325

LINEAR PROGRAMMING WITH MULTIPLE OBJECTIVES AS AN

INTRODUCTION TO RATIONING DECISION-MAKING IN THE

INDUSTRIAL ENTERPRISES.

CASE STUDY OF THE ANTIBIOTICAL BRANCH (DEPARTMENT)

MEDEA

MEKID ALI-

Pr Université de Médéa

mekid_a@yahoo.fr

LARBI Benhora Amel

Doctorante Université de Médéa

nabilalarbi0@gmail.com

Abstract:

All challenges caused by globalization and capitalism economic system in

the development countries imposed radical changes in the economic

activities, which need redirecting in the Methods and in work tools in the

economic enterprises, to improve its positive merger in the national and

international economic environment.

To realize this object, manager should build up the competitiveness tools

of the local economic enterprises, especially to develop management

systems.

This research paper tries to study the possibilities of adopting the

multiple-objectives optimization in the industrial enterprises.

Keywords: the multiple objectives, the rationalization of the decision, linear

programming with multiple objectives.

Introduction:

The modernization of management systems in the Algerian economies

enterprises has taken a place due to its direct and strong impact on the results

and performance of companies, also improving merger conditions in the

national and international economic environment.

This object must be realized by providing possibilities of rationing

decisions in order to optimize the enterprise activity by maximizing

revenues, minimizing costs and limiting efforts.

One of the most important conditions to apply this method is the best

controlling of the quantitative analysis tools, especially the estimation, and

the optimization. That is to say the efficient decisions made by mangers

which are based on its possibility to use those methods.

There is a big importance in using the economic modeling within the

rationing decision in enterprise through providing a quantitative relationship

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between the economic index which allows manager to realize the different

policies based on objectivity and accuracy.

In the other hand the manager’s effort contributes the development of the

enterprise decisions, and to build up its competitiveness, also helping

company to confront all challenges in the external environment, which is

characterized by the existence of foreign companies with a considerable

experience, technological abilities and perfect managing.

During the complexity of modern enterprise environment which is

characterized by high competitiveness and high levels of probability,

markets extended by similar products, decision maker doesn’t depended on a

single standard with a single objective. but with multiple standards with

different requirement in a new enterprise fact, in order to keeping up with

environment results and changes in the global business environment.

This fact pushes researcher to find new methods in order to reach linear

programming with single objective, because enterprises always seek to

realize the integrated and inconsistent objectives.

The linear programming with multiple objectives is one of the most

important quantitative tools that enable decision makers to rationalize the

distribution of the available resources, through the activity modeling in the

mathematic program with a special interest in achieving multiple objectives

in the same time.

Study problem: Developments in the business environment have led to the

emergence of several objectives to be achieved at the same time. The

increase of profits or the decrease of costs is no longer the main objective of

the Foundation, but we find other vital goals must be taken into account, and

therein lies the problem of our research through the answer on the following

question: To what scope could the economic application of the method of

programming goals in the branch of antibiotics, leads to rationalize their

decisions?

Study Importance: The importance of our study is derived from the

importance of the industry and the rationalization of administrative decisions

at various levels and in various sectors and systems, and especially in light

of the institutions that are characterized as big as the size and complexity of

its internal systems ,and depends its decisions on systems operations

research and decisions-making science to get the system’s optimization in

the light of many of the goals and objectives that may be conflicting

sometimes and restricted at other times.

Study Objectives: This study aims to evaluate the reality of the productive

activity of the Antibiotical –Medea- and highlight the possibility of using

linear programming with multiple objectives and its role in improving the

level of efficient and efficiency of enterprise decisions, in order to have an

optimal performance within the circumstance.

1. Theoretical approach of decision making.

The operation of decision making is considered as the basic of the

economic activity, that is why the decision maker aims to achieve rationing

327

in the decision to get the optimal results, rationing here means looking for

the case of rationality in all human behavior, but the rationalization of

decision making means the rationalization of decision in which the decision

maker can get the best result and the most acceptable. 1

Decision making process is a series of steps in order to achieve whatever

goals decision maker may want to have, such as defining purpose, data

collection, analysis and comparison, chosen alternatives2.

Also decision making is that organized process using logical steps within

limited resources. 3

Andersson D.R said to make decision is the analysis and organized way

which focus on the scientific logic using all available data in order to get the

best alternative to solve problems. 4

In general, we can describe making decision as a study of different data

related to the problem raised by using the relevant method to choose the best

choices after the preferences.

But the making decision is the trend chosen by the manager because it is the

best way to achieve goals.5. And Thomas said decision making translates a

specific desire into a real fact. 6

Decision making is to choose one of the multiple choices to achieve the

enterprise goals, by solving problem and taking advantages. 7

We can say that decision making is the optimal way to solve problems.

2. Theoretical approach of linear programming with multiple

objectives: The requirements of capitalism impose a lot of rationality to

resolve problems in the enterprise, and to be in the optimal case. This

situation requires a rationalization taken by manager in order to achieve

tasks in the enterprise. We need to apply the quantitative techniques.

Linear programming with multiple objectives is the best the technical way

which response with an objectivity, because enterprise seeks to find the best

case to maximize profit within the multiple objectives and limitation of

resources.

The using of linear programming with multiple objectives provides and

advances step with the processing of resources available, to find the optimal

economic activity compared with the traditional linear programming with

single objective.

2.1. The definition of linear programming with multiple objectives.

Since the previous years, many attempts to illustrate the principle of these

methods and reach its using.

Linear programming with multiple objectives is a flexible mathematical tool,

directed to treat the complex decision-making which constrains multiple

objective and limitation. 8

Linear programming with multiple objectives is a mathematical model seeks

to find the typical solution in the enterprise. It aims to minimize the total

deviations from the previous objectives. The mathematical model identifies

the principle element of the model as decision variable, technical limitation

and the objective function. 9

328

According to (Lee S.M and Olson D. L) the linear programming with

multiple objectives is one of the administrative tools which directed to find

solutions of the decision-making with multiple objectives. 10

Linear programming with multiple objectives is a mathematical way in the

linear programming allows manager to put the multiple objectives function

priority. 11

In general, Linear programming with multiple objectives is one of the linear

programming tools used during the different objectives to choose the best

decision alternative in the face of limited resources.

2.2. The importance of linear programming with multiple objectives: Most of the decisions cases do not characterize by a single objective but by

many principles and secondary objectives which are integrated or

inconsistent. The importance of Linear programming with multiple

objectives is the possibility to treat the different objectives:12

- The conflicting objectives: Enterprise tries to achieve many inconsistent

objectives which are relevant with its desire and increasing the maximum

service provided to customer. There is a direct relationship between costs

and services.

- The various dimensions objectives: In general, all objectives

characterized by different dimensions can be measured by different units

dependent with many aspects, can affect in other element such as

maximizing profit, extending market share.

- Uncertainly quantification objectives: There are many objectives can’t

be measured with quantitative way, that is to say we can’t included in the

linear programming. This case requires a special treatment.

The Linear programming with multiple objectives is a complement of the

traditional linear programming that deals with different optimal objectives

within the constraints of problem, here we change any objective of problem

by constraints.

The function included deviation variables that measure the amount of

deviation according to the previous objectives, or real target values.

2.3. Formulation of the linear programming with multiple objectives: The formulation more commonly used in the Linear programming with

multiple objectives presented in the following function: 13

ƒi: represents objectives;] ƒi(𝑥)= 𝑎𝑖𝑗 𝑥𝑗 (𝑖 = 1 , 2, … . . 𝑝)]. gi:requested Goals ;(i=1,2,…..p).

𝑥j: the decision variable; (j=1, 2….., n).

aij :Technological factors.

C𝑥: Factors matrix.

C: Resources vector.

Minimize |ƒi (𝒙 ) – gi|

Under technical constraints:

C≤C𝒙

𝑥j ≥ 0 (j=1,2,…….,n)

329

This model can be written on the following linear form: 14

𝑴𝒊𝒏 𝒁 = 𝜹𝒊+ + 𝜹𝒊

−

𝒑

𝒊=𝟏

𝒂𝒊𝒋 𝒙𝒋 − 𝜹𝒊+

𝒏

𝒋=𝟏

+ 𝜹𝑰− = 𝒈𝒊

𝒄𝒙 ≤ 𝒄. 𝒙𝒋 ≥ 𝟎 𝒋 = 𝟏. 𝟐…𝒏 .

𝜹𝒊+ 𝒆𝒕 𝜹𝒊

− ≥ 𝟎 𝒊 = 𝟏. 𝟐…𝒑

Whereas:

𝜹𝒊+: Positive deviation from the objective (Goal) i.

𝜹𝒊−:Negative deviation from the objective (Goal) i.

The multiplied of the positive and negative deviation is zero (nil) because

both 𝜹𝒊+,𝜹𝒊

−vectors can’t achieved in the same time.

We can’t find a fewer value then the optimal value gi for the objective i.

Linear programming with multiple objectives adopted on The solution which

is less than the absolute value of the target values. It is represented in the

following table: 15

3. The empirical part:

To complete the theatrical part we choose the case of Antibiotical (Medea).

3.1. General presentation of the Antibiotical (Medea)

Antibiotical is the branch of the Antibiotics of Saidal Group; it is located in

MEDEA, 15 km far of the Wilaya seat, 80 km south of Algiers, an area of 25

hectares.

After the restricting of the public enterprise, it became the Antibiotical

branch belonging to SAIDAL.

In 1998 ANTIBIOTICAL started the independence in management. Medea

site specializing in the production of penicillanic and non-penicillanic

antibiotics.

It has two units of semi-synthesis for oral and injectable products, a unit for

pharmaceutical specialties and two buildings: one devoted to penicillanic

products, other to non-penicillanic.

The deviations which

appeared in the economic

function

Constraint function Type of

constraint

𝛿𝑖+ 𝑓𝑖 𝑥 − 𝛿𝑖

+ + 𝛿𝑖− = 𝑔𝑖 . 𝑓𝑖 𝑥 ≤ 𝑔𝑖 .

𝛿𝑖− 𝑓𝑖 𝑥 − 𝛿𝑖

+ + 𝛿𝑖− = 𝑔𝑖 . 𝑓𝑖 𝑥 ≥ 𝑔𝑖 .

𝛿𝑖+ + 𝛿𝑖

− 𝑓𝑖 𝑥 − 𝛿𝑖+ + 𝛿𝑖

− = 𝑔𝑖 . 𝑓𝑖 𝑥 = 𝑔𝑖 .

330

3.2. Linear programming with multiple objectives in the Antibiotical

(Medea).

In the case of the Antibiotical (Medea), we try to formulate a model of

producing quantities of different pharmaceutical product using the different

resources.

To define the amount of those different pharmaceutical products, we use the

linear programming with multiple objectives.

3.2.1. Mathematical model to produce pharmaceutical products in the

Antibiotical (Medea) .

3.2.1.1. Production hypothesis: We adopted seven distinct products by the

following:

- Tablets: included the following products: Orapen1-Mul, Amoxypen 1g.

- Capsules: included the following products: Amoxypen 500 mg, Oxymed

250mg.

- Bottles: included the following products: Gectapen 1mul, Oxaline 1g.

- Pomade/ointment: included the following products: Betasone 0.1%

Clomycine 3%. - Syrups: included one product: Ximalex (Danalise).

- Ampoules: included one product: Clofenal B/2.

- Injectable powder preparation: included the following products:

Amoxypen 500mg, Ampiline 1g.

The following table explains the products symbols and the active

substance for each product.

Table 01: The following chart represents products Symbols and the

Active substance for each products

Products Type

Products

Symbol

(p.s)

Active substance

ORAPEN1MUI comprimés X1

Phénoxyméthylpénicilli

ne

AMOXYPEN 1g comprimés X2 Amoxiciline

OXYMED 250mg gélules X3 Oxytéracycline

AMOXYPEN 500 mg gélules X4 Amoxiciline

GECTAPEN 1mul flacons X5 Benzylpénicilline

OXALINE 1g flacons X6 Oxacilline

BETASONE 0.1 % pommade X7 bétaméthasone

dipropionate

CLOMYCINE 3% pommade X8 Chlortétracycline

XIMALEX sirop X9 alpha_ amylase

CLOFENAL B/2 ampoule X10 declofénac sodique

AMOXYPEN 250mg poudre pour p

inj X11 Amoxiciline

AMPILINE 1g poudre pour p

inj X12 Ampiciline

Source: Data of the production department, Antibiotical-Médéa.

3.2.1.2. The hypothesis of determining constraints:

331

The linear programming with multiple objectives constraints can be

divided into two types: objective functions constraints, technological

constraints.

3.2.1.2.1. Objective constraint:

According to an advanced study in the ANTIBIOTICAL (MEDEA)

activity and the general policies priority of its management, we can describe

the following objectives and constraints: Maximize profit objective,

maximize revenues objective, and maximize production quantity objective

and the cost reducing objective.

3.2.1.2.2. Technological constraints:

- The exploitation (using) of active substance constraints:

symbol The constraints of using

the active substance symbol

The constraints of using

the active substance

b6 bétaméthasone dipropionate b1 Phénoxyméthylpénicilline

b7 Chlortétracycline b2 Amoxiciline

b8 alpha_ amylase b3 Oxytéracycline

b9 declofénac sodique b4 Benzylpénicilline

b10 Ampiciline b5 Oxacilline

- Production capacity constraints:

The production of any type of production takes different time according to

the type of production, type of pharmaceutical, therapeutic form, preparation

method…..etc in the condition of the available time for each product.

The work time divided into seven constraints according to the products

lines, here we can distinguish between three production workshops:

Workshop A: included three production lines:

The first production line: syrup line included the product: X9

The second production line: tablets line included the products: X1, X2.

The third production line: capsules line included the products: X3, X4.

Workshop B: included three production lines:

The first production line: bottles line included the products: X5, X6.

The second production line: Ampoules line included the product: X10.

The third production line: pomade line included the products: X7, X8.

Workshop C: included a single product line:

Injectable powder preparation included two products: X11, X12.

- Satisfying (fulfilling) demand constraints: Antibiotical (Medea) meets

all customers’ orders.

- None zero constraints: Antibiotical (Medea) always produces certain

amount of products; it means enterprise can’t get negative value in the

production variable.

3.2.1.3. Measure units hypothesis: All profit in the Antibiotical (Medea)

measured by the Algerian Dinar, the quantities of production are measured

in grams, time of work in minutes of work, production quantity by number

of cans, production capacity by work hours.

3.2.2. Mathematical formulation Model of Antibiotical (Medea).

3.2.2.1. Mathematical formulation of objectives:

332

We can summarize the model in the following table:

Table 02: production quantity, global Revenues, profit, cost during

2014. produ

ct

Productio

n quantity

Price per

unit (AD)

Total

Revenues

Profit

per

unit(AD

)

Actual(Real)

profit

Cost

price

(AD)

Total costs

X1 2267295 124.172 281534554.74 21.2 48066654 102.972 233467900.74

X2 779790 90.18 70321462.2 9.01 7225907.9 81.17 63295554.3

X3 426451 115.12 49093039.12 19.57 8345646.07 95.55 40747393.05

X4 928427 86.2 80030407.4 14.65 13601455.55 71.55 66428951.85

X5 3806741 71.82 273400138.62 9.33 35516893.53 62.49 237883245.09

X6 2989184 80.76 241406499.84 10.5 31386432 70.26 210020067.84

X7 1459673 73.2 106848063.6 9.49 13852296.77 63.71 92995766.83

X8 526245 63.58 33458657.1 8.27 4352046.15 55.31 29106610.95

X9 2478290 108.25 268274892.5 14.07 34869540.3 94.18 233405352.2

X10 1754145 54.36 95355322.2 9.26 16243382.7 45.1 79111939.5

X11 616840 105.2 64891568 7.7 4749668 97.5 60141900

X12 1000794 80.25 80313718.5 10.07 10077995.58 70.18 70235722.92

Tota

l

1903387

5

- 1644928323.8

2

- 228087918.5

5

- 1416840405.27

source: data processed by the audit and analysis department, Antibiotical

Medea) All resources values will appear as an approximate value in the

linear model QM for Windows).

- Maximize profit objective: Antibiotical (Medea) aims to find the typical

production variety in order to maximize profit much possible, according to

the previous year (2014) 228087918. 55, it means 9 % as the minimum, that

is to say: 228087918.55 × 1.09 = 248615831.21

Through the above, we can define the target value to maximize profit by

the following constraints:

21.2X1+9.01X2+19.57X3+14.65X4+9.33X5+10.5X6+9.49X7+8.27X8+14.07X

9+9.26X10+ 7.7X11+10.07X12 ≥ 248615831.21.

- Maximize Revenues constraints: Antibiotical (Medea) aims to find the

typical production variety which maximizes revenues of the forward

revenues of 2015. By 9% as the minimum, that is to say:

1644928323.82×1.09 = 1792971872.96.

According to this value, we can define the target value to maximize revenues

by the following constraints:

124.172X1+90.18X2+115.12X3+86.2X4+71.82X5+80.76X6+73.2X7

+63.58X8 +108.25X9 + 54.36X10+105.2X11+80.25X12 ≥ 1792971872.96.

333

- Maximize production quantity constraints: Antibiotical (Medea) aims to

increase the level of forward production of 2015, according to the previous

production of 2014 for a minimum of 9%.

19033875 × 1.09 = 20746923.75

According to this value, we can maximize the total production quantity by

the following constraint:

X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12 ≥ 20746923.75

- Minimize production cost objective: Antibiotical (Medea) tries to

minimize the forward production costs. It was at the level of 1416840405.27

in 2014, but Antibiotical Medea aims to decrease cost for a maximum of 9

%, So the maximum of the allowed cost for 2015 is:

1416840405.27 × 1.09 = 1544356041.74.

According to the above details the constraints is:

102.972X1+81.17X2+95.55X3+71.55X4+62.49X5+70.26X6+63.71X7+55.31

X8+94.18X9+ 45.1X10 +97.5 X11+70.18X12 ≤ 1544356041.74

3.2.2.2. Mathematical formulation of technical constraints:

- Active substances constraints: Antibiotical (Medea) focuses on the active

substances to produce the variety products used in our study.

The following table describes the different consumed quantities of the active

substances to produce a single unit of the product.

Table 03: The quantity of active substances of single can of product

annual quantities available

symb

ol Product

The used active

substances

The

quantity of

the active

substances)

g)

Annual

available

quantity

(g)

X1 ORAPEN1MUI Phénoxyméthylpénicill

ine

3 30514000

X3 AMOXYPEN 1g Amoxiciline 0.25 22128000

X3 OXYMED 250mg Oxytéracycline 4 31221000

X4 AMOXYPEN 500

mg Amoxiciline 8 22128000

X5 GECTAPEN 1mul Benzylpénicilline 0.7 29718000

X6 OXALINE 1g Oxacilline 0.5 26553600

X7 BETASONE 0.1 % bétaméthasone 0.014 362320

X8 CLOMYCINE 3% Chlortétracycline 0.45 20136400

X9 XIMALEX alpha_ amylase 0.5 24000000

X10 CLOFENAL B/2 declofénac sodique 0.075 01964600

X11 AMOXYPEN 250mg Amoxiciline 1 22128000

X12 AMPILINE 1g Ampiciline 1 27668000

Source: data treated by production directory of Antibiotical Medea.

According to the table 3, we can define active substance constraints as the

following:

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Technological

constraint

The

exploitation(using)

of active substance

constraints

Technological

constraint

The exploitation(using)

of active substance

constraints

0.014 X7 ≤ 362320 bétaméthasone 3 X1 ≤ 30514000 Phénoxyméthylpénicilline

0.45 X8 ≤ 20136400 Chlortétracycline 0.25 X2 +8 X4 + X11

≤ 22128000 Amoxiciline

0.5 X9 ≤ 24000000 alpha_ amylase 4 X3 ≤ 31221000 Oxytéracycline

0.075 X10 ≤ 1964600. declofénac sodique 0.7 X5 ≤ 29718000 Benzylpénicilline

1 X12 ≤ 27668000 Ampiciline 0.5 X6 ≤ 26553600. Oxacilline

- Capacity production exploitation constraint:

There is a maximum work to produce any product, we define the time of

production by the following way, For example to produce the product X5, we

can use 14 hours of work per day. It is about 840 Minute that is to say the

total work is: 840×253=212520

Total product time capacity equal: 45000 cans per day of the product X5.

Time of producing one can of X5 is 840/45000 =0.019 minute.

The constraint related to the available time for the product X5 as following:

212520 ≤ 5X 0.019

The following chart represents the available work used to produce, and the

available productive capacity during 2014:

Table 04: the available work used to produce and the available

productive capacity during 2014

Source: data treated by production directory of Antibiotical (Medea).

Using the data table (4), it can determine the constraints of the available

time labor:

product

Estimated

work day

per

annum

Available

work time

per day

Available

work

time per

annum

Daily

productive

capacity

Time

using to

produce

a single

can in

minutes

The

maximum

available

work time

in each

production

line in

minutes

X1 253 900 227700 22000 0.04 455400

X2 253 900 227700 23000 0.039

X3 253 900 227700 32000 0.028 425040

X4 253 780 197340 20000 0.039

X5 253 840 212520 45000 0.019 440220

X6 253 900 227700 30000 0.03

X7 253 720 182160 25000 0.029 364320

X8 253 720 182160 25000 0.029

X9 253 600 151800 25000 0.024 151800

X10 253 720 182160 31000 0.023 182160

X11 253 960 242880 33000 0.029 485760

X12 253 960 242880 40000 0.024

335

0.024 X9 ≤ 151800 0.04x1 +0.039x2 ≤ 455400

0.023 X10 ≤ 182160. 0.028x3 +0.039x4 ≤ 425040.

0.029x11 +0.024x12≤ 485760. 0.019x5+0.03x6 ≤ 440220

0.029x7 +0.029x8≤ 364320.

- Satisfying demand constraint.

Antibiotical (Medea) involves many orders for its customers. The quantities

of those orders are included in the constraints:

X1 ≥ 3420000,X2 ≥ 830200,X3 ≥ 526451,X4 ≥ 1200300,X5 ≥ 4800000,X6

≥ 4320000, X7 ≥ 1652430,X8 ≥ 780000,X9 ≥ 2650000,X10 ≥2000600,X11 ≥

943000,X12 ≥ 1542000.

- Non-zero constraint:

X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12 ≥ 0.

According to the below constraint, we can define the linear mathematical

formulation of Antibiotical (Medea) by the following:

- Objective constraints:

- Technological constraints:

3X1 ≤ 30514000. 0.023 X10 ≤ 182160.

0.25 X2 +8 X4 +1 X11 ≤

2212800.

0.029X11 +0.024X12 ≤ 485760.

4 X3 ≤ 31221000. X1 ≥ 3420000.

0.7 X5 ≤ 29718000. X2 ≥ 830200.

0.5 X6 ≤ 26553600. X3 ≥ 526451.

0.014 X7 ≤ 362320. X4 ≥ 1200300.

0.45 X8 ≤ 20136400. X5 ≥ 4800000.

0.5 X9 ≤ 24000000. X6 ≥ 4320000.

0.075 X10 ≤ 1964600. X7 ≥ 1652430.

1 X12 ≤ 27668000. X8 ≥ 780000.

0.04X1+0.039X2 ≤ 455400. X9 ≥ 2650000.

0.028X3 +0.039X4 ≤ 425040. X10 ≥ 2000600.

0.019X5+0.03X6 ≤ 440220. X11 ≥ 943000.

0.029X7 +0.029X8 ≤ 364320. X12 ≥ 1542000.

0.024 X9 ≤151800.

3.2.3. Linear model solution using (QM for windows) program.

To solve linear model solution, we should transform constraints to the

standard form where:

21.2X1+9.01X2+19.57X3+14.65X4+9.33X5+10.5X6+9.49X7+8.27X8+14.07X9+

9.26X10 +7.7X11+10.07X12 ≥ 248615831.21.

124.172X1+90.18X2+115.12X3+86.2X4+71.82X5+80.76X6+73.2X7+63.58X8+

108.25X9+54.36X10+105.2X11+80.25X12 ≥ 1792971872.96.

X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12 ≥ 20746923.75

102.972 X1+ 81.17 X2+95.55 X3 +71.55 X4+62.49 X5 +70.26 X6+63.71 X7

+55.31X8 +94.18 X9+45.1 X10+97.5 X11 +70.18 X12 ≤ 1544356041.74.

336

- Constraints (1-4): the objective constraints are classed by priority, we

enter the deviation variable as the following:

- In the case of: objective constraints (≤), we should enter the deviation

variable which maximize (d+) to minimize objective function.

- In the case of: objective constraints (≥) we should enter the deviation

variable which (d-) to minimize objective function.

- The constraint (5-29): it is technological constraints transformed into

function such as resources constraints in the linear programming.

Whereas:

Min Z= dˉ1+d

ˉ2+d

ˉ3+ d

+4.

21.2X1+9.01X2+19.57X3+14.65X4+9.33X5+10.5X6+9.49X7+8.27X8+14.07X

9+9.26X10+7.7X11+10.07X12+ dˉ1- d

+1=248615831.21.

124.172X1+90.18X2+115.12X3+86.2X4+71.82X5+80.76X6+73.2X7+63.58X8

+108.25X9+54.36X10+105.2X11+80.25X12+dˉ2 - d

+2=1792971872.96.

X1+X2+X3+X4+X5+X 6+ X7+ X8+ X9+ X10 +X11 +X12 + dˉ3-d

+3=20746923.23

102.972X1+ 81.17 X2+95.55X3 +71.55 X4+62.49X5 +70.26 X6+63.71X7

+55.31X8+94.18X9+45.1X10 +97.5X11 +70.18 X12+ dˉ4-d

+4= 1544356041.74.

3 X1 +S1= 30514000.

0.25 X2 +8 X4+1X11 +S2=22128000

4 X3+S3=31221000.

0.7 X5 +S4=29718000.

0.5 X6 +S5 = 26553600.

0.014 X7+S6 = 362320.

0.45 X8+S7 =20136400.

0.5 X9 +S8=24000000.

0.075 X10+S9=1964600.

1 X12 +S10= 27668000.

0.04X1 +0.039X2 + S11= 455400.

0.028X3+0.039X4 +S12=425040.

0.019X5+0.03X6 + S13= 440220.

0.029X7 +0.029X8+S14=364320.

0.024 X9 +S15=151800.

0.023 X10 +S16 =182160.

0.029X11 + 0.024X12+ S17 = 485760.

X1 –S18 = 3420000

X2 –S19= 830200

X3 – S20= 526451

X4 – S21 = 1200300

X5 – S22 = 4800000

X6 – S23 = 4320000

X7 – S24= 1652430

X8 – S25 = 780000

X9 – S26= 2650000

X10 – S27 =2000600

X11 – S28 = 943000

337

X12 – S29= 1542000

X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+ X12≥ 0 .

3.2.4. Reading and analysis of linear programming with multiple

objectives using (QM for windows)

3.2.4.1. Analysis of decisions variables:

According to the typical solution table (index 01) we find the following

quantities:

X1 = 3420000, X2 = 830200, X3 = 526451, X4 = 1200300, X5 = 4800000,

X6 = 4320000, X7 =1652430, X8 =780000, X9 =2650000, X10 =2000600, X11 = 943000, X12 = 1542000.

3.2.4.2. Psychiatry of objective value:

- Maximize profit objective: The profit plan of each product and its total

profit summarized in the following table:

Table 05: Production quantity and total profit according to the proposed

Source: data of program outputs.

Through the previous table, we can observe that proposed profit is more

than target profit, and the deviation obtained is a positive deviation, thus the

typical solution is about: 50129730 AD. Here we can say that profit of the

Antibiotical (Medea) will be increases in the case of using the proposed plan

presented by 50129730 AD, it is about 20.16%.

- Maximize revenues objective: We can describe the revenues plan for each

product in the following table:

Table6: Obtained production quantity and obtained revenues according

to the proposed plan.

product

Proposed

production

quantity

Profit unit (AD) Proposed profit

X1 3420000 21.2 72504000

X2 830200 9.01 7480102

X3 526451 19.57 10302646.07

X4 1200300 14.65 17584395

X5 4800000 9.33 44784000

X6 4320222 10.5 45362331

X7 1652430 9.49 15681560.7

X8 780000 8.27 6450600

X9 2650000 14.07 37287047.7

X10 2000600 9.26 18523704

X11 943000 7.7 7261100

X12 1542000 10.07 15507800

total 24663113 - 298729286.5

Product Proposed production

quantity Unit price

Proposed

Revenues

X1 3420000 124.172 424668240

X2 830200 90.18 74867436

X3 526451 115.12 60605039.12

338

Sourc

e:

data

of

progr

am

outpu

ts.

The analysis of data below can describe a positive deviation in revenues,

where the proposed revenues are more than target revenues. The typical

solution presented by: 353357300 AD.

There is a possibility of revenues increase in the same of using the proposal

plan, it is about 19.70%.

- Maximize production quantity objective: The quantity proposed

production for each product and total production is described in the

following table:

Table 7: Production quantity according to the proposed plan

Product Proposed

production quantity Product

Proposed

production

quantity

X1 3420000 X7 1652430

X2 830200 X8 780000

X3 526451 X9 2650000

X4 1200300 X10 2000600

X5 4800000 X11 943000

X6 4320222 X12 1542000

total 24663113

Source: data of program outputs

We can observe a positive deviation where the proposed plan was more than

the target quantity.

The deviation equal 3917860 AD, 18,87%, it is the result of the increase in

the production quantities X1,X2,X3,X4,X5,X6,X7,X8,X9,X10,X11,X12 in 2015

according to 2014. It means that the proposed plan is the most efficient.

- Minimize production cost objective: The following table explains the

proposed production plan for each product related to the total costs.

X4 1200300 86.2 103465860

X5 4800000 71.82 344736000

X6 4320222 80.76 348901128.7

X7 1652430 73.2 120957876

X8 780000 63.58 49592400

X9 2650000 108.25 286874407.5

X10 2000600 54.36 108741744

X11 943000 105.2 99203600

X12 1542000 80.25 123585000

total 24663113 - 2146198731

Product Proposed

production quantity Cost price (AD)

Proposed

production costs

X1 3420000 102.972 352164240

X2 830200 81.17 67387334

X3 526451 95.55 50310992.55

X4 1200300 71.55 85881465

X5 4800000 62.49 299952000

X6 4320222 70.26 303538797.7

X7 1652430 63.71 105276315.3

339

Tabl

e 08 :

prod

uctio

n

quan

tity and costs according to proposed plan.

Source: data of program outputs

The production cost in the proposed plan was 1847478044, the deviation

was negative, where the production cost increase above the allowed amount

1544356000 AD, with difference of 303122044 according to the typical

solution table.

3.2.4.3. Technical constraints value analysis.

These constraints divided into the following:

- Active substance constraints: The typical solution table describes [dˉ

(rowi)] as the non used active substance quantities (b1, b2, b3……….b10)

they are the following:

(20254000),(11375050),(29115200),(26358000),(24393600),(339186),

(19785400), (22675000),(1814570),(26126000).

The exploitation of the following quantities:

(12260000), (10752950), (2105800), (3360000), (2160000), (23134),

(351000), (1325000), (150030),(1542000).

- Production capacity constraints: The corresponding values of the

available production capacity presented by the column [d+(rowi)] which is

nil(zero), whereas it is not zero in the column [dˉ(rowi)] .

In other word, there is an available capacity which is not exploited in the

production lines.

The excess production capacity in the production lines:

286222.2 AD in the first production line, 363487.7 AD in the second

production line, 219420 AD in the third production line, 293779.5 AD in the

fourth productive line, 88200 AD in the fifth productive line, 136150.8 in the

sixth productive line and 421405 AD in the seventh production line.

The decision maker must rethink how to exploit the excess in production

capacity in order to improve the Antibiotical (Medea) profit.

- Satisfying (fulfilling) demand constraints: There is a zero (nil)

corresponding values represented by the column [dˉ(rowi)], it means we

have a decrease in the quantity demanded presented by [d+(rowi)], that is to

say no increase in the quantity demanded.

4. Sensitivity Analysis: To analyze the linear programming with multiple

objectives sensitivity, we will mention the following points:

4.1. Effect of objective levels variations:

Through this study we find that antibiotical (Medea) determines four 04

main objectives in the same time:

X8 780000 55.31 43141800

X9 2650000 94.18 249587359.8

X10 2000600 45.1 90218040

X11 943000 97.5 91942500

X12 1542000 70.18 108077200

total 24663113 - 1847478044

340

- Non basic deviation variable: From the typical solution table we can

determine the non basic deviation variables which appeared with nil (zero)

values in the typical solution table: d-1, d

-2, d

-3, d

-4.

There is a possibility to determine the maximum and minimum change in the

following table:

Table09: Maximum and minimum variation for the non basic variables Objective Non basic

variable

Minimum

possible

Maximum

possible

1 dˉ1 -1 0

2 dˉ2 -1 0

3 dˉ3 -1 0

4 dˉ4 -1 0

Source: data of program outputs

From the table 9, we observe: without any impact on the production plan,

the antibiotical (Medea) can change the target values under the allowed

limits for each objective.

- The basic deviation variables: The typical solution table indicates that the

variable d+

1 appeared as a positive deviation presented by 50129730 cans,

that is to say the total production increase by 50129730 cans, or be decrease

any amount without any changes under the typical case in the basic variables

in the typical solution table: d+

2, d+

3, and d+

4.

4.2. The relative exchange between objectives: Although the linear

programming with multiple objectives is not the way to exchange objective,

but there is a possibility to determine the relative values of different

objectives by analyzing and testing the last solution table, because it will

make decision maker able to get the important data and the flexibility to

make the best decisions.

The relative exchange between objectives means the determination of the

impact caused by decreasing negative deviation to the low priority objective

than the negative deviation to the top priority objective.

Through the typical solution table, we find the negative deviation for the

fourth objective (minimize cost of production) represented by 303227800

cans. It is possible to decrease this deviation?

From the fourth objective, we can say that there is possibility to improve

the objective function. through selecting the variable d+

4 as anon basic

variable, where it takes a positive exchange (+1), this selection will decrease

deviation in this objective to a less value.

In the case of d+

4 is non basic variable, its value equal the value of

achieving objective with the top priority (maximizing production quantity

objective (objective 03). It means that for any unit of decrease from the

maximizing production quantity objective. We get a decrease in production

costs by one single unit of money. The exchange between the two objectives

(cost of production and quantity of production) is one single unit of money.

341

4.3. Order priorities variation: Objectives for the decisions maker haven’t

the same importance, with different levels from one objective to another.

Decision maker can’t decide definitely about the objectives priority, where

he rethinks about priority orders variation due to the new changes in this

enterprise environment or the changes in objectives.

According to this big importance, we can explain it by reordering

objectives for this enterprise antibiotical (Medea) as the following:

Table 10: typical solution table after the reordering of objectives

Source: data of program outputs

Through the previous table, we find that typical solution may have a small

change after reordering objectives, that is to say after changing priorities the

typical table will be change to a new production variety. This can make

decisions maker more carefully in ordering objectives.

Study results:

Antibiotical (Medea) can increase its production variety, according to the

same period but with minimizing deviation to the less possible value.

After analyzing objectives levels, linear programming with multiple

objectives presents the distinct results in the antibiotical (Medea). This way

can determine the typical solution which reconcile between objectives and

its efficiency. Antibiotical (Medea) plan to produce large quantities of

production to get more profit in order to obtain more revenues under the

constraint of high costs.

Through sensitivity analysis, we determine different deviation change for

the non basic variables in the mathematical model, this result allows the

decision maker to change the target values (rising/reducing), without

affecting the typical production variety.

Study recommendation:

There is a need to use the linear programming with multiple objective and

the quantitative tools, especially operational research, moreover the

coordination with universities to solve enterprise problems.

Antibiotical (Medea) could follow the proposed plan because it affords

more advantages.

Linear programming with multiple objectives describes the non used

production capacity; there is a need to reconsider it to realize enterprise

objectives.

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