An efficient linearization approach for mixed-integer problems

European Journal of Operational Research 123 (2000) 652±659

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Short Communication

An ecient linearization approach for mixed-integer problems Ching-Ter Chang

*

Department of Business Education, National Changhua University of Education, Paisa Village, Changhua 50058, Taiwan, ROC Received 15 September 1998; accepted 25 January 1999

Abstract Oral and Kettani previously developed a linearization technique, published in Management Science in 1990 and in Operational Research in 1992, for solving quadratic and cubic mixed-integer problems. For a quadratic problem with n 0±1 variables, their method would introduce n additional continuous variables and n auxiliary constraints. For a cubic problem with n 0±1 variables, their method would introduce 3n additional continuous variables and 3n auxiliary constraints. This linearization approach of Oral and Kettani has been accepted as the most ecient linearization technique published, requiring the least number of additional continuous variables and auxiliary constraints. However, their method is dicult to extend for linearizing higher order polynomial terms that appear in mixed-integer problems, and in addition, all constraints should be kept as linear. This note proposes a new general model for linearizing various orders of mixed-integer problems which cannot be solved by Oral and Kettani's model when the order is higher than three. Some computational results show that the proposed model is more ecient than Oral±Kettani's method because it uses less additional variables and auxiliary constraints to linearize the same size of mixed-integer problems. In addition, the proposed model can be easily applied to polynomial mixed-integer terms that appear in the objective function and/or constraints. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Linearization; Polynomial; Binary; Mixed-integer

1. Introduction A variety of decision problems in areas such as facility layout, job assignment and even communication network designs are most often formulated as pure or mixed-integer problems with a set of linear constraints (e.g. [1,14,13,2]). Sherali and Tuncbilek [11] derived a reformulation linearization technique (RLT) which generated polynomial *

E-mail address: [email protected] (C.T. Chang).

implied constraints, and then linearized the resulting problem by de®ning new variables. This construct was then used to obtain lower bounds in the context of a proposed branch and bound scheme. Later Sherali et al. [12] proposed a hierarchy of relaxations for mixed 0±1 integer problems which is the extension of the RLT. Although the RLT process is promising for converging to a global solution, this process is practically very dicult to implement for the following reasons: (i) Several types of implied constraints, or subsets, or surrogates need to be generated in a linearized

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 0 6 - X

C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

form. Tightening its representation step by step at the expense of an exponential constraint is a long trial-and-error process. (ii) An RLT algorithm always requires the generation of huge amounts of bounded constraints, and many of these constraints are redundant. (iii) There are considerable variants in designing an RLT process, depending on the actual structure of the problem being solved, so a user needs to formulate a special RLT scheme corresponding to each of his programs. Li and Chang [6] proposed a binary approximate approach for polynomial programming problems, which can solve general polynomial problems close to a global optimum with a prespeci®ed tolerance. A major diculty in implementing Li and Chang's method is that if the range of a variable x is large, and the bound of the error tolerance of x is small, then it requires the addition of binary. Tabu search with target analysis (TA) as a learning approach to identify e€ective search parameters and choice rules for mixed-integer problems was proposed by Lùkketangen and Glover [7]. Later a probabilistic measures was used to examine how the learning tool TA can be applied to identify better control structures and decision rules (Lùkketangen and Glover [8]). However, these methods can obtain the best solutions in only some special cases. Formulations proposed in Glover [3] have the feature of yielding signi®cantly smaller problem sizes, in which it is desired to minimize P x subject to xi ˆ 0 or 1 for all i;j2N i dij xj i 2 N ˆ f1; . . . ; ng, with all remaining constraints linear. In thisP representation, the nonlinear P objective function i;j2N xi dij xj is replaced by i;j2N ni , subject to the following set of constraints: Dÿ i xi X j

6 ni 6 D‡ i xi ;

dij xj ÿ D‡ i …1 ÿ xi † 6 ni 6

…1:1† X j

dij xj ÿ Dÿ i …1 ÿ xi †; …1:2†

D‡ i

ˆ the sum (over j) of the positive dij 's, where Dÿ ˆ the sum (over j) of the negative dij 's. i The above linearization process of Glover [3] requires only P adding n additional continuous variables (i.e. i ni ) and 4n auxiliary constraints (i.e., Eqs. (1.1) and (1.2)). Subsequently, Kettani

653

and Oral [5] proposed a linearization process which substantially improved the technique of Glover [3]. In Kettani±Oral's [5] quadratic model, the number of auxiliary constraints will be reduced from 4n down to 2n (plus n sign constraints), while the number of additional continuous variables will be kept at n, but constrained in sign. A technique similar to this had also previously been proposed by Glover [4]. Oral and Kettani [10] later proposed another linearization process for quadratic and cubic mixed-integer problems having n of 0±1 variables. This method adds n additional continuous variables and n auxiliary constraints to linearize the quadratic mixed-integer problems, and it also requires the addition of 3n additional continuous variables and 3n auxiliary constraints to linearize the cubic mixed-integer problems. Oral± Kettani's model is shown below: Oral±Kettani's cubic model (OKCM): X

Minimize

…ct S ‡ xt ‡ ut † ‡

t:ct <0

X

…ct S ÿ xt ‡ ut †

t:ct >0

subject to n X ‡ ÿ ut P ct …D‡ i xi ÿ pi † ÿ ct S xt ÿ ct S …1 ÿ xt † iˆ1

for all t: ct < 0; ut P ct

…1:3†

n X ÿ ‡ …Dÿ i xi ÿ qi † ÿ ct S xt ÿ ct S …1 ÿ xt † iˆ1

for all t: ct > 0;

…1:4†

ÿ pi P ÿ Di …x; y† ‡ D‡ i xi ‡ Di …1 ÿ xi †;

i ˆ 1; . . . ; n;

…1:5†

‡ qi P Di …x; y† ÿ Dÿ i xi ÿ Di …1 ÿ xi †;

i ˆ 1; . . . ; n; …1:6†

pi P 0; qi P 0; xi and xt are binary variables; i ˆ 1; . . . ; n; L…x; y† and ut P 0;

…1:7† t ˆ 1; . . . ; n;

…1:8†

where Di …x; y† is a linear function of x1 ; . . . ; xn and y1 ; . . . ; ym ; and L…x; y† is a set of linear constraints. ‡ ÿ Furthermore, Dÿ i 6 Di …x; y† 6 Di , Di is a lower ‡ bound and Di is an upper bound of Di …x; y†;

654

C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

P similarly, i:xi ˆ1 Di …x; y† Pmust also lie between certain bounds S ÿ 6 i:xi ˆ1 Di …x; y† 6 S ‡ , with S ÿ and S ‡ being constants. From constraints (1.3) and (1.4), we know that pi s are related only to the case where ct < 0, while qi s are related only to ct > 0. The constraints (1.5) and (1.6) thus reduce to:  0 if xi ˆ 0; …1:9† pi P if xi ˆ 1; ÿDi …x; y† ‡ D‡ i  qi P

0 Di …x; y† ÿ Dÿ i

if xi ˆ 0; if xi ˆ 1:

…1:10†

When we replace constraints (1.3) and (1.4) with (1.9) and (1.10), the constraints on ut are reduced as 8 if ct < 0 and xt ˆ 0; > ! <0 P ut ˆ ‡ if ct < 0 and xt ˆ 1; > : ct i:x ˆ1Di …x; y† ÿ ct S

ables and auxiliary constraints in the linearization process for linearzing quadratic or cubic mixedinteger problems. The author proposes a general model for linearizing any order of polynomial mixed-integer problems, which cannot be solved by OKCM when the problem's order is higher than three. In addition, some computational results show that the proposed model is more ecient than OKCM. 2. Proposed model (PM) In this note, we propose a new model for a mixed-integer problem, which may contain, quadratic cubic, or even higher order polynomial terms. A polynomial mixed-integer problem should minimize mixed-integer terms in the objective, subject to a set of linear constraints expressed below:

i

…1:11†

ut ˆ

8 > <0 > : ct

P i:xi ˆ1

if ct > 0 and xt ˆ 0;

! Di …x; y†

ÿ ct S ‡

if ct > 0 and xt ˆ 1: …1:12†

Substituting u1 ; . . . ; un in the objective P function by (1.11) and (1.12) to obtain t:xt ˆ1 ct P … i:xt ˆ1 Di …x; y†† implies that OKCM and cubic mixed-integer problems are equivalent in the sense that they have the same optimal solutions. The linearization approach of OKCM seems to be the most ecient linearization technique proposed to date, requiring the least number of additional continuous variables and auxiliary constraints. OKCM, however, has two restrictions which prohibit it from being applied to linearize more general polynomial mixed-integer problems. These restrictions are described below: (i) It can only linearize quadratic and cubic mixed-integer problems where all constraints should be kept linear. (ii) It becomes too complicated when applied to a polynomial term with order higher than three. This paper notes that it is still possible to reduce the number of added additional continuous vari-

Problem h0 . Q Minimize ct Di …x; y† i2J xi subject to L…x; y†; ÿ where ct , Di …x; y†, D‡ i , Di , and L…x; y† are de®ned as in OKCM; and xi is a 0±1 variable, indexed by some set J.

Program h1 . In the following inequalities, h0 can be replaced with a continuous variable z: ! X xi ÿ j J j ‡ ct Di …x; y†; …2:1† z P jct Di j i2J

z P ÿ jct Di jx1



…2:2†

z P ÿ jct Di jxn ; ÿ where Di ˆ Max{|D‡ i |, |Di |} and other variables are de®ned as h0 .

Proposition 1. h0 and h1 are equivalent in the sense that they have the same optimal solutions.

C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

655

which cannot be solved by any of the previous methods above.

Proof. Observe that z in h1 is subject to two constraints: Qn xi ˆ 0 then z P 0 (from (2.2)), and (i) If iˆ1P z P jct Di j… i2J xi ÿ j J j† ‡ ct Di …x; y† 6 0 (from (2.1)) Q will force z to be zero. (ii) If niˆ1 xi ˆ 1 then z P ÿ jct Di j (from (2.2)), z P ÿ jct Di j (from (2.1)) will force z to be ct Di …x; y†. The constraints on z thus reduce to 8 n Q > > xi ˆ 0; if < 0; iˆ1 zˆ n Q > > xi ˆ 1 : ct Di …x; y†; if

3. An illustrative example Consider a general mixed-integer problem with four 0±1 variables, which can be solved by the proposed model, but cannot be solved by the OKCM. Example 1. Minimize ÿ2x1 x2 x3 D1 …x; y† ‡ 3x1 x2 x3 x4 D2 …x; y† subject to

iˆ1

Q which is obviously the same as ct Di …x; y† i2J xi ˆ z. This completes the proof of Proposition 1. The number of additional continuous variables and auxiliary constraints used in Refs. [3,5,10], and the proposed model are listed in Table 1. Table 1 indicates that the proposed model, in the case of quadratic mixed-integer problems having n of 0±1 variables, requires n additional continuous variable and 2n auxiliary constraints. When applied to cubic mixed-integer problems having n of 0±1 variables, it requires n additional variables and 3n auxiliary constraints. These values are better than existing models such as Glover [3], Kettani and Oral [5] and Oral and Kettani [10], since, under the case of a cubic mixed-integer problem, the proposed model requires fewer additional variables and fewer auxiliary linear constraints to linearize the same order of mixedinteger problems, and the model also has more compact form in the representation of linearization process. In addition, it can easily be applied to higher order polynomial mixed-integer problems

D1 …x; y† ˆ x1 ÿ 2x2 ‡ 5x3 ‡ 4y1 ;

…3:1†

D2 …x; y† ˆ 2x1 ‡ x2 ÿ 2x3 ÿ y2 ;

…3:2†

x1 ‡ 2x2 ‡ x3 ‡ y1 P 3:4;

…3:3†

x2 ‡ 3y2 ‡ x4 P 2:5;

…3:4†

x1 ; x2 ; x3 ; x4 ˆ 0 or 1;

…3:5†

ÿ 8 6 D1 …x; y† 6 10; ÿ9 6 D2 …x; y† 6 6:

…3:6†

The proposed model linearizes this example as follows: Minimize z1 ‡ z2 subject to z1 P 20…x1 ‡ x2 ‡ x3 ÿ 3† ÿ 2…x1 ÿ 2x2 ‡ 5x3 ‡ 4y1 †; z1 P ÿ 20x1 ; z1 P ÿ 20x2 ;

Table 1 Size of linearizing constraints and variables

No. of additional continuous variables No. of auxiliary constraints

Glovel's model (1975)

Kettani±Oral's model (1990)

Oral±Kettani's model (1992)

Proposed model

Quadratic

Quadratic

Cubic

Polynomial

Quadratic

Cubic

Polynomial

n

n

3n

±

n

n

n

4n

2n (plus n sign constraints)

3n

±

2n

3n

nm

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C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

z1 P ÿ 20x3 ; z2 P 27…x1 ‡ x2 ‡ x3 ‡ x4 ÿ 4† ‡ 3…2x1 ‡ x2 ÿ 2x3 ÿ y2 †; z2 P ÿ 27x1 ; z2 P ÿ 27x2 ; z2 P ÿ 27x3 ; z2 P ÿ 27x4 ; constraints (3.1)±(3.6). Solve this problem using LINDO [9] to obtain the solution as x1 ˆ x2 ˆ x3 ˆ 1, x4 ˆ 0, y1 ˆ 1:5, y2 ˆ 0:5, and the objective function value ˆ ÿ20. This is the global optimal solution. For relaxation of ®rst restriction (all constraints should be kept as linear) in Oral±Kettani's [10] model, we propose a more general strategy. Suppose weQwant to minimize a function f ˆ ct Di …x; y† i2J xi which can appear in the objective function and/or constraints, then this function can be replaced by a continuous variable z, where z satis®es the following inequalities: ! X xi ÿ j J j ‡ ct Di …x; y† jct Di j J

X xi 6 z 6 jct Di j j J j ÿ

! ‡ ct Di …x; y†;

J

ÿxi jct Di j 6 z 6 xi jct Di j for all i; where all of the variables are de®ned as in the proposed model. We know from the above inequality constraints that the use of a linear function unrestricted in sign is allowed. 4. Computational experience 4.1. Randomly generated data The superiority of the proposed method can also observed through some test examples. These

test examples have the same pattern as in Problem h0 . Five groups of the test examples are characterized by the number of 0±1 variables (n ˆ 6, 12, 18) and the number of constraints (m ˆ 5, 10, 20) with positive and negative coecients appearing in the objective function alternately. For each of the ®ve groups, 10 cubic mixedinteger problems are randomly generated. Thus, a set of 50 test problems is formed. Each of the test examples is formulated as `cubic mixed-integer problem' by the proposed model and OKCM and then solved by LINDO [9] on a PC/586. The average relative performance of the proposed model and OKCM, measured by CPU time and number of iterations, is compared in Tables 2 and 3. From Tables 2 and 3, we can see that, compared with the proposed model, OKCM takes 174% of the average CPU time and 498% of the number of iterations. The performance of the proposed model becomes much better when the number of variables is increased. This is an expected outcome since the proposed method reduces the number of additional 0±1 variables and auxiliary constraints. Consider Table 2, for instance, the average CPU time required by OKCM is 24% more for 6-variables, 139% more for 12variables, 233% more for 18-variables. A similar

Table 2 Relative performance of PM compared to OKCM (CPU time) Number of constraints

Number of 0±1 variables 6

12

18

5 OKCM/PM 10 OKCM/PM 20 OKCM/PM

1.04 1.04 1.64

2.36 2.03 2.80

4.40 2.20 3.40

Table 3 Relative performance of PM compared to OKCM (no. of iterations) Number of constraints

Number of 0±1 variables 6

12

18

5 OKCM/PM 10 OKCM/PM 20 OKCM/PM

4.77 2.44 3.33

7.69 4.53 7.84

12.60 7.40 9.26

Road no.

4 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 1 15 15 15 15 15 15 15 15 15

Project no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

21k+10021k+400 21k+80021k+870 19k+30019k+500 14k+40014k+800 40k+60041k+450 41k+40041k+450 42k+15042k+300 44k+30044k+400 46k+85047k+150 48k+00048k+100 48k+75048k+950 49k+70051k+400 48k+20048k+700 21K+00021k+670 44k+40044k+550 45k+25045k+400 35k+00035k+500 37k+00038k+250 39k+00039k+500 48k+00048k+220 45k+60045k+800 44k+62044k+680 40k+00041k+450 39k+75039k+900 39k+50039k+700 36k+00036k+500

Section

L R L R R R R R R R R L L L L L R R R L L L L L L L

300 70 200 400 100 50 150 100 300 100 200 1700 500 670 150 150 500 1250 500 220 200 60 1450 150 200 200

Direction Defect length (m)

Table 4 Highway pavement distress in Taoyuan county, Taiwan

3790 3790 3790 3860 38501 38501 38501 4565 4565 4565 4565 4565 4565 2100 4565 4565 4989 2339 2339 1360 2339 2339 2339 2339 2339 2339

Trac (vpd) 0.5 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.1 0.5 0.5 0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

Trac factor Re¯ection cracking Old patching Rutting Alligator cracking Old patching Old patching Re¯ection cracking Alligator cracking Rutting Alligator cracking Re¯ection cracking Rutting Re¯ection cracking Rutting Re¯ection cracking Old patching Alligator cracking Rutting Rutting Re¯ection cracking Re¯ection cracking Old patching Alligator cracking Open potholes Alligator cracking Alligator cracking

Defect

Rehabilitation Overlay Overlay Rehabilitation Overlay Overlay Rehabilitation Rehabilitation Overlay Rehabilitation Rehabilitation Overlay Rehabilitation Overlay Rehabilitation Overlay Rehabilitation Overlay Overlay Rehabilitation Rehabilitation Overlay Rehabilitation Rehabilitation Rehabilitation Rehabilitation

Treatment

21975 68320 390000 855000 213750 73125 109688 375000 614250 375000 97500 4271250 937500 2889375 320625 146250 543375 5625000 243750 181500 390000 64125 5437500 562500 1005000 1875000

Cost (NT$) 0.20 0.50 0.30 0.15 0.50 0.50 0.20 0.15 0.30 0.15 0.20 0.30 0.15 0.30 0.20 0.50 0.15 0.30 0.30 0.20 0.20 0.50 0.15 0.10 0.15 0.15

Defect factor 65±72 63±74 66±74 86±93 76±83 70±79 62±73 80±88 75±85 78±84 61±76 81±87 83±88 84±95 65±78 60±68 74±84 80±88 66±74 72±81 70±82 85±94 82±90 85±93 89±98 80±90

3000 280 1333 5333 200 100 750 1333 2000 1333 2000 11333 6667 22333 1500 600 6667 41667 16667 11000 10000 1200 96667 15000 13333 33333

User Priority dissatisindex faction(%)

C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659 657

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C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

superiority can also be observed by studying Table 3, where the number of iterations is taken as the criterion for computational eciency. The PM far outperforms OKCM, ranging from 144% to 1160%. Again, the PM becomes more ecient as the number of 0±1 variables increases, as one expects since the advantage of the PM is directly related to the number of 0±1 variables. In other words, this superiority become more pronounced as the number of 0±1 variables increases.

4.2. Real-world data Consider an actual case of project level optimization/prioritization of pavement rehabilitation from Taoyuan county in Taiwan. The related data for pavement distress in this case is shown in Table 4. In this case, we try to minimize the problems for motorists by following the priority index, and limited by the total pavement rehabilitation budget for this case. To clarify how the proposed model solves this practical problem, the following explains Table 3: To establish a priority measure, a treatment index associated with each district was calculated as Priority index ˆ

defect length : traffic factor  defect factor

The trac factor used values of 0.1, 0.5, and 1.0 for average daily trac levels of less than 2500 vehicles per day (vpd), between 2500 and 10 000 vpd, and more than 10 000 vpd, respectively. The user dissatisfaction index was obtained from questionnaires and then evaluated by six pavement experts. The defect factor is assigned to each section based on the defect types and required treatment, as shown in Table 5. The priority index for the entire section was then calculated as the sum of the priority indexes for both the highway and the shoulder. The total budget for this phase of pavement repair is NT$20 000 000. For simplicity without loss of generality, above problem is then formulated as follows:

Table 5 Assignment of defect factor Defect

Treatment

Defect factor

Open potholes Alligator cracking Re¯ection Rutting Old patching Lean surface texture Edge fretting Low shoulder

Rehabilitation Rehabilitation Rehabilitation Reshape and overlay Overlay Surface dressing Edge patching Shoulder works

0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.00

Maximize

n X

p i x i ui

iˆ1

subject to n X ci xi 6 B;

…4:1†

iˆ1

xi ˆ 0 or 1; where n is the number of projects in need of speci®ed treatment, xi equal to 1 if ith project is implemented in this phase, pi the priority index of ith project, ui the index of user dissatisfaction, which is a bounded continuous variable, ci the cost of the ith project is implemented and B the total budget of this phase. Constraint (4.1) ensures that the summation of all projects does not exceed total budget of this phase. Clearly, this is a typical quadratic mixedinteger problem. This problem is formulated by the proposed model and OKCM and then solved by LINDO [9] on a PC/586. The average performance of the proposed model and OKCM, measured by CPU time and number of iterations again, is compared in Table 6. From Table 6, we can see that, compared with the proposed model, OKCM takes 77% of the average CPU time and 295% of the number of itTable 6 Relative performance of PM compared to OKCM Number of projects

CPU time

No. of 0±1 iterations

5 OKCM/PM 15 OKCM/PM 26 OKCM/PM

1.03 1.78 2.50

2.65 3.42 5.78

C.-T. Chang / European Journal of Operational Research 123 (2000) 652±659

erations. The performance of the proposed model becomes much better again when the number of projects is increased. In other words, the superiority of computational performance of the proposed model becomes more pronounced as the number of projects increases (i.e., number of 0±1 variables is increased).

5. Concluding remarks Decision making problems such as facility layout, job assignment, or communication network designs are often formulated as mixed-integer problems. This note proposes a new linearization model for linearizing mixed-integer problems. This model requires fewer additional variables and auxiliary constraints than some well-known methods, as proposed by Glover [3], Kettani and Oral [5], and Oral and Kettani [10]. The analytical superiority of this proposed model is also supported by a computational experiment conducted on a small scale and it can easily be applied to more general mixed-integer problems in which the order is higher than three. In addition, this model allows the use of a linear function Di …x; y† that is unrestricted in sign. In other words, it can linearize mixed-integer terms that appear in the objective function and/or constraints.

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