Teep

However, that routine returns L := [d[0]*x[i, 0, 1]+d[0]*x[i, 1, 1]+d[1]*x[i, 0, 1]+d[1]*x[i, 1, 1], d[0]*x[i, 0, 2]+d[0]*x[i, 1, 2]+d[1]*x[i, 0, 2]+d[1]*x[i, 1, 2]].

As you can see, the values, i=0 and i=1 are not carried through to the full series; we only receive unknown terms, for instance x[i, 1,2] and so on. This is what I am trying to protect against. As the equation is stated, one would think that your routine would work ... but you see the problem. If we then take the values i=0 and i=1, then we will obtain two sequences ... however, I need to be sure.

Any other suggestions?

Model Detail...

Answered: Teep 190

January 17 2017

0 0

Constraint Problem

Given the sets:

•	C = {1,2, ..., n}

•	K = {1,2, ...., m}

•	Arcs, A ={i,j}, representing connections between locations, i and j. For each arc, (i,j), i≠j, i≠n+1 and j≠0.

For each arc, {i,j}, i≠j, there are two matrices namely and and for each point, i, there is a specified positive quantity,

Further, we impose a positive integer value on

There are two decision variables under consideration; one of which is binary.

•

" z=(∑) (∑) (∑)c[i,j] x[i,j,k]"

subject to the constraints

"(i) (∑) (∑)x[i,j,k ] =1, ∀i in C"

"(ii) (∑)d[i ] (∑)()[ ]x[i,j,k ]<= q, ∀k in K"

"(iii) (∑)x[0,j,k ] = 1, ∀ k in K"

"(iv) (∑)x[i,h,k ] - (∑)x[h, j,k ]= 0, ∀ h in C, ∀ k in K"

"(v) (∑)x[i, n+1, k ] = 1, ∀ k in K"

(vi)

(vii)

(viii)

>

The following model values are set-up for simplicity.

>

The values for

>

c[0, 0] := 0:

>

t[0, 0] := 0:

>

S[0, 0] := 0:

The vector, :

>

The quantitity q:

>

The objective function:

>

Constraint (1)

>

Constraint (7)

>

For

Constraint (8): Setting the condition i ≠ j

OPEN

Constraint (9): Setting the condition i≠ n+1

OPEN

Constraint (10): Setting the condition j ≠ 0

OPEN

Constraint (11)

The final constraint require binary outputs, zero or one.

Excluding constraints (8) to (10), the consolidated constraints are:

>

The list, L, of all terms is given by the sequence:

>

The set of elements binary variables. The binaryvariables option may be given by the indets syntax.

The solution is:

>

(1)

>

(2)

>

Download Constraint.mw

Many thanks for the effort and response - I appreciate it.

As you recommended, I have included the worksheet. The constraints are incomplete (missing the final three). If you could review the structure, I'd be interested in your comments / feedback.

Thanks again!

indets...

Answered: Teep 190

January 07 2017

0 0

Thanks so much!

@vv

Great idea (and attitude)!...

Answered: Teep 190

December 22 2016

0 0

>

This is a combinatorial (multi-objective) optimization problem in which the minimization of the following is sought:

(a) No of vehicles

(b) Total travel time

(c) Waiting time

(d) Total distance incurred by the fleet

This is an extension of the classic vehicle routing problem. It accounts for the following features.

Let

V = Fleet of (homogeneous) vehicles

C = Customers (1,2,..., n)

= Cost

= Time (includes service time at customer j)

q = vehicle capacity

Each customer has a time window,

Assume the service time is in direct proportion to the demand at the location

Unused vehicle is modeled by driving the empty route, (0, n+1)

k 2 V

i 2 N

j 2 N

There are two decision variables, namely:

1.

2.

The objective to be minimized is:

>

sum(sum(sum(c[i*j]*x[i*j*k], j), i), k);

(1)

subject to the following 8 constraints

Each customer is visited once only:

>

sum(sum(x[i*j*k], j), k) = 1;

(2)

The vehicle load limits are defined

>

sum(d[i]*(sum(x[i*j*k], j)), i) <= q;

(3)

Each vehicle leaves depot 0

>

sum(x[(0*j)*k], j) = 1;

(4)

After arriving at a customer, the vehicle departs

>

sum(x[i*h*k], i)-(sum(x[h*j*k], j)) = 0;

(5)

Finally, the vehicle arrives at the depot, n+1

>

sum(x[i, n+1, k], i) = 1;

(6)

Vehicle k cannot arrive at j before

>

(7)

The time windows must be observed (assume

>

(8)

Integrality constraint

>

(9)

>

Download Optimization_Problem.mw

@Carl Love

Consolidated matrix form...

Answered: Teep 190

December 11 2016

0 0

The use of matrices is really useful here.

Many thanks!

@Carl Love

it works now...

Answered: Teep 190

December 11 2016

0 0

That's great!

Thanks for your interest and help.

@brian bovril

Reconsidered constraints...

Answered: Teep 190

December 10 2016

0 0

Thanks for your timely response .... much appreciated.

I'll review the formulation in light of your advice.

@vv

The saga continues ... feasibility toler...

Answered: Teep 190

December 10 2016

0 0

Hi again.

The procedure you sent worked fine.

However, I need to alter the first constraint to account for a realistic situation. This involved switching the order of i and j. I get an error message to adjust the feasibility tolerance; when I try to do that, I am prompted to alter the depth limit. I have tried various values .... with no luck (worksheet is uploaded).

>

(1)

>

To minimize the objective function, z:

>

The first constraint is:

>

(2)

>

constraint1 := `union`(`union`(`union`(`union`(`union`(`union`(`union`(`union`({constraint[1]}, {constraint[2]}), {constraint[3]}), {constraint[4]}), {constraint[5]}), {constraint[6]}), {constraint[7]}), {constraint[8]}), {constraint[9]}):

The second constraint is:

>

The third constraint is:

>

The constraints (4 and 5) require binary outputs; zero or one. This is automatically accounted for, so removed, since all variables will be assumed binary.

Thus, the consolidated constraints are:

>