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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. Patte-chain Formulation

As one of our main contributions, we next present the patte-chain formulation for the RMCPR. Before presenting the patte-chain formulation, we first introduce concepts of patte and patte chain, respectively. A directed subpath from nodes i to j is referred to as a patte, 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 pattes (patte 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 pattes ps,b, pb,k, pk,h, ph,e, and pe,t. These pattes form a chain of pattes (patte chain). We refer to a node pair < i, j > as “directly connected node pair (DCNP)”, if there exists at least one patte from i to j in network G. Let U be the set of all DCNPs. Given one DCNP < i, j >, there may exist different pattes between nodes i and j. For example, consider DCNP < k, h > in Figure 1. Associated with this DCNP, there are two pattes: (k, h) with relay h, and (k, g, h) with relay h. Let Pi,j be the set of all pattes associated with DCNP < i, j >. Let Ap be the set of arcs included in patte p. Since the ending point of one patte may be the starting point of another one, we exclude from calculation of the patte cost the relay cost of the starting point. For each patte p ∈ Pi,j , define δ p i,j as patte cost, which is calculated as δ p i,j = X (u,v)∈Ap cu,v + gj . Following the above definition, the patte cost consists of two parts: arc cost and relay cost. A patte p with cost of min{δ p 0 i,j : p 0 ∈ Pi,j} is called the minimal-cost patte associated with DCNP < i, j >.

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