model for long range capacity expansion and production planning has been extensively studied (Sahinidis et al., 1989; Sahinidis and Grossmann, 1991), much less work has been devoted to the short term inventory and production planning problem of chemical processing network including temporary shut-down plan and product change-over policy for flexible processes. Unlike long range capacity expansion problems, the short term planning problem is more focused on how best to operate existing facilities. Often short term plans are the beginning of long-term plans such as the plan for the next 6 months (Bodington, 1995), and capacity expansion is not discussed in a short term problem since building a process usually takes longer than the considered planning horizon.

This work first proposes an effective multiperiod MILP model and its solution method for short term production and shut-down planning of the chemical processing network. The major objective in this model is to determine the temporary shut down policies for all processes, inventory levels at the end of each time period, purchase/production/sales amount of each chemical, and product mix or change-over policy for flexible processes under given forecasts of prices and demands of chemicals which show seasonality or growth/reduction trend. Based on the developed model, supplementary robust contingency plan is presented to cope with the multiple price forecast scenarios which cause conflict between relevant staffs who are responsible for the forecast of the respective price parameters such as raw material/product price. For example, management in the head office predicts the price of the specific chemical raw material/products decrease, sales department expects the demand urge for some products from the specific market can increase the price of the products, while staffs from raw material procurement department expect the gradual price increase in some raw material. The job of planner or production department thus is to build a planning model which compromises among diverse forecast scenarios and goals while reflecting the operation limitation and shut-down plan. In this study using the pay-off matrix (Masch and Baker, 1991) and the simple robust optimization technique, a contingency planning model for the short term production and shut-down plan is also proposed.

Mathematical formulation of the base short-term planning model

The qualitative descriptions of the MILP model for the production planning problem is as follows.

Maximize

A) Profit	=	Revenue by chemical sales - Cost of chemical purchase

- Process unit operating cost - Inventory cost

- Maintenance cost during shut-down

subject to

B) Production/Inventory capacity limitations

C) Availability/Demand limitation of each chemical

D) Material balance equations for each chemical in process network

E) Shut-down operation rules

F) Production limitations for flexible processing units

Robust contingency plan under multiple price forecasts

Based on the model developed in the previous section, a robust contingency plan is developed using the pay-off matrix. The pay-off matrix alerts the decision-maker to both hidden dangers and potential opportunities associated with different strategies and scenarios (Masch and Baker, 1991). To implement the pay-off matrix, it is first required to define the strategic variables. The strategic variables, according to Masch and Baker, are associated with immediate and irrevocable decisions. In the chemical industry, usually raw materials procurement and product sales are contracted and it is hardly revocable to modify or cancel the contract during the time period in which sale/purchase activity is planned to occur. In that sense, variables related to raw material supply and sales activity are defined as strategic variables in this study. The elements of the pay-off matrix include parameter sets, strategies, and payoff matrix entries. Matrix entries are optimized objective values (Profit) under the given strategy and parameter set. Optimization under a given strategy means finding the solutions for the non strategic variables, while strategic variables have been already defined. The algorithmic flow for the strategy selection using the pay-off matrix is described in the following steps. As described in this approach, the strategy selection is an interactive process since the planner is required to iteratively find the most compromising solution which will be robust to all parameter sets.

Step 1.	Generate multiple forecast parameter sets according to the forecast scenarios respectively by management, sales, and/or material staff.

Step 2.	Solve the independent MILP model with each forecast parameter set, and get the upper bound for each MILP model under each forecast parameter set.

Step 3.	Based on the upper bounds obtained from step 2, the planner interactively decide the allowable lower bound for the planning model under each forecast parameter set.

Step 4.	Solve the resulting extended MILP model which is based on the model developed in the previous section.

Maximize

Objective function = Summation of PV(s) over s

subject to

A(s)x+C(s)nsv(s)=0	s=1,...,NS	(1)

lowb(s)<=PV(s)<=upperb(s)	s=1,...,NS	(2)

Where,

(a)	sets:

s=1,NS	forecast parameter set

(b)	variables:

sv	strategic variables

nsv(s)	non strategic variables under parameter set s

PV(s)	Profit as described in the previous section under	parameter set s

(c)	parameters:

lowb(s)	Lower bound of PV under parameter set s

upperb(s)	Upper bound of PV under parameter set s

Step 5.	If the MILP model in Step 4 is proved to be infeasible, relax the constraint (2) by reducing the lower bounds. Iterate the relaxation process until the compromising solution under multiple forecast scenarios has been found.

Test Example

The processing network shown in Figure 1 has 3 processing units and 1 flexible processing unit with 6 chemicals. Process 4 is a flexible process in this example, and it can change its operating mode to produce three different chemicals. Three price forecasts have been considered in this example. The first price forecast assumes slow price growth of all three products, while the second forecast assumes rapid price increase of chemical 5. Forecast 3 predicted the price decrease of chemical 3 which is the intermediate product.

Figure 1 Chemical processing network

Table 1 Pay-off matrix

	Strategy 1 based on Forecast 1	Strategy 2 based on Forecast 2	Strategy 3 based on Forecast 3	Strategy 4	Strategy 5
Forecast 1	66296	58556	61149	65343	64647
Forecast 2	66296	67946	61148	65343	67000
Forecast 3	66296	58556	68562	67000	64647

As shown in Table 1, strategy 2 resulted very poor solution under forecasts 1 and 3. Even if strategy 1 seems to be the most compromising solution among the first three strategies, planner may put more emphasis on the specific forecast scenario. Strategies 4 and 5 were respectively obtained by putting more emphasis on forecast 3 and 2, respectively. Strategy 4 was obtained by imposing lower bound 67000 on the NPV by forecast 3, while strategy 5 was obtained by lower bound 67000 on NPV by forecast 2. In this way, the compromising short-term production plan under multiple price forecasts can be obtained, interactively.

Conclusions

The short term production and shut-down planning problem of a chemical processing network with supplementary contingency plan under multiple price forecast  has been discussed in this study. This problem can be modeled as a multiperiod MILP model in which time is uniformly discretized. Pay-off matrix and interactive solution robust optimization method has been implemented and developed to deal with the multiple price forecast scenarios. The LP-based branch and bound method was used to solve the model, and logic cut techniques were applied to reduce the high computational expense due to making shut-down decision variables. Finally, a test example has been considered to show the economic potential by using the formulation and solution method proposed.

Acknowledgment

Financial support from the Korea Science and Engineering Foundation through the Automation Research Center at Postech is gratefully acknowledged. This research was also supported by Korea Research Foundation.

References

Sahinidis, N. V.; Grossmann, I. E.; Fornari, R. E.; Chathrathi, M. Optimization model for long range planning in the chemical industry. Computers chem. Engng. 1989, 13, 1049-1063.

Sahinidis, N. V.; Grossmann, I. E. Multiperiod investment model for processing networks with dedicated and flexible plants. Ind. Eng. Chem. Res. 1991, 30, 1165-1171.

Bodington, C. E. Planning, scheduling, and control integration in the process industries; McGraw-Hill, Inc.: NY, 1995.

Masch, V.A.; Baker, T.E. Robust Optimization. Presented at the American Institute of Chemical Engineers, Spring National Meeting, Houston, Texas 1991.