# Operations research ><

1. May 10, 2010

### operationsres

Hi. I really need help with a problem I've been given @ university regarding optimisation. I have little idea how to solve it.

We are given an employee class, say "Senior Manager", and the skills that need training, given as a vector:
0 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1

or perhaps "Assistant Manager"
1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0

Then we are given numerous different programs that cover different skills, some that will obviously overlap with the ones we need to cover, and each program has a different cost.

P4 has skill X at $0.10. We should've picked P2 in the first iteration, but the problem is our first iteration is BLIND to future iterations, and thus cannot make an optimal allocation!>!>!>!?>!?!? 8. May 11, 2010 ### Mark44 ### Staff: Mentor Isn't all that taken care of in your formula? Cost(of program) + ... 9. May 12, 2010 ### operationsres ------------------------------------ HERE'S A FORMULATION THAT MAKES SENSE. Let's say these are our thingys: Senior Manager needs..... : 1 1 1 1 1 0 1 1 1 1 1 0 1 Program 1(($1/dot-skill))..: 1 1 1 1 1 0 0 0 0 0 0 0 0
Program 2($1.50/dot-skill): 1 1 1 1 1 0 0 0 0 0 0 0 1 Program 3($999/dot-skill).: 0 0 0 0 0 0 0 0 0 0 0 0 1
Program 4($50/dotskill)....: 1 0 0 0 0 0 0 0 0 0 0 0 1 If we follow the model, we'd pick Program 1, THEN program 2 ASWELL, when we should just be picking Program 2!!! How do we fix this dilemma >< Last edited: May 12, 2010 10. May 12, 2010 ### Mark44 ### Staff: Mentor I get the sense that you didn't understand what I said back in post #2 about dot products. What I wrote "dot" that meant the dot product of two vectors. For example, I had The sr. manager vector dotted with the program 1 vector gives <0 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1> $\cdot$ <0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1> = 4. I'm assuming you knew what I meant by the dot product, but I'm not sure that you do. In post #3 you have [Cost(of program) + (Days of program/250 * Yearly Salary)]/DOT What do you mean by DOT? In post #9 you have Program 3($999/dot-skill).: 0 0 0 0 0 0 0 0 0 0 0 0 1
What do you mean by dot-skill?

11. May 12, 2010

### imiyakawa

Yeah I did get what you meant :)

The reason why I said $$/dot-skill is that I thought$$$/skill would be misleading, because that could've been interpreted as the cost per all skills the program offers, not just the skills that are matched up with what the Senior Manager needs (which I what I was intending to convey with /dot-skill). Let's say that for the example I provided, the rows 2-5 (i.e. Programs 1-4) are dot products of row 1. So the dot-value that I'd be inputing into my equation for Program 4 would be "2", as long as no programs before it have been assigned that cover skill 1 or skill 13. _______________________________________________ My use of "/" when saying "$50/dot-skill" or whatever is NOT a divided sign, it's simply saying $50 per dot-skill. ------------------------------------------------------ If we run our formula on the 4 programs provided, we get a suboptimal allocation, right? Last edited: May 12, 2010 12. May 12, 2010 ### Mark44 ### Staff: Mentor I'm glad that one of us does!$/skill conveys to me the amount of dollars per (one) skill, not for all skills. When someone writes km/hr they mean kilometers per each hour, not the number of kilometers for all the hours.
Please, no more "dot-skill." That's terrible! It adds complexity and reduces understandability!
Then you need to examine the formula you came up with. You can build the per-skill costs into the program vectors by using the cost, rather than just a 1. E.g., here is program 4 revised: Program 4($50/skill)....: 50 0 0 0 0 0 0 0 0 0 0 0 50 13. May 12, 2010 ### imiyakawa How would one output a solution though? Last edited: May 12, 2010 14. May 12, 2010 ### Mark44 ### Staff: Mentor Since you are doing this in Excel, you could have some logic that chooses the optimal plan and displays the decision results in some cell. 15. May 12, 2010 ### operationsres Thankyou so much for your help Mark. Perhaps there's a way, but I can't fathom a way of incorporating future iterations into the current iteration without some kind of mass simulation. Oh well, I guess I could try and formulate the problem in LINGO, somehow... What do you think of this, for Senior Manager, where x1, x3, x8, x10, x12 are the skills he needs to be covered, his salary is$250,000, 250 working days. Of course, in the constraints we can ignore the extra skills that the senior manager doesn't need.

MINIMIZE z = x1+x3+x8+x10+x12;

s.t.

(500 - program cost) + (2 - number of days running)(250000 - salary)/(250) = x1 + x3 + x8
600 + (3)(250000)/(250) = x1 + x8
(800) + (4)(250000)/(250) = x10 + x12
(1000) + (2)(250000)/(250) = x1 + x3 + x12
.
.

I haven't got LINGO on this computer so I haven't had a chance to try it out xD

The left side of the constraints are the total cost of the program, and the right side are the skills it covers that matches with the senior managers skills.

Will this work?

Last edited: May 13, 2010
16. May 13, 2010

### operationsres

^^^
Nope, that model's totally wrong.

17. May 18, 2010

### ych22

Define Xij = 1 if employee class i is assigned to program j, 0 otherwise.
Let Ai be the cost of program i.
Let Bi be the salary of employee i.
Let Di be the length of program i.
Let Kij = 1 if employee i's jth skill must be trained.
Let Qij = 1 if program i trains the jth skill.

minimize $$\Sigma$$ $$\Sigma$$Xij(Aj+DjBi/250)

subject to

XijQjk>= Kij for all i,k
Xij= 0 or 1

Last edited: May 18, 2010
18. May 18, 2011

### bijaya

- Use the dual simplex method to solve the following LP:
Max Z = -2X1 - X3
s.t. X1 + X2 – X3 >= 5
X1 - 2X2 + 4X3 >=8
X1, X2, X3 >=0