A method for minimizing the loss of optimization of the construction period assignment

Abstract

This invention proposes an optimization method for minimizing downtime losses by considering schedule assignment. By establishing a scheduling model that minimizes downtime losses while considering schedule assignment, the characteristics of the optimal solution are analyzed, narrowing the range of optimal schedule assignment values ​​and obtaining the optimal schedule assignment scheme under a given production schedule. Based on the optimal schedule assignment scheme, the complexity of the problem is determined, and optimal algorithms in polynomial time are given for both common schedule scenarios and scenarios with equal processing time. Starting from practical needs, this invention extends the scope from known schedules to schedule decision-making, and from traditional delay penalties to downtime loss penalties, providing a theoretical basis for schedule decision-making and downtime loss optimization in actual production.

Description

Technical Field

[0001] This invention relates to the field of scheduling optimization in product manufacturing processes, specifically to an optimization method that minimizes downtime losses while considering project schedule assignment. Background Technology

[0002] The Due Date Assignment problem extends the assumption of a known parameter in classic scheduling problems by treating the due date as a decision variable. It optimizes the process along with the scheduling scheme to minimize penalties related to due date, such as late work, early arrival, and tardiness. A typical application scenario is Just-in-Time (JIT) manufacturing. A closer due date attracts more orders but carries the risk of late payment fees; a further due date guarantees higher completion rates but increases inventory costs. Therefore, due date assignment needs to balance the benefits of due date commitments with the penalties for late arrival, aiming to shorten order durations as much as possible while avoiding late payment costs. Based on the correlation between the project duration and the workpiece, the project duration assignment problem is divided into common due date assignment model and different due date assignment model: different due date assignment means that each workpiece has a different due date and the allocation of the due date is not restricted in any way; common due date assignment means that all workpieces have the same due date.

[0003] In a production system, downtime loss is used to measure the processing loss of the workpiece after the completion date. Essentially, it defines an upper bound for the traditional tardiness penalty, which is equal to the workpiece processing time. Unlike tardiness, downtime loss is at most the length of the workpiece being processed and does not increase with the completion time, while tardiness increases with the completion time. Typical application scenarios for downtime loss include: (1) Compensation for delayed orders in production delivery. Orders that are completed late need to pay a certain penalty fee. This penalty fee will increase with the extension of the delay, but will not exceed the value of the overdue order itself; (2) Loss of delayed information in computer control systems (CCS). Information collected by sensors that arrives before the start of control calculation is considered valid information. If there is input delay information, it is considered invalid information. Delay information leads to the loss of calculation information, and the amount of loss does not exceed the carrying capacity of the information itself; (3) Loss of expired products in the sale of perishable goods. Products that have exceeded their shelf life will be directly removed from the shelves, and the resulting economic loss is the value of the expired products. Summary of the Invention

[0004] To address the problems existing in the prior art, this invention proposes an optimization method that minimizes downtime losses by considering schedule assignment. Given a production sequence π, the optimal schedule assignment scheme is determined, and the optimal schedule assignment and workpiece scheduling schemes are given for the cases of common schedule and equal processing time, respectively.

[0005] The technical solution of this invention is as follows:

[0006] The aforementioned optimization method for minimizing downtime losses considering project schedule assignment includes the following steps:

[0007] Step 1: Establish an optimization model that minimizes downtime losses while considering project schedule assignment:

[0008] Given n independent workpieces J = {J1, J2, ..., J...} n Processed on a single machine; each workpiece J j They all have a fixed processing time p j Acceptable lead time A when the customer submits the order j And the promised lead time d from the manufacturer based on production capacity feedback. j j = 1, ..., n; When the promised construction period exceeds the customer's acceptable construction period, it will cause reputational damage due to the inability to meet customer needs, resulting in construction period assignment costs R. j =max{d j -A j If the promised construction period is too short, the workpiece may not be completed within the scheduled time, resulting in lost work time and losses. j =min{max{C j -d j ,0},p j The optimization objective of the model is to minimize the weighted total duration assignment cost and downtime loss; C j For workpiece J j Completion time;

[0009] Step 2: Given the production scheduling scheme π, determine the optimal schedule assignment scheme:

[0010] For a given production scheduling scheme π, there exists an optimal schedule assignment scheme such that for any workpiece J j The optimal project duration d for ∈J j A is 0 j Or C j ;in

[0011] If C j ≤A j Then workpiece J j The optimal project duration d j =C j ;

[0012] If Aj <C j j +p j If α j ≥β j workpiece J j The optimal project duration d j =A j If α j <β j workpiece J j The optimal project duration d j =C j ;where α j For workpiece J j The weight of the time assignment cost, β j For workpiece J j Weighting of lost work time;

[0013] If A j +p j ≤C j If Workpiece J j The optimal project duration d j =0, if Workpiece J j The optimal project duration d j =C j ;

[0014] By merging the schedule assignment schemes under different scenarios, the optimal schedule assignment scheme D is obtained. * (π):

[0015] If α j ≥β j workpiece J j The optimal construction period is

[0016]

[0017] If α j <β j workpiece J j The optimal construction period is

[0018]

[0019] Step 3: For situations with shared project durations:

[0020] For a given common project duration d, the optimal order of workpiece production is π. * For β j Non-incremental order;

[0021] And for the optimal workpiece scheduling sequence π * ,if Then the optimal common construction period​ Otherwise, d * =l k For the optimal common construction period, where l k Satisfy f(l) k-1 )+g(l k-1 )≤0 and f(l k )+g(l k )≥0;

[0022] f(d) represents The slope, g(d) represents Let the slope be A′1,…,A′. n For A1,…,A n The non-subtractive order, α′ j For A′ j Given the weights of the corresponding workpieces, then:

[0023]

[0024] and

[0025]

[0026] For situations involving equal processing time:

[0027] The problem is solved using a linear assignment model: if workpiece J j If processing is performed at the i-th position, its completion time is C. j =ip, and the corresponding objective function value is w. ij =min{max{α j (ip-A j ),0},β j p j Define decision variable x ij :

[0028]

[0029] Linear assignment model:

[0030]

[0031] st

[0032]

[0033]

[0034] By solving the linear assignment model, the optimal workpiece scheduling sequence for the case of equal processing time is obtained.

[0035] Beneficial effects

[0036] This invention proposes a method for minimizing downtime losses by considering schedule assignment. By establishing a scheduling model that minimizes downtime losses while considering schedule assignment, the characteristics of the optimal solution are analyzed, narrowing the range of optimal schedule assignment values ​​and obtaining the optimal schedule assignment scheme under a given production schedule. Based on the optimal schedule assignment scheme, the complexity of the problem is determined, and optimal polynomial-time algorithms are given for both common schedule scenarios and scenarios with equal processing time. Starting from practical needs, this invention extends the scope from known schedules to schedule decision-making, and from traditional delay penalties to downtime loss penalties, providing a theoretical basis for schedule decision-making and downtime loss optimization in actual production.

[0037] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0038] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0039] Figure 1 Z(π) * The V-shaped relationship between ,d) and the construction period d. Detailed Implementation

[0040] In a production system, to minimize downtime losses, the following optimization process needs to be considered:

[0041] Determining a reasonable promised lead time is crucial. If a company promises a lead time far exceeding the customer's acceptable lead time, it risks losing that customer. Conversely, promising a lead time that is too short might attract customers but incur penalties for late delivery. The key is optimizing the workpiece scheduling to minimize these losses. In practice, optimizing workpiece scheduling seems to favor earlier processing for shorter lead times and later processing for longer lead times. However, to quantify this trade-off, numerous influencing factors must be considered, such as workpiece processing time, the customer's acceptable lead time, and the size of the promised lead time. Furthermore, this optimization problem requires assessing its complexity—theoretically determining if it is NP-hard—and then designing approximate / optimal solution algorithms.

[0042] Specifically, the optimization method for minimizing downtime losses considering schedule assignment proposed in this invention starts from actual needs, extending the scope from known schedule to schedule decision-making, and from traditional delay penalties to downtime loss penalties, in order to solve the difficult problems of schedule decision-making and downtime loss optimization in actual production. For the two-layer scheduling optimization problem considering schedule assignment decision-making and workpiece processing scheduling, a scheduling model for minimizing downtime losses considering schedule assignment costs is established. Through problem characteristic analysis, the optimal schedule assignment scheme is given, and the complexity of the problem is assessed. For cases with shared schedules and equal processing times, the optimal workpiece scheduling schemes are given respectively.

[0043] The specific problem model is described as follows:

[0044] Given n independent workpieces J = {J1, J2, ..., J...} n Processed on a single machine. Each workpiece J j They all have a fixed processing time p j Acceptable lead time A when the customer submits the order j And the promised lead time d from the manufacturer based on production capacity feedback. j Let j = 1, ..., n. When the promised construction period exceeds the customer's acceptable timeframe, it will cause reputational damage due to the inability to meet customer needs, resulting in a construction period assignment cost R. j =max{d j -A j However, if the promised lead time is too short, the workpiece may not be completed within the scheduled time, resulting in lost work time and losses. j =min{max{C j -d j ,0},p j The objective is to minimize the weighted total duration assignment cost and downtime.

[0045] Specifically, for the schedule decision problem, for any given production scheduling plan, analyze the workpiece (i.e., C) that is completely ahead of schedule. j ≤d j ), partially pre-workpiece (i.e., d) j <C j <d j +p j ) and completely misoperated workpieces (i.e., d) j +p j ≤C j By finding the extreme values ​​of the project schedule assignment for |DIF|Z(π,D), we obtain the optimal project schedule assignment scheme D for Problem 1. * .

[0046] The problem is then transformed into minimizing the weighted total delay penalty, and the complexity of problem 1|DIF|Z(π,D) is determined.

[0047] Finally, regarding the workpiece scheduling problem, specifically addressing the common schedule issue 1|d j =d|Z(π,D) and the problem of equal processing time 1|p j =p|Z(π,D), which gives the optimal schedule assignment and workpiece production plan respectively.

[0048] The symbols required in this article are shown in Table 1.

[0049] Table 1. Symbol Explanation

[0050]

[0051]

[0052] The detailed analysis process is given below:

[0053] (I) Providing theoretical basis:

[0054] For any given production scheduling scheme, there exists an optimal schedule assignment scheme that ensures that for any workpiece J... j The optimal project duration d for ∈J j A is 0 j Or C j .

[0055] Proof: Let π be any given production scheduling scheme. Let D = (d1, ..., d2) n The optimal schedule assignment scheme is J. We will discuss J separately. j For a fully pre-built workpiece (in d) j (Completed previously), some pre-workpieces (in d) j and d j +p j Completed between d) and completely misused workpieces (in d) j There are three scenarios: (after completion)

[0056] Case 1. Workpiece J j For a completely pre-built workpiece.

[0057] If workpiece J j For a fully pre-planned workpiece (C) j <d j ), then d j -C j =Δ>0. Let D ′ To modify only workpiece J j The new schedule assignment scheme, new schedule d ′ =C j We have Z(π,D)-Z(π,D)′ ) = Z j (d j =C j +Δ)-Z j (d j =C j )=α j max{C j +Δ-A j ,0}-max{C j -A j ,0}. If C j ≥A j Z(π,D)-Z(π,D) ′ )=α j Δ≥0. If C j j Then Z(π,D)-Z(π,D) ′ )=α j max{C j +Δ-A j ,0}≥0. In both of the above cases, Z(π,D)-Z(π,D) ′ Since all ) are non-negative, we let d j =C j It will not increase the value of the objective function.

[0058] Scenario 2. Workpiece J j For a part of the workpiece to be prepared in advance.

[0059] If workpiece J j For a partially completed workpiece, then d j ∈[0,C j -p j Project schedule assignment penalty α j max{d j -A j ,0} about d j In [0, C j -p j [The inner is not reduced. And because when d...] j In [0, C j -p j Lost work time Y j =β j min{max{C j -d j},0},p j} is a constant, so we let d j =0 will not increase the value of the objective function.

[0060] Scenario 3. Workpiece J j This is a completely unusable workpiece. ​

[0061] If workpiece J j For a completely misoperated workpiece (C) j >d j +p j ), making 0 <C j -d j =Δ <p j We assert that C j -p j ≤A j <C j We will prove this by contradiction. If A j ≥C j We have Z j (d j =C j ) = 0 <Z j (d j =C j -Δ)=β j Δ contradicts the fact that D is the optimal schedule assignment scheme. If A j <C j We have Z j (d j =A j )=β j p j Z j (d j =C j -Δ)=α j (C j -Δ-A j )+β j Δ and Z j (d j =C j )=α j (C j -A j When β j ≤α j Z j (d j =C j -Δ)≥β j (C j -Δ-A j )+β j Δ=β j (C j -A j )>β j p j =Z j (d j =A j This contradicts the fact that D is the optimal schedule assignment scheme. When β j >α j Zj (d j =C j -Δ)=α j (C j -Δ-A j )+β j Δ>α j (C j -A j ) = Z j (d j =A j This contradicts the fact that D is the optimal schedule assignment scheme.

[0062] Therefore, the assertion is true.

[0063] Furthermore, we assert that for partially advanced workpiece β j ≤α j By contradiction, if β j >α j So Z j (d j In the interval [C j -p j C j [Within] about d j Decreasing, therefore Z(d) j =C j ) <Z j (d j =C j -Δ), which contradicts the fact that D is the optimal schedule assignment scheme.

[0064] Therefore we have C j -p j ≤A j <C j and β j ≤α j Z j (d j In the interval [C j -p j A j [Within] about d j Decreasing, in the interval [A] j C j [Within] about d j Non-subtractive. Thus, let d... j =A j It will not increase the value of the objective function.

[0065] Based on the above theoretical basis, compare Z j (d j =0), Z j (d j =A j ) and Zj (d j =C j The optimal project schedule can be obtained by calculating the value of ).

[0066] (two) Given a production scheduling plan Through discussion of completion time C j With A j and A j +p j Z under different relationships j (d j =0), Z j (d j =A j ) and Z j (d j =C j The value of ) is used to provide the corresponding schedule assignment scheme:

[0067] (1) If C j ≤A j Then workpiece J j The optimal project duration d j =C j .

[0068] (2) If A j <C j j +p j If α j ≥β j workpiece J j The optimal project duration d j =A j If α j <β j workpiece J j The optimal project duration d j =C j .

[0069] (3) If A j +p j ≤C j If Workpiece J j The optimal project duration d j =0; if Workpiece J j The optimal project duration d j =C j .

[0070] prove:

[0071] Property (1). When C j ≤A j We have Z j (d j ​=0)=β j p j Z j (d j =C j ) = 0 and Z j (d j =A j Z = 0. j (d j =C j ) = Z j (d j =A j ) <Z j (d j =0), so let d j =C j It is the optimal one.

[0072] Property (2). When A j ≤C j ≤A j +p j We have Z j (d j =0)=β j p j Z j (d j =C j )=β j (C j -A j ) and Z j (d j =C j )=α j (C j -A j ).

[0073] If α j ≥β j Z j (d j =A j ) <Z j (d j =0) and Z j (d j =A j )≤Z j (d j =C j Therefore, let d j =A j Optimal.

[0074] If α j <β j Z j (d j =C j)≤Z j (d j =A j ) <Z j (d j =0), therefore let d j =C j Optimal.

[0075] Property (3). When A j +p j ≤C j We have Z j (d j =0)=Z j (d j =A j )=β j p j Z j (d j =C j )=α j (C j -A j ).

[0076] like Z j (d j =0)=Z j (d j =A j ) <Z j (d j =C j Therefore, let d j =0 is optimal.

[0077] If α j <β j Z j (d j =C j )≤Z j (d j =0) <Z j (d j =A j Therefore, let d j =C j Optimal.

[0078] (III) Based on the analysis in step (II), by combining the schedule assignment schemes under different scenarios, we can obtain that for problem 1|DIF|Z(π,D), for any given workpiece scheduling scheme π, there exists an optimal schedule assignment scheme.

[0079] If α j ≥β j workpiece J jThe optimal assignment period is

[0080]

[0081] If α j <β j workpiece J j The optimal assignment period is

[0082]

[0083] (iv) Based on the optimal schedule assignment scheme D in step (iii) * (π), we can obtain the objective function value in,

[0084] If α j ≥β j ,

[0085]

[0086] If α j <β j ,

[0087]

[0088] (V) The problem is transformed into in Or it can be written as a piecewise function:

[0089]

[0090] Proof: In step (iv), it is observed that for α j ≥β j The workpiece, α j The value of α does not affect the target value, therefore we can apply it to all α values. j ≥β j The workpiece is reset α j ′ =β j Thus, we obtain a new instance that satisfies α. j ′ ≤β j For j = 1, ..., n. Because these two problems have the same objective function value, they can be simplified to the formula in step 5.

[0091] (vi) Because of the problem It is strongly NP-hard, and as a corollary problem, 1|DIF|Z(π,D) is also strongly NP-hard.

[0092] Proof: Consider the problem Examples: We have so,

[0093]

[0094] Because in this scenario, the problem is equivalent to minimizing the weighted delay penalty on a single machine, i.e. Where T j =max{0,C j -d j}.because It is strongly NP-hard, so our problem is also NP-hard.

[0095] (vii) Study on the case of shared construction period 1|d j =d|Z(π,D), that is, d j Given d, j = 1, ..., n, and provide the optimal algorithm with time complexity O(n log n).

[0096] (1) Optimal production scheduling scheme:

[0097] For any given common duration d, the optimal order of workpiece scheduling π * For β j The order of non-incrementing.

[0098] Proof: The objective function we are considering For any given common duration d, Since it is a constant, our objective is to minimize Because of the problem The optimal order is β j Since the order is not incremental, the optimal order for workpiece scheduling is β. j The order of non-incrementing.

[0099] (2) Nature of the problem:

[0100] If the workpiece production schedule is based on β j If production is not incrementally ordered, then Z(π) * ,d) in the interval The inner decreases with respect to d, or as... Figure 1 The figure shows a V-shape that first decreases and then increases, where l1,…,l p (p≤2n) are C1,…,C n ,A1,…,A n The non-decreasing order.

[0101] Proof: Assume the workpiece follows β j Relabel the non-decreasing orders, f(d) represents The slope, g(d) represents The slope of the slope. Let A′1,…,A′ n For A1,…,A n The non-subtractive order, α′ j For A′ j The weights of the corresponding workpieces. We have property 1:

[0102]

[0103] Property 2:

[0104]

[0105] Regarding property 1, if d∈[A′] j ,A′ j+1 ), j=1,…,n-1, then Therefore, for d∈[A′ j ,A′ j+1 ), j=1,…,n-1,have If d∈[0,A′1), f(d)=0. If d∈[A′1), f(d)=0. n ,∞), Therefore, property 1 holds true.

[0106] Regarding property 2, if d∈[C] j C j+1 ), j=1,…,n-1, then Therefore, for d∈[C] j C j+1 For j = 1, ..., n-1, we have g(d) = -β j+1 If d∈[0,C1), g(d)=-β1. If d∈[C1,C2,C3,C4,G(d)=-β1. n Since (∞) and g(d) = 0, property 2 holds.

[0107] According to properties 1 and 2, we have f(0) + g(0) = -β1, and f(d) + g(d) in the interval [0, ... The interior is non-decreasing with respect to d. Therefore, if Then Z(π) * ,d) in the interval The inner value decreases with respect to d; if Then Z(π) * ,d) in the interval The inner diameter of d is V-shaped.

[0108] (3) Optimal assigned construction period.

[0109] For the optimal workpiece scheduling sequence π * ,if Then the optimal common construction period Otherwise, d * =l k For the optimal common construction period, where l k Satisfy f(l) k-1 )+g(l k-1 )≤0 and f(l k )+g(l k )≥0.

[0110] (4) Algorithm complexity analysis.

[0111] The optimal production scheduling scheme determined in (1) is arranged in non-increasing order for β. j The time taken is O(nlog n). The optimal common duration is determined in (2) and (3). Here, the binary search method is used to find the optimal common duration, which takes O(log n). Therefore, the time complexity of solving the problem is O(nlog n).

[0112] (viii) Study on the processing time of workpieces, etc. 1|p j =p|Z(π,D), that is, p j =p,j=1,…,n, and give O(n) 3 The optimal algorithm for time.

[0113] (1) Problem transformation.

[0114] Based on step (5), we will address the problem. Transform to Solve the problem.

[0115] (2) The problem is characterized as a linear assignment model.

[0116] Since the processing time for all workpieces is p, the completion time of the workpiece is C. j ∈{p,2p,…,(n-1)p,np}. If workpiece J j If processing is performed at the i-th position, its completion time is C. j =ip, and the corresponding objective function value is w. ij =min{max{α j (ip-A j ),0},β j p j Define the decision variable x. ij :

[0117]

[0118] The problem is transformed into a linear assignment model:

[0119]

[0120] satisfy

[0121]

[0122]

[0123] x ji ∈{0,1},1≤j≤n,1≤i≤n.

[0124] (3) Because the linear assignment model can achieve O(n 3 The problem can be solved within the range of ) and p, so problem 1|p j =p|Z(π,D) can also be achieved in O(n 3 Solve within a time limit of (n). 3 Solve the linear assignment model within the range of ) to obtain the problem The optimal production sequence for workpieces.

[0125] Through the above research process, the present invention mainly achieves:

[0126] (1) Given the production scheduling sequence π, the optimal schedule assignment scheme;

[0127] (2) Determine if the problem is strongly NP-hard;

[0128] (3) For the cases of shared project duration and equal processing time, O(nlog n) and O(n log n) time are given respectively. 3 The optimal algorithm for time.

[0129] Specifically, for (1), the completion time C of each workpiece can be calculated based on the production scheduling sequence π. j ,j=1,…,n, and then execute the optimal project schedule allocation according to the formula in step 3.

[0130] Regarding (2), the problem is reduced to a known strongly NP-hard problem of minimizing weighted delay penalties, i.e. Based on this, it was determined that 1||Z(π,D) is strongly NP-hard.

[0131] Regarding (3), for the case of the same construction period 1|d j =d|Z(π,D), first according to β j The non-increasing order is used to sort the data, and then the optimal common duration is determined using a binary search method. The correctness of this method is proven. For the case of equal processing time 1|p j =p|Z(π,D), first transform it to Then the problem was reduced to a linear assignment model for solution, and its correctness was proven.

[0132] 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. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method of minimizing lost work optimization considering schedule assignment, characterized by: Includes the following steps: Step 1: Establish an optimization model that minimizes downtime losses while considering project schedule assignment: Given Individual workpieces Processed on a single machine; each workpiece They all have fixed processing times. Acceptable lead time when the customer submits the order And the promised lead time from the manufacturer based on production capacity feedback. , When the promised construction period exceeds the customer's acceptable timeframe, it will result in reputational damage due to the inability to meet customer needs, and incur construction period assignment costs. If the promised lead time is too short, the workpiece may not be completed within the scheduled time, resulting in lost work time and losses. The optimization objective of the model is to minimize the weighted total duration assignment cost and downtime loss. For workpiece Completion time; Step 2: Given a production scheduling plan Next, determine the optimal schedule assignment scheme: For a given production scheduling plan There exists an optimal schedule assignment scheme that ensures that any workpiece... Optimal project duration for or ;in if Then the workpiece Optimal project duration ; if If workpiece Optimal project duration ,like workpiece Optimal project duration ;in For workpiece Weighting of project schedule allocation costs For workpiece Weighting of lost work time; if If workpiece Optimal project duration ,like workpiece Optimal project duration ; By merging the schedule assignment schemes, the optimal schedule assignment scheme is obtained. ; Step 3: For situations with shared project durations: For a given common duration Optimal order of workpiece production for Non-incremental order; And for the optimal workpiece scheduling sequence ,if Then the optimal common construction period ; otherwise, For the optimal common construction period, among which satisfy and ; represent The slope, represent The slope, denoted as for The order of non-subtraction, for Given the weights of the corresponding workpieces, then: and For situations involving equal processing time: Solving the problem using a linear assignment model: If the workpiece In the If each location is processed, its completion time is The corresponding objective function value is Define decision variables : Linear assignment model: st By solving the linear assignment model, the optimal workpiece scheduling sequence for the case of equal processing time is obtained.

2. The optimization method for minimizing downtime loss considering project schedule assignment as described in claim 1, characterized in that: In step 2, the optimal schedule assignment scheme for: if workpiece The optimal construction period is if workpiece The optimal construction period is 。

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