Polyhedral study of simple plant location problem with order

Operations Research Letters 41 (2013) 153–158

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Polyhedral study of simple plant location problem with order Igor Vasilyev a , Xenia Klimentova a,∗ , Maurizio Boccia b a

Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (ISDCT SB RAS), Lermontov str., 134, Post Box 292, 664033, Irkutsk, Russia

b

Dipartimento di Ingegneria, Università del Sannio, Palazzo ex INPS - Piazza Roma, 21, 82100, Benevento, Italy

article

info

Article history: Received 21 May 2012 Received in revised form 13 December 2012 Accepted 13 December 2012 Available online 28 December 2012

abstract This paper is addressed to the generalization of simple plant location problem where customer’s preferences are taken into account. Some basic polyhedral studies and a new family of facet-defining inequalities are given. The effectiveness of the proposed approach is illustrated by the computational experience. © 2012 Elsevier B.V. All rights reserved.

Keywords: Simple plant location problem with order Facet-defining inequality Branch-and-cut algorithm

1. Introduction The discrete location problems are of great importance in the combinatorial optimization and are widely addressed in the literature [1–3]. The basic models of this type are the simple plant location problem (SPLP) and the p-median problem. In the SPLP one needs to choose places from the given set to locate some plants and assign each client to one of the located plants minimizing the total cost for opening plants and servicing clients. Clients are supposed to have the same goal as the supplier. Actually the clients may have their own preferences about what plant to choose according to habits, profession, income etc. First, a model which takes into account clients’ preferences was considered in [4]—the so-called Simple Plant Location Problem with Order (SPLPO). In this model each client ranks the different plants and it is supposed that the clients choose the most preferable plants to be served from the set of open plants. The SPLPO can be considered as a bilevel programming model [5,6]. Indeed, on the upper level supplier chooses the set of plants to be open, and on the lower level clients choose optimal plants according to their preferences from the set of plants opened on the upper level. Opposite to the SPLP, the SPLPO is not widely addressed. In [4] a simple heuristic for small problems was considered together with integer linear formulation. The bilevel formulation and its reductions to the integer linear programs were studied in [5,6]. Also, in [5] a heuristic algorithm was developed, which combine

genetic and local search approaches and it was proven to be effective on some special test instances [7]. Different types of constraints modeling closest assignment of clients were investigated in [8–12]. In [13] the review of these constraints is presented and new set of inequalities is proposed. Also the interconnections of different sets of constraints are investigated. A strengthened SPLPO formulation with new families of valid inequalities was suggested in [14]. The formulation was used in a branch-and-cut algorithm for finding optimal solutions on randomly generated test instances. A family of valid inequalities was also proposed in [15] and its effectiveness was tested in [16]. In this paper the polyhedron of SPLPO is studied. We consider a related full dimensional polytope and study which inequalities from the basic formulation are facet-defining. A new family of facet-defining inequalities is presented as well. These new inequalities strengthen the inequalities from [15]. On the base of these inequalities and an upper bound heuristic, a branch-and-cut algorithm is developed and its effectiveness is illustrated by the computational comparison with the results from [14]. The rest of this paper is organized as follows. In Section 2 the problem statement and some known inequalities from [14] are given. Basic polyhedral properties of a polytope associated to the problem are studied in Section 3. In Section 4 the new family of facet-defining inequalities is proposed. Finally, computational results are reported in Section 5. 2. Problem statement and some valid inequalities

∗

Corresponding author. E-mail addresses: [email protected] (I. Vasilyev), [email protected], [email protected] (X. Klimentova), [email protected] (M. Boccia). 0167-6377/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2012.12.006

The formal problem statement of SPLPO can be given by the following way [14]. We are given with a set of plants I = {1, . . . , m}, a set of clients J = {1, . . . , n}, the cost of supplying the whole

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I. Vasilyev et al. / Operations Research Letters 41 (2013) 153–158

demand of customer j from plant i cij ≥ 0 and the cost of opening plant i fi ≥ 0. They say that plant k is j-worse than i if customer j prefers plant i to plant k and it is denoted by k


yi +



y ,x

i∈I

cij xij +

j∈J



fi y i ,

i ∈ I, j ∈ J,

xij = 1,

j ∈ J,

i ∈ I, j ∈ J,

xij , yi ∈ {0, 1},

(5)

One can see that the only difference between SPLPO and SPLP is the inequalities (2), which guarantee that each client is served by the most preferable open plant. Indeed, if plant i is open, each customer j will not be served by any plant which is j-worse than i. These inequalities are called as 1-customer preference inequalities. We will denote the feasible set of this problem by X = {(y, x) : (y, x) satisfies (2)–(5)} and the problem polytope by P = conv(X ). Cánovas et al. [14] proposed some families of valid inequalities for P which strengthen the initial formulation and which are effective for finding the optimal solution of the problem. Let us remind them here. 1. 2-customers preference inequalities. Let j1 , j2 ∈ J , i ∈ I. The following inequalities are valid for the polytope P:

 k∈Wij

xkj1 +

 k∈Wij ∩Bij 2

1

xkj2 + yi ≤ 1.

k∈Wij

xkj1 +

1

s  t =2

(6)

  k∈Wijt ∩

t −1 q=1

xkjt + yi ≤ 1

(7)

 Bijq

are valid for the polytope P. This family of inequalities induces a formulation with the exponential number of constraints. Therefore it is reasonable to use only a part of these inequalities to strengthen the formulation. For example, in [14] the authors proposed to choose elements j1 , . . . , js ∈ J , i ∈ I, with empty intersections of the sets Wijt , t = 1, . . . , s. 3. Dominance inequalities. Let j1 , j2 ∈ J and i ∈ I. If Bij2 ⊆ Bij1 , then the inequality xij1 ≤ xij2

(8)

is valid for the polytope P and dominates the inequality xij1 ≤ yi . If Bij1 = Bij2 , the inequality (8) is transformed into an equality xij1 = xij2 .

(9)

One can obtain the equivalent problems with equations and inequalities by modifying the coefficients of the objective function (1). We will denote the feasible set of new problem with inequalities by X ≤ = {(x, y) | (x, y) satisfies (2), (4)–(5), (9)}

Proof. By the definition, to prove the proposition it is sufficient to construct m + mn + 1 affinely independent feasible points in X ≤ . Let us consider points (yu , xu ) for all u ∈ I such that

∀i ∈ I

yui

1, if i = u , 0, if i ̸= u

 =

∀i ∈ I , j ∈ J xuij = 0

(10)

and points (yuv , xuv ) for all u ∈ I , v ∈ J such that

∀i ∈ I

yui v

 =

1, 0,

∀i ∈ I , j ∈ J xuijv =

if i = u , if i ̸= u



1, 0,

if i = u and j = v otherwise.

(11)

It is evident that these points together with the origin create a system of m + mn + 1 affinely independent feasible points.  It is known that xij ≥ 0, i ∈ I , j ∈ J, and xij ≤ yi , i ∈ I , j ∈ J are facet-inducing for the polytope of SPLP. They are also facetinducing for SPLPO polytope.

1

As it can be seen, these inequalities dominate initial inequalities (2) which ensure clients’ preferences. 2. s-customers preference inequalities. The previous inequalities can be generalized to constraints with any number s of clients. Let j1 , . . . , js ∈ J and i ∈ I. Then the inequalities



j ∈ J.

Proposition 1. The polytope P ≤ is full dimensional, i.e. dim(P ≤ ) = m + mn.

(4)

i ∈ I, j ∈ J.

xij ≤ 1,

i∈I

(3)

i∈I

xij ≤ yi ,



(2)

k∈Wij



Following after [17,18], who investigated polyhedral properties of SPLP, we consider the formulation where the Eq. (3) is replaced by the inequalities

and the polytope of this problem will be denoted by P ≤ = conv (X ≤ ). The motivation for considering the formulation of SPLP with inequalities is that the polytope of such a problem is full dimensional [17,18]. This is also true for the SPLPO.

(1)

i∈I

xkj ≤ 1,

3. Polyhedral properties of the formulation

(8′ )

Proposition 2. The inequalities (1) xij ≥ 0, i ∈ I , j ∈ J, (2) xij ≤ yi , i ∈ I , j ∈ J are facet-defining for the polytope P ≤ . Proof. (1) The origin and all points (10)–(11) except (yij , xij ) satisfy xij ≥ 0 as equality. (2) The origin, all points (10) except (yi , xi ) and points of set (11) with except (yuv , xuv ) for u ̸= i satisfy xij ≤ yi as equality. For u = i the points of set (11) can be modified to belong to the facet as follows: yiiv = 1, xiivv = 1

and

xiijv = 1, v ∈ J . 

The more interesting question concerns the properties of inequalities (2) which ensure that all customer’s preferences are  satisfied. Note that for polytope P the constraint yi + k∈Wij xkj ≤ 1 with Bij = ∅ becomes xij ≥ yi and, taking into account (5), we obtain xij = yi . So the inequalities (2) cannot be facet-defining for P in general case, but for P ≤ the following theorem takes place. Theorem 3. If Bij = ∅ for some i ∈ I , j ∈ J then inequality yi +  ≤ k∈Wij xkj ≤ 1 is facet-defining for the polytope P .

I. Vasilyev et al. / Operations Research Letters 41 (2013) 153–158

Proof. For the sake of simplicity let us suppose that B11 = ∅ and consider the equality y1 +



xk1 = 1.

(12)

k∈W11

To prove the theorem we are going to find m + mn affinely independent points, which lies in P ≤ and satisfy (12). Let elements of any point (y, x) be presented in the following order:

(y, x) = (y1 , y2 , . . . , ym , x11 , x21 , . . . , xm1 , x12 , x22 , . . . , xm2 , . . . , x1n , x2n , . . . , xmn ),

y1 + x21 + x31 + · · · + xm1 ≤ 1.

xi1 ,i=1,...,m

xi2 ,i=1,...,m

   (1, 0, 0, .. , 0, (1, 1, 0, .. , 0, (1, 0, 1, .. , 0, .. (1, 0, 0, .. , 1,

   0, .. , 0, 0, .. , 0, 0, .. , 0, .. 0, .. , 0,

0, .. , 0, 0, .. , 0, 0, .. , 0,

  

.. 0, .. , 0,

.. .. .. .. ..

1, 0, 0, .. , 0, 0, 1, 0, .. , 0, 0, 0, 1, .. , 0,

0, .. , 0, 0, .. , 0, 0, .. , 0,

0, 0, 0, .. , 1,

0, .. , 0,

..

.. .. .. .. ..

..

1, 0, 0, .. , 0, 0, 1, 0, .. , 0, 0, 0, 1, .. , 0,

1, 0, 0, .. , 0, 0, 1, 0, .. , 0, 0, 0, 1, .. , 0,

0, 0, 0, .. , 1,

0, 0, 0, .. , 1,

0, 0, 0, .. , 0)

:

:

:

:

:

(1, 0, 0, .. , 0, (0, 1, 0, .. , 0, (0, 0, 1, .. , 0, .. (0, 0, 0, .. , 1,

1, 0, 0, .. , 0, 0, 1, 0, .. , 0, 0, 0, 1, .. , 0,

0, 0, 0, .. , 0, 0, 0, 0, .. , 0, 0, 0, 0, .. , 0,

1, 0, 0, .. , 0) 0, 1, 0, .. , 0) 0, 0, 1, .. , 0)

0, 0, 0, .. , 1,

0, 0, 0, .. , 0,

.. .. .. .. ..

..

..

t where t = bv (I \ Wiu ) and Uiu = I \ (Wiu ∪ {t }), are valid for the polytope P.

Proof. In the proof we will always use one evident property of SPLPO: i ̸= k then Wij ⊂ Wkj .

..

..

(17)

Assume the contradiction, i.e. there exists (¯y, x¯ ) ∈ X , i ∈ I , u, v ∈ J and t = bv (I \ Wiu ) such that x¯ ku +



x¯ kv + y¯ t > 1.

t k∈Uiu

k∈Wiu

Taking into account the constraints (4), this assumption can hold only in the following three cases:



y¯ t +

x¯ ku > 1.

So we have obtained that (¯y, x¯ ) violates (2) and the contradiction that (¯y, x¯ ) is feasible. ¯ kv = 1, i.e. there exists q ∈ Uiut such that (2) y¯ t = 1 and k∈U t x iu

x¯ qv = 1. By the definition of t we have that Wqv ⊂ Wt v thus q ∈ Wt v and



y¯ t +

(14)

0, .. , 0).

.. .. .. .. ..

(16)

k∈Wtu

It is easy to see that for all i ∈ I (yi , xi ) ∈ P ≤ and satisfy (12) since W11 = {2, . . . , m}. ij ij ij (iii) In the next (n − 1)m points we set yi = 1, xi1 = 1, xij = 1 for each i ∈ I , j ∈ J \ {1} and we will (1, 0, 0, .. , 0, (0, 1, 0, .. , 0, (0, 0, 1, .. , 0, .. (0, 0, 0, .. , 1,

xkv + yt ≤ 1,

t k∈Uiu

k∈Wiu

(13)

.. 0, .. , 0).

0, .. , 0) 0, .. , 0) 0, .. , 0)





  

0, .. , 0) 0, .. , 0) 0, .. , 0)

xku +

(1) y¯ t = 1 and k∈Wiu x¯ ku = 1, i.e. there exists q ∈ Wiu such that x¯ qu = 1. By the hypothesis that t ∈ I \ Wiu and using (17) we have that q ∈ Wtu and

xin ,i=1,...,m

It is evident that (yi , xi ) ∈ P ≤ and satisfy (12) for all i ∈ I. (ii) In the next m points we set yii = 1, xii1 = 1 for each i ∈ I and we will have

..





Let us consider points constructed by the following rules: (i) The first m points are such that yii = 1 and yi1 = 1 for each i = 1, . . . , m, all the other elements of (yi , xi ) equal to 0, i.e. we get

(1, 0, 0, .. , 0, (0, 1, 0, .. , 0, (0, 0, 1, .. , 0, .. (0, 0, 0, .. , 1,

Theorem 4. For all i ∈ I , u, v ∈ J the inequalities

if i ∈ I , j ∈ J and k ∈ I \ Wij then Wij ⊆ Wkj and if

and 1 >1 2 >1 · · · >1 m, i.e. the order of the first customer’s preferences coincides with the indices of plants, hence constraint (12) have the following form:

yi ,i=1,...,m

155

x¯ kv > 1.

k∈Wt v

(3)

Again  we have obtained  the contradiction that (¯y, x¯ ) is feasible. ¯ ku = 1 and k∈U t x¯ kv = 1, i.e. there exists q ∈ Wiu k∈Wiu x iu

and s ∈ I \ Wiu such that x¯ qu = 1 and x¯ sv = 1. This implies by (4) that y¯ s = 1 and by (17) that q ∈ Wsu thus



y¯ s +

x¯ kv > 1.

k∈Wsu

The contradiction that (¯y, x¯ ) is feasible has been obtained as well.

0, 0, 0, .. , 0) 0, 0, 0, .. , 0) 0, 0, 0, .. , 0)

It follows that the inequalities (16) are valid for the polytope P.

..

(15)

..

0, 0, 0, .. , 1).

It is easy to see again that all these points belong to P ≤ and satisfy (12). Rows from (13)–(15) form a triangle matrix with 1 on the diagonal, so these rows are linearly independent. As the set of points do not contain origin, they are also affinely independent and the theorem is proved.  4. New family of valid inequalities In this section we study a new family of valid inequalities. For some client j and subset of plants S ⊆ I let us denote by bj (S ) ∈ S the best plant from S for client j, i.e. i


The next theorem clarifies the question whether the new inequalities define facets. Theorem 5. The inequalities (16) are facet-defining for the polytope P ≤. Proof. If u = v we have that t = bu (I \ Wiu ). It means Wtu = Wiu ∪ t Uiu = Wiu ∪ I \ (Wiu ∪ {t }) = I \ {t } and Btu = ∅. In this way (16) becomes yt +



xku ≤ 1

k∈Wtu

and by Theorem 3 this inequality is facet-defining. Let us consider the case when u ̸= v . Without loosing generality we suppose that u = 1 and v = 2, then for i ∈ I , t = b2 (I \ Wi1 ) and the elements of any point (y, x) are presented in the same way as in Theorem 3. Remind that the order of the first customer’s preferences coincides with the indices of plants. With these assumptions inequalities (16) can be re-written as follows: yt + xi+1,1 + xi+2,1 + · · · + xm1 + x1,2 + x2,2 + · · · , +xt −1,2

+ xt +1,2 + · · · + xi,2 ≤ 1.

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I. Vasilyev et al. / Operations Research Letters 41 (2013) 153–158

Fig. 1. Constructed points in the proof of Theorem 5.

We construct points by the following rules: (i) The first m points such that yhh = 1 and yht = 1 for all h ∈ I, all other elements (yh , xh ) equal to 0 (see Fig. 1, (18)). (ii) In the next m points yhh = 1 for all h = 1, . . . , m; yht = 1 if h = 1, . . . , t − 1; xhh1 = 1 if h = 1, . . . , t and h = i + 1, . . . , m; xhh2 = 1 if h = t + 1, . . . , i (see Fig. 1, (19)). (iii) In the next m points yhh = 1 for h = 1, . . . , m; xhh1 = 1 if h = t + 1, . . . , m; xhh2 = 1 if h = 1, . . . , m (see Fig. 1, (20)). (iv) In the next m(n − 2) points we modify the points of (iii) in the following way: yhk = yh , xhk = xh , xhk hk

but also

= 1 for all h ∈ I and k ∈ J \ {1, 2}.

(21)

It is not difficult to note that all the points defined in (18)–(21) are feasible, satisfy (16) as equality and form a matrix, which can be easily transformed into the triangle matrix with 1 on the main diagonal by column permutation (columns (xt +1,1 , . . . , xi,1 ) and (xt +1,2 , . . . , xi,2 )). So we have m(n + 1) linearly independent points and the theorem is proved.  We have mentioned in the proof, that in case of u = v inequality (16) becomes (2) with Btu = ∅. Let us also discuss the case when t = i for some u and v . In this case inequality (16) becomes yi +

 k∈Wiu

xku +



xkv ≤ 1

(22)

k∈Biu

and we have that Biu ⊂ Wiv . It means that Wiv ∩ Biu = Biu and Biu ∩ Biv = ∅, thus inequality (22) coincides with the 2-customers preference inequality (6) and it cannot be strengthened by any other customer in the form of s-customers preference inequality (7).

5. Computational experiments In this section we present the computational results of the branch-and-cut (B&C ) algorithm with the new family of facets (16) on the test instances from [14]. The experiments have been carried out on a PC with Intel Core 2 Duo 1.8 GHz processor and 1Gb of RAM. The MIP solver FICO Xpress [19] callable library has been used as B&C framework. Xpress provides functionalities for managing the search tree. Users can implement separation routines, add cutting planes and can implement a problem specific heuristic etc. We have embedded our cutting plane method and an upper bound heuristic as described below. Our code is public available and can be found at http://iv.icc.ru/Papers.html. The number of inequalities in our family of facets is O(n2 m). If we add all of them to the formulation, we obtain very large problem even for ‘‘small’’ instances. Therefore to compute lower bounds with new facets we need to implement the Cutting Plane method (CP) [20,21]. At first, we tried to add all the violated inequalities (16), but after the preliminary computational experiments we found out that it was not effective because the number of inequalities which were added to the formulation was still too large. Therefore we used some computational tricks to reduce the number of added inequalities, considering the two following parameters: 1. Let aT x ≤ 1 be a violated inequality, then the distance between the current fractional solution x¯ and the hyperplane corresponding to the inequality is calculated as follows: r (¯x, a) =

aT x¯ − 1

∥ a∥

and it measures the violation of the corresponding cut.

I. Vasilyev et al. / Operations Research Letters 41 (2013) 153–158

2. If aT x ≤ 1 and bT x ≤ 1 are two violated inequalities. Cosine of the angle between the corresponding hyperplanes is computed ⟨a,b⟩ as follows: cos(a, b) = ∥a∥ ∥b∥ . We define two parameters for CP: M is the maximum number of inequalities that can be added on each iteration of the method and 0 < η < 1 is the maximum value of a cosine of the angle between hyperplanes of violated inequalities. In our implementation of CP we add no more than M of the most violated inequalities on each iteration with a cosine between all of them not larger than η, i.e. we prevent the inequalities which are almost parallel from adding. To find the upper bounds of optimal value we implemented a standard scheme of Simulated Annealing (SA) algorithm [22,23], so we do not describe it in details. We used a flip-neighborhood which is defined as the set of feasible points being reachable from the current one by means of just opening or closing one plant. All other implementation details can be found in our available code. The test instances presented in [14] are randomly generated and can be divided in the following way:

• Small instances consist of 12 instances with 50 plants and clients.

• Medium instances include 12 instances with 50 plants and 75 clients. On the basic of these initial instances 36 additional instances were generated as well [14]. The only difference among them is the values of the opening costs fi , which are varied depending on some parameter α (the more value the bigger fi ). Three values of α are considered (1, 0.75 and 1.25). So, this set contains 48 instances. • Large instances are 12 instances with 75 plants and 100 clients. To check the effectiveness of the proposed approach we compare the three following cases: XPR — FICO Xpress with the default settings is run for the formulation from [14], whose code was kindly provided by authors. XPRSA — the same as XPR but optimizer is run with given upper bound of optimal value obtained by SA. C &B — where our cutting plane algorithm is used only at the root node of enumeration tree (cut-and-branch method), starting from the formulation obtained in [14], also with given upper bound. B&C — where our cuts are generated at each node (branch-andcut method) and the upper bound of optimal value is also given. The computational results are presented in Tables 1–6, where the following notations are used: Name is the instance name; Opt is the optimal value; GAP is the integrality gap computed as GAP = Opt −LP × 100%, where LP is the value of linear programming reOpt laxation of corresponding formulation; Time is the computational time in seconds. The small instances are very easy and they can be solved in a few seconds by any approach. We just want to mention that the integrality gap is significantly decreased by our family of valid inequalities and the gap is closed for a couple of instances (133-2 and 134-2). The integrality gap for the medium instances was reduced as well. It is 13% at the average in comparison with 24% from [14]. The computational time XPR was reduced by all three methods XPRSA , B&C and C &B, and it is difficult to say which of them is more effective. The improvement of the computational time is on average 37% for the all medium instances, with the worst improvement of 30% for the tests with α = 1.25 and the best performance of

157

Table 1 Small instances. Name 132-1 132-2 132-3 132-4 133-1 133-2 133-3 133-4 134-1 134-2 134-3 134-4

Opt 1 122 749.5 1 157 721.9 1 146 301.4 1 036 779.4 1 103 272.0 1 035 443.0 1 171 331.3 1 083 636.5 1 179 639.4 1 121 633.0 1 171 408.6 1 210 465.9

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

8.7 11.8 10.1 6.1 8.5 5.2 11.7 6.3 7.8 5.4 12.2 11.8

0.9 3.8 1.9 1.1 0.1 0 2.4 1.5 0.4 0 3.1 4.5

2.5 3.7 2.4 1.5 1.4 1.3 3.4 2.2 3.3 1.5 6.4 6.6

2.7 2.6 2.4 1.2 0.8 0.9 2.1 1.4 2.2 1.2 3.8 3.1

2.3 3.0 2.7 1.4 1.5 1 2.4 1.4 2.2 1.4 4.1 3.7

1.7 2.9 2.6 1.3 1.1 0.9 2.1 1.5 2.3 1.2 3.8 3.2

8.8

1.6

3.1

2.0

2.3

2.1

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

24.5 23.7 23.7 21.8 24.5 27.2 24.4 22.7 26.8 25.2 22.3 26.6

15.4 15.1 15.1 13.4 12 16.2 12.9 10.3 13 10.6 8.3 14.3

165.7 147.3 162.8 143.1 87.6 157.9 143.9 80.0 110.8 68.2 41.8 143.9

114.3 97.1 117.4 76.8 49.0 137.6 71.1 41.7 80.1 50.3 18.0 76.3

125.4 99.8 111.0 79.3 54.9 133.5 76.8 38.6 73.9 47.0 23.5 77.5

128.1 96.9 116.3 73.7 46.9 128.3 73.6 40.3 84.6 50.5 19.2 70.7

24.5

13.1

121.1

77.5

78.4

77.4

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

24.5 24.1 22.5 22.5 24.4 26.7 23.2 23.5 27.2 24.5 22.5 27.2

15 15.5 14.4 14.1 12.2 15.9 11.6 11.4 13.3 9.9 8.4 14.9

170.1 125.9 150.1 156.2 54.9 188.1 128.4 89.6 120.3 65.2 47.5 142

104.6 100.9 94.8 94.7 39.2 146.5 61.6 43.7 92.1 34.8 20.6 93.7

101.9 98.8 85.1 90.9 48.9 119.4 61.5 47.4 86.4 42.5 25.0 90.2

111.1 107.4 92.8 93.3 38.7 144.5 59.6 42.8 97.0 40.4 21.2 94.1

24.4

13.1

119.9

77.2

74.8

78.6

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

24.6 25 22.7 22.6 24.5 26 23.2 23.5 26 24.5 22.4 27.2

14 15.7 13.4 13.5 11.7 14.2 10.7 10.1 11.2 9.1 7.4 14.4

177.6 216.9 155.6 140.0 60.3 166.1 93.9 98.4 84.2 65.3 33.7 144.9

87.2 114.3 76.9 79.8 40.1 99.5 48.9 48.5 53.6 31.2 19.2 81.1

98.1 109.1 80.2 79.6 56.8 81.3 45.1 48.0 41.0 33.6 21.6 76.2

86.3 111.7 75.3 81.3 41.3 99.8 48.3 51.9 48.4 32.0 22.7 90.6

24.4

12.1

119.7

65.0

64.2

65.8

Average

Table 2 Medium instances. Name a-75-50-1 a-75-50-2 a-75-50-3 a-75-50-4 b-75-50-1 b-75-50-2 b-75-50-3 b-75-50-4 c-75-50-1 c-75-50-2 c-75-50-3 c-75-50-4

Opt 1 661 269.3 1 632 907.0 1 632 212.6 1 585 027.7 1 252 803.6 1 337 446.4 1 249 750.2 1 217 508.1 1 310 192.7 1 244 255.0 1 201 706.4 1 334 782.5

Average

Table 3 Medium instances with α = 1. Name a-75-50-1 a-75-50-2 a-75-50-3 a-75-50-4 b-75-50-1 b-75-50-2 b-75-50-3 b-75-50-4 c-75-50-1 c-75-50-2 c-75-50-3 c-75-50-4

Opt 1 658 472.6 1 643 852.8 1 619 966.0 1 603 735.5 1 259 251.4 1 339 678.7 1 227 927.1 1 238 553.3 1 316 414.5 1 238 473.3 1 209 913.2 1 348 663.0

Average

Table 4 Medium instances with α = 0.75. Name a-75-50-1 a-75-50-2 a-75-50-3 a-75-50-4 b-75-50-1 b-75-50-2 b-75-50-3 b-75-50-4 c-75-50-1 c-75-50-2 c-75-50-3 c-75-50-4 Average

Opt 1 538 065.1 1 540 646.4 1 499 715.0 1 485 204.2 1 198 430.6 1 257 997.4 1 160 398.5 1 166 420.8 1 236 600.4 1 183 939.6 1 143 874.9 1 293 171.7

158

I. Vasilyev et al. / Operations Research Letters 41 (2013) 153–158

Table 5 Medium instances with α = 1.25. Name a-75-50-1 a-75-50-2 a-75-50-3 a-75-50-4 b-75-50-1 b-75-50-2 b-75-50-3 b-75-50-4 c-75-50-1 c-75-50-2 c-75-50-3 c-75-50-4

Opt 1 711 566.2 1 721 127.4 1 692 029.9 1 695 894.1 1 315 014.0 1 403 516.3 1 290 660.0 1 298 265.6 1 371 537.2 1 288 357.1 1 274 762.2 1 403 535.2

Average

Acknowledgments

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

22.2 22.5 20.8 21.7 24.4 26.8 23.4 23.4 27.3 24.5 23 27.3

13.2 14.6 13.3 13.9 12.6 16.7 12.4 12.1 14 10.5 9.6 15.7

99.3 135.6 95.4 134.1 95.5 193.8 100.3 84.7 108.5 82.7 49.8 161.9

68.5 98.2 74.9 87.1 46.2 150.1 62.9 49.8 97.6 44.4 27.8 110.1

67.7 93.1 68.0 92.8 60.1 165.3 75.8 54.5 103.5 46.1 27.1 118.6

70.9 99.1 69.2 89.9 50.0 151.5 64.5 51.8 105.6 42.2 28.5 109.3

23.9

13.2

111.8

76.5

81.1

77.7

Table 6 Large instances. Name a-100-75-1 a-100-75-2 a-100-75-3 a-100-75-4 b-100-75-1 b-100-75-2 b-100-75-3 b-100-75-4 c-100-75-1 c-100-75-2 c-100-75-3 c-100-75-4 Average

Opt 2 286 397.5 2 389 326.7 2 415 835.8 2 380 149.7 1 950 231.4 2 023 097.4 2 062 595.0 1 865 322.8 1 843 620.5 1 808 867.0 1 820 587.3 1 839 007.2

GAP XPR

C &B

Time XPR

XPRSA

C &B

B&C

19 25.2 24.2 22.3 28.5 30.5 31 27.1 28 28.4 28.9 29.4

11.2 17.9 17.1 15.3 20.6 22.1 22.5 18 19 18.6 18.8 19.6

1040.3 1678.3 1507.5 1210.1 5204.9 7797.8 7947.0 3399.8 4712.8 4257.7 2961.6 4451.7

322.64 567.2 1123.2 830.2 3129.91 4792.94 5812.5 1704.61 2699.34 3145.05 1814.69 2740.13

319.1 533.7 1036.7 821.9 3103.2 4119.6 5634.2 1492.1 2574.5 2568.5 1676.3 2732.2

318.9 586.7 1129.6 827.4 3130.1 4671.0 5737.8 1674.9 2548.8 2701.2 1796.7 2603.3

26.9

18.4

3847.5

2390.2

2217.7

2310.5

44% at the average for the instances with α = 0.75. It can be concluded that for medium instances the reduction of time was obtained mainly due to using of SA method. The duality gap for the large instances was 27% at the average and it is reduced to 18% by using our facet-defining inequalities. Concerning the computational time, our cut-and-branch method is a little bit more effective than the branch-and-cut, it outperforms XPRSA for all 12 instances and it is approximately two times faster at the average than Xpress on the formulation from [14] (XPR). Thus the improvement has been obtained here thanks to the implemented CP. Making the conclusions, we can say that the proposed family of valid inequalities improves the known formulations and, together with the upper bounding heuristic, the implemented branch-and-bound algorithm improves the state-of-the-art exact approaches presented in [14].

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