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I would like to incorporate a constraint to my model, this constraint is related to supplier quality or/ reliability selection with respect to efficiency, furthermore, I would like to make the parameter $ q_{s} $ as an uncertain and deal with it using robust optimization where $ {1}\geq q_{s}\geq {0} $ and $ \varepsilon _{s} $ is a lower bound parameter, for instance 0.95.So I am wondering if the below constraint is correct? I would like also to know if considering the parameter $q_{s}$ as an uncertain parameter in robust optimization is a valid assumption ?
S : Set of suppliers.
W : Set of warehouses
C : Set of customers.
$ Q{sw}$ : decision variable: amount of product shipped from s to w.
$ Q{wc}$ : decision variable: amount of product shipped from w to c.
$ q_{s}$ : non-defective rate of ssuplier.
$ \varepsilon _{s} $ : minimum acceptable non defective rate.
Constraint:
$ q_{s}* \sum_{s}\left(Q_{sw} \right) \geq \sum_{s}\left(Q_{wc} \right)* \varepsilon _{s} $ for all s
The original constraint was nonlinear given as;
$ \frac{q_{s}* \sum_{s}\left(Q_{sw} \right)}{\sum_{s}\left(Q_{wc} \right)} \geq \varepsilon _{s} $ for all s

Abde
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 02 '23 at 09:30
  • Thank you for the feedback, I have updated my questions clearly, hope this help get feedback. – Abde Feb 02 '23 at 09:58

1 Answers1

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RO is still in academic research area and several methods have been proposed. What I would do is this( not using any math programming model as you have already formulated already)

  1. Solve for q=0
  2. Solve for q=1: in other words best and worst case scenario.
  3. Use max (or min) of the two optimal values
    Or
  4. Then use the values of the decision variables as bounds and re-solve without q.
    Or
  5. Goal programming: use the optimal values of the above two cases as constraint ( upper and lower bound), on the objective function with slack variables ($s^+$,$s^-$): max and min deviation and solve with objective as min ($s^++s^-$).
    You can use both 4&5.
    Below are couple of materials. I'd add one more that I found in this forum couple of days back.

Reading 1
Reading 2 as mentioned here
Reading 3

Sutanu Majumdar
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  • Thank Sutanu for your feedback, my point is I am planning to utilize RO in my model, which does consist of uncertain parameters already, however, I would like to extend it a little bit more, because it seems little bit trivial. I am planning to apply RO, with interval, ellipsoid and polyhedral sets, so I am wondering, if adding this constraint, and do some analysis on the supplier selection along with other parameter may be valid . – Abde Feb 02 '23 at 13:35
  • The constraint is saying, if the some over all the suppliers non defective rate of the shipped amount is less than the lower bound epsilon (i.e minimum acceptable non defective rate). – Abde Feb 02 '23 at 13:41
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    Understood, its a box set. Added one more reading. Hope it's helpful. – Sutanu Majumdar Feb 02 '23 at 14:07
  • Thank you for the links, my question is about the constraint it self, if it is correct or not ? – Abde Feb 02 '23 at 14:48
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    In the 2nd paper from pages (3-5) using Box set (Table 1) I'd try a Robust counterpart as $ \sum_s Q_{wc}\epsilon - \sum_s Q_{sw} \vert q_s \vert_{2} $ – Sutanu Majumdar Feb 02 '23 at 15:35
  • Thank you so much, first the constraint was nonlinear, $\frac{q_{s}* \sum_{s}\left(Q_{sw} \right)}{\sum_{s}\left(Q_{sw} \right)} \geq \varepsilon {s}$, then, I linearize it like that, $ q{s}* \sum_{s}\left(Q_{sw} \right) \geq \sum_{s}\left(Q_{sw} \right)* \varepsilon {s}$ which I am not sure if it is correct, and then if the constraint is correct, is it safe to handle $q{s}$ as uncertain param utilizing RO ? – Abde Feb 02 '23 at 16:40
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    Linearization is correct. That's how you avoid division to make it convex. As for the forms to use, couple of papers should be useful to determine the robust counterpart. – Sutanu Majumdar Feb 02 '23 at 16:42
  • Let us continue this discussion in chat. – Abde Feb 02 '23 at 16:42