دانلود رایگان مقاله لاتین هزینه مسدود منابع رله از سایت الزویر
عنوان فارسی مقاله:
مدل ها و رویکرد تولید ستون برای کمترین مسائل هزینه مسدود هزینه منابع با رله
عنوان انگلیسی مقاله:
Models and Column Generation Approach for the Resource-constrained Minimum Cost Path Problem with Relays
سال انتشار : 2017
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بخشی از مقاله انگلیسی:
2.2. Pattern-chain Formulation
As one of our main contributions, we next present the pattern-chain formulation for the RMCPR. Before presenting the pattern-chain formulation, we first introduce concepts of pattern and pattern chain, respectively. A directed subpath from nodes i to j is referred to as a pattern, pi,j , if the following two conditions are satisfied: (1) For each resource k, the total resource consumption along this subpath does not exceed the given upper bound Wk , and (2) Only the starting and ending nodes i and j are relay nodes. We further assume that two relays are respectively located at source s and destination t, and they do not produce any cost, i.e., gs = gt = 0. Then, a feasible solution to the RMCPR can be viewed as a chain of patterns (pattern chain). For example, in Figure 1, one feasible solution corresponds to path (s, a, b, f, k, g, h, d, e, t) with nodes b, k, h, and e being relays. Obviously, we have patterns ps,b, pb,k, pk,h, ph,e, and pe,t. These patterns form a chain of patterns (pattern chain). We refer to a node pair < i, j > as “directly connected node pair (DCNP)”, if there exists at least one pattern from i to j in network G. Let U be the set of all DCNPs. Given one DCNP < i, j >, there may exist different patterns between nodes i and j. For example, consider DCNP < k, h > in Figure 1. Associated with this DCNP, there are two patterns: (k, h) with relay h, and (k, g, h) with relay h. Let Pi,j be the set of all patterns associated with DCNP < i, j >. Let Ap be the set of arcs included in pattern p. Since the ending point of one pattern may be the starting point of another one, we exclude from calculation of the pattern cost the relay cost of the starting point. For each pattern p ∈ Pi,j , define δ p i,j as pattern cost, which is calculated as δ p i,j = X (u,v)∈Ap cu,v + gj . Following the above definition, the pattern cost consists of two parts: arc cost and relay cost. A pattern p with cost of min{δ p 0 i,j : p 0 ∈ Pi,j} is called the minimal-cost pattern associated with DCNP < i, j >.
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