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IE 518 Discrete Optimization Course Project

Huseyin Gurkan 20701694

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Outline

◮ MIP Model ◮ Robust Optimization Model ◮ Heuristic Method and Evaluation ◮ Numerical Examples ◮ Conclusion and Furhter Developments

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

Motivation:”Assign each surgery to an order of the surgeon while deciding on the operating rooms.”

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

Parameters N: the set of the surgeries to be scheduled |N| = n tpi: the pre-incision time of the surgery i ∈ N tsi: the incision of the surgery i ∈ N tci: the post-incision time of the surgery i ∈ N Pi: sum of the pre-incision, incision and post-incision duration

• f the surgery i ∈ N

Pi = tpi + tsi + tci ∀i ∈ N T: the length of normal working day cv : cost per hour of having an OR vacant (idle) cw: cost per hour of having the surgeon waiting (inactive) co: cost per hour of using OR staff in any of the ORs beyond their normal time shift of length T; that is, during overtime

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

Decision Variables yij ∈ 0, 1 xj ∈ 0, 1 Sj: the start time of the surgery on the jth order of the surgeon ∀j = 1, ..., n Cj: the completion time of the surgery on the jth order of the surgeon ∀j = 1, ..., n Rj: the time when the surgeon starts the surgery at the jth order f : idle time in OR1 and OR2 between surgeries t1: end of the whole process in OR1 t2: end of the whole process in OR2 d1: overtime in OR1 d2: overtime in OR2

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

min

n

• j=1

(Cj−Sj−

• i∈N

Pi.yij+f ).cv+(Rn−

n−1

• j=1
• i∈N

yij.tsi).cw+(d1+d2).co

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

s.to

• i∈N

yij = 1 ∀j = 1, ..., n (1)

I

• j=1

yij = 1 ∀i ∈ N (2) Rj ≥ Sj +

• i∈N

yij.tpi ∀j = 1, ...n (3) Cj ≥ Rj +

• i∈N

yij.(tsi + tci) ∀j = 1, ..., n (4) Sk ≥ Cl − M.(2 − xk − xl) ∀k, l st. k > l (5) Sk ≥ Cl − M.(xk + xl) ∀k, l st. k > l (6)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

t1 ≥ Cj − M.(1 − xj) ∀j = 1, ..., n (7) t2 ≥ Cj − M.(xj) ∀j = 1, ..., n (8) d1 ≥ t1 − T (9) d2 ≥ t2 − T (10) f ≥ t1 + t2 −

n

• j=1

Cj − Sj (11) Rj+1 ≥ Rj +

• i∈N

yij.tsi ∀j = 1, ..., n − 1 (12)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

MIP Model

Sj, Rj, Cj ≥ 0 ∀j = 1, ..., n (13) t1, t2, d1, d2 ≥ 0 (14) yij, xj ∈ {0, 1}∀i ∈ N, j = 1, ..., n (15)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

Motivation:

◮ ”Change objective function in a way that we get rid of

randomness, the completion time of the last surgery”

◮ ”Handle randomness by the chance constraints ” [1]

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

Figure : Plot of sample data for ts of surgery A

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

P(Rj ≥ Sj +

• i∈N

yij.tpi) ≥ 0.9 ∀j = 1, ...n (16) P(Cj ≥ Rj +

• i∈N

yij.(tsi + tci)) ≥ 0.9 ∀j = 1, ..., n (17) P(Rj+1 ≥ Rj +

• i∈N

yij.tsi) ≥ 0.9 ∀j = 1, ..., n − 1 (18)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

Rj − Sj −

i∈N yij. ˜

tpi

• i∈N y2

ij .σ2 i

≥ φ−1(0.9) ∀j = 1, ..., n (19) Cj − Rj −

i∈N yij.( ˜

tsi + ˜ tci)

• i∈N y2

ij .(β2 i + η2 i )

≥ φ−1(0.9) ∀j = 1, ..., n (20) Rj+1 − Rj −

i∈N yij. ˜

tsi

• i∈N y2

ij .β2 i

≥ φ−1(0.9) ∀j = 1, ..., n − 1 (21)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

In order to approximate the value of

• i∈N y2

ij .σ2 i and the

corresponding terms in the other constraints: Cauchy-Schwarz inequality,

• i∈N

yij σi √n ≤

• i∈N

y2

ij .σ2 i

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Robust Optimization Model

Rj − Sj ≥ φ−1(0.9).

• i∈N

yij

• ˜

tpi + σi √n

• ∀j = 1, ..., n

(22) Cj −Rj ≥ φ−1(0.9).

• i∈N

yij

• ˜

tsi + ˜ tci + βi + ηi √n

• ∀j = 1, ..., n (23)

Rj+1 − Rj ≥ φ−1(0.9).

• i∈N

yij

• ˜

tsi + βi √n

• ∀j = 1, ..., n − 1 (24)

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Heuristic Method

Figure : Cn times for the sample data of Case 1 Figure : Cn times for the sample data of Case 5

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Numerical Examples

MIP Model

Table : Cost, Idle Time and Overtime Data

Case Cost IT of OR’s IT of the Srgn Total OT 1 $568.102 16.646 13.302 2 $648.425 14.364 20.529 3 $348.502 7.412 11,383 4 $830.129 10.470 35.415 5 $1258.668 13.887 55.99 6 $2423.124 28.716 105.504 7 $6638.309 2.281 147.139 299.132 8 $1070.404 16.547 42.152 9 $2265.513 129.606 10 $4513.932 17.298 131.140 139.351

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Numerical Examples

Table : Deterministic Approach-The order of surgeries in OR’s and for the surgeon

Case OR1 OR2 Surgeon 1 A C J A A J C A 2 A G A H J A H G J A 3 A G J D G A D G G J 4 A G G H F B A H G F G B 5 J F J J C D H J C F D J H J 6 D A A J G J J I A C D A A J I G A J C J 7 F I J H J I G A F H A F I I G A J F H H J A 8 A G G E J D B A J G D G B E 9 A I B D J G A I G J A G A I I B G D J J 10 A C E G H I J F I A E A J C F E I G A H E I

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Numerical Examples

Table : Stochastic Approach-The order of surgeries in OR’s and for the surgeon

Case OR1 OR2 Surgeon Cn 1 A C J A A J C A 178.322 min. 2 A G A J H A J G H A 224.681 min. 6 A J I J J D C G A A A J J D C I G A J A 644.310 min.

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Numerical Examples

Table : The Comparison of two models over 100 instance for Cases 1,2,6

Case Average Cn Stochastic Approach Deterministic Approach 1 122.83 min. 122.83 min. 2 166.5 min. 171.93 min. 6 499.99 min. 510 min.

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

Conclusion and Furhter Development

◮ Heuristic method is sensitive against standard deviation of the

parameters.

◮ The randomness of the objective function can be handled

without changing it.

◮ If you have questions...

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project

References

Goyal, Vineet and Ravi, R., ”Chance Constrained Knapsack Problem with Random Item Sizes” (2009).Tepper School of Business.Paper 367. http://repository.cmu.edu/tepper/367

Huseyin Gurkan 20701694 IE 518 Discrete Optimization Course Project