外文翻譯--混合整數非線性規(guī)劃（MINLP）方式的滾子和滑門結構

外文翻譯原文 ABSTRACT Part III of this three-part series of papers describes the synthesis of roller and sliding hydraulic steel gate structures performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP approach enables the determination of the optimal number of gate structural elements (girders, plates), optimal gate geometry, optimal intermediate distances between structural elements and all continuous and standard crossectional sizes. For this purpose, special logical constraints for topology alterations and interconnection relations between the alternative and fixed structural elements are formulated. They have been embedded into a mathematical optimization model for roller and sliding steel gate structures GATOP. GATOP has been developed according to a special MINLP model formulation for mechanical superstructures (MINLP-MS), introduced in Parts I and II. The model contains an economic objective function of self-manufacturing and transportation costs of the gate. As the GATOP model is non-convex and highly non-linear, it is solved by means of the Modified OA/ER algorithm accompanied by the Linked Two-Phase MINLP Strategy, both implemented in the TOP computer code. An example of the synthesis is presented as a comparative design research work of the already erected roller gate, the so-called Intake Gate in Aswan II in Egypt. The optimal result yields 29)4 per cent of net savings when compared to the actual costs of the erected gate. . 1998 John Wiley & Sons, Ltd. KEY WORDS: structural synthesis； optimization； topology optimization； discrete variable optimization； Mixed-Integer Non-linear Programming； MINLP； the Modified OA/ER algorithm； MINLP strategy； hydraulic gate； sliding gate； roller gate； Aswan 1. INTRODUCTION This paper describes the Mixed-Integer Non-linear Programming (MINLP) approach to the synthesis of roller and sliding gate structures, i.e. the simplest types among vertical-lift hydraulic steel gates, see Figure 1. Roller and sliding gates are also regarded as the most frequently manufactured types of hydraulic steel gates for headwater control. They are used to regulate the water stream on hydro-electric plants, dams or spillways. As hydraulic steel gates are very special structures, only a few authors have discussed their optimization, e.g. Kravanja et al.,1D. Jongeling and Kolkman. as well as Almquist et al. Particular interest was shown in the optimization not of these (roller and sliding gates) but of similar structures. In such investigations, Vanderplaats and Weisshaar. as well as Gurdal et al. optimized stiffened laminated composite panels, Butler. and Ringertz. optimized stiffened panels, Farkas and Jarmai1. optimized welded rectangular cellular plates, Finckenor et al.1. treated skin stringer cylinders and Gendy et al.1. stiffened plates. Almost all authors used Non-linear Programming (NLP) techniques. Gurdal et al. proposed the genetic algorithm, while Kravanja et al.1D. introduced MINLP algorithms and strategies to the simultaneous topology and parameter optimization of the gate. In Parts I of this three-part series of papers, a general view of the MINLP approach to the simultaneous topology and parameter optimization of structures is presented. Part II describes the extension to the simultaneous standard dimension optimization. Based on the superstructure approach defined in Parts I and II, the main objective of this paper (Part III) is the MINLP synthesis of roller and sliding hydraulic steel gate structures, obtained at minimal gate costs and subjected to defined design, material, stress, deflection and stability constraints. As the MINLP approach enables simultaneous topology, parameter and standard dimension optimization, a number of gate structural elements (girders and plates), the gate global geometry, intermediate distances between structural elements and all continuous and standard dimensions are obtained simultaneously. This last part of the three-part series of papers is divided into three main sections: 1. Section 2 describes how different topology and standard dimension alternatives are postulated and how their interconnection relations are formulated by means of explicit logical constraints in order to perform topology and standard dimension alterations within the optimization procedure. 2. Section 3 represents a general MINLP optimization model for roller and sliding gate structures GATOP. 3. Finally, in Section 4, the proposed superstructure MINLP approach is applied to the synthesis of an already erected roller gate, the so-called Intake Gate in Aswan II in Egypt. 2. SUPERSTRUCTURE ALTERNATIVES AND THE MODELLING OF THEIR DISCRETE DECISIONS 2.1. Postulation of topology and standard dimension alternatives The first step in the synthesis of the gate is the generation of an MINLP superstructure in which different topology/structure and standard dimension alternatives are embedded to be selected as the optimal result. The gate superstructure also contains the representation of structural elements which may construct each possible structure alternative as well as different sets of discrete values, defined for each standard dimension alternative. As both the roller and sliding gates have almost the same structure, it was reasonable to propose a superstructure, which could well be useful for both of them. 2.1.1. Topology alternatives The gate superstructure typically includes a representation of main gate elements, where each gate element is composed of various structural elements, such as horizontal girders, vertical girders, stiffeners and plate elements of the skin plate, see Figure 2. The superstructure comprises n main gate elements, n3N, each containing m horizontal girders, m3M, the (3#2v) number of vertical girders through the entire gate, v3., and the corresponding (m!1)(2#2v) number of skin-plate elements. To each mth horizontal girder of the nth main gate element an extra binary variable yn,m is assigned. The number of horizontal girders and corresponding plate elements of the skin plate, distributed over the nth main gate element, can therefore be determined simply by,m n my. Note that the proposed minimal number of identical vertical girders is 3 and that they can take only odd numbers. If a binary variable yv is assigned to each v, v V., the number of vertical girders can be obtained by (3+2 vyv). In the same way an even number (2+2 . vyv) of equal horizontal partitions of the entire gate is proposed. In the case of vertical girders, we can see that the structural elements can also be determined by suitable linear combinations of binary variables. Among the maximal number Mmax of horizontal girders per each main gate element, the upper and lower girders together with the minimal number Mminof intermediate horizontal girders and the adjoining skin-plate elements are treated as fixed structural elements, which are always present in the optimization. All other (Mmax Mmin) intermediate horizontal girders with the corresponding number of adjoined skin-plate elements are then regarded as alternative structural elements, which may either disappear or be selected. Since only alternative structural elements participate in the discrete optimization, the size of the discrete decisions is significantly reduced. Each possible combination between selected alternative structural elements and fixed structural elements forms an extra gate topology/structure alternative. 2.1.2. Standard dimension alternatives Four standard dimensions are additionally defined to represent the standard thicknesses of sheet-iron plates: the thickness of the skin-plate tsn for each nth main gate element, the .flange thickness of the horizontal girder tfn, the web thickness of the outer horizontal girder,outwnmt m=1or m=M ,and the web thickness of the inner horizontal girder int,wnmt,1(m(M. Since the thickness tsn has a common value for the entire skin-plate of the nth main gate element and the tare the same for all horizontal girders of the nth main gate element, i.e. they correspond to the common standard design variables for the superstructure or its part ,st comtd from the special MINLP-MS model formulation in Part II. Similarly, the web thicknesses ,outwnmt, which take a common value for both outer horizontal girders of the nth main gate element, and,innwnmt , which are the same for all the inner horizontal girders, correspond to the common standard design variables ,st comacd of the alternative structural elements. An extra set of discrete values of standard dimension alternatives and a special set of the same size of binary variables are introduced for each mentioned standard dimension. Each standard dimension tsn shall be expressed within the given i standard dimension alternatives, i I, standard dimension tfnby k alternatives, k K, standard dimension ,outwnmtby p alternatives, p P, and standard dimension ,innwnmtby r alternatives, r R. The vector of i binary variables yn,i and the vector of i discrete values qn,I are assigned to the variable tsn, the vectors of k elements yn,k and qn,k to the variable tfn, the vectors of p elements yn,pand qn,kto the variable,innwnmtand the vectors of r elements yn,vand qn,v to the variable ,innwnmtConsequently, the overall vector of binary variables assigned to the gate superstructure is y=yn,m ,y,v,yn,I,yn,p,yn,v 2.2. Modeling of discrete decisions The postulated gate superstructure of topology and standard dimension alternatives can be formulated as an MINLP problem using a special MINLP model formulation (MINLP-MS) for simultaneous topology, parameter and standard dimension optimization of mechanical superstructures, described in Part II. As can be seen from the (MINLP-MS), the objective function is typically subjected to structural analysis and logical constraints. While structural analysis constraints represent a mathematical model of a synthesized structure, logical constraints are used for the explicit modeling of logical decisions. Modeling of discrete decisions to determine topology alternatives is an objective of the highest importance for the synthesis. In order to perform these decisions within the MINLP optimization, interconnection logical constraints Dy+R(d, p) r are proposed. While variables, their bounds and most of the constraints of the MINLP-MS model formulation are represented in Part II, interconnection logical constraints and the constraints defining topology alterations are described in this section. The latter ones are derived from the following basic integer or mixed-integer logical constraints: (a) Multiple choice constraints for selecting among a set of units I: Select exactly M units: . iiIyM (1) Select M units at the most: . iiIyM (2) Select at least M units: . iiIyM (3) (b) If then conditions: if unit k is selected then unit i must be selected: yk -yi 0 (4) (c) Activation or deactivation of continuous variables, inequalities or equations: 1. example to relate continuous variable x to the scalar value U: x=Uy (5) if 1 , 0 0y x u i f y x 2. an opposite relation: X=U（ 1-y） (6) 1 0 , 0y x i f y x u 3. example for the bounds of continuous variable x: x1,0y x xupy (7) if 1 , 01 , 0 0 0upy x x x i f y x (d) Integer cuts constraint eliminates unnecessary integer combinations yk= , 1, .,kiy i m 0,1me.g. those found at previous MINLP iterations: k| B I | - 1k kiii B I i N Iyy (8) where kkiB I |y 1i , kkiN I |y 0i In order to describe the modeling of discrete decisions, a general gate superstructure from Figure 2 is addressed in which the defined structural elements are typically horizontal and vertical girders. As the modeling of vertical girders is simplified and needs no special interconnection logical constraints, the modeling of discrete decisions regarding horizontal girders proved to be more sophisticated. 2.2.1. Modeling of topology alterations Let us consider the vertical cross-section of the gate element superstructure with fixed and alternative horizontal girders, see Figure 3(a). The number of fixed and alternative girders and their locations in the superstructure can be described by the following logical constraints: m i n m a xmmMM y M (9) 1 , 2 , 3 , . . . , 1mmy y m M (10) max 1My (11) Logical constraint (9) defines the minimal (Mmin) and maximal (Mmax) number of structural elements (girders). While number Mmin represents the number of fixed structural elements, the difference between the maximal and minimal number of elements (Mmax-Mmin) gives thenumber of alternative structural elements. Constraint (10) defines the direction of the removal of alternative elements: from the top down the superstructure. From Figure 3 is evident that the most upper element is the fixed one, which is set by the constraint (11). It then further follows from constraint (10) that all the rest fixed elements are located at the bottom of the superstructure. Hence, constraints (9)-(11) represent the explicit model for topology alterations of horizontal girders. 2.2.2. Modeling of interconnection relations Interconnection relations between alternative and fixed structural elements within the superstructure require special attention paid to the structural synthesis performed by the MINLP approach. Interconnection relations either restore the connections between currently selected (existing) structural elements or cancel the relations between currently rejected (disappearing) structural elements. Since MINLP methods optimize the topology and parameters simultaneously, it is necessary to define these interconnection relations in an explicite equational form, so that they can enable interconnections and disconnections between the elements during the optimization process. Special interconnection logical constraints for interconnection relations between the alternative and fixed structural elements have been proposed. They will be embedded into the MINLP optimization model of the gate structure, enabling the latter to thus become self-sufficient for automatic topology and parameter optimization. The modeling of interconnection logical constraints, however, requires additional effort, since most element constraints include functions not only of their own variables but also of the variables belonging to their adjoining structural elements. Such an example is, e.g. the constraint of the moment of inertia In,mof the mth horizontal girder of the nth main gate element (see equation (23) in the following section), which includes the substituted expression (S6) of the skin-plate effective width ,nmsb with the heights (between girders and the sill) hm-1and hm+1of both adjoining girders. The constraints of the mth intermediate horizontal girder are typically functions of three heights: hm-1, hm and hm+1, and two vertical distances between horizontal girders: 1mhd,mhdand1mhd The distance mhdis simply defined by the constraint 1mh m md h h m=2,3 ,M-1 (12) The problem arises if hm+1is not defined when the adjoining upper alternative girder to the mth horizontal girder does not exist. For example, let us consider the third girder in Figure 3(a) which is the uppermost existing intermediate element. In order to define h4so as to fulfil the constraints of girder 3, h4should temporarily become equal to the height of the uppermost fixed girder h6 =hM (Figure 3(b). The main idea is to set all heights of non-existing intermediate girders (girders 4 and 5 in Figure 3(a) to the value of h6by means of the logical constraints .mmuph h md h y , m=2,3, ,M-1 (13) , .mmlow exh h md h y , m=2,3, ,M-1 (14) Note, that constraints (13) and (14) restore the upper muphdand lower ,mlowexhd bounds of the distancemhdwhen the corresponding girder exists (ym=1) and set it to zero, otherwise. When the distance is zero, it follows from constraint (12) that hm becomes equal to h. In this way all distances and heights are defined for any girder that becomes the uppermost selected intermediate one and re-establishes its connection to the uppermost fixed girder. As the uppermost selected intermediate girder is connected to the uppermost fixed girder (e.g. girder 3 to girder 6 in Figure 3(a), the latter should also, in the similar way, be connected to the former one (girder 6 to girder 3 in Figure 3(a). Constraints for the uppermost fixed girder are then just functions of two heights: hM and hM-1and a distance1Mhd . The problem arises if some intermediate girders do not exist, e.g. girders 4 and 5 in Figure 3(a). In such cases, hM-1should not be considered. Instead, the height hS(h3in Figure 3(a) of the upper selected intermediate girder should be defined and substituted for hM-1. The vertical distance dhs of the uppermost selected intermediate girder is then defined by the constraint: hs M sd h h (15) The selection of the height hs among all hm can be performed by the following logical constraints: 11(1 ) .u p u ps m s m s mh h h y h y , m=2,3, ,M-2 (16) 11(1 ) .u p u ps m s m s mh h h y h y , m=2,3, ,M-2 (17) 11(1 )ups M s Mh h h y (18) 11(1 )ups M s Mh h h y (19) Constraints (16) and (17) set hS to the height hm of that mth existing horizontal girder (ym=1), which has the existing adjoining lower girder (ym-1=1) and the disappearing adjoining upper girder (ym+1=0). However, for mM!1 the next upper girder always exists, since it is fixed, i.e. (yM =1). In this case we need additional constraints, i.e. (18) and (19), which set hS to the height hM-1. 3. THE MINLP OPTIMIZATION MODEL FOR ROLLER AND SLIDING HYDRAULIC STEEL GATE STRUCTURESDGATOP An MINLP mathematical optimization model for roller and sliding hydraulic steel gate structures GATOP (GATe OPtimization) has been developed. The model has proven efficient for the synthesis of roller and sliding gates. As an interface for mathematical modeling and data inputs/outputs GAMS (General Algebraic Modeling System) by Brooke et al.14 ,a high-level language has been used. The first version of GATOP was developed to perform NLP optimization problems of fixed gate structures, while the dead weight of the gate structure was considered in the objective function, see Reference 15. The new GATOP version is a much more general one: many alternative horizontal girders, vertical girders and plate elements of the skin-plate are now simultaneously represented in a composite form of the gate superstructure. Thus, the new GATOP model is formulated as an MINLP problem performing gate synthesis. In the process of the development of the GATOP model, the following assumptions and simplifications have been considered: 1. A simplified static system for roller and sliding gates is to be used. It includes independent simply supported horizontal girders that are combined with independent clamped skin-plate elements. Such a static system is convenient for gates in which the horizontal girders are much longer than their intermediate vertical distances. Vertical girders have the same height as horizontal ones. 2. In the above case, the horizontal girders transmit almost all the water load, so that the participation of vertical girders can be neglected. Although the vertical girders are not analysed, they are nevertheless considered as a geometrical and economic fact in the objective function. 3. Only the water load, i.e. the hydrostatic pressure on the skin plate, is taken into consideration, while the dead weight, friction and buoyancy are neglected. 外文翻譯譯文 摘要 執(zhí)行的水力 鋼門結構綜合 非線性規(guī)劃 方法使能門結構元素 (大梁，板材 )，優(yōu)選的門幾何、結構元素和所有連續(xù)和標準 剖面圖 大小之間的結構元素和所有連續(xù)和標準尺寸的最佳 優(yōu)選 中間距離。 為此，拓撲結構改變和互聯聯系的特別邏輯限制在選擇和被修理的結構元素之間被公式化。 他們被埋置了入路輾和滑的鋼門結構 GATOP 一個數理優(yōu)化模型。 GATOP 根據已經制定了一個特殊的 MINLP 機械上層 上層 建筑（ MINLP - MS）的第一和第二部分介紹了模型的制定。該模型包含一個自我制造和運輸費用的經濟大門的目標函數。由于 GATOP 模型非凸，高度非線性的，它是 由鏈接解決兩相的 MINLP 戰(zhàn)略，無論是在頂級計算機代碼中實現辦公自動化陪同的 改性 推理算法的手段。一個綜合的例子是作為一個比較設計已經豎立滾子門，所謂的二進水口閘門埃及阿斯旺的研究工作。最優(yōu)結果產生凈儲蓄占百分之 二十九點 四的時候相比，在門口豎立的實際成本。 關鍵詞：結構合成，優(yōu)化，拓撲優(yōu)化，離散變量優(yōu)化，混合整數非線性規(guī)劃 ；的 MINLP；修正 辦公 /推理算法 ；MINLP（非線性規(guī)劃）的 戰(zhàn)略液壓啟閉閘門，推拉門 ；輥閘門 ；阿斯旺 1。簡介 本文介紹了混合整數非線性規(guī)劃（ MINLP）的方式來滾子和滑門 結構，即在垂直升降水工鋼閘門合成最簡單的類型，見圖 1。滾子和閘也被視為最頻繁對水源的控制水工鋼閘門制造類型。它們用來規(guī)管水力發(fā)電廠，大壩或溢洪道的水流。 由于水工鋼閘門是非常特殊的結構，只有一個已經討論了一些作者的優(yōu)化，例如 卡爾文尼伽、蔣格林 和 柯克曼 以及阿爾姆基斯特等人。特別有趣的是表現在沒有這些（輥閘），但類似的結構優(yōu)化。在這種調查， 萬德普拉斯 和 維斯哈爾 以及 戈達爾 等 。 加筋復合材料層合板優(yōu)化 ， 巴特勒和 瑞格提斯 加筋板的優(yōu)化，法卡斯和 加麻依 優(yōu)化焊接矩形蜂窩板， 菲克肯 等。斯特林格氣瓶和治療皮膚 茛蒂 等。加勁 板。幾乎所有使用的非林程式學（ NLP）技術作家。 茛蒂 等。提出了遺傳算法，而 科若文 等 .-推出的 MINLP 算法和拓撲結構的同時和 參數 的閘門優(yōu)化策略。 圖 1。垂直升降水工鋼閘門結構 在我對這個文件三個部分組成的系列部分，一個是的 MINLP 方法和參數的同步拓撲結構優(yōu)化的一般看法是主辦。第二部分描述了標準尺寸的同時優(yōu)化擴展。根據在第一和第二部分所定義的上層建筑的方法，本文件（第三部分）是合成和滑動滾輪的 MINLP 水工鋼閘門結構，取得了以最小的成本，受到 明確的 設計，材料，應力的主要目標， 偏轉 和穩(wěn)定性約束。 隨著的 MINLP 方法能夠同時拓撲結構，參數和標準尺寸的優(yōu)化，一門結構構件（梁，板），門全球性的幾何形狀，結構之間的所有連續(xù)元素和中間的距離和數量的標準尺寸可同時得到。 這種對論文三部分組成的系列的最后一部分，分為三個主要部分： 1。第 2 節(jié)介紹了如何 不同 拓撲結構和標準尺寸的替代品 是假設 以及如何聯網關系是由明確的邏輯約束手段，制定以執(zhí)行在優(yōu)化過程的拓撲結構和標準尺寸的改變。 2。第 3 代表滾子，滑動閘門結構 GATOP 一般的 MINLP 優(yōu)化模型。 3。最后，在第 4 節(jié)，建議上層建筑的 MINLP 方法應用到一個已經豎立滾筒門 ，所謂的二進水口閘門埃及阿斯旺合成 。 2。上部結構及其替代品和離散決策建模 2.1。推導的拓撲結構和標準尺寸的替代品 在合成大門 的第 一步是一個混合整數非線性規(guī)劃上層建筑中 不同 拓撲 /結構和標準尺寸的嵌入式替代品被選定為最佳結果的產生。選定為最佳效果。門上層建筑也包含了結構元素可建造每一個可能的結構替代以及 不同 套離散值，每個標準尺寸替代 不同 代表性。但由于該輪和滑動閘門幾乎相同的結構，它是合理的建議上層建筑，這很可能是為他們兩個非常有用。 2.1.1。拓撲結構的替代品 門上層建筑通常包括一門主要元素，其中每個元素 是門各種結構元素，如梁橫向，縱向梁， 加勁 和 面 板塊內容，代表組成，見圖 2。上層建筑包括 N 邁門元素， n N，每個包含米的水平梁， m M，縱向大梁（ 3+2V）通過整個大門， v V, v3 的數量，以及相應的（ m-1） （ 2+2V）的數目皮板元素。 圖 2。門的上層建筑，由三個主要元素門，建造水平和每個包含六縱九梁 對每一個 m 個第 n 個元素的水平梁大門額外二進制變量 yn,m是勁分配。橫向梁和相應的 面 板單元板，在第 n 個元素的分布大門，因此被確定數量可以簡單地用 mYn,m 表示。 附注 k 勁，該垂直梁相同數量最 少為 3，他們可以只需要奇數。如果一個二進制變量 yv是分配給每個 v， v V 的，縱向大梁可以 用（ 3+2 v yv ）表示。 以同樣的方式一個偶數（ 2+2 v yv）的同等水平將得到整個分區(qū)的建議。在縱向大梁的情況下，我們可以看到，結構構件，也可確定合適的二元變量的線性組合 。 其中最大數量 Mmax 的水平梁，每門各主要元素，上下梁連同中間水平梁和毗鄰的 面 板元素的最小數 Mmin 是當作固定結構的部分，它們總是在優(yōu)化本。所有其他（ Mmax -Mmin）的中間與鄰接 面 橫板單元對應的數字，然后把梁結構元素作為替代，這 可能不是消失或者被選中。由于唯一的選擇結構元素的離散優(yōu)化參與，離散決定大小 顯著 減少。選定的替代結構之間的每個元素和 固定 結構元素可能組合形式一門額外的拓撲 /結構的選擇。 2.1.2。標準尺寸的替代品 四個標準尺寸此外 定義 代表鐵皮板的標準厚度： 面 板噸 tsn 每 n 個元素的正門厚度， 水平梁 的 法蘭 厚度 tfn，外層的水平梁腹板厚度噸,outwnmt， m=1 或 m=M，和內部橫向梁腹板厚度噸,innwnmt， 1mM 自厚度 tsn 有一個對整個 面 的第 n 個元素的大門板共同的價值和分別為第 n 個元素的所有大門梁的水平相同，即它們對應到上層建筑或部分從特殊噸的 MINLP - MS 在第二部分制定模型的通用標準設計變量。同樣， 門 頁厚度噸,outwnmt， 它 為第 n 個元素外正門水平梁的共同價值，這是所有內部橫向梁相同，對應的共同的 ,st comaed可替代的標準設計變量結構元件。一種額外的標準離散值設置一個替代品和一維二元變量的大小相同，介紹了一套特別的標準尺寸為每個提及。 每個標準 尺寸 tsn 應表示在給定的標準尺寸 i 中選擇替代尺寸 ， i I，被 K的替代， k K，標準尺寸噸標準尺寸由 P 選擇 ， p P 的，由 R 所替代 ， r R。 i 的二進制變量載體 yn,r 和離散值 qn,r 分配給變量載體 tfn時， p 元素 yn,p 和 qn,p的變量載體,outwnmt和 R 元素 的載體 yn,r 和 qn,k 分配給 變量,innwnmt。因此，分配到門上層建筑整體向量變量是 y= yn,m ,y1,v ,yn,I yn,k ,yn,p ,yn,r 2.2。離散決策建模 該拓撲結構和標準尺寸的替代假設門上層建筑可歸結為一個混合整數非線性規(guī)劃 MINLP 模型采用一種特殊配方（的 MINLP - MS）的拓撲結構的同時，參數和機械超標準尺寸的優(yōu)化結構第二部分所述的問題。由于可以從（的 MINLP - MS）的出現，其目標函數通常受到結構分析和邏輯約束。雖然結構分析制約代表了一個綜合結構的數學模型，邏輯約束是用于顯式建模的邏輯判斷 決定確定的離散拓撲建模是一種替代品為合成最為重要的目標。為了執(zhí)行內部的 MINLP 優(yōu)化 這些決定，互連邏輯約束 Dy+R（ d,p） r 是建議。而變量，其范圍和對的 MINLP - MS 模型制定的限制大多數是在第二部分，互連邏輯約束和約束丹寧拓撲結構的改變代表介紹本節(jié)。后者的是來源于以下基本的整數或混合整數邏輯約束： （一）在我設置的單位選擇多個選擇的限制： 精確地選擇測繪單位： 選擇 M 單位至多： 選擇至少 M 單位： （二）如果當時條件： 如果單位 k 為單位選定，