Integer programming models for hierarchical workforce scheduling problems including excess off-days and idle labour times

Applied Mathematical Modelling 37 (2013) 9117–9131

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Integer programming models for hierarchical workforce scheduling problems including excess off-days and idle labour times Cemal Özgüven ⇑, Banu Sungur Erciyes University, Faculty of Economics and Administrative Sciences, Department of Business, 38039 Kayseri, Turkey

a r t i c l e

i n f o

Article history: Received 28 September 2011 Received in revised form 15 March 2013 Accepted 6 April 2013 Available online 3 May 2013 Keywords: Integer programming Hierarchical workforce Scheduling Divisibility of the works

a b s t r a c t The decision problem considered in this paper is a hierarchical workforce scheduling problem in which a higher qualified worker can substitute for a lower qualified one, but not vice versa, labour requirements may vary, and each worker must receive n off-days a week. Within this context, five mathematical models are discussed. The first two of these five models are previously published. Both of them are for the case where the work is indivisible. The remaining three models are developed by the authors of this paper. One of these new models is for the case where the work is indivisible and the other two are for the case where the work is divisible. The three new models are proposed with the purpose of removing the shortcomings of the previously published two models. All of the five models are applied on the same illustrative example. Additionally, a total of 108 test problems are solved within the context of two computational experiments. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Workforce scheduling is the process of balancing the number of employees to be employed with the number of works demanded over the planning horizon. A comprehensive survey of literature on this subject can be found in [1,2]. A hierarchical workforce is a workforce structure in which a higher qualified worker can substitute for a lower qualified one, but not vice versa. Scheduling of hierarchical workforce using combinatorial methods began with the research of Emmons and Burns [3]. They considered a facility that must be staffed 7 days a week. There are m-types of workers to be scheduled and each worker must be given 2 days off a week. They developed algorithms which generate a feasible schedule, and provide work stretches between 2 and 5 days. Narasimhan [4] presented an algorithm for single shift scheduling of hierarchical workforce in seven-days-a-week industries. Each employee must be given two days off a week. Narasimhan [5] presented an optimal algorithm for multiple shift scheduling of hierarchical categories of employees on four-days or three-days workweeks. Hung [6] considered a hierarchical workforce problem, in which daily labour requirements within a week may vary but each worker must receive n (n = 2,3,4) off-days a week, and presented a simple one-pass method. Billionnet [7] developed an integer programming model for the problem studied by Hung. The integer programming model developed by Seçkiner et al. [8], which is a modification of Billionet’s Model addresses the same hierarchical workforce problem except for the following difference. In Billionet’s Model where the works are assumed to be indivisable, the workers can be assigned to only one shift (with a length of 8 h). Whereas in Seçkiner’s model, the workers can be assigned to either one of the three alternative shifts in a day. The alternative three shifts in Seçkiner’s model differ in terms of working hours a

⇑ Corresponding author. Tel.: +90 352 4374901/30155; fax: +90 352 4375239. E-mail addresses: [email protected] (C. Özgüven), [email protected] (B. Sungur). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.006

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day (8,10,12) and off-days a week (2,3,4). Both of these integer programming models aim at determining the schedule that minimizes the total labour cost. Five mathematical models are presented in this paper: The model developed by Billionnet (Billionnet [7]) and its four extentions. Billionnet’s Model, Model A and Seçkiner’s Model (Ulusam Seçkiner S, Gökçen H, Kurt M. [8]) are for the case where the work is indivisible. Model B1 and Model B2 are for the case where the work is divisible. Model A, Model B1 and Model B2 are the models developed by the authors of this paper. In Model A and Model B2 a mechanism is added to Billionnet’s Model to make sure that the workers are paid on the basis of the days they work during the week. In models B1 and B2, additonal shifts with diverse durations are incorporated into Billionnet’s Model not under the indivisibility assumption as is done in Seçkiner’s Model but under the correct assumption that the work is divisible. These two models had to be based on the assumption that the work is divisible due to the fact that when additional shifts with daily working hours longer than the fixed duration of the works come into play, the works must be assumed to be divisible. The remainder of this paper is organized as follows: Problem formulation is given in Section 2. Billionnet’s Model, Model A and Seçkiner’s Model, which are all based on the assumption of indivisibility, are described in Section 3. Section 4 is devoted to Model B1 and Model B2 developed under the assumption of divisibility. In Section 5, all of the five models are applied on Billionnet’s illustrative examples. In Section 6 total of 108 test problems are solved within the context of two experiments. Concluding remarks are made in Section 7. 2. Presentation of the decision problems Under this heading the decision problems addressed by the models given in this paper are presented. 2.1. Problems in which the works are indivisible In Billionnet’s Problem, in Problem A and in Seçkiner’s Problem the works are assumed to be indivisible, meaning that each work requires only one worker. Billionnet’s problem The workforce scheduling problem addressed by Billionnet’s Model is the core problem. Starting with the notations and the parameters. Notation The following indices are used in the general presentation: k l j The parameters are. ck dlj n the problem is described in

workers type ðk 2 f1; . . . ; mgÞ. work type ðl 2 f1; . . . ; mgÞ. days ðj ¼ 1; . . . ; 7Þ. the cost of a type-k worker. the number of works of type l that must be executed on day j. the number of off-days that each worker must receive. (see Billionnet 1999) as follows:

1. A facility is staffed 7 days a week, Monday through Sunday. The days are abbreviated to be 1,2,3,4,5,6 and 7. 2. All workers are full timers and they are classified into m types, with type 1 the most qualified, type 2 the next most qualified, and so on. The cost of a type k worker is ck and c1 > c2 > . . . > cm . This cost takes into account the number of off-days received by the worker each week. 3. For each day labor requirements are defined in terms of numbers of type-1 works to be executed, type-2 works to be executed,. . ., and type-m works to be executed. Specifically, dlj works of type l must be executed on day j; j= 1,. . .,7. Each of the works completely requires one worker and type-l work can be executed by a type-k worker provided that l P k (a higher qualified worker can substitute for a lower qualified worker, but not vice versa). 4. Each worker must receive n-off-days each week, where n = 2,3,4 for 5-day, 4-day and 3-day workweeks, respectively. 5. The objective is to find minimal labour cost and a corresponding shedule that satisfies the labour and off-days requirements. Problem A Problem A is basically the same as Billionnet’s Problem with only one exception. In Billionnet’s Problem, the workers who receive excess off-days (i.e.more than n off-days a week) are paid full weekly wage. Whereas, in the workforce scheduling problem addressed by Model A, the workers who receive excess off-days are not paid full weekly wage.

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Table 1 Objective Function Coefficients. Models

Objective function coefficients

(1) (2) (3) (4) (5)

c1 ¼ 12; c2 ¼ 8; c3 ¼ 6 c1 ¼ 12; c2 ¼ 8; c3 ¼ 6; e1 ¼ 2:4; e2 ¼ 1:6; e3 ¼ 1:2 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6; c21 ¼ 10; c22 ¼ 6; c23 ¼ 4; c31 ¼ 8; c32 ¼ 5; c33 ¼ 3 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6; c21 ¼ 12; c22 ¼ 8; c23 ¼ 6; c31 ¼ 10:8; c32 ¼ 7:2; c33 ¼ 5:4 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6; c21 ¼ 12; c22 ¼ 8; c23 ¼ 6; c31 ¼ 10:8; c32 ¼ 7:2; c33 ¼ 5:4; e11 ¼ 2:4; e12 ¼ 1:6; e13 ¼ 1:2; e21 ¼ 3; e22 ¼ 2; e23 ¼ 1:5; e31 ¼ 3:6; e32 ¼ 2:4; e33 ¼ 1:8

Billionnet’s Model Model A Seçkiner’s Model Model B1 Model B2

The indices and parameters of Model A are identical with those used in the core problem presentation. Only a parameter is added. The additional parameter ek daily cost for each excess off day of the type k worker. The additional parameter ek is calculated as ek ¼ ck /(7-nÞ. [See Objective Function Coefficients (2) in Table 1]. Seçkiner’s problem The indices and parameters of Seçkiner’s Problem are identical with those used in Billionnet’s Problem. Only an indice is added and the parameter ck is replaced by cbk . The additional indice b

the shift types ðb 2 f1; 2; 3gÞ.

The parameter cbk

the cost of a type-k worker on shift type b.

Some assumptions are changed in or added to Billionnet’s Problem: 1. Some workers work for fewer days a week, but work for a longer day. The rest of the workers work for more days a week, but work for a shorter day. 2. A worker can work on only one shift type. 3. If a worker works on shift 1, he/she takes 2 days-off. Similarly, if worker works on shift 2 and 3, he/she takes 3 and 4 days-off, respectively. 4. All workers are full timers and they are classified into m types, with type 1 the most qualified, type 2 the next most qualified, and so on. The cost of a type k worker on shift b is cbk and b = 1 c11 > c12 > . . . > c1m ; b = 2, c21 > c22 > . . . > c2m ; b = 3, c31 > c32 > . . . > c3m . In addition, k = 1, c11 > c21 > c31 ; k=2, c12 > c22 > c32 and k = 3, c13 > c23 > c33 . This cost takes into account the number of off-days and shift types received by the worker each week. 5. Each worker receives n-off-days each week, where n = 2,3,4 for 5-day (8 h), 4-day (10 h) and 3-day (12 h) workweeks, respectively. 6. The objective is to find minimal labour cost and a corresponding shedule that satisfies the labour and off-days requirements. 2.2. Problem in which the works are divisible (Problem B) Problem B is an extensively modified form of Billionnet’s problem. In this decision problem, the workers who receive excess off-days are not paid full weekly wage, the work is assumed to be divisible and the workers can be assigned to three different shifts with 8 h working day (Shift 1), 10 h working day (Shift 2) and 12 h working day (Shift 3) respectively. Incorporating into the problem working hours longer than 8 h becomes meaningful only under the assumption that the work is divisible. Whenever a Shift 2 worker or a Shift 3 is assigned to a work, it gives rise to an idle time of 2 h or 4 h respectively, because every day is 10 h long for a Shift 2 worker and 12 h long for a Shift 3 worker, whereas the duration of each work is 8 h. In order to avoid the idle times of the Shift 2 and Shift 3 workers, they must be assigned to a new work upon completing the work they started with. The indices and parameters are same as those in Seçkiner’s Problem, except for two parameters. The Additional parameters hb

X ebk

the daily working hours of shift b workers ðb 2 f1; 2; 3gÞ. working hours needed for completing a work. (X is set equal to 8 h). daily cost for each excess off day on shift b.

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Paying the workers for staying home is avoided in the objective function by deducting the daily cost for each excess off day (ebk Þ from the total weekly cost. ebk is calculated as ebk ¼ cbk /(7-abn Þ where b ¼ n; a11 =2, a22 =3, a33 =4 if shift type b calls for days-off alternative n and zero otherwise. [See Objective Function Coefficients (5) in Table 1]. In order to remove the excessive cost advantages for the Shift 2 and Shift 3 workers artificially created by Seçkiner with the purpose of reducing the weekly cost, the values of (cbk Þ parameters are revised in Problem B by taking into account the total weekly working hours. Since the weekly working Shift 1 and Shift 2 workersis 40 h and  hours of both  since that of a Shift 3 worker is 36 h, for k = 1, c11 ¼ c21 > c31 c31 ¼ 36 c ; k=2, c12 ¼ c22 > c32 c32 ¼ 36 c and k = 3, 40 11 40 12   c13 ¼ c23 > c33 c33 ¼ 36 c . [See Objective Function Coefficients (4) in Table 1]. 13 40 3. The Models for the case where the work is indivisible Billionnet’s Model, Model A and Seçkiner’s Model are presented under this section. 3.1. Billionnet’s integer programming formulation The indices and parameters of Billionnet’s model are identical with those used in the above Billionnet’s Problem presentation. The variables the number of workers of type k ðk 2 f1; . . . ; mgÞ the number of workers of type k assigned to a work of type l ðl P kÞ on day j ðk 2 f1; . . . ; mg; l 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ the number of type-k workers who take day j off ðk 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ

wk xklj ykj

The constraints Billionnet’s Model is composed of three constraint sets. Constraint set (I)

X xklj þ ykj ¼ wk

ð1Þ

lPk

ðk 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ Constraint set (I) determines the number of type k (k = 1,. . .,mÞ workers to be employed. Constraint set (II)

X ykj P wk n

ð2Þ

j

ðk 2 f1; . . . ; mgÞ Constraint set (II) makes sure that each type k worker receives at least n off-days. Constraint set (III)

X xklj ¼ dlj

ð3Þ

k6l

ðl 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ By means of constraint set (III) one worker of type k (k = 1,. . .,mÞ is assigned to each one of the dlj works of type l to be executed on day j, provided that l P k. The objective function

Minimize

X

ck wk

k¼1;m

The objective is to find minimal total weekly labour cost. 3.2. Integer programming formulation (Model A) The indices and parameters of Model A are identical with those used in the above Problem A presentation. The decision variables additional to the ones in Billionnet’s Model are given below.

ð4Þ

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The additional variables yk

the total weekly number of the off-days received by type k workers ðk 2 f1; . . . ; mgÞ the total weekly number of excess off-days over wkn received by the type k workers ðk 2 f1; . . . ; mgÞ

zk

The constraints Model A is composed of five constraint sets. The first three constraint sets in Model A are the same as those in Billionnet’s Model (1)–(3). Constraint set (IV)

X ykj ¼ yk

ð5Þ

j

ðk 2 f1; . . . ; mgÞ The constraint set (IV) is added for determining the yk variables to be used in constraint set (V). Constraint set (V)

yk  wk n ¼ zk

ð6Þ

ðk 2 f1; . . . ; mgÞ The constraint set (V) is added for determining the zk variables to be used in the objective function. The objective function

Minimize a

X

ck wk 

k¼1;...;m

X

ek zk

ð7Þ

k¼1;...;m

The objective is to find minimal total weekly labour cost. Since constraint set II (2) is written in (PÞ form both in Billionnet’s Model and in Model A, a worker may receive more than n off-days a week. Whenever a worker receives an excess off day, he/she is paid a full weekly wage in Billionnet’s Model. Model A is built with the purpose of not paying full weekly wage to the workers who receive excess off-days i.e.more than n off-days a week. Paying the workers for staying home is avoided in the objective function of Model A by deducting the daily cost for each excess off day (ek Þ from the total weekly cost. P In the above formulated objective function, k¼1;...;m ck wk is weighted by a sufficiently large positive number a. This is done to make sure that the total labor cost is minimized without giving rise to an undue increase in the total number of workers. In all of the forhcoming applications of Model A, a is set equal to 10. 3.3. Seçkiner’s integer programming formulation The indices and parameters of Seçkiner’s Model are identical with those used in the above Seçkiner’s Problem presentation. The variables wbk xbklj ybkj

the number of workers of type k on shift type b ðb 2 f1; 2; 3gk 2 f1; . . . ; mgÞ the number of workers of type k on shift type b assigned to a work of type l ðl P kÞ on day j ðb 2 f1; 2; 3g; k 2 f1; . . . ; mg; l 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ the number of type k workers who take day joff on shift type b ðb 2 f1; 2; 3g; k 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ

The constraints Seçkiner’s Model is composed of three constraint sets. Constraint set (I)

X xbklj þ ybkj ¼ wbk lPk

ðb 2 f1; 2; 3g; k 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ Constraint set (I) determines the number of type k (k = 1,. . .,mÞ workers on shift type b to be employed.

ð8Þ

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Constraint set (II)

X ybkj P Abn wbk

ð9Þ

j

ðb 2 f1; 2; 3g; k 2 f1; . . . ; mg; n 2 f2; 3; 4gÞ Constraint set (II) makes sure that each type k worker on shift type b receives at least n off-days. Abn , an (Bi  N i Þ matrix, where b ¼ n; a11 =2, a22 =3, a33 =4 if shift type b calls for days-off alternative n and zero otherwise. Constraint set (III)

X X xbklj ¼ dlj

ð10Þ

b¼1;2;3 k6l

ðl 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ By means of constraint set (III) one worker of type k (k = 1,. . .,mÞ on shift type b is assigned to each one of the dlj works of type l to be executed on day j, provided that l P k. The objective function

Minimize

X X

cbk wbk

ð11Þ

b¼1::B k¼1::m

The objective is to find minimal total weekly labour cost. 4. The models for the case where the work is divisible We developed two models under the assumption that the work is divisible, namely Model B1 and Model B2. Model B2 is a continuation of Model B1. They are solved sequentially. The optimum number of workers which is the output of Model B1 is used as an input into Model B2. 4.1. Integer programming formulation (Model B1) The indices and parameters of Model B are identical with those used in the above Problem B presentation. The decision variable additional to the ones in Seçkiner’s Model is given below. The additional variable W

the total number of workers.

Model B1 is composed of four constraint sets. The first two constraint sets in Model B1 are the same as those in Seçkiner’s Model (8) and (9). Constraint set (III)

X X hb xbklj P Xdlj

ð12Þ

b¼1;2;3 k6l

ðl 2 f1; . . . ; mg; j 2 f1; . . . ; 7gÞ Constraint Set (III) represents day j’s requirements not in terms of the number of works but in terms of working hours needed. Day j’s requirement of dlj works of quality l may be viewed as Xdlj working hours needed for completing these works. Constraint set (IV)

W¼

XX wbk b

ð13Þ

k

Constraint (IV) is added to the model as a counter in order to determine the resulting total number of workers when the total weekly cost is minimized. W is not a decision variable. The objective function

Minimize

X

X

cbk wbk

ð14Þ

b¼1;2;3k¼1;2;;::;m

The objective is to find minimal total weekly labour cost. 4.2. Integer programming formulation (Model B2) The indices and parameters of Model B2 are identical with those used in the above Problem B presentation. The decision variables of Model B2 are same as those in Model B1. Two variables are added.

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The additional variables ybk

the total weekly number of the off-days of type k workers on shift b ðb 2 f1; 2; 3gk 2 f1; . . . ; mgÞ total weekly number of off-days over Abn wbk (excess off-days) of the type k workers on shift b ðb 2 f1; 2; 3gk 2 f1; . . . ; mgÞ

zbk

Model B2 is composed of six constraint sets. The first three constraint sets in the Model B2 are the same as those in Model B1 (8), (9) and (12). Constraint set (IV)

X ybkj ¼ ybk j

ðb 2 f1; 2; 3g; k 2 f1; . . . ; mgÞ

ð15Þ

The constraint set (IV) is added for determining the ybk variables to be used in constraint set (V). Constraint set (V)

ybk  Abn wbk ¼ zbk

ð16Þ

ðb 2 f1; 2; 3g; k 2 f1; . . . ; mg; n 2 f2; 3; 4gÞ The constraint set (V) is added for determining the zbk variables to be used in the objective function. Constraint set (VI)

XX wbk ¼ W  b

ð17Þ

k

The total number of workers is fixed at W  by means of constraint set (VI). W  is transferred from the optimal solution of Model B1. The objective function

XX XX Minimize cbk wbk  ebk zbk b

k

b

ð18Þ

k

The objective of Model B2 is to find the minimal total weekly labour cost under the conditions that the total number of workers is fixed at its optimal value W  and that the payment of full weekly wage to the workers who receive excess off-days is avoided. 5. Evaluation on illustrative examples Billionnet’s Model is applied in Billionnet [7] on two example problems that consider a hierarchical workforce scheduling scenario for a 7 day operation, m = 3 (i.e. 3 worker types), n = 2 (5 days a workweek) and c1 ¼ 12; c2 ¼ 8; c3 ¼ 6. The duration of each work is 8 h. In Example 1 the dlj parameters are

d11 ¼ 2 d12 ¼ 3 d13 ¼ 1 d14 ¼ 4 d15 ¼ 2 d16 ¼ 5 d17 ¼ 5 d21 ¼ 2 d22 ¼ 1 d23 ¼ 2 d24 ¼ 2 d25 ¼ 2 d26 ¼ 1 d27 ¼ 1 ðdlj Set 1Þ d31 ¼ 4 d32 ¼ 4 d33 ¼ 4 d34 ¼ 1 d35 ¼ 6 d36 ¼ 4 d37 ¼ 4 In Example 2 the dlj parameters are

d11 ¼ 4 d12 ¼ 0 d13 ¼ 5 d14 ¼ 3 d15 ¼ 3 d16 ¼ 3 d17 ¼ 3 d21 ¼ 0 d22 ¼ 3 d23 ¼ 5 d24 ¼ 4 d25 ¼ 5 d26 ¼ 2 d27 ¼ 2 ðdlj Set 2Þ d31 ¼ 5 d32 ¼ 0 d33 ¼ 4 d34 ¼ 2 d35 ¼ 0 d36 ¼ 0 d37 ¼ 1 All of the following applications rest on the same scenario either with dlj Set 1 or with dlj Set 2. The objective function coefficients are all derived from c1 ¼ 12; c2 ¼ 8; c3 ¼ 6 as indicated in Table 1. 5.1. Solutions of the models for the case where the work is indivisible Billionnet’s Model and Seçkiner’s Model are both applied on dlj Set 1 in Billionnet [7] and in Seçkiner et.al. [8]. To compare these two models with Model A and to illustrate the shortcomings of Seçkiner’s Model we also applied Model A on the very same dlj set. The solutions are given in Table 2.

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Table 2 Solution of the models based on dlj set 1. Models

(1) Billionnet’s Model (2) Model A (3) Seçkiner’s Model (4) Seçkiner’s Model

Objective function coefficients

Solutions

c1 ¼ 12; c2 ¼ 8; c3 ¼ 6 c1 ¼ 12; c2 ¼ 8; c3 ¼ 6; e1 ¼ 2:4; e2 ¼ 1:6; e3 ¼ 1:2 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6c21 ¼ 10; c22 ¼ 6; c23 ¼ 4 c31 ¼ 8; c32 ¼ 5; c33 ¼ 3 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6c21 ¼ 12; c22 ¼ 8; c23 ¼ 6c31 ¼ 10:8; c32 ¼ 7:2; c33 ¼ 5:4

Cost

Number of type k workers

Total number of workers

Total idle time

Excess offdays

106 106

w1 ¼ 5; w2 ¼ 2; w3 ¼ 5 w1 ¼ 5; w2 ¼ 2; w3 ¼ 5

12 12

0 0

0 0

98

w11 ¼ 3; w21 ¼ 1; w22 ¼ 2; w23 ¼ 6; w31 ¼ 1; w32 ¼ 1; w33 ¼ 1 w11 ¼ 5; w12 ¼ 2; w13 ¼ 5

15

108

0

12

0

0

106

Table 3 Solution of the models based on dlj set 2. Models

Objective function coefficients

Solutions Cost

(5) Billionnet’s Model (6) Model A

Number of type k workers

Total number of workers

Total idle time

Excess off-days of the workers Type 1

Type 2

Type 3

c1 ¼ 12; c2 ¼ 8; c3 ¼ 6

124

w1 ¼ 5; w2 ¼ 5; w3 ¼ 4

14

0

2

-

14

c1 ¼ 12; c2 ¼ 8; c3 ¼ 6; e1 ¼ 2:4; e2 ¼ 1:6; e3 ¼ 1:2

98.8

w1 ¼ 5; w2 ¼ 5; w3 ¼ 4

14

0

4

3

9

Since dlj set 1 yields no excess off-days, the solution of Billionnet’s Model and that of Model A are the same, with a total weekly cost of 106. Whereas in the first solution of Seçkiner’s Model [Solution (3)/Table 2] the total weekly cost is reduced to 98, due to the weekly costs of the Shift 2 and 3 workers that are deliberately kept in Seçkiner et. al. [8] much lower than warranted by the their weekly working hours. The total weekly cost in the second solution of Seçkiner’s Model [Solution (4)/Table 2] is the same as the total weekly cost in the solutions of Billionnet’s Model [Solution (1)/Table 2] and of Model A [Solution (2)/Table 2]. This is because Seçkiner’s Model is solved in this case using the objective function coefficients calculated on the basis of working hours a week. If Seçkiner’s Model had been solved in this manner [See Objective Function Coefficients (4) in Table 1], it would assign only the Shift 1 workers as Billionnet’s Model does. [See solution (1) and solution (4) in Table 2]. Seçkiner’s Model must be eliminated from the list of alternative models to be used under the case of indivisibility, because the reduction in the total cost is artificially achieved and due regard is not given to the increases in the total number of workers and in the total idle time. In order to be able to compare the remaining two models, namely Billionnet’s Model and Model A, we use dlj set 2 which yields excess off-days. The solutions of these two models are given in Table 3. To facilitate the comparison of solutions yielded by the models in question, we start with scheduling the off-days for each worker of each type. For doing so, the values of the ykj variables can be used. For a given type first off-day is assigned to each worker, then a second off-day is assigned to each worker and so on. In this way each type k worker will receive at least n offdays. It is possible for a worker to receive more than n off- days because of the Constraint Set II (2) common to Billionnet’s Model and Model A. In all of the following schedules, each worker on each day is either assigned to a work of type-l (l) or is on an off-day ðXÞ. An excess off-day received by a worker is indicated by ðX  Þ. The schedule generated by the solution of Billionnet’s Model is given in Table 4. The schedule generated by the solution of Model A is given in Table 5. The total weekly cost is reduced by Model A from 124 to 98.8 [Solutions (5) and (6)/Table 3]. This reduction of 25.2 monetary units is achieved, because Model A makes sure that the workers who receive excees off-days are not paid for staying home. If the workers had not been paid for their excess off-days (16 days in total), the total weekly cost found by Billionnet’s Model would have been not 124 but (124-2e1 -14e3 =)102.4. Total weekly cost is further reduced from 102.4 to 98.8 when Model A is applied directly. Model A achieves an additonal reduction of 3.6 in the total cost by altering the distribution of the excess off-days among the three types of workers[See Table 3]. Of the two remaining alternative models to be used under the case of indivisibility, Model A must be preferred to Billionnet’s Model.

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Worker

Mo

Tu

We

Th

Fr

Sa

Su

k=1

1 2 3 4 5 1 2 3 4 5 1 2 3 4

X 1 1 1 1 3 3 3 3 3 X X X X

X X X X X X X 2 2 2 X X X X

1 1 1 1 1 2 2 2 2 2 3 3 3 3

1 X X 1 1 2 2 X 2 2 X X 3 3

1 1 1 2 2 2 2 2 X X X X X X

1 1 1 X X X X X 2 2 X X X X

X X 1 1 1 2 2 3 X X X X X X

Mo

Tu

We

Th

Fr

Sa

Su

1 1 1 X X 2 2 2 2 2 X X X X



1 1 X X 1 X X X 2 2 X 3 X X

k=2

k=3

Table 5 The Corresponding Schedule (Model A) dlj Set 2. Worker type k=1

k=2

k=3

Worker 1 2 3 4 5 1 2 3 4 5 1 2 3 4

X 1 1 1 1 X X X X 3 3 3 3 3

X X X X X X 2 2 2 X X X X X

1 1 1 1 1 2 2 2 2 2 3 3 3 3

1 X X 1 1 2 X 2 2 2 X X 3 3

X X 1 1 1 2 2 X X X X X X X

Table 6 Solution of the models based on dlj set 1. Models

(7) Model B1

(8) Model B2

Objective function coefficients

c11 ¼ 12; c12 ¼ 8; c13 ¼ 6; c21 ¼ 12; c22 ¼ 8; c23 ¼ 6; c31 ¼ 10:8; c32 ¼ 7:2; c33 ¼ 5:4 c11 ¼ 12; c12 ¼ 8; c13 ¼ 6; c21 ¼ 12; c22 ¼ 8; c23 ¼ 6; c31 ¼ 10:8; c32 ¼ 7:2; c33 ¼ 5:4 e11 ¼ 2:4; e12 ¼ 1:6; e13 ¼ 1:2; e21 ¼ 3; e22 ¼ 2; e23 ¼ 1:5; e31 ¼ 3:6; e32 ¼ 2:4; e33 ¼ 1:8

Solutions Cost

Number of type k workers

Total number of workers

Total idle time

Excess off-days

105.4

w11 ¼ 2; w12 ¼ 2; w13 ¼ 1; w31 ¼ 3; w33 ¼ 5

13

8

0

103.6

w11 ¼ 3; w12 ¼ 2; w13 ¼ 4; w31 ¼ 2; w33 ¼ 2

13

0

3

5.2. Solutions of the models for the case where the work is divisible Under this subsection, we apply only Model B1 and Model B2 which are the models presented above under the assumption of divisibility. Even under the same objective function coefficients and the same dlj set, the solutions of Model B1 and Model B2 should not be compared with the solutions of the three models given above under the assumption of indivisibility. This is because the assumption of divisibility offers an undue advantage in reducing the total cost to Model B1 and Model B2 over the three other models. The solutions of Model B1 and Model B2 are given in Table 6. The schedule corresponding to solution (7) is given in Table 7. In terms of dlj Set 1, 60 works of various qualities must be completed within a week. Since each one of works is 8 h long, 480 working hours are required to complete all of the 60 works. As w11 þ w12 þ w13 =5 and w31 þ w32 þ w33 =8 and as a Shift 1

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Table 7 The Corresponding Schedule (Model B1). Shift

Worker type

Worker

Mo

Tu

We

Th

Fr

Sa

Su

k=1

1 2 1 2 1 – – – 1 2 3 – 1 2 3 4 5

1 1 2 2 3 – – – X X X – X X 3 3 3

X X X 2 3 – – – X 1 1 – 3 3 X X X

X 1 2 2 3 – – – X X X – X X X X 3

2 1 2 X X – – – 1 X 1 – X X X 3 X

1 1 2 2 X – – – X X X – 3 3 3 X 3

1 1 X 2 3 – – – 1 1 X – X X 3 3 X

1 X 2 X 3 – – – 1 1 1 – 3 3 X X X

b=1 k=2

b=2

b=3

k=3 k=1 k=2 k=3 k=1

k=2 k=3

Table 8 The Corresponding Schedule (Model B2). Shift

Worker type

Worker

Mo

Tu

We

Th

Fr

Sa

Su

b=1

k=1

1 2 3 1 2 1 2 3 4 – – – 1 2 – 1 2

X 1 1 2 2 3 3 3 3 – – – X X – X X

1 1 1 X 2 X X X 3 – – – X X – 3 3

1 X X 2 2 3 3 3 3 – – – X X – X X

X X 1 2 2 X X 3 X – – – 1 1 – X X

1 1 X 2 2 3 3 X 3 – – – X X – 3 3

X 1 1 2 X X X 3 X – – – 1 1 – 3 3

2 1 1 X X 3 3 3 3 – – – 1 1 – X X

k=2 k=3

b=2

b=3

k=1 k=2 k=3 k=1 k=2 k=3

Table 9 Test problems for Billionnet’s Model and for Model A (Experiment 1). Worker and work types (m) Number of off-days (n) Number of type l works on day j (dlj Þ

3,5,7,10 2, 3 Uniform [0,5], Uniform [0–10], Uniform [0–20]

Table 10 Test problems for Models B1 and Model B2 (Experiment 2) Worker and work types (m) Number of type l works on day j (dlj Þ

3,5,7,10 Uniform [0,5], Uniform [0–10], Uniform [0–20]

worker works (5⁄8=)40 h a week and a Shift 3 worker works (3⁄12=)36 h a week, the sum total of working hours of the 13 workers amounts to (40⁄5 + 36⁄8=)488 h a week. This means that the total idle time is ð488  480 ¼Þ8 h a week. The schedule corresponding to solution (8) is given in Table 8. As w11 þ w12 þ w13 =9 and w31 þ w32 þ w33 =4, the sum total of working hours of the 13 workers amounts to (40⁄9 + 36⁄4=)504 h a week. If excess off-days had not been given, the total idle time would have been ð504  480 ¼Þ24 h a week. Model B2 reduces the total idle time to zero by giving an excess off-day to each one of the 3 Shift 1 workers. By doing so, it reduces the total weekly cost by 1.8 monetary units as well.

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6. Computational results The integer programming model is coded in AIMMS 3.11 optimization software. By using the solver CPLEX 12.3, the ALBHW instances are solved on a computer with Intel Core 2 Duo 2.53 GHz processor and 2.96 GB memory. Two computational experiments are done: Experiment 1 on Billionnet’s Model and Model A, Experiment 2 on Model B1 and Model B2. Both of the experiments are conducted on four problem groups in terms of m. The problem groups in Experiment 1 are furher categorized into subgroups in terms of the number of off-days and in terms of the number of works to be executed. The problem groups in Experiment 2 are similarly categorized in terms of the number of works to be executed. The details are given in Table 9 and 10. Experiment 1: In all of the applications, optimal solutions are found. A total of 72 problem instances are solved. In 63 of the instance problems total cost yielded by Model A is lower than total cost yielded by Billionnet’s model. In 9 of the instance problems, total costs are equal. In all of the solutions found by the models in question, total number of workers is equal. However, as it has been in case of Problem P1-131, distribution of the total among the worker types is not the same. There are five other similar cases. The model sizes of Billionnets’ Model and Model A for problem instances in Experiment 1 are given in Table A1. The CPU times for both of the models are very low, almost negligible.[See Tables A2, A3, A4 and A5 in the Appendix]. Experiment 2: The running time is limited to 7200 s. A total of 36 problem instances are solved. In 26 of the 27 problem instances where the number of work types are taken as 3, 5 and 7 optimal solutions are found within reasonable CPU times. In 4 of the 9 problem instances where the number of work types is increased to 10, optimal solutons are found within reasonable CPU times. But, in the remaining 5 instances positive gaps, albeit small, occurred. [See Table A6 in the Appendix]. Both in Experiment 1 and in Experiment 2, 3 problem instances are solved for each subgroup. 7. Conclusion Five mathematical models for hierarchical workforce scheduling problem are presented in this paper. Billionnet’s Model, Model A and Seçkiner’s Model are for the case where the work is indivisible. Model B1 and Model B2 are for the case where the work is divisible. Model A, Model B1 and Model B2 are the models developed by the authors of this paper. The three models based on the assumption of indivisibility are applied on the same illustrative example. Seçkiner’s Model is eliminated from the list of models to be used under the assumption of indivisibility due to its shortcomings explained in Section 5.1. As Model A gives the workers unpaid excess off-days instead of keeping them idle during the whole work day, it is therefore preferred to Billionnet’s Model. Three alternative shifts in Seçkiner’s Model with different daily working hours (8, 10 and 12) does not make any sense under the assumption of indivisibility when the duration of each work is 8 h. The alternative shifts can meaningfully be incorporated into mathematical models only under the assumption of divisibility. To this end, we developed two models to be used sequencially under the assumption of divisibility, namely Model B1 and Model B2, and applied them on an illustrative example problem. In terms of five models presented in this paper, Model A must be used under the assumption of indivisibility, Model B1 and Model B2 must be used sequentially under the assumption of divisibility. The computational results on a total of 72 problem instances in Experiment 1 and 36 problem instances in Experiment 2 are presented. In all of the 72 problem instances within Experiment 1 where the number of work types are taken as 3, 5, 7 and 10, optimal solutions are found and the CPU times have turned out to be quite low. As for Experiment 2, in 26 out of 27 problem instances where the number of work types is taken as 3, 5 and 7 optimal solutions are found within short CPU times. In 5 of the 9 problem instances where the number of work types is increased to 10, positive gaps, albeit small, occurred. Developing the relevant mathematical models under the following assumptions may be thought of among the possible future work: The planning horizon is longer than one week, 2 off-days are consecutive for each worker and at least one off-day of a worker is placed into the week-end. Appendix A (See Tables A1,A2,A3,A4,A5,A6). Table A1 Model Sizes of Billionnets’ Model and Model A for problem instances in Experiment 1. Problem No

Model Sizes Billionnets’ Model

P1 P2 P3 P4

(3 types) (5 types) (7 types) (10 types)

Model A

#integer variables

#constraints

_ #Integer variables

#constraints

66 145 252 465

46 76 106 151

72 155 266 485

53 87 121 172

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Table A2 Computational results for problem instances in Experiment 1 (3 Types). Problem intances Problem No

Solution Types

Off-Days

dlj

Cost

# type-h workers

BM

MA

Billionnet y1 y2

CPU (sec) y3

Model A y1 y2

y3

BM

MA

P1–111 P1–112 P1–113

3 3 3

2 2 2

0–5 0–5 0–5

130 96 112

125,6 77,6 86,4

5 4 4

5 3 5

5 4 4

5 4 4

5 3 5

5 4 4

0.015 0.00 0.00

0.00 0.00 0.016

P1–121 P1–122 P1–123

3 3 3

2 2 2

0–10 0–10 0–10

244 172 180

191,2 158,8 172,8

10 9 9

8 5 9

10 4 0

10 9 9

8 5 9

10 4 0

0.00 0.00 0.031

0.015 0.016 0.00

P1–131 P1–132 P1–133

3 3 3

2 2 2

0–20 0–20 0–20

442 460 402

372 404,8 395,6

19 20 18

11 14 15

21 18 11

19 20 18

17 14 15

15 18 11

0.00 0.016 0.016

0.015 0.016 0.015

P1–211 P1–212 P1–213

3 3 3

3 3 3

0–5 0–5 0–5

160 102 112

160 99 109

7 4 4

5 3 5

6 5 4

7 4 4

5 3 5

6 5 4

0.062 0.00 0.00

0.015 0.016 0.00

P1–221 P1–222 P1–223

3 3 3

3 3 3

0–10 0–10 0–10

248 194 208

239 192,5 204,5

10 9 19

10 7 11

8 5 2

10 9 9

10 7 10

8 5 13

0.016 0.00 0.016

0.00 0.015 0.015

P1–231 P1–232 P1–233

3 3 3

3 3 3

0–20 0–20 0–20

458 506 482

455 504 482

19 20 20

16 16 19

17 23 15

19 20 20

16 16 19

17 23 15

0.016 0.00 0.00

0.015 0.078 0.031

Table A3 Computational results for problem instances in Experiment 1 (5 Types). Problem intances Problem No

Solution

Types

Off-Days

dlj

Cost BM

# type-h workers MA

CPU (sec)

Billionnets’ Model y1 y2 y3 y4

y5

Model A y2 y3

y4

y1

y5

BM

MA

P2–111 P2–112 P2–113

5 5 5

2 2 2

0–5 0–5 0–5

147 140 131

146.4 122.4 126.6

5 5 5

5 3 4

4 4 3

5 5 3

1 4 3

5 5 5

5 3 4

4 4 3

5 5 3

1 4 3

0.015 0.016 0.015

0.016 0.016 0.016

P2–121 P2–122 P2–123

5 5 5

2 2 2

0–10 0–10 0–10

258 220 262

239.4 211.2 255.4

8 9 8

9 3 7

6 8 8

9 10 8

6 0 10

8 9 8

9 3 7

6 8 8

9 10 8

6 0 10

0.015 0.016 0.016

0.031 0.015 0.016

P2–131 P2–132 P2–133

5 5 5

2 2 2

0–20 0–20 0–20

447 530 510

436.8 526.4 507.6

17 20 18

19 11 15

5 18 12

7 10 15

11 18 14

17 20 18

19 12 15

5 16 12

7 11 15

11 18 14

0.00 0.032 0.031

0.015 0.015 0.047

P2–211 P2–212 P2–213

5 5 5

3 3 3

0–5 0–5 0–5

175 153 152

172 151 151.5

5 5 5

6 5 5

6 3 5

7 5 4

1 5 2

5 5 5

6 5 5

6 3 5

7 5 4

1 5 2

0.031 0.063 0.016

0.046 0.078 0.016

P2–221 P2–222 P2–223

5 5 5

3 3 3

0–10 0–10 0–10

294 254 316

293.5 254 314.5

9 9 9

9 6 10

12 8 8

6 11 11

6 2 12

9 9 9

9 6 10

12 8 8

6 11 11

6 2 12

0.016 0.016 0.062

0.015 0.016 0.078

P2–231 P2–232 P2–233

5 5 5

3 3 3

0–20 0–20 0–20

500 633 612

500 629 611

17 21 20

19 16 18

8 20 17

15 16 15

12 23 22

17 21 20

19 16 18

8 20 17

15 16 15

12 23 22

0.00 0.047 0.015

0.016 0.046 0.062

Table A4 Computational results for problem instances in Experiment 1 (7 Types). Problem intances Problem No

Types

Solution Off-Days

dlj

Cost BM

P3–111 P3–112

7 7

2 2

0–5 0–5

161 148

# type-h workers MA 159.8 137.6

CPU (sec)

Billionnets’ Model y1 y2 y3 y4

y5

y6

5 4

1 1

4 5

5 5

4 4

5 5

y7

Model A y1 y2

y3

y4

y5

y6

y7

4 2

5 4

4 4

5 5

1 1

4 5

4 2

5 5

BM

MA

0.032 0.015

0.031 0.031

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C. Özgüven, B. Sungur / Applied Mathematical Modelling 37 (2013) 9117–9131 Table A4 (continued) Problem intances Problem No

Solution

Types

Off-Days

dlj

Cost BM

# type-h workers MA

CPU (sec)

Billionnets’ Model y1 y2 y3 y4

y5

y6

y7

Model A y1 y2

y3

y4

BM y5

y6

MA

y7

P3–113

7

2

0–5

136

123.6

4

4

4

2

5

3

2

4

4

4

2

5

3

2

0.015

0.016

P3–121 P3–122 P3–123

7 7 7

2 2 2

0–10 0–10 0–10

278 283.5 296

275 281.9 293

10 9 10

8 10 6

10 5 9

0 10 5

3 2 9

8 6 9

6 5 6

10 9 10

8 10 6

10 5 9

0 10 5

3 2 9

8 6 9

6 5 6

0.031 0.031 0.062

0.016 0.031 0.109

P3–131 P3–132 P3–133

7 7 7

2 2 2

0–20 0–20 0–20

553 574.5 477.5

530.1 572.1 476

20 20 17

9 16 7

18 12 10

13 13 16

14 12 16

12 12 16

10 15 9

20 20 17

7 16 7

18 12 10

11 13 16

15 12 16

13 12 16

12 15 9

0.062 0.032 0.032

0.031 0.031 0.047

P3–211 P3–212 P3–213

7 7 7

3 3 3

0–5 0–5 0–5

192.5 160 146.5

188 160 146.5

5 4 4

6 5 4

6 4 4

6 5 5

3 4 4

4 5 3

5 4 3

5 4 4

6 5 4

6 4 4

6 5 5

3 4 4

4 5 3

5 4 3

0.093 0.016 0.031

0.109 0.015 0.016

P3–221 P3–222 P3–223

7 7 7

3 3 3

0–10 0–10 0–10

307 350 354.5

303.5 348.6 351.5

10 12 11

8 12 8

10 5 10

1 9 8

9 7 11

10 7 10

8 6 9

10 12 11

8 12 8

10 5 10

1 9 8

9 7 11

10 7 10

8 6 9

0.031 0.078 0.062

0.046 0.109 0.063

P3–231 P3–232 P3–233

7 7 7

3 3 3

0–20 0–20 0–20

658.5 690 563.5

658.1 688.8 561.6

20 22 17

16 19 11

22 17 13

13 17 22

20 12 11

12 16 25

15 24 15

20 22 17

16 19 11

22 17 13

13 17 22

20 12 11

12 16 25

15 24 15

0.046 0.062 0.047

0.062 0.063 0.062

Table A5 Computational results for problem instances in Experiment 1 (10 Types). Problem No

Solution Cost BM

# type-h workers MA

CPU (sec)

Billionnets’ Model y1 y2 y3 y4 y5

y6

y7

y8

y9

y10

Model A y1 y2 y3

y4

y5

y6

y7

y8

y9

y10

BM

MA

P4–111 P4–112 P4–113

138.750 153.250 135.500

138.050 147.600 133.400

5 5 5

5 4 2

1 3 3

0 3 5

4 3 0

3 4 3

4 4 4

4 3 4

3 5 4

5 3 5

5 5 5

5 4 2

1 3 3

0 3 5

4 3 0

3 4 3

4 4 4

4 3 4

3 5 4

5 3 5

0.047 0.078 0.047

0.032 0.032 0.312

P4–121 P4–122 P4–123

306.750 292.500 288.500

305.050 290.300 288.200

9 9 8

8 8 9

8 5 8

7 7 6

6 8 3

8 7 9

9 6 5

4 6 6

7 6 6

4 10 7

9 9 8

8 8 9

8 5 8

7 7 6

6 8 3

8 7 9

9 6 5

4 6 6

7 6 6

4 10 7

0.156 0.078 0.140

0.140 0.125 0.187

P4–131 P4–132 P4–133

675 604.750 532.500

672.150 604.350 531.600

19 17 17

20 17 19

17 14 5

21 17 7

7 10 11

11 16 15

17 18 14

12 8 18

18 15 10

14 9 18

19 17 17

20 17 19

17 14 5

21 17 6

7 10 13

11 16 14

17 18 14

12 8 18

18 15 10

14 9 18

0.078 0.109 0.110

0.093 0.109 0.172

P4–211 P4–212 P4–213

157 177.250 158.750

155.250 176.625 158.500

5 5 5

5 5 3

1 5 4

3 4 5

4 2 1

5 4 5

4 5 4

5 6 5

4 3 5

6 3 6

5 5 5

5 5 3

1 5 4

3 4 5

4 2 1

5 4 5

4 5 4

5 6 5

4 3 5

6 3 6

0.110 0.125 0.109

0.109 0.094 0.140

P4–221 P4–222 P4–223

375.750 352 341.500

375.750 351.374 341.125

11 10 10

10 10 8

9 7 10

10 8 6

6 11 8

10 7 11

11 9 7

6 6 7

9 8 6

5 11 11

11 10 10

10 10 8

9 7 10

10 8 6

6 11 8

10 7 11

11 9 7

6 5 7

9 10 6

5 10 11

0.062 0.140 0.140

0.078 0.187 0.203

P4–231 P4–232 P4–233

810.750 745.500 609.500

810.750 744.875 608.562

20 21 17

26 20 19

23 19 8

24 19 14

9 13 14

17 20 20

20 21 16

15 14 20

19 18 16

17 11 23

20 21 17

26 20 19

23 19 8

24 19 14

9 13 14

17 20 20

20 21 16

15 14 20

19 18 16

17 11 23

0.140 0.125 0.266

0.141 0.156 0.203

Table A6 Computational results for problem instances in Experiment 2 . Problem intances Problem No

Model Sizes Types

dlj

#Int. B1

Solution

B2

#Cons. B1

W B2

Cost B1

B2

P1–111

3

0–5

199

216

95

113

15

127

125.6

P1–112

3

0–5

199

216

95

113

10

79.6

78.4

P1–113

3

0–5

199

216

95

113

11

92

89.6

CPU (sec) Gap (%) B1

B2

2.761 0.00 1.108 0.00 0.453 0.00

0.094 0.00 0.141 0.00 0.905 0.00 (continued on next page)

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Table A6 (continued) Problem intances Problem No

Model Sizes Types

dlj

#Int. B1

Solution

B2

#Cons. B1

W B2

Cost B1

B2

P1–121

3

0–10

199

216

95

113

23

192.8

189.6

P1–122

3

0–10

199

216

95

113

18

151.4

148.4

P1–123

3

0–10

199

216

95

113

19

164

163.2

P1–131

3

0–20

199

216

95

113

44

354.4

350.8

P1–132

3

0–20

199

216

95

113

49

404

403.2

P1–133

3

0–20

199

216

95

113

45

380.8

380.8

P1–211

5

0–5

436

465

157

187

21

139.8

137.7

P1–212

5

0–5

436

465

157

187

19

121.9

120.0

P1–213

5

0–5

436

465

157

187

18

120.4

119.2

P1–221

5

0–10

436

465

157

187

35

233.0

231.8

P1–222

5

0–10

436

465

157

187

31

201.2

199.6

P1–223

5

0–10

436

465

157

187

42

251.8

250

P1–231

5

0–20

436

465

157

187

61

376.1

369.8

P1–232

5

0–20

436

465

157

187

80

502

501.6

P1–233

5

0–20

436

465

157

187

77

483.8

483.2

P1–311

7

0–5

757

798

219

261

29

153.35

150.8

P1–312

7

0–5

757

798

219

261

26

131.800

129.400

P1–313

7

0–5

757

798

219

261

23

115.900

114.500

P1–321

7

0–10

757

798

219

261

48

232.800

228.500

P1–322

7

0–10

757

798

219

261

48

275.700

274.900

P1–323

7

0–10

757

798

219

261

58

284.800

284.200

P1–331

7

0–20

757

798

219

261

100

517.950

516.400

P1–332

7

0–20

757

798

219

261

107

546.500

546.100

P1–333

7

0–20

757

798

219

261

96

438.900

434.900

P1–411

10

0–5

1396

1455

312

372

35

125.600

118.200

P1–412

10

0–5

1396

1455

312

372

35

140.850

139.800

P1–413

10

0–5

1396

1455

312

372

36

129.350

126.050

P1–421

10

0–10

1396

1455

312

372

74

298.575

298.250

P1–422

10

0–10

1396

1455

312

372

73

278.450

277.850

P1–423

10

0–10

1396

1455

312

372

72

271.475

267.750

P1–431

10

0–20

1396

1455

312

372

161

642.050

640.449

CPU (sec) Gap (%) B1

B2

13.401 0.00 3.229 0.00 17.191 0.00

0.577 0.00 0.109 0.00 7.270 0.00

0.515 0.00 11.092 0.00 12.200 0.00

2.839 0.00 7.363 0.00 6.661 0.00

39.842 0.00 20.826 0.00 8.081 0.00

0.234 0.00 18.221 0.00 1.950 0.00

35.506 0.00 37.705 0.00 203.597 0.00

25.818 0.00 79.483 0.00 7200 0.22

169.058 0.00 14.305 0.00 448.238 0.00

177.202 0.00 3.401 0.00 5.616 0.00

1601.63 0.00 462.637 0.00 133.287 0.00

10.312 0.00 16.052 0.00 96.237 0.00

2855.006 0.00 390.377 0.00 1466.472 0.00

704.329 0.00 457.395 0.00 1244.18 0.00

5.585 0.00 489.313 0.00 28.221 0.00

114.661 0.00 300.084 0.00 47.705 0.00

1152.52 0.00 183.988 0.00 121.525 0.00

29.094 0.00 83.289 0.00 48.251 0.00

25.834 0.00 1287.74 0.00 4587.817 0.76 0.00

584.099 0.00 7200 0.08 7200 0.17

177.155 0.00

7200 0.01

9131

C. Özgüven, B. Sungur / Applied Mathematical Modelling 37 (2013) 9117–9131 Table A6 (continued) Problem intances Problem No

Model Sizes Types

dlj

#Int. B1

Solution

B2

#Cons. B1

W B2

Cost B1

B2

P1–432

10

0–20

1396

1455

312

372

149

592.050

592.049

P1–433

10

0–20

1396

1455

312

372

141

463.000

460.050

CPU (sec) Gap (%) B1

B2

211.631 0.00 7200 0.01 0.00

7200 0.03 7200 1.77

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