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Volume 13, Issue 1, 2020, Pages 727 - 733

Optimizing Production Mix Involving Linear Programming with Fuzzy Resources and Fuzzy Constraints

Authors
B.O. Onasanya1, 4, Y. Feng2, 6, *, ORCID, Z. Wang3, O.V. Samakin4, ORCID, S. Wu5, X. Liu3
1Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China
2Chongqing Engineering Research Center of Internet of Things and Intelligent Control Technology, Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China
3School of Three Gorges Artificial Intelligence, Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China
4University of Ibadan, Ibadan, Oyo State, 200284, Nigeria
5School of Three Gorges Big Data, Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China
6School of Computer Science and Engineering, Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China
*Corresponding author. Email: yumingfeng25928@163.com
Corresponding Author
Y. Feng
Received 26 February 2020, Accepted 17 May 2020, Available Online 11 June 2020.
DOI
10.2991/ijcis.d.200519.002How to use a DOI?
Keywords
Fuzzy resources; Fuzzy constraints; Fuzzy objective function; Fuzzy optimization
Abstract

In this paper, Fuzzy Linear Programming (FLP) was used to model the production processes at a university-based bakery for optimal decisions in the daily productions of the bakery. Using the production data of five products from the bakery, a fuzzy linear programme was developed to help make decisions when fuzzy resources were involved. As usual, classical linear programme gave only one feasible solution. However, while it was established that solving the linear programme from the production mix when fuzzy constraints were introduced (Verdegay's model) gave a more robust and alternative set of decisions than the classical model, it was equally established that a better result could be obtained when all the constraints and the objective function were fuzzy (Werners' Model).

Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Linear Programming (LP) has its origin during the second world war (1939-1945) when a balance between man and material (resources) had to be maintained. Hence, LP was developed in order to plan expenditures and returns to reduce costs of the army and increase losses imposed on the enemy. During that time, Marshall K. Wood worked on the allocation of the resources for the United States and methods were developed to allocate resources in such a way that will minimize or maximize the desired objective of the problems as the case may require. As time went on, economists formulated classical economic problems, transportation problems and assignment problems. Then the simplex method of solving linear programmes was introduced by George B. Dantzig [1] which, for the first time, efficiently tackled the LP problem in most cases. Many industries found the use of LPvaluable, hence, accelerating its development.

LP is the mathematical technique which involves the use of limited resources in an optimal manner. LP problems requiring such judicious use of resources are called optimization problems. In such classical optimization problems, a solution which is feasible is the goal.

It is also important to note that optimization has gone beyond allocating resources. Various optimization methods have also been developed to solve problems that occur in other physical sciences. Some of such can be found in [6,5,12,15,17,18] just to mention a few.

Most of the traditional tools for formal modeling, reasoning and computing are precise in nature, that is, they are the yes-or-no type and not more-or-less type. Unfortunately, many real life problems encountered from time to time suggest the need for more robust mathematical tools. Take, for an instance, if an investor hopes to make profit “around” an amount using “not more” than “around certain quantity” of raw materials, the usual LP method cannot model the situation properly. This is as a result of the vague language “around” which introduces uncertainty into the problem. Also, to be able to develop model which classifies goods and services as being “expensive” or “affordable” and the like, it is necessary for the existence of a mathematical modeling of vague knowledge to capture elements that cannot be precisely said to be in a set or not but are in between.

Mathematical study of vagueness of this sort (fuzzy sets) began with a professor of electrical engineering, Lofti A. Zadeh, in the University of California at Berkeley, when he published the first paper Fuzzy Sets in 1965 [20]. In it, he noted that “The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual frame-work which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables.”

Fuzzy set theory provides a mathematical framework in which vague conceptual phenomena can be rigorously studied. It can also be considered as a modeling language well suited for situations in which fuzzy linguistic variables such as “much,” “very much,” “high,” “very high” and the like occur. Zadeh and others continued to develop the fuzzy logic at that time. The idea of fuzzy sets and fuzzy logic, though were not well accepted then, have now been extended to many areas of discipline and studies.

Extension of fuzziness to LP has led to Fuzzy Linear Programming (FLP) in which some of the parameters constitute fuzzy set(s). One case is to have a crisp objective but fuzzy constraints. Another case is to have a fuzzy objective but crisp constraints. However, it is also possible to have a FLP in which both the constraints and the objective are fuzzy. Precisely, in this paper, an approach used by [4], crisp objective but fuzzy constraints, is improved on.

2. PRELIMINARIES

Definition 1.

[20] A fuzzy set A in X is a set of ordered pairs A={(x,μA(x)):x∈X}, where μA(x) is the grade of membership of x∈A and μA:X→[0,1].

Definition 2.

[20] The support of a fuzzy set A, S(A)={x∈A:μA(x)>0}.

Definition 3.

([20] A fuzzy set A is empty if and only if μA(x)=0,∀x∈X.

Definition 4.

([20] Two fuzzy sets A and B are equal if and only if μA(x)=μB(x),∀x∈X.

Definition 5.

[20] A fuzzy set A is contained in a fuzzy set B, written as A⊆B, if and only if μA(x)⩽μB(x),∀x∈X.

Definition 6.

[20] The intersection of two fuzzy sets A and B is denoted by A∩B and is defined as the largest fuzzy set contained in both A and B. The membership function of A∩B is given by μA(x)∧μB(x)=min{μA(x),μB(x),∀x∈X}.

Definition 7.

[20] The union of A and B, denoted by A∪B, is defined as the smallest fuzzy set containing both A and B. The membership function of A∪B is given by μA(x)∨μB(x)=max{μA(x),μB(x),∀x∈X}.

Definition 8.

[20] If A is a fuzzy subset of X, then an α-level set of A is a nonfuzzy set Aα which comprises all elements of X whose grade of membership is greater than or equal to α. It is denoted by Aα={x∈X:μA(x)⩾α,∀x∈X}.

3. LINEAR PROGRAMMING

A LP is an optimization problem in which the objective function and constraints are given as mathematical functions and functional relationships. Linear programmes are of the form

Optimizez=f(x1x2x3…,xn)Subjectto
g1(x1,x2,x3,…,xn)  g2(x1,x2,x3,…,xn)  .  .  .  gm(x1,x2,x3,…,xn)⩽=⩾b1b2...bn   (1)
where b1,b2,b3,⋯,bm are real numbers and x1,x2,x3,⋯,xn⩾0.

The programme attempts to find the best solution, called an optimal solution, for a problem under consideration. It seeks to find values of the decision variables that optimize (that is, maximize or minimize) an objective function among a set of values.

Solving LP by simplex method according to [1] involves using matrix procedures to solve standard LP of the form

Optimize   z=CTXSubject to   AX=B    X⩾0(2)
where
C=c1c2...cn ,X=x1x2...xn ,B=b1b2...bn ⩾0(3)
and
A=a11a12⋯a1na21a22⋯a2n⋮⋮ ⋮am1am2⋯amn (4)

In what follows, X0 denotes the matrix of slack variables, C0 designates the cost matrix associated with X0. For minimization programs and maximization programmes, the simplex method utilizes Tables 1 and 2 respectively.

XT
CT
X0C0 A B
CT−C0TA −C0TB
Table 1

Simplex table for minimization programmes.

XT
CT
X0C0 A B
C0TA−CT C0TB
Table 2

Simplex table for maximization programmes.

4. METHODOLOGY: FLP

In FLP, the fuzziness of available resources is characterized by the membership function over the tolerance range. Then, the membership function of the optimal solutions is constructed by the convolution of the membership function of the constraints. The general model of linear programmes with fuzzy resources is

Maximize   z=CTXSubject to   (Ax)i⩽b˜i    i=1,2,⋯,m    xi∈X    xi⩾0(5)
where b˜i∈[bi,bi+pi] are the fuzzy resources and z is the objective function. (Ax)i⩽b˜i is equivalent to (Ax)i⩽(bi+θpi), where θ is in [0, 1], given that the tolerance pi and the actual resources bi for each fuzzy constraint are known.

Verdegay's Approach—A Nonsymmetric Model: Verdegay in [19] considered that the membership functions of the fuzzy constraints in (5) are

μi(x)=1,if (Ax)i⩽bi1−(Axi)−bipi,if bi<(Ax)i<bi+pi0,if (Ax)i⩾bi+pi(6)

Furthermore, if these μis are continuous and monotonic functions, and trade-off between them are allowed, then (5) is equivalent to

Maximise   z=CTXSubject to   x∈Xα(7)
where Xα={x|μi(x)⩾α, x⩾0} is the α-cut of X, for each α∈[0,1]. The α-cut concept is based on the works of [13] and [11].

In the membership functions, the degree of satisfaction of the constraints is depending on whether (Ax)i⩽bi (ith constraint is absolutely satisfied when μi(x)=1) or (Ax)i⩾bi+pi, where pi is the maximum tolerance from bi (in which case the degree of satisfaction approaches 0 as b˜i approaches bi+pi). Thus, (Ax)i∈(bi,bi+pi) means that the membership functions are monotonically increasing. Hence, the more resources consumed, the less satisfaction the decision-maker feels.

Now, the membership function of (6) is substituted into (7) to get a parametric programme.

Maximize   z=CTXSubject to   (Ax)i⩽bi+(1−α)pi    x∈X⩾0    α∈[0,1](8)
where α=1−θ. Note that for each α, there is an optimal solution, so the solution with α grade of membership is actually fuzzy.

But, further than having a FLP with fuzzy constraints, the objective function can altogether be made fuzzy and the results of these two can be compared.

Werners' Approach—A Nonsymmetric Model [16]: Given the linear programme with fuzzy resources

Maximize   z=CTXSubject to   (Ax)i⩽b˜i    i=1,2,⋯,m    xi∈X    xi⩾0(9)
where b˜i, are in [bi,bi+pi]. Assume that tolerance pi for each fuzzy constraint is known, then, (Ax)i⩽b˜i is equivalent to (Ax)i⩽(bi+θpi), where θ is in [0,1].

Werners proposed that the objective function of Equation (9) should be fuzzy because of fuzzy total resources or fuzzy inequality constraints. Let it be assumed that the tolerances pis for the fuzzy resources are available and given. Then solving Equation (9), Werners first defined z0 and z1 as follows:

z0=maximize   cxsubject to   (Ax)i⩽bi    x⩾0(10)
z1=maximize   cxsubject to   (Ax)i⩽bi+pi    x⩾0(11)

Thus, a continuously nondecreasing linear membership function can be constructed for the objective function by use of z0 and z1. Since the optimal solution will be in between z0 and z1, the satisfaction of the optimal solution will increase when its value increases. The membership function μ0(x) of the objective function is

μ0(x)=1,if cx<z11−z1−cxz1−z0,if z0⩽cx⩽z10,if cx>z0(12)

With the above membership function, the max-min operator can be used to obtain optimal decision. Then, Equation (9) can be solved by solving

maximize   αsubject to   μ0(x)⩾α   μi(x)⩾α   α∈[0,1]   x⩾0(13)

5. RESULTS AND DISCUSSION

The data used in this study was collected from an institution-based bakery. The production data of five different kinds of bread stated below was used for the purpose of this project. The linear programme was solved, using the Simplex method to get the crisp optimal solution.

Furthermore, fuzzy elements were introduced to the resources and the new Fuzzy linear programme was solved using the Verdegay's Approach—A Nonsymmetric Model, where only constraints were fuzzy. Then, the Werners' Approach—A Nonsymmetric Model, where both the objective function and the constraints were fuzzy was also applied to the data. The results of the Classical Linear Programme and these Fuzzy linear Programmes were compared.

Products Considered: Let UC represent Unit of Currency

  1. 70UC bread

  2. 100UC bread

  3. Chocolate bread

  4. Coconut bread

  5. White bread

Note also, f1(x), f2(x), f3(x) and f4(x) will denote the inputs flour, sugar, salt and butter respectively. P is the profit function and P∗ the optimal profit.

From the data collected, the bakery made a profit of 24.88UC on a unit of 70UC bread, 35.18UC on a unit of 100UC bread, 119.39UC on a unit of chocolate bread, 126.84UC on a unit of coconut bread and 90.84UC on a unit of white bread. The bakery used 200kg of flour with tolerance level of 50kg, 16kg of sugar, 3.85kg of salt with tolerance level of 0.8kg, and 1kg of butter per production time. The data used are given in Table 3, where all inputs are multiplied by 10−3.

Bread Kind Flour Sugar Salt Butter Unit Profit
70UC 126 7.5 2.0 0.0 24.88
100UC 174 10.0 2.8 0.0 35.18
Chocolate 420 59.0 5.5 4.2 119.39
Coconut 420 25.0 6.7 4.2 126.84
White 420 25.0 6.7 0.0 90.84
Available resources ⩽200 ⩽16 ⩽3.85 ⩽1 Maximize
Table 3

The input data for the Bakery problem.

5.1. Obtaining Crisp Solution

The associated linear programme is

MaximizeP(x1,x2,x3,x4,x5)=24.88x1+35.18x2 +119.39x3+126.84x4+90.84x5Subject tof1(x)=(126x1+174x2+420x3 +420x4+420x5)10−3⩽200 f2(x)=(7.5x1+10x2+59x3+25x4 +25x5)10−3⩽16 f3(x)=(2x1+2.8x2+5.5x3+6.7x4 +6.7x5)10−3⩽3.85 f4(x)=(4.2x3+4.2x4)10−3⩽1 (14)
where x1,x2,x3,x4,x5⩾0, x1 is the 70UC bread, x2 is the 100UC bread, x3 is the chocolate bread, x4 is the coconut bread and x5 is the white bread.

From the simplex table we have the crisp optimal solution

x∗=(0,0,0,500021,500021) =(0,0,0,238.095,238.095)
and
P∗=3628007UC=51,828.57UC

The actual resources used are 200kg,11.905kg,3.191kg and 1kg of flour, sugar, salt and butter respectively.

5.2. Obtaining Fuzzy Solution When Only Some Constraints are Fuzzy

Given that the flour used has a tolerance of 50kg and the salt has a tolerance of 0.8kg, the following are the membership functions for flour and salt, using the Verdegay's Approach—A Nonsymmetric Model. Figures 1 and 2 for flour,

μ1(x)=1,if f1(x)⩽2001−f1(x)−20050,if 200<f1(x)<2500,if f1(x)⩾250(15)

Figure 1

The membership function of flour constraint.

Figure 2

The membership function of salt constraint.

For salt,

μ3(x)=1,if f3(x)⩽3.851−f3(x)−3.850.8,if 3.85<f3(x)<4.650,if f3(x)⩾4.65(16)

The fuzzy linear programme of (14) is

MaximizeP(x1,x2,x3,x4,x5)=24.88x1+35.18x2 +119.39x3+126.84x4+90.84x5Subject tof1(x)=(126x1+174x2+420x3+420x4 +420x5)10−3⩽200+50(1−α) f2(x)=(7.5x1+10x2+59x3+25x4 +25x5)10−3⩽16 f3(x)=(2x1+2.8x2+5.5x3+6.7x4 +6.7x5)10−3⩽3.85+0.8(1−α) f4(x)=(4.2x3+4.2x4)10−3⩽1(17)
where x1,x2,x3,x4,x5⩾0 and α∈[0,1]. Setting θ=1−α (θ gives the level set for each of the membership function), the following fuzzy programming problem is obtained
MaximizeP(x1,x2,x3,x4,x5)=24.88x1+35.18x2 +119.39x3+126.84x4+90.84x5Subject tof1(x)=(126x1+174x2+420x3+420x4 +420x5)10−3⩽200+50θ f2(x)=(7.5x1+10x2+59x3+25x4 +25x5)10−3⩽16 f3(x)=(2x1+2.8x2+5.5x3+6.7x4 +6.7x5)10−3⩽3.85+0.8θ f4(x)=(4.2x3+4.2x4)10−3⩽1(18)
where x1,x2,x3,x4,x5⩾0 and θ∈[0,1] is a parameter of the level set. Using the parametric technique and the simplex method, the optimal solution is then
x∗=(0,0,0,500021,500021+250021θ) =(0,0,0,238.095,238.095+119.048θ)
and
P∗=3628007+757007θ=51828.571+10814.286θ

The solutions of the FLP (18) is given in the Table 4 below.

θ P∗ f1 f2 f3 f4
0.00 51828.52 200.00 11.91 3.19 1.00
0.10 52910.00 205.00 12.20 3.27 1.00
0.20 53991.43 210.00 12.50 3.35 1.00
0.30 55072.66 215.00 12.80 3.43 1.00
0.40 56154.29 220.00 13.06 3.51 1.00
0.50 57235.71 225.00 13.39 3.59 1.00
0.60 58317.14 230.00 13.69 3.67 1.00
0.70 59398.57 235.00 13.99 3.75 1.00
0.80 60480.00 240.00 14.29 3.83 1.00
0.90 61561.43 245.00 14.58 3.91 1.00
1.00 62642.86 250.00 14.88 3.99 1.00
Table 4

The solutions of the fuzzy liner programing (FLP) (18).

5.3. Obtaining Fuzzy Solution When Both the Objective Function and All Constraints are Fuzzy

From the crisp LP problem in (14), it can be seen that 16kg of sugar and 3.85kg of salt will have too many ideal resources. So, given that 12kg of sugar and 3.45kg of salt are used with tolerances 50kg, 4kg, 0.4kg and 0.2kg for flour, sugar, salt and butter respectively. Then, the problem becomes

MaximizeP(x1,x2,x3,x4,x5)=24.88x1+35.18x2 +119.39x3+126.84x4+90.84x5Subject tof1(x)=(126x1+174x2+420x3+420x4 +420x5)10−3⩽200~ f2(x)=(7.5x1+10x2+59x3+25x4 +25x5)10−3⩽12~ f3(x)=(2x1+2.8x2+5.5x3+46.7x4 +6.7x5)10−3⩽3.45~ f4(x)=(4.2x3+4.2x4)10−3⩽1~(19)
where x1,x2,x3,x4,x5⩾0 and the membership function μi(x) for the ith fuzzy constraints are
μ1(x)=1,if f1(x)⩽2001−f1(x)−20050,if 200<f1(x)<2500,if f1(x)⩾250(20)
μ2(x)=1,if f2(x)⩽121−f2(x)−124,if 12<f2(x)<160,if f2(x)⩾16(21)
μ3(x)=1,if f3(x)⩽3.451−f3(x)−3.450.4,if 3.45<f3(x)<3.850,if f3(x)⩾3.85(22)
μ4(x)=1,if f4(x)⩽11−f4(x)−10.2,if 1<f4(x)<1.20,if f4(x)⩾1.2(23)

Setting θ=1−α, θ∈[0,1] is a parameter. Using the parametric technique and the simplex method, we get,

x∗=(0,0,0,500021+100021θ,500021+150021θ)
and
P∗=3628007+877007θ

Then, solving Equation (11),

  • P0=P∗(θ=0)=51828.57

  • P1=P∗(θ=1)=64357.14

where θ is the parameter in the new parametric programming problem. The symmetric LP problem can then be formulated as follows:
Maximize   αSubject to   μ0(x)⩾α    μi(x)⩾α,i=1,2,3,4    α∈[0,1](24)
where x⩾0 and the membership function μ0 of the fuzzy objective μ0(x) is defined as
1,if P⩾64357.141−64357.14−P64357.14−51828.57,if 51828.57<P<64357.140,if P⩽51828.57(25)

The problem is actually equivalent to

MinimizeθSubject to24.88x1+35.18x2+119.39x3+126.84x4 +90.84x5⩾64357.14−12528.57 f1(x)=(126x1+174x2+420x3+420x4 +420x5)10−3⩽200+50θ f2(x)=(7.5x1+10x2+59x3+25x4 +25x5)10−3⩽12+4θ f3(x)=(2x1+2.8x2+5.5x3+6.7x4 +6.7x5)10−3⩽3.45+0.4θ f4(x)=(4.2x3+4.2x4)10−3⩽1+0.2θ(26)
where x1,x2,x3,x4,x5⩾0 and θ=1−α∈[0,1]. The solution is
x∗=(0,0,0,238.095+47.62θ,238.095+71.43θ).

6. COMPARISON OF RESULTS

As can be seen from Section 5.1, using the classical LP produced one feasible point

x∗=(0,0,0,500021,500021+500021θ) =(0,0,0,238.095,238.095+119.048θ)
with the optimal profit
P∗=3628007+757007θ =51828.571+10814.286θ

The result of Verdegay's model can be seen in Table 4, giving various optimal points inclusive of the one obtained by classical LP. More importantly, the result of Werner's model can be seen in Table 5. It also gave as many optimal solutions as the Verdegay's model but, with as much resources as used under Verdegay's model, it gives greater profit margin.

θ P∗ f1 f2 f3 f4
0.00 51828.52 200.00 11.91 3.19 1.00
0.10 53081.40 205.00 12.20 3.27 1.02
0.20 54334.28 210.00 12.50 3.35 1.04
0.30 55587.17 215.00 12.80 3.43 1.06
0.40 56840.05 220.00 13.06 3.51 1.08
0.50 58092.93 225.00 13.39 3.59 1.10
0.60 59345.81 230.00 13.69 3.67 1.12
0.70 60598.70 235.00 13.99 3.75 1.14
0.80 61851.58 240.00 14.29 3.83 1.16
0.90 63104.46 245.00 14.58 3.91 1.18
1.00 64357.34 250.00 14.88 3.99 1.20
Table 5

The solutions of the fuzzy linear programing (FLP) (26).

7. CONCLUSIONS

The study has revealed that, using FLP gives a more robust information on making decision. More importantly, it has also shown that, with almost the same resources, modeling production mix with both the objective function and the constraints being fuzzy yields a better result and can enhance better performance and production output more than crisp modeling and modeling with only the resources being fuzzy.

CONFLICT OF INTEREST

The authors declare that there are no conflicts of interest.

AUTHORS' CONTRIBUTIONS

All the authors have contributed fairly well.

ACKNOWLEDGMENTS

This work is supported by Foundation of Chongqing Municipal Key Laboratory of Institutions of Higher Education ([2017]3), Foundation of Chongqing Development and Reform Commission (2017[1007]), and Foundation of Chongqing Three Gorges University.

REFERENCES

1.G.B. Dantzig, Maximization of a Linear Function of Variables Subject to Linear Inequalities, Wiley and Chapman-Hall, New York, NY, USA, 1951. London, England
8.L.V. Kantorovich, A new method of solving some classes of extremal problems, Proc. USSR Acad. Sci., Vol. 28, 1940, pp. 211-214.
9.D. Mamoni, Cardinality of fuzzy sets: an overview, Int. J. Energy Inf. Commun., Vol. 4, 2013, pp. 15-22. https://pdfs.semanticscholar.org/6e10/e504e0761f60bc957ebfddae592d6efbef30.pdf
10.M. Milan, Computations Over Fuzzy Quantities, CRC-Press, Boca Raton, FL, USA, 1994.
14.J.L. Verdegay, Fuzzy Information and Decision Processes, North-Holland Publishing, Amsterdam, 1982.
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International Journal of Computational Intelligence Systems
Volume-Issue
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727 - 733
Publication Date
2020/06/11
ISSN (Online)
1875-6883
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DOI
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risenwbib
TY  - JOUR
AU  - B.O. Onasanya
AU  - Y. Feng
AU  - Z. Wang
AU  - O.V. Samakin
AU  - S. Wu
AU  - X. Liu
PY  - 2020
DA  - 2020/06/11
TI  - Optimizing Production Mix Involving Linear Programming with Fuzzy Resources and Fuzzy Constraints
JO  - International Journal of Computational Intelligence Systems
SP  - 727
EP  - 733
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200519.002
DO  - 10.2991/ijcis.d.200519.002
ID  - Onasanya2020
ER  -