JPH07262173A - Quasi optimization method for scheduling problem - Google Patents

Abstract

PURPOSE:To decide a production process schedule in a comparatively short time and to find a solution satisfied by a planner i.e., a quasi optimization solution by evading the nonexistence of the solution by performing self adjustment by varying the constraint condition of the production process schedule. CONSTITUTION:For example, in a manufacturing system of cement, a cement clinker from a calcining system 1 is stored transiently in an intermediate silo 2, and after it is milled in a milling system 3 with gypsum, cement as a final product is sent out from a product silo 4 to shipping equipment 5. In such a case, a state in which a targeted soultion can be obtained by clearing the constraint condition is generated when unmatching occurs between input data and the constraint condition. Therefore, a penalty constant and a penalty variable to relax the limiting value of a constraint equation are added. In this way, it is possible to set a state to which constraint condition importance should be given depending on the value of the penalty constant, and simultaneously, the solution of a linear programming problem is outputted by performing the self adjustment by supplying the excess share of a constraint value as the penalty variable.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a semi-optimization method for a scheduling problem using linear programming. The plan referred to in this specification includes general plans such as a resource distribution plan, a production process plan, and a device arrangement plan, but for convenience of description, the production process plan will be described as an example.

[0002]

2. Description of the Related Art In a production process plan, it is necessary to decide a plan for allocating resources so that a given production amount can be minimized with a limited processing capacity. The operation schedule was decided based on experience and simple calculation. It is well known that linear programming is used for that purpose. However, in a large-scale and complicated production process, there are many types of decision variables in the objective function of the linear programming method, and the number of solutions becomes extremely large due to the relation with the constraint conditions. It is well known that it is difficult to determine a large-scale production process plan. Further, it has been attempted to apply the branch and bound method to determine such a production process plan. The branch and bound method can be divided into two types: superiority relation and upper and lower bounds. The superiority relationship indicates the order relationship of the jobs to be processed, and by doing so, the number of solution candidates can be reduced, and a solution considered to be effective is extracted from a large number of solutions for planning. Further, the upper and lower bounds are the upper bound and the lower bound of the evaluation value of the solution, and the number of solution candidates can be similarly reduced based on these values.

[0003]

The method for determining the production process profile is as described above. However, a problem in using the linear programming method is that there is no feasible solution as a result of calculation. (When all constraints cannot be met). In such cases, conventionally,
The constraint formula was modified and the calculation was repeated until a feasible solution existed. Therefore, there is a problem that an appropriate solution is often not obtained within a practical time.

In some cases, the operation schedule is determined based on the experience of the operator and a simple calculation without using the linear programming method. In such a case, it is difficult to make a plan satisfying the objective function. There was a problem.

The present invention has been made in order to solve the above-mentioned problems, and it is necessary to determine a production process plan in a relatively short time and to obtain a solution which is satisfactory to the planner, that is, a suboptimal solution. The goal is to provide a semi-optimization method for the scheduling problem.

[0006]

[Means for solving the problems] Each constraint condition of the production process plan is a bottleneck, and the production process plan is determined by avoiding the absence of a solution by making the constraint condition variable and self-adjusting. Decided to do efficiently.

[0007]

In the calculation of the linear programming problem, an appropriate solution can be found within a practical time, so that the production process planning work can be performed in a relatively short time. In addition, since it is possible to always find a solution, it is not necessary to modify the constraint formula and repeatedly calculate until a feasible solution exists, and the planner can obtain a satisfactory solution.

[0008]

Embodiments of the present invention will now be described with reference to the drawings, taking a cement manufacturing system as an example. FIG. 1 is a block diagram of a cement manufacturing system to which the present invention is applied. As shown in the figure, the cement clinker from the firing series 1 is temporarily stored in the intermediate product silo 2, and the crushing series 3
After being crushed together with gypsum in, cement as a final product is sent from the product silo 4 to the shipping facility 5.

In such a configuration, it is assumed that the shipment amount of cement D _2t is requested from the shipping facility 5, and the scheduling problem that minimizes the power cost in which the supplied power amount fluctuates within the range of E ^{∧ is} as follows. explain.

When the schedule problem for determining the amount of electric power distribution in the cement manufacturing process in the time interval t = 1 to P is formulated into a linear programming problem, it is usually as follows as an example. First, as a variable, the power distribution amount: E _ft (f = 1 to N, t = 1 to P) can be mentioned. Electric power cost coefficient: _Ct mix coefficient: H Production coefficient: A _f Intermediate product initial stock: L ₁₀ Intermediate product production: Y _t Cement product initial stock: L ₂₀ Cement product shipment: D _2t Intermediate product silo upper and lower _{limit: M 1max,} M _1min product silo on the lower _{limit: M 2max,} M _2min and the like.

The objective function is equation 1, the constraint conditions are power consumption limitation 2, equation 3 silo upper and lower limit constraint, cement product silo upper and lower constraint equation 4, intermediate product silo model equation 5, cement The product silo model is shown in Eq.

[0012]

[Equation 1]

[0013]

[Equation 2]

[0014]

[Equation 3]

[0015]

[Equation 4]

[0016]

[Equation 5]

[0017]

[Equation 6]

In the linear programming problem formulated as described above, inconsistency between the input data and the constraint condition may occur, and the target optimum solution may not exist. That is, a situation occurs in which a solution cannot be obtained unless the constraint conditions shown in the equations 2 to 6 are exceeded. On the other hand, the linear programming problem to which the present invention is applied can be formulated by making the following changes (additions, etc.).

In the present development, constants (referred to as penalty constants) for relaxing the limit values (upper limit value, lower limit value, etc.) of the constraint equation: a, b, c, d, e and variables for reducing (penalty constant). Called variables): Add α, β, δ, ζ, θ. The objective function is Eq. 7, the power consumption limit is Eq. 8, the intermediate product silo upper / lower limit constraint is Eq. 9, the cement product silo upper / lower limit constraint is Eq. 10, and the intermediate product silo model is Eq.
1. The number of cement product silo models is 12.

[0020]

[Equation 7]

[0021]

[Equation 8]

[0022]

[Equation 9]

[0023]

[Equation 10]

[0024]

[Equation 11]

[0025]

[Equation 12]

In the present invention thus formulated, the action of the penalty constants: a, b, c, d, e and the penalty variables: α, β, δ, ζ, θ on the linear programming problem is , It depends on the setting of the penalty constant. That is, since the objective function (Equation 7) including the penalty constant and the penalty variable is minimized while satisfying the constraint condition (Equation 8 to 10) including the penalty variable,
When the penalty constant is set to a very large value, the constraint condition including the penalty constant is bound by the increased penalty constant. Therefore, the penalty variable used in the constraint condition has to take the smallest possible value as the penalty constant increases.

In summary, the relation among the penalty constant, the penalty variable, and the linear programming problem can be summarized as follows. It is possible to set which constraint condition should be weighted depending on how the set value of the penalty constant is taken, and at the same time, the constraint condition can be set. The excess of the constraint value of is given as a penalty variable and self-adjusted to output the solution of the linear programming problem. In this way, although the cost increases by the amount exceeding the limit value of the constraint condition, it is possible to find a feasible solution and capture it as scheduling data.

2 and 3 show the flow of processing up to planning in accordance with the conventional formulations shown in Formulas 1 to 6 and the formulations shown in Formulas 7 to 12 of the present invention. Data input 6,
In the processing from 12 to the planning 11 and 15, in the conventional example, the feasible solution 10 does not always exist as described above in solving the linear programming problem 7, and when there is no feasible solution, the constraint is re-constrained in the case of 8. It is necessary to consider the modification 9 of the formula. That is, in the present invention, calculation of linear programming problem 1
In step 3, a variable that allows an excess is added to the upper limit value or the lower limit value of the constraint equations shown in the equations 8 to 10, and a penalty constant multiple of the excess is added to the objective function equation shown in the equation 7. Further, since this example of the linear programming problem is a problem that minimizes the objective function value, the calculation is performed so as to satisfy the constraint condition and also minimize it. By this method, the penalty variable does not have a value when the constraint condition is satisfied with the default upper limit value or the lower limit value, and the penalty variable is value when the constraint condition is not satisfied unless the default upper limit value or the lower limit value is exceeded. And the objective function value becomes large, but the feasible solution 14 is always derived.

The invention also applies to the maximization problem,
It can be applied by subtracting the penalty value from the objective function value. Needless to say, the present invention can be applied to various scheduling problems, not limited to cement production planning.

[0030]

As described above, according to the present invention, when a large-scale scheduling system that solves a linear programming problem by linking with another system is operated and processed, a method that always derives an feasible solution is used. Therefore, it is superior in terms of calculation time and system consistency.

[Brief description of drawings]

FIG. 1 is a diagram showing the processing steps of an example linear programming problem embodying the present invention.

FIG. 2 is a diagram showing a flow of processing up to conventional planning.

FIG. 3 is a diagram showing a flow of processing up to planning of the present invention.

[Explanation of symbols]

 1 Firing series 2 Intermediate product silo 3 Grinding series 4 Product silo 5 Shipping facility 6 Data input 7 Calculation of linear programming problem 8 Feasible solution (none) 9 Correction of constraint equation 10 Feasible solution (yes) 11 Planning 12 Data input 13 Calculation of linear programming 14 Feasible solution (Yes) 15 Planning

Claims (2)

[Claims] 1. In a large-scale scheduling problem having a complex operation pattern, when various constraints are formulated into a linear programming problem, the constraints are made variable and self-adjusted to avoid the absence of a solution. A sub-optimization method for scheduling problems characterized by the following. 2. A linear programming problem is formulated by adding a variable that relaxes the limit value of the constraint condition to the constraint expression and adding a penalty equivalent to the added variable to the objective function. A sub-optimization method for a scheduling problem according to claim 1.