The Role of Software in Optimization and Operations Research

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$University of Colorado Denver Math 1070 Topics Updated: Fall 2009 ...$

$Math 3000 Running Glossary$

OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J. Greenberg Optimization software begins to process this problem by solving its Phase I: 16 ∑ min v : v ≥ 0, i= 1 Supply at node 1: x14 + v1≤ 20 Supply at node 2: x24 + x25 + v2≤ 20 Supply at node 3: x35 + v3≤ 20 Supply at node 4: x14 + x24 −x46 − x47 + v4= 0 Supply at node 5: x25 + x35 −x57 − x58 + v5= 0 Supply at node 6: x46 + v6≥10 Supply at node 7: x47 + x57 + v7≥ 20 Supply at node 8: Flow bounds: x58 + v8≥30 0 ≤ x14 + v 9 ≤ 30 0 ≤ x24 + v10 ≤ 20 0 ≤ x25 + v11 ≤ 10 0 ≤ x35 + v12 ≤ 10 0 ≤ x46 + v13 ≤ 10 0 ≤ x47 + v14 ≤ 2 0 ≤ x + v ≤ 20 0 ≤ x + v ≤ 30 i 57 15 58 16 The analyst must now debug this infeasible LP by diagnosing its cause. Conventional solvers and most modeling languages provide only displays of values, such as what you see in Figure 5. In a realistic problem this table would be far too large to be useful, so the analyst needs some screening aids. Figure 5: Display of Solution Values for Network in Figure 4 ©Encyclopedia of Life Support Systems (EOLSS) (14)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J. Greenberg There are several approaches to help analysts diagnose an infeasibility. One is to use the Phase I dual prices to form one infeasible aggregate constraint: x + 2x + x ≥ 70. (15) 25 35 47 This inequality is implied by the original system by taking a weighted sum. Letting pi denote the dual price of the i-th supply constraint, and noting that p ≤ 0, we first obtain the following: p ( x + p ( x + x ) + p x ≥ ( p + p + p )20. (16) 1 14 2 24 25 3 35 1 2 3 Since p = (−1,−1, 0), this becomes the aggregate supply limit: −x14 −x24 −x25 ≥ − 40 . Combining the balance equations in the same manner, and using its price vector (1,2) (shown in Figure 1: Display of Solution Values for Network in Figure 4) we obtain the aggregate balance equation: x + x −x − x + 2( x + x − x − x = 0 . 14 24 46 47 25 35 57 58 The demand requirements use the price vector (1, 2, 2) to obtain the following aggregate demand requirement: x + 2( x + x ) + 2x ≥ 1× 10 + 2× 20 + 2× 30 = 110 . 46 47 57 58 Adding these three aggregates produces the overall aggregate constraint, and this approach notes that the greatest aggregate value on the left-hand side, for the given bounds, is less than the aggregate requirement: max{ x + 2 x + x : 0 ≤ x ≤10,0 ≤ x ≤10,0 ≤ x ≤ 2} = 32 < 70. (17) 25 35 47 25 35 47 The ANALYZE system does this and can provide an English response, such as what is shown in Figure 6. The system can continue to help the analyst by explaining the aggregate coefficients and right-hand side. Figure 6: ANALYZE Diagnosis Using Phase I Price Aggregation An IIS approach could produce the subnetworks shown in Figure 7. In 7(a), the only constraints included are the flow bounds on x25, x35 and x4t . The sources of those arcs are not shown because it does not matter where those flows come from, only that they are limited to the values shown. Nodes 5, 7 and 8 are shown because the constraints associated with them are in this IIS. Only the demand arcs are oriented; the others are undirected, which means that the non-negativity constraints on those flows are not in the ©Encyclopedia of Life Support Systems (EOLSS)
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