OperationsResearchReport_2019

Application of Linear Programming in
Manpower Planning Problem
May 10, 2019
Contents
1 Introduction 2
2 Description of Data 2
3 Methods and Analysis 4
3.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.5 Additional Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Results and Discussion 8
5 Conclusion 12
6 References 13
7 Appendix 13
1

1 Introduction
Manpower, also referred to as human resources, is a workforce composed of individuals of an
economy, company or organisation. The simplest and most conventional definition used to
define ‘manpower planning’ is “getting the right number of people with the right competencies
in the right jobs at the right time” (Sinclair, 2004, p.2). The term ‘manpower planning’ is
synonymous to other terms, for example, ‘succession planning’, ‘human resource planning’
and ‘workforce planning’ (Sinclair, 2004).
With the increasing economic integration and economic growth, manpower planning has
become a crucial problem for the human resources (HR) department, particularly for the
multinational enterprises, or worldwide corporation. The development of an optimal manpower
planning modus operandi, or strategy, is necessitated to satisfy the manpower requirements
and the unoccupied vacancies for attaining specific objectives. Hence, manpower planning
allows for a more productive and well-organised workforce in an organisation. Economically,
an increase in workforce productivity may reduce the unit costs and prices of output. Besides
that, a rise in productivity would enhance the competitiveness of the firms and international
trade performance. Consequently, these advantages would generate higher efficiency gains
and increase profits, which could be reinvested in physical capital to sustain long run growth.
This report seeks to explicate and model a linear programme to achieve the objectives
of the company: redundancy minimisation, and cost minimisation. This is because the
company is experiencing a number of structural changes, which will influence its manpower
planning in future years. In addition, the plans on recruitment, retraining, redundancy, and
short-time working contract, concerning the company workforce in the three types of jobs
(unskilled, semi-skilled, and skilled) over the next three years, will be provided. Finally, useful
information, which might aid the decision making, will be provided with additional analyses.
2 Description of Data
Table 1 tabulates the data for the manpower planning. The first row shows the name of the
three types of jobs, for instance, unskilled, semi-skilled and skilled. The second row shows
the current number of employees on each type of jobs in the company. These figures refer to
the full-time equivalent (FTE) employees. An employee may work on short-time employment
contracts. The productivity of an employee on short-time employment contract is a fraction
of an employee full-time employment contract, so is counted as a fraction of a FTE employee.
However, unless stated otherwise, an employee always means a FTE employee. In addition,
the current workforce has no short-time employees. The third row presents the manpower
requirements, or the number of workers needed on each type of jobs, for three years (Year 1,
Year 2 and Year 3).
Besides that, there is a natural wastage of labour owing to employees leaving the profession
voluntarily. A fairly high number of employees leave during the first year in service. After
that, the wastage rates reduces. Considering such a scenario, the wastage rates for both the
2

less than and more than one year’s service are illustrated in the fourth row. It is known
that all employees in the current workforce (at the beginning of the three-year period) have
been in service for more than a year. Therefore it is assumed that the wastage occurs at the
beginning of each year. Hence the figures provided in the in the first line of the table are
fixed in a year. It is possible for the company to recruit a limited number of workers from
outside the company. The fifth row provides the maximum number of external recruits each
year. The recruitment is part of the normal operations, therefore the budget of the company
has been allocated and so, no extra cost is incurred. Furthermore, the sixth row presents the
overstaffing cost (per employee per year), in pound sterling (£).
The cost of short-time employment (per employee per year), in pound sterling (£), is given
in the last row of the table. The short-time employment contracts last only for a year;
the workers automatically obtain or regain full time employment, unless they are put on
short-time employment contracts again. The employees taking the retraining scheme are
counted as part of the workforce at the upper skill level. The decisions are implemented and
made effective at the beginning of each year. For example, on the 1st of January, the new
recruits will start their jobs, the redundant employees will leave, etc.
Types of jobs Unskilled Semi-skilled Skilled
Current number of
employees in each
type of jobs
2000 1000 1000
Manpower
requirements
Year 1 1000 1400 1000
Year 2 500 2000 1500
Year 3 0 2500 2000
Wastage rate
Less than one year’s service 25% 20% 10%
More than one year’s service 10% 5% 5%
Maximum number of
external recruits
each year
500 800 500
Over-staffing cost
(per employee per
year)
£1420 £2000 £3180
Cost of short-time
employment (per
employee per year)
£516 £400 £382
Table 1: The data sheet for manpower planning.
3

3 Methods and Analysis
3.1 Linear Programming
The method utilised is this research is the linear programming method. Linear programming,
also known as linear optimisation, is a method to attain the optimal outcome by maximising
or minimising the objective function(s) of a linear programming problem, and the constraints
are represented by linear functions of the variables (Li, 2018). According to Li (2018), the
linear programmes with d independent variables can be written in conanical form as:
Max (or Min) z = c x
subject to Ax b
and x 0,
where z denotes the objective function, c = (c1, c2, ..., cd) ∈ Rd
and b = (b1, b2, ..., bm) ∈ Rm
are constant vectors, x = (x1, x2, ..., xd) ∈ Rd
, and A is a constant m × d coeﬃcient matrix.
x is the decision variable vector, c is the cost coeﬃcient vector, and b is the recourse vector.
The linear programming model is solved using the IBM ILOG CPLEX Optimization Studio
(simply known as CPLEX), an optimisation software package in AMPL (A Mathematical
Programming Language). The operating system used is macOS Majave Version 10.14.4.
3.2 Variables
Let i denotes the three years (Year 1, Year 2 and Year 3) covered by the plans, where i = 1, 2, 3.
In addition, i = 0 is used to represent the year before the three-year period. The ‘jth type of
jobs’, where j = 1, 2, 3, represents the three type of jobs, notably unskilled, semi-skilled and
skilled, respectively. To formulate the linear programming model, the variables introduced
are the following:
• tij denotes the total number of employees on the jth type of jobs in year i,
• uij denotes the number of newly recruited employees for the jth type of jobs in year i,
• vij denotes the number of employees moved into the jth type of jobs through retraining
in year i,
• wij denotes the number of employees on the jth type of jobs made redundant at the
beginning of year i,
• xij denotes the number of employees on the jth type of jobs on the short-time employ-
ment contracts in year i, and
• yij the number of surplus employees on the jth type of jobs in year i,
where tij 0, uij 0, vij 0, wij 0, xij 0 and yij 0. Moreover, uij ∈ Z, vij ∈ Z,
wij ∈ Z, xij ∈ Z and yij ∈ Z, where (·) ∈ Z means that (·) is an element in the set of integer.
4

3.3 Objective Functions
The first objective is to minimise the employee redundancy, which can be computed by
summing up the number of employees on the jth type of jobs made redundant at the
beginning of each year (Year 1, Year 2 and Year 3). The objective function can be formulated
as follows:
Min (w11 + w12 + w13 + w21 + w22 + w23 + w31 + w32 + w33).
The second objective is to minimise the cost. The objective function is computed by
summing up the costs of short-time employment, redundancy costs, overstaffing costs, and
retraining costs. The retraining of a semi-skilled employee, to make him/her skilled, is £500.
Additionally, if an unskilled employee is declared redundant, a one-off payment of £200 is
made to the employee. If a semi-skilled or skilled employee is declared redundant, a payment
of £500 is made to the employee. Furthermore, the cost to retrain an unskilled employee,
to make him/her semi-skilled, is £400. The overstaffing costs and the cost of short-time
employment are given in Table 1. The cost function can be formulated as follows:
Min [516(x11 + x21 + x31) + 400(x12 + x22 + x32) + 382(x13 + x23 + x33)
+ 200(w11 + w21 + w31) + 500(w12 + w22 + w32) + 500(w13 + w23 + w33)
+ 1420(y11 + y21 + y31) + 2000(y12 + y22 + y32) + 3180(y13 + y23 + y33)
+ 400(v11 + v21 + v31) + 500(v12 + v22 + v32)].
3.4 Constraints
After analysing the data, six types of constraints are considered:
(i) Manpower requirements: Due to the implementation of the new artificial intelligence (AI)
technologies, relatively more semi-skilled and skilled, but less unskilled employees will be
necessitated. The productivity requirement of an employee on short-time employment
contract is half of that of a full-time employee, thus is considered as half of a full-time
employee. The approximate requirements for the next three years are computed by
subtracting the number of employees on short-time employment contracts (xij) and the
number of surplus employees (yij), from the total number of employees (tij):
t11 − 0.5x11 − y11 = 1000,
t21 − 0.5x21 − y21 = 500,
t31 − 0.5x31 − y31 = 0,
t12 − 0.5x12 − y12 = 1400,
t22 − 0.5x22 − y22 = 2000,
t32 − 0.5x32 − y23 = 2500,
t13 − 0.5x13 − y13 = 1000,
t23 − 0.5x23 − y23 = 1500,
t33 − 0.5x33 − y33 = 2000.
5

(ii) Work continuity: The work continuity constraints calculate the total number of employees
who will continue working on the jth type of jobs in year i. Regarding unskilled jobs
(j = 1), the total number of employees who will continue working on unskilled jobs
in year i (ti1) can be calculated by adding up the current number of employees on
the unskilled jobs (t01 = 2000), the number of newly recruited employees on unskilled
jobs in year i (ui1), and subtracting the number of employees moved into semi-skilled
jobs through retraining in year i (vi2), as well as the number of employees on unskilled
jobs, who are made redundant, at the beginning of year i (wi1). According to Table
1, the wastage rate for less than a year’s service and more than a year’s service on
unskilled jobs are 0.25 (25%) and 0.10 (10%), respectively. Hence, the work continuity
constraints for the unskilled jobs can be formulated as:
t11 = 0.9(2000) + 0.75u11 − v12 − w11,
t21 = 0.9t11 + 0.75u21 − v22 − w21,
t31 = 0.9t21 + 0.75u31 − v32 − w31.
In respect of semi-skilled jobs (j = 2), the total number of employees who will continue
working on semi-skilled jobs in year i (ti2) can be calculated by summing up the
current number of employees on the semi-skilled jobs (t02 = 1000), the number of newly
recruited employees on semi-skilled jobs in year i (ui2), the number of employees moved
into semi-skilled jobs through retraining in year i (vi2), and subtracting the number
of employees moved into skilled jobs through retraining in year i (vi3), as well as the
number of workers on semi-skilled jobs, who are made redundant, at the beginning
of year i (wi2). As shown in Table 1, the wastage rate for less than a year’s service
and more than a year’s service on semi-skilled jobs are 0.20 (20%) and 0.05 (5%),
respectively. Hence, the work continuity constraints for the semi-skilled jobs are:
t12 = 0.95(1000) + 0.8u12 + 0.95v12 − v13 − w12,
t22 = 0.95t12 + 0.8u22 + 0.95v22 − v23 − w22,
t32 = 0.95t22 + 0.8u32 + 0.95v32 − v33 − w32.
With regard to skilled jobs (j = 3), the total number of employees who will continue
working on skilled jobs in year i (ti3) can be calculated by summing up the current
number of employees on the skilled jobs (t03 = 1000), the number of newly recruited
employees on skilled jobs in year i (ui3), the number of employees moved into semi-
skilled jobs through retraining in year i (vi3), and subtracting the number of workers
on skilled jobs, who are made redundant, at the beginning of year i (wi3). As shown in
Table 1, the wastage rate for less than a year’s service and more than a year’s service
on skilled jobs are 0.10 (10%) and 0.05 (5%), respectively. Hence, the work continuity
constraints for the skilled jobs can be formulated as:
t13 = 0.95(1000) + 0.9u13 + 0.95v13 − w13,
t23 = 0.95t13 + 0.9u23 + 0.95v23 − w23,
t33 = 0.95t23 + 0.9u33 + 0.95v33 − w33.
6

(iii) External recruitment: The company is possible to recruit a limited number of employees
from outside the company. The maximum number of employees that can be recruited
in each category are given in Table 1. The external recruitment constraints can be
written as:
u11 500,
u21 500,
u31 500,
u12 800,
u22 800,
u32 800,
u13 500,
u23 500,
u33 500.
(iv) Short-time working: The company can put up to 50 employees on each skill level on
short-time employment contracts annually. The short-time employment constraints can
be written as the following:
x11 50,
x21 50,
x31 50,
x12 50,
x22 50,
x32 50,
x13 50,
x23 50,
x33 50.
(v) Retraining: It is possible for the company to retrain up to 200 unskilled employees
annually to make then semi-skilled. The retraining of semi-skilled employees to make
then skilled is limited to a number of no more than a quarter of the skilled workforce
at the time as some retraining is done on the job. The retaining constraints can be
formulated as follows:
v12 200,
v22 200,
v32 200,
v13 − 0.25t13 0,
v23 − 0.25t23 0,
v33 − 0.25t33 0.
7

(vi) Surplus employees: The company is allowed to employed up to 155 more employees than
the total number required across all three skills level, but there will be an incurrence of
extra cost. The surplus employees constraints can be formulated as the following:
y11 + y12 + y13 155,
y21 + y22 + y23 155,
y31 + y32 + y33 155.
3.5 Additional Analyses
This subsection provides some possible additional analyses that are performed. These analyses
may provide useful information for the decision making. The company’s main objective is to
minimise employee redundancy and cost. It is worth looking into other aspects, such as the
cost on external recruitment minimisation, cost on employee retraining minimisation, cost
on short-time employees minimisation, and cost on surplus employees minimisation. The
objective functions considered are as follows:
External recruitment: Min (u11 + u12 + u13 + u21 + u22 + u23 + u31 + u32 + u33),
Employee retraining: Min (v11 + v12 + v13 + v21 + v22 + v23 + v31 + v32 + v33),
Short-time employees: Min (x11 + x12 + x13 + x21 + x22 + x23 + x31 + x32 + x33),
Surplus employees: Min (y11 + y12 + y13 + y21 + y22 + y23 + y31 + y32 + y33),
subject to the constraints in Section 3.4.
4 Results and Discussion
The first objective of the company is to minimise employee redundancy. It is vital for a
company to decide the appropriate number and the most efficient use of the employees. The
development of a modus operandi for manpower management is significant in enhancing
workforce productivity, and minimising or avoiding the disappearance of jobs. Based on the
AMPL output tabulated in Table 2, the total number of employees on unskilled jobs, who
will be made redundant in Year 1, Year 2 and Year 3, will be 420, 282 and 142, respectively.
This gives a total of 844, and it costs £1,380,740 if these policies are followed.
The second aim of the company is to minimise cost. Table 3 tabulates the AMPL output
for the policies for cost minimisation. The policies for cost minimisation cost £714,106.
As shown in Table 2, the adoption of these policies lead to a total redundancy of 1233
employees. For instance, the amount of employees on unskilled jobs, who will be made
redundant in Year 1, Year 2 and Year 3, will be 744, 236 and 250, respectively. In addition,
the number of employees on unskilled jobs, who will be made redundant in Year 3, will
be 3 employees. By comparison, the policies for employee redundancy minimisation cost
(£1, 380, 740 − £714, 106 =) £666,634 more than the policies for cost minimisation. However,
the policies for cost minimisation lead to (1233 − 844 =) 389 more number of employees to
be made redundant, as compared to the policies for redundancy minimisation.
8

Variable Year Unskilled Semi-skilled Skilled
External recruitment
Year 1 0 325 60
Year 2 0 665 717
Year 3 0 500 498
Retraining
Year 1 0 200 0
Year 2 0 200 22
Year 3 0 200 14
Redundancy
Year 1 420 0 0
Year 2 282 0 0
Year 3 142 0 0
Short-time employees
Year 1 50 0 8
Year 2 50 50 50
Year 3 50 9 1
Surplus employees
Year 1 155 0 0
Year 2 55 5 95
Year 3 155 0 0
Table 2: Policies for employee redundancy minimisation, AMPL output.
Year 1 0 496 60
Year 2 0 799 639
Year 3 0 499 498
Retraining
Year 1 0 56 0
Year 2 0 164 1
Year 3 0 200 107
Redundancy
Year 1 744 0 3
Year 2 236 0 0
Year 3 250 0 0
Year 1 0 0 2
Year 2 0 50 50
Year 3 0 0 1
Surplus employees
Year 1 0 0 0
Year 2 0 99 2
Year 3 0 0 0
Table 3: Policies for cost minimisation, AMPL output.
9

Year 1 0 325 60
Year 2 0 635 756
Year 3 0 489 489
Retraining
Year 1 0 200 0
Year 2 0 200 4
Year 3 0 200 4
Redundancy
Year 1 600 0 0
Year 2 200 0 0
Year 3 70 0 0
Year 1 0 0 8
Year 2 0 18 0
Year 3 50 0 0
Surplus employees
Year 1 0 0 0
Year 2 0 15 138
Year 3 155 0 0
Table 4: Policies for external recruitment minimisation, AMPL output.
Year 1 0 500 70
Year 2 0 800 790
Year 3 0 499 470
Retraining
Year 1 0 60 0
Year 2 0 53 0
Year 3 0 187 0
Redundancy
Year 1 740 0 13
Year 2 322 0 1
Year 3 285 0 0
Year 1 0 0 0
Year 2 50 26 50
Year 3 1 3 0
Surplus employees
Year 1 0 7 0
Year 2 0 14 135
Year 3 0 1 0
Table 5: Policies for employee retraining minimisation, AMPL output.
10

Year 1 0 495 500
Year 2 0 496 728
Year 3 8 499 500
Retraining
Year 1 0 200 0
Year 2 0 200 0
Year 3 0 198 31
Redundancy
Year 1 600 0 396
Year 2 200 0 0
Year 3 258 0 0
Year 1 0 0 0
Year 2 0 0 0
Year 3 0 0 0
Surplus employees
Year 1 0 136 4
Year 2 0 46 109
Year 3 0 0 8
Table 6: Policies for short-time employees minimisation, AMPL output.
Year 1 500 500 500
Year 2 800 650 730
Year 3 500 500 498
Retraining
Year 1 0 200 0
Year 2 0 200 0
Year 3 0 200 28
Redundancy
Year 1 975 140 400
Year 2 800 0 1
Year 3 625 0 0
Year 1 0 0 0
Year 2 0 50 50
Year 3 0 0 1
Surplus employees
Year 1 0 0 0
Year 2 0 15 81
Year 3 0 0 0
Table 7: Policies for surplus employees minimisation, AMPL output.
11

As aforementioned, it is worth looking into other policies before deciding on which policy to
be adopted. Table 4 tabulates the AMPL output for the policies for external recruitment
minimisation. Regarding external recruitment minimisation, the policies cost £1,143,000
over the three year period and generate a total of (325 + 60 + 635 + 756 + 489 + 489 =)
2754 external recruited employees. These figures can be found in Table 4. The policies for
external recruitment minimisation cost (£1, 143, 000 − £714, 106 =) £428,894 more than the
cost for minimisation policies, and result in (2991 − 2754 =) 237 less number of employees
to be recruited externally. The human resources department should define the recruitment
procedures clearly, concisely and precisely. Besides that, it is essential to review the job
description, roles requirements and skills requirements, because this could minimise the risk
of hiring unsuitable employees.
Retraining and reskilling programmes help employees to develop and keep their skills and
abilities up-to-date with technological change and business innovation. The employees will
perform the jobs or tasks more effectively and efficiently after the retraining programme,
which improve productivity. Team morale may improve and employee turnover may decrease,
which in turn results in cost reduction for external recruitment. However, it may be costly to
run the retraining program. Table 5 tabulates the AMPL output for the policies for employee
retraining minimisation. The cost retraining minimisation policies is £926,716, and this will
lead to (60 + 53 + 187 =) 300 employees to be retrained in the three year period. These
policies cost (£926, 716 − £714, 106 =) £212,070 more than the cost of the policies for cost
minimisation.
Whilst employing short-time employees could reduce unemployment rates in an economy and
decrease workloads of other employees, it could require recruitment costs and extra costs
in terms of benefits and training. It would be sensible to look into short-time employees
minimisation. Table 6 tabulates the AMPL output for the policies for employee retraining
minimisation. These policies cost £1,413,080, which cost (£1, 413, 080 − £714, 106 =)
£698,974 more than the policies for cost minimisation. These policies allow the company
to minimise the short-time employees to zero, but this is not practical to be conducted.
Furthermore, Table 7 tabulates the AMPL output for the policies for surplus employees
minimisation. Regarding surplus employees minimisation, the policies cost £1,331,560, which
cost (£1, 331, 560 − £714, 106 =) £617,454 more the policies for cost minimisation. The
minimum number of surplus employees will be (15 + 81 =) 96 employees. On average, it cost
the company approximately £13,870 to reduce a surplus employee.
5 Conclusion
As elucidated thus far, it is difficult to make a conclusion regarding the best policy to be
adapted. The policies for cost minimisation cost the least, as compared to other policies
suggested. However, the other policies would generate advantages to the company, in terms
of cost saving, skills improvement, and so on. It would be reasonable to suggest that the
company should consider all of the policies suggested, to achieve a minimum cost.
12

6 References
• Li, Y., (2018). MAS322 and MAS423: Operations Research [online]. The University of
Sheffield: School of Mathematics and Statistics. [Viewed 6 May 2019]. Available from:
MOLE.
• Sinclair, A., (2004). Workforce planning: A literature review [online]. Brighton:
Institute for Employment Studies. [Viewed 3 May 2019]. Available from: https://www.
employment-studies. co.uk/system/files/resources/files/mp37.pdf.
7 Appendix
reset;
var w11 >=0 integer;
var y11 >=0 integer;
var x11 >=0 integer;
var v11 >=0 integer;
13

var t11 >=0;
var t12 >=0;
var t13 >=0;
var t21 >=0;
var t22 >=0;
var t23 >=0;
var t31 >=0;
var t32 >=0;
var t33 >=0;
var u11 >=0 integer;
minimize w: w11 + w12 + w13 + w21 + w22 + w23 + w31 + w32 + w33;
minimize c: 400 * ( v12 + v22 + v32 ) + 500 * ( v13 + v23 + v33 ) + 516 * ( x11 + x21 +
x31 ) + 400 * ( x12 + x22 + x32 ) + 382 * ( x13 + x23 + x33 ) + 200 * ( w11 + w21 + w31
) + 500 * ( w12 + w22 + w32 + w13 + w23 + w33 ) + 1420 * ( y11 + y21 + y31 ) + 2000 *
( y12 + y22 + y32 ) + 3180 * (y13 + y23 + y33);
subject to
c1: y11 + y12 + y13 <= 155;
c2: y21 + y22 + y23 <= 155;
c3: y31 + y32 + y33 <= 155;
c4: v12 <= 200;
c5: v22 <= 200;
c6: v32 <= 200;
c7: v13 - 0.25 * t13 <= 0;
c8: v23 - 0.25 * t23 <= 0;
c9: v33 - 0.25 * t33 <= 0;
14

c10: t11 - y11 - 0.5 * x11 = 1000;
c11: t21 - y21 - 0.5 * x21 = 500;
c12: t31 - y31 - 0.5 * x31 = 0;
c13: t12 - y12 - 0.5 * x12 = 1400;
c14: t22 - y22 - 0.5 * x22 = 2000;
c15: t32 - y32 - 0.5 * x32 = 2500;
c16: t13 - y13 - 0.5 * x13 = 1000;
c17: t23 - y23 - 0.5 * x23 = 1500;
c18: t33 - y33 - 0.5 * x33 = 2000;
c19: t11 = 0.9 * 2000 + 0.75 * u11 - v12 - w11;
c20: t21 = 0.9 * t11 + 0.75 * u21 - v22 - w21;
c21: t31 = 0.9 * t21 + 0.75 * u31 - v32 - w31;
c22: t12 = 0.95 * 1000 + 0.8 * u12 + 0.95 * v12 - v13 - w12;
c23: t22 = 0.95 * t12 + 0.8 * u22 + 0.95 * v22 - v23 - w22;
c24: t32 = 0.95 * t22 + 0.8 * u32 + 0.95 * v32 - v33 - w32;
c25: t13 = 0.95 * 1000 + 0.9 * u13 + 0.95 * v13 - w13;
c26: t23 = 0.95 * t13 + 0.9 * u23 + 0.95 * v23 - w23;
c27: t33 = 0.95 * t23 + 0.9 * u33 + 0.95 * v33 - w33;
c28: x11 <=50;
c29: x12 <=50;
c30: x13 <=50;
c31: x21 <=50;
c32: x22 <=50;
c33: x23 <=50;
c34: x31 <=50;
c35: x32 <=50;
c36: x33 <=50;
c37: u11 <=500;
c38: u12 <=500;
c39: u13 <=500;
c40: u21 <=800;
c41: u22 <=800;
c42: u23 <=800;
c43: u31 <=500;
c44: u32 <=500;
c45: u33 <=500;
option solver ‘./cplex’;
solve;
display w, c, w11, w12, w13, w21, w22, w23, w31, w32, w33, y11, y12, y13, y21, y22, y23,
y31, y32, y33, x11, x12, x13, x21, x22, x23, x31, x32, x33, v11, v12, v13, v21, v22, v23, v31,
v32, v33, u11, u12, u13, u21, u22, u23, u31, u32, u33;
display w, c, w11, w12, w13, w21, w22, w23, w31, w32, w33, y11, y12, y13, y21, y22, y23,
15

y31, y32, y33, x11, x12, x13, x21, x22, x23, x31, x32, x33, v11, v12, v13, v21, v22, v23, v31,
v32, v33, u11, u12, u13, u21, u22, u23, u31, u32, u33 > RM1.res;
For other objectives:
minimize u: u11 + u12 + u13 + u21 + u22 + u23 + u31 + u32 + u33;
minimize v: v11 + v12 + v13 + v21 + v22 + v23 + v31 + v32 + v33;
minimize x: x11 + x12 + x13 + x21 + x22 + x23 + x31 + x32 + x33;
minimize y: y11 + y12 + y13 + y21 + y22 + y23 + y31 + y32 + y33;
16

OperationsResearchReport_2019

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