CN105913147B - Product combination optimization problem solution based on residual capacity redistribution - Google Patents

Product combination optimization problem solution based on residual capacity redistribution Download PDF

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CN105913147B
CN105913147B CN201610220621.XA CN201610220621A CN105913147B CN 105913147 B CN105913147 B CN 105913147B CN 201610220621 A CN201610220621 A CN 201610220621A CN 105913147 B CN105913147 B CN 105913147B
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王军强
徐建利
周先华
杜向阳
范国强
胥军
杨宏安
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Abstract

The invention provides a method for solving a product combination optimization problem based on residual capacity redistribution, which adopts the principle that after the product is subjected to priority sorting, the product sequence is 1, 2. And then allocating the bottleneck resources to the products according to the priority order, calculating the number of each product, and in the calculation process, if the number of the product J is a decimal, asking a certain product X (i) in front of the product J for resource processing J, thereby obtaining a better product combination scheme. The existing product combination is adjusted according to the adjusting scheme, and the method has advantages over the traditional TOCh method and the revised TOCh method in certain situations, so that the waste of enterprise resources can be effectively reduced, the enterprise income is improved, and the effective output of enterprises is increased.

Description

Product combination optimization problem solution based on residual capacity redistribution
Technical Field
The invention relates to the technical field of product combination optimization, in particular to a method for solving a product combination optimization problem based on residual capacity redistribution.
Background
The product mix optimization (product mix optimization) problem is how to configure the product types and corresponding quantities that meet the resource capacity constraints to maximize the system profit for a given market demand. The traditional TOC product combination optimization model is generally expressed as: assume that an enterprise has m device resources, ready to produce n different products. The time that the product i occupies the equipment j is tijAnd the capability of device j is limited to CPjThen the number y of products planned to be processed needs to be determinediSo as to achieve the maximum total effective output of the system. It is to be noted that yiIt is also required to satisfy: (ii) must not exceed market demand DiThe production of finished product stock is avoided; ② in view of processing economy, not below a certain throughput li；③The product quantity is an integer, and the semi-finished product stock is avoided.
A mathematical model for the combination optimization of the traditional TOC product is shown as formulas (1) and (2):
Figure BDA0000961691990000011
Figure BDA0000961691990000012
wherein i is product number, i is 1,2, …, n, note: i belongs to N; j is the device serial number, j is 1,2, …, m, note: j is an element of M; y isiNumber of planned processed products i; TPiIs the effective output of the product i; t is tijThe time that a unit product i occupies the equipment j; CP (CP)jNormal available processing time for equipment j; diIs the market demand for product i; liThe minimum production amount of the product i is not lower than 0; z+Is a non-negative integer set.
The item 1 in the formula (2) is a resource capacity limit; item 2 is the limit of the upper and lower limits of the product quantity, the maximum is the market demand, and the minimum is the minimum production; item 3 is the product quantity integer limit.
The traditional TOCh method:
goldritt adopts TOC operation logic to develop a traditional TOCh method to solve a P-Q two-product combination optimization problem in 1990, and the specific steps are as follows:
1) identifying system bottlenecks
For resource (the machine, equipment, etc. used in production can be called resource) j, the difference between the required capacity and the available capacity is calculated, if djIf < 0, then the resource j is considered to be the bottleneck resource.
Figure BDA0000961691990000021
In the formula, CPjIndicates the available capacity of resource j, DiIndicating the market demand for the ith product,tijrepresenting the time that resource j is occupied by unit product i.
2) Determining product priority
Calculating the effective output of the product i and the bottleneck resource RsTime t of machiningiThe ratio of (a) to (b), i.e. the effective yield per unit time ri,BN：
Figure BDA0000961691990000022
And sequencing the products according to the principle that whether the products are processed on the bottleneck resources or not to obtain the production priority of the products based on the bottleneck resources. The specific method comprises the following steps:
product priority rules using bottleneck resources: if tiIf > 0, according to ri,BNArranging in descending order; if ri,BNSame, then press TPiAnd (5) arranging in descending order.
Product priority rules without using bottleneck resources: if tiWhen it is 0, then press TPiThe descending order is behind the products that use the bottleneck resource.
3) Allocating bottleneck resources according to product priority
Sequentially distributing bottleneck resources according to the product priority, namely giving all resources to the product with the highest priority, and if the quantity of the resources is enough to meet the market demand of the product, sequentially transmitting the rest resources to the product with the low priority after meeting the market demand of the product; if the quantity of the resources is not enough to meet the market demand, the quantity of the products obtained after the resources are exhausted is the optimal product combination. The flow chart of the conventional TOCh method is shown in FIG. 1.
Revising the TOCh method:
fredendall et al (1997) suggested that the conventional TOCh method sometimes did not yield optimal solutions, and therefore proposed a revision TOCh method. The TOCh method is revised to solve the problem of single-bottleneck product combination optimization, 4 steps are total, the first 3 steps are the same as the traditional TOCh method, and the 4 th step adjusts the initial solution obtained by the traditional TOCh method by using a neighborhood search method, wherein the specific adjustment method comprises the following steps:
1) and determining a candidate set of products to be adjusted. In the product sequence obtained by determining the product priority, the first product and all subsequent products which simultaneously satisfy the two judgment conditions of the formulas (5) and (6) form a candidate set of products to be adjusted.
Figure BDA0000961691990000031
Qi＜Dior Qi+1＜Di+1 (6)
In the formula, tleftThe residual capacity refers to the resource capacity which is not utilized on the bottleneck resource after the initial solution of the product combination is determined,
Figure BDA0000961691990000032
TPiis the effective output of unit product i, tiTime of bottleneck resource occupied by unit product i, QiFor the actual production quantity of product i, DiIs the market demand quantity of product i.
The physical meaning of equation (5) is to determine whether the accumulation of the resources obtained by reducing the high priority products by 1 unit and the remaining resources to produce the low priority products is beneficial to increasing the effective output of the production system, if so, the corresponding adjustment is performed, and if not, the adjustment is not performed.
2) And reducing the 1 st product of the candidate set of 1 unit, and distributing the obtained resources and the residual resources to the rest products of the candidate set again according to the product priority.
3) And repeating the adjusting process, and calculating the total effective output of all products after each adjustment until the end condition of the adjustment is reached. The determination conditions for the cycle termination are that the sum of the fluctuation amounts of the effective output is calculated:
Gain＝ΔCMk+ΔCMk+1+...+ΔCMk+n (7)
when Gain is less than zero or Qk＞DkThe cycle is terminated.
4) And comparing the total effective output of all the products in each adjusting step, wherein the maximum effective output is the optimal product combination scheme.
The flow chart of the modified TOCh method is shown in FIG. 2.
The revised TOCh method is obtained by performing neighborhood search adjustment on the basis of the initial solution of the product combination obtained by the traditional TOCh method. The final result of this adjustment makes the enterprise profit more than the solution obtained by the traditional TOCh method. However, revising the TOCh method still presents failure scenarios when solving some product portfolio optimization problems. In order to effectively analyze and explain the failure reason of the revised TOCh method, an example is described, in this example, three products A, B and C are assumed to be produced by a factory, the effective output (i.e. profit) of each product is different, the number of products has respective market demand limitation, one equipment resource R has, and the equipment resource R has production capacity limitation, and the specific example is described in Table 1.
TABLE 1 description of the examples
Figure BDA0000961691990000041
The example process shown in Table 1 is solved by the revised TOCh method as follows:
1) identify bottleneck resources of the production system, by formula (3)
Figure BDA0000961691990000042
I.e. the device resource R is a bottleneck resource.
2) Determining product priority
Represented by formula (4):
Figure BDA0000961691990000043
rB,R＝2.6833，rC,R＝4.2877，r D,R1. Because, rA,R＞rC,R＞rB,R＞rD,RSo the product priority order is: a. the>C>B>D。
3) Allocating bottleneck resources according to product priority
And scheduling the products according to the priority ratio in a non-ascending order to obtain the results that the product A meets the market demand of 50, 10C, 0B and 2D.
The solving process is as follows:
A:50，
Figure BDA0000961691990000046
Figure BDA0000961691990000047
Figure BDA0000961691990000053
the effective output at this time is: 50 × 146+10 × 313+9 × 2 ═ 10448, and the remaining time is 2400-50 × 33-10 × 73-2 × 9 ═ 2.
This set of solutions is also the solution obtained by the traditional TOCh method. On the basis of the solution set, the revision TOCh method adjusts the product combination by judging whether the initial solution obtained by the traditional TOCh method meets the adjustment condition of neighborhood search, thereby obtaining a better product combination.
4) Adjusting an initial solution using a neighborhood search method
According to the adjustment judgment condition formula (5), all products do not meet:
Figure BDA0000961691990000052
therefore, the candidate set can be adjusted to be X ═ Φ, and the current product combination is the optimal solution found by the algorithm.
And the ILP (integer programming) is adopted for solving, and the obtained product quantity is respectively as follows: 46A,12C,0B and 0D, meet resource capacity constraints and market demand constraints, and the total effective yield is: 10472.
the results obtained in the above groups of different processes are shown in table 2:
TABLE 2 comparison of product combinations solved by different methods
Traditional TOCh method Revising TOCh method ILP (integer programming)
Product A amount 50 50 46
Quantity of product B 0 0 0
Product C amount 10 10 12
Product D number 2 2 0
Total effective output 10448 10448 10472
From the solving process and the result, it can be seen that there is 750 units of resource left after 50a products are produced, regardless of the conventional TOCh method or the modified TOCh method. If this part of the resource is used to produce B, 12.5B products will be produced, however, the number of products must be an integer according to the market demand, so the end result is 12B products, and 30 units of resource capacity remain. And the resource capacity of 30 units is the remaining capacity. Although the conventional TOCh method and the revised TOCh method are different, the same product combination is finally obtained through the processing of the remaining capacity. However, this result is worse than the ILP solution. The conventional TOCh method and the revised TOCh method have the advantage that the result obtained by the conventional TOCh method and the revised TOCh method is not as good as the result obtained by the ILP when some problems are solved, namely the conventional TOCh method and the revised TOCh method have failure conditions in the process of solving the integer solution.
Disclosure of Invention
In order to solve the failure condition of the traditional TOCh method and the revised TOCh method in the solving process, the invention provides a product combination optimization problem solving method based on residual capacity redistribution.
The principle of the method is as follows: assume that after prioritization, the product sequence is 1, 2. And then allocating the bottleneck resources to the products according to the priority order, calculating the number of each product, and in the calculation process, if the number of the product J is a decimal, asking a certain product X (i) in front of the product J for resource processing J, thereby obtaining a better product combination scheme.
The technical scheme of the invention is as follows:
the product combination optimization problem solution based on residual capacity redistribution is characterized in that: the method comprises the following steps:
step 1: identifying system bottleneck resources:
for the resource j, j is 1,2, …, m, m is the resource in the systemThe number of sources; calculating the difference d between the required capacity and the available capacityjIf d isjIf < 0, then the resource j is considered to be the bottleneck resource:
Figure BDA0000961691990000061
in the formula, CPjIndicates the available capacity of resource j, DiIndicates the market demand of the ith product, i is 1,2, …, n, n is the number of product types, tijRepresenting the time of the unit product i occupying the resource j;
step 2: determining the product priority:
calculating the effective output TP of product iiAnd the processing time t of the bottle neck resourceiTo obtain the effective output r per unit timei,BN
Figure BDA0000961691990000062
And (3) according to whether the product is processed on the bottleneck resource and the effective output of the product in unit time, carrying out priority sequencing on the product to obtain a product sequence P:
for tiThe effective output r of the product using bottleneck resource is more than 0 according to unit timei,BNIn descending order, if ri,BNSame, press TP againiArranging in descending order; for tiProduct without bottleneck resource, per TP ═ 0iArranged in descending order after a product using a bottleneck resource;
and step 3: sequentially distributing bottleneck resources according to the product priority:
giving all resources to the product with the highest priority, and if the quantity of the resources is enough to meet the market demand of the product with the highest priority, transmitting the rest resources to the products with low priorities in sequence after meeting the market demand; if the resource is not enough to produce the quantity required by the market when the resource is transmitted to a certain product, the real number solution of the product can be obtained; when the production quantity of all products reaches the quantity required by the market or the resources are exhausted, obtaining a product combination real number solution;
and 4, step 4: determining whether a real solution contains a non-integer solution:
if the product combination real number solution does not have a non-integer solution, the obtained product combination real number solution is an optimal solution, and the method is ended, otherwise, the following steps are carried out;
and 5: solving the residual capacity:
the non-integer solution in the product combination real number solution corresponds to a product J, and after the non-integer solution of the product J is rounded downwards, the residual capacity t on the bottleneck resource is calculatedleft；
Step 6: constructing a feasible set X:
for the first product P from the sequence P1To J-1 product pJ-1Calculating
Figure BDA0000961691990000071
Figure BDA0000961691990000072
Get the first to satisfy
Figure BDA0000961691990000073
Product I of (1), then X ═ pI,pI+1,pI+2…pJ-1}; if X is an empty set, the resource t is leftleftDistributing the products after the product J according to the sequence in the sequence P and the market demand, and ending the method, otherwise, performing the following steps;
and 7: residual capacity reallocation:
for each product X (k) in the rowset X, the following calculation is performed:
the production quantity of the product X (k) is reduced by s, s is 1,2, …, DX(k)The excess resource s x tX(k)And remaining resources tleftThe sum is distributed to a product J and products behind the product J in an integer solution mode according to the sequence in the sequence P and market requirements to obtain a redistributed product combination integer solution; wherein DX(k)Is the market demand, t, of product X (k)X(k)Is processed on bottleneck resources for unit product X (k)A (c) is added;
in total obtain
Figure BDA0000961691990000081
Combining the products after redistribution into an integer solution;
Figure BDA0000961691990000082
is a product pk(ii) market demand;
calculating the system effective output corresponding to each redistributed product combination integer solution; and solving the corresponding product combination integer which is redistributed by the maximum system effectively into the optimal product combination.
Advantageous effects
The traditional TOCh method and the revised TOCh method directly discard the decimal part appearing in the product combination solving process, and transfer the bottleneck resource occupied by the decimal part, namely the residual capacity low-priority product. The invention designs the residual capacity redistribution scheme, and can obtain a better residual capacity processing mode than the traditional TOCh method and the revised TOCh method, thereby obtaining the optimal product combination optimization result and obtaining higher effective output.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1: traditional TOCh method flow diagram;
FIG. 2: revising the TOCh method flow diagram;
FIG. 3: the method of the invention is a flow chart.
Detailed Description
The following detailed description of embodiments of the invention is intended to be illustrative, and is not to be construed as limiting the invention.
In this embodiment, the enterprise production algorithm still uses the algorithm described in table 1:
description of the examples
Figure BDA0000961691990000083
Aiming at the above calculation, the method for solving the product combination optimization problem based on residual capacity redistribution comprises the following steps:
step 1: identifying system bottleneck resources:
for the resource j, j is 1,2, …, m, m is the number of resources in the system; calculating the difference d between the required capacity and the available capacityjIf d isjIf < 0, then the resource j is considered to be the bottleneck resource:
Figure BDA0000961691990000091
in the formula, CPjIndicates the available capacity of resource j, DiIndicates the market demand of the ith product, i is 1,2, …, n, n is the number of product types, tijRepresenting the time that resource j is occupied by unit product i.
Figure BDA0000961691990000092
I.e. the device resource R is a bottleneck resource.
Step 2: determining the product priority:
calculating the effective output TP of product iiAnd the processing time t of the bottle neck resourceiTo obtain the effective output r per unit timei,BN
Figure BDA0000961691990000093
And (3) according to whether the product is processed on the bottleneck resource and the effective output of the product in unit time, carrying out priority sequencing on the product to obtain a product sequence P:
for tiProduct using bottleneck resource greater than 0, effective in unit timeYield ri,BNIn descending order, if ri,BNSame, press TP againiArranging in descending order; for tiProduct without bottleneck resource, per TP ═ 0iThe descending order is behind the products that use the bottleneck resource.
Figure BDA0000961691990000094
rB,R＝2.6833，rC,R＝4.2877，r D,R1. Because of this, it is possible to reduce the number of the,
Figure BDA0000961691990000095
d is in R-priority order: a. the>C>B>D. The sequence P is P ═ (a, C, B, D).
And step 3: sequentially distributing bottleneck resources according to the product priority:
giving all resources to the product with the highest priority, and if the quantity of the resources is enough to meet the market demand of the product with the highest priority, transmitting the rest resources to the products with low priorities in sequence after meeting the market demand; if the resource is not enough to produce the quantity required by the market when the resource is transmitted to a certain product, the real number solution of the product can be obtained; and when the production quantity of all products reaches the quantity required by the market or the resources are exhausted, obtaining a product combination real number solution.
And scheduling the products according to the priority ratio in a non-ascending order to obtain the result that the product A reaches the market demand of 50 and 10 products C. The solving process is as follows:
A:50，
Figure BDA0000961691990000106
the number of C is 10.274, which is less than 30 market demands. Thus, definition of J-2, QJ＝10。
And 4, step 4: determining whether a real solution contains a non-integer solution:
and if the product combination real number solution does not have the non-integer solution, the obtained product combination real number solution is the optimal solution, and the method is ended, otherwise, the following steps are carried out.
The real number of the product C obtained from step 3 is 10.274, i.e. there is a decimal solution, so the next step is performed.
And 5: solving the residual capacity:
the non-integer solution in the product combination real number solution corresponds to a product J, and after the non-integer solution of the product J is rounded downwards, the residual capacity t on the bottleneck resource is calculatedleft(ii) a I.e. the part of the bottleneck resource occupied by the fractional part of the real solution.
And rounding the decimal solution downwards, and calculating the bottleneck resource occupied by the original decimal part. Taking C as 10, the product combination is changed from 50A, 0B, 10.274C, 0D to 50A, 0B, 10C, 0D. The remaining time is tleft＝20。
Step 6: constructing a feasible set X:
for the first product P from the sequence P1To J-1 product pJ-1Calculating
Figure BDA0000961691990000102
Figure BDA0000961691990000103
Get the first to satisfy
Figure BDA0000961691990000104
Product I of (1), then X ═ pI,pI+1,pI+2…pJ-1}; if X is an empty set, the resource t is leftleftThe products following product J are assigned according to the order and market demand in sequence P, the method ends, otherwise the following steps are performed.
Figure BDA0000961691990000105
Thus, I is 1, the set of feasible solutions X is P1}。
And 7: residual capacity reallocation:
for each product X (k) in the rowset X, the following calculation is performed:
the production quantity of the product X (k) is reduced by s, s is 1,2, …, DX(k)The excess resource s x tX(k)And remaining resources tleftThe sum is in the order of the sum in the sequence PThe market demand is distributed to the product J and the products behind the product J in an integer solution mode to obtain a product combination integer solution after redistribution; wherein DX(k)Is the market demand, t, of product X (k)X(k)Is the processing time of unit product X (k) on the bottleneck resource;
in total obtain
Figure BDA0000961691990000111
Combining the products after redistribution into an integer solution;
Figure BDA0000961691990000112
is a product pk(ii) market demand;
calculating the system effective output corresponding to each redistributed product combination integer solution; and solving the corresponding product combination integer which is redistributed by the maximum system effectively into the optimal product combination.
Table 3 gives the integer solutions of all redistributed product combinations and their effective outcomes:
Figure BDA0000961691990000113
from the results in Table 3, the final product portfolio optimization run changed from 50A, 0B, 10C, 0D to 46A, 0B, 12C, 0D. The effective output is 10472, which is better than 10448 of the revised TOCh algorithm and is consistent with the solving result of ILP. The optimal product combination obtained by the method is 46A, 0B, 12C and 0D. Consistent with the results obtained by ILP, the proposed method is shown to be advantageous over the conventional TOCh method and the modified TOCh method in some cases.
The inventor firstly realizes that the traditional TOCh method and the revised TOCh method have failure conditions in the solving process, provides a brand-new residual capacity redistribution algorithm aiming at the failure conditions, can obtain the optimal product combination adjustment scheme through the residual capacity redistribution scheme provided in the solving algorithm, and adjusts the existing product combination according to the adjustment scheme, thereby effectively reducing the resource waste of enterprises, improving the income of the enterprises and increasing the effective output of the enterprises. Therefore, the residual capacity redistribution algorithm method provided by the invention is a good method for solving the combination of single bottleneck products.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.

Claims (1)

1. A solution to the problem of product combinatorial optimization based on residual capacity redistribution is characterized in that: the method comprises the following steps:
step 1: identifying system bottleneck resources:
for the resource j, j is 1,2, …, m, m is the number of resources in the system; calculating the difference d between the required capacity and the available capacityjIf d isjIf < 0, then the resource j is considered to be the bottleneck resource:
Figure FDA0000961691980000011
in the formula, CPjIndicates the available capacity of resource j, DiIndicates the market demand of the ith product, i is 1,2, …, n, n is the number of product types, tijRepresenting the time of the unit product i occupying the resource j;
step 2: determining the product priority:
calculating the effective output TP of product iiAnd the processing time t of the bottle neck resourceiTo obtain the effective output r per unit timei,BN
Figure FDA0000961691980000012
And (3) according to whether the product is processed on the bottleneck resource and the effective output of the product in unit time, carrying out priority sequencing on the product to obtain a product sequence P:
for tiThe effective output r of the product using bottleneck resource is more than 0 according to unit timei,BNIn descending order, if ri,BNSame, press TP againiArranging in descending order; for tiProduct without bottleneck resource, per TP ═ 0iArranged in descending order after a product using a bottleneck resource;
and step 3: sequentially distributing bottleneck resources according to the product priority:
giving all resources to the product with the highest priority, and if the quantity of the resources is enough to meet the market demand of the product with the highest priority, transmitting the rest resources to the products with low priorities in sequence after meeting the market demand; if the resource is not enough to produce the quantity required by the market when the resource is transmitted to a certain product, the real number solution of the product can be obtained; when the production quantity of all products reaches the quantity required by the market or the resources are exhausted, obtaining a product combination real number solution;
and 4, step 4: determining whether a real solution contains a non-integer solution:
if the product combination real number solution does not have a non-integer solution, the obtained product combination real number solution is an optimal solution, and the method is ended, otherwise, the following steps are carried out;
and 5: solving the residual capacity:
the non-integer solution in the product combination real number solution corresponds to a product J, and after the non-integer solution of the product J is rounded downwards, the residual capacity t on the bottleneck resource is calculatedleft；
Step 6: constructing a feasible set X:
for the first product P from the sequence P1To J-1 product pJ-1Calculating
Figure FDA0000961691980000021
i-1, 2, …, J-1, the first one being obtained
Figure FDA0000961691980000022
Product I of (1), then X ═ pI,pI+1,pI+2…pJ-1}; if X is an empty set, then resources will remaintleftDistributing the products after the product J according to the sequence in the sequence P and the market demand, and ending the method, otherwise, performing the following steps;
and 7: residual capacity reallocation:
for each product X (k) in the rowset X, the following calculation is performed:
the production quantity of the product X (k) is reduced by s, s is 1,2, …, DX(k)The excess resource s x tX(k)And remaining resources tleftThe sum is distributed to a product J and products behind the product J in an integer solution mode according to the sequence in the sequence P and market requirements to obtain a redistributed product combination integer solution; wherein DX(k)Is the market demand, t, of product X (k)X(k)Is the processing time of unit product X (k) on the bottleneck resource;
in total obtain
Figure FDA0000961691980000023
Combining the products after redistribution into an integer solution;
Figure FDA0000961691980000024
is a product pk(ii) market demand;
calculating the system effective output corresponding to each redistributed product combination integer solution; and solving the corresponding product combination integer which is redistributed by the maximum system effectively into the optimal product combination.
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