Assignment Problems

The methods discussed in this section extend the original concept of deriving structures from experimental NMR data in two ways. First, during the structure calculation, part of the assignment problem is solved automatically. This allows specification of the NOE data in a fonn closer to the raw data, which makes the refinement similar to X-ray refinement. Second, the quality of the data is assessed. The methods have been recently reviewed in more detail [64,67]. 

This memory erasure problem is sometimes called the credit assignment problem [peret92l. Fortunately, there is an easy way out. We merely generalize the binary (on/off) McCulloch-Pitts neuronal values to continuous variables by smoothing out the step-function threshold. 

Manne, Alan S., A Target-Assignment Problem, Operations Research, 6, No. 3, 307-466 (1958). 

Note that if we relax the t binary variables by the inequalities 0 < y < 1, then (3-110) becomes a linear program with a (global) solution that is a lower bound to the MILP (3-110). There are specific MILP classes where the LP relaxation of (3-110) has the same solution as the MILP. Among these problems is the well-known assignment problem. Other MILPs that can be solved with efficient special-purpose methods are the knapsack problem, the set covering and set partitioning problems, and the traveling salesperson problem. See Nemhauser and Wolsey (1988) for a detailed treatment of these problems. 

Aboudi, R., Hallefjord, A. and J0rnsten, K. (1991) A facet generation and relaxation technique applied to an assignment problem with side constraints. EurJ Oper Res, 0 (3), 335-344. 

Aboudi, R. and Hallefjord, K. (1990) Resource constrained assignment problems. Discr Appl Math, 26 (2/3), 175-191. 

Bertsekas, D., Castanon, D. and Tsak-nakis, H. (1993) Reverse auction and the solution of inequality constrained assignment problems. SIAM J Optim,... 

An efficient cost scaling algorithm for the assignment problem. Math Prog, 71, 153-178. 

The integer solution property is particularly important in assignment problems. These are transportation problems (like the problem just described) with n supply nodes and n demand nodes, where each supply and demand is equal to 1.0, and all constraints are equalities. Then the model in Equations (7.41) through (7.44) has the following interpretation Each supply node corresponds to a job, and each demand node to a person. The problem is to assign each job to a person so that some measure of benefit or cost is optimized. The variables xi are 1 if job i is assigned to person ./, and zero otherwise. 

First let us write the problem statement. The total number of streams n is 4. Let Cy be an element of the cost matrix, which is the cost of assigning stream i to exchanger j. Then we have the following assignment problem ... 

The two types of equality constraints ensure that each city is only visited once in any direction. We define yu = 0 because no trip is involved. The equality constraints (the summations) ensure that each city is entered and exited exactly once. These are the constraints of an assignment problem (see Section 7.8). In addition, constraints must be added to ensure that the ytj which are set equal to 1 correspond to a single circular tour or cycle, not to two or more disjoint cycles. For more information on how to write such constraints, see Nemhauser and Wolsey (1988). 

The structure assignment problem, which offers difficulties similar to those discussed for the alkylation products, was solved by means of mass spectrometry in combination with isotope labeling at position 2 with 15N. ESCA (X-ray photoelectron) spectroscopy and IR spectra are in agreement with the assignment as a 3-oxide.23... 

The I3C NMR spectra of quinine, quinidine, dihydroquinine, dihydro-quinidine, and their A -oxides, (2) cinchonidine, 9-epiquinine, 9-epiquinidine, and dihydro-9-epiquinidine (189) have been described. Some of the assignment problems are illustrated with reference to cinchonidine [297] and quinidine [298]. Assignment of the C(2) and... 

These suggestions are illustrated by the following solved problems presented in increasing order of difficulty. The assigned problems of Chapter 8, again in increasing order of difficulty, will provide the essential practice. 

---