Optimization Models
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1. Introduction
1.1. Overview
1.2. Optimization Models
2. Linear Algebra
2.1. Vectors
2.2. Matrices
2.3. Linear Equations
2.4.  Least-Squares
2.5. Eigenvalues
2.6  Singular Values
3. Convex Models
3.1. Convexity
3.2. LP and QP
3.3. SOCP
3.4. Robust LP
3.5. GP
3.6. SDP
3.7. Non-Convex Models
4. Duality
4.1. Weak Duality
4.2. Strong Duality
4.3.  Applications
5. Case Studies
5.1. Senate Voting
5.2. Antenna Arrays
5.3. Localization
5.4. Circuit Design

Robust LP Solution to the Drug Production Problem

Return to the drug production problem described here, and the corresponding uncertainty issue discussed here.

Robust LP model

The only constraint affected by uncertainty was the active agent balance inequality. In the nominal problem, this constraint was

a_1 cdot x_{rm Raw I} + a_2 cdot x_{rm Raw II} - 0.05 cdot x_{rm Drug I} - 0.600 cdot x_{rm Drug II} ge 0 ,

where the nominal values of the coefficients were set to a_1 = 0.01 and a_2 = 0.02.

The uncertainty affects the coefficients a_1,a_2, which are assumed to be only known within intervals: a_1 in [0.00995, 0.01005] and a_2 in [0.0196, 0.0204].

To obtain a robust solution, we can solve a new problem where we insist that the above constraint holds for every possible value of the coefficients within their respective intervals.

In this problem, since the variables are non-negative, the worst-case value of the coefficients corresponds to a particular scenario where both coefficients are set to their lowest possible value: we set a_1=0.00995 and a_2 = 0.0196 and solve the corresponding LP.

Robust LP solution

The new (robust) solution is then

p^ast = -8,294.566, ;; x_{rm Raw I} = 877.7319, ;; x_{rm Raw II} = 0, ;;x_{rm Drug I} = 17.4669, ;;x_{rm Drug II} = 0.

This is to be compared with the nominal solution, which was:

p^ast = -8819.658, ;; x_{rm Raw I} = 0, ;; x_{rm Raw II} = 438.789, ;;x_{rm Drug I} = 17.552, ;;x_{rm Drug II} = 0.

The profit goes down, by a relatively small amount of 6%. This is to be compared with the expected loss of 11% (and worst-case loss of 22% if we had stuck to our original solution, and tried to adjust it to uncertainty (see here).

Note also that the qualitative behavior of the robust solution is very different from that of the nominal one. In the latter, we only purchase the raw material II, and produce only Drug I. This makes sense as XXX. The robust solution, in contrast, recommends to purchase only raw material II and produce Drug II. This makes sense, as the uncertainty in the active ingredient amount in the raw material II is higher.