# Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy Sets

Chapter

## Abstract

In this chapter, we show an alternative to estimate the demand for the inventory control by fuzzy set for its calculation under uncertainty, and in this way, it is incorporated the subjective knowledge and administrative experience in its determination. One of the problems in Mexican companies with regard to the definition and calculation of the demand has to be with the maintenance provided by those companies, the updated data, and the system variables considered in the inventories. This action is considered as an extra cost and unnecessary. The most used models for inventory controls in the Mexican industry are Material Requirements Planning (MRP) and Economic Order Quantity (EOQ), models raised and solved in this work considering the demand as a fuzzy number. To solve EOQ with fuzzy demand, it is presented a methodology that will facilitate its use in the industry; for MRP model, it is developed to solve it through a problem of linear programming with flexible restrictions and making calculations of approximation to fuzzy numbers, using α-cuts. Finally, it is presented illustrative examples to understand the way of work with these models and fuzzy sets and also graphics and analysis of the results to know the advantages and disadvantages of using fuzzy numbers.

## Keywords

EOQ fuzzy MRP fuzzy Inventory and fuzzy demand Fuzzy optimization

## 8.1 Introduction

The management of inventories is one of the most basic links in the supply chain, and the optimization forms part of a greater planning process within the chain. While this may not be the only one, it is the most important because the rest of the processes (distribution, limits, production, and materials) mostly depend on the inventory strategy chosen.

If we consider the main components of a system of inventories to be demand pattern, supply pattern, operation restrictions, request policies, and total inventory cost, it is possible to establish that due to these components, inventories are used as buffers between supply processes and demand.

The main differences between these two processes are internal factors such as customer service, scale economies, and easiness of operation, which depend on the decisions taken by the administrators or inventory, production, and sales managers, and external factors such as demand, supply process, and delivery time, which generally are buried under uncertainty.

The easiest way to avoid uncertainty of these processes, which has been for the demand, is to keep more units than those anticipated in inventory (security inventory). For restocking supply, keeping a security inventory may justify minimizing risk. As for the delivery time, that is, the time lapse between issuing an order and receiving the product, also keeping a security inventory may guarantee minimizing uncertainty.

Considering the importance of determining demand in inventory control, since its behavior is little known, it is possible to establish that uncertainty originates in the lack of information or historical data about the behavior of the same, as well as there is difficulty because of this lack to estimate a distribution of possibilities function, which represents it.

As a result, the effective determination of demand is one of the main problems of enterprises, which generally is specified based on the experience and judgments subject by the administration, linguistically described as “the approximate demand is of b units” (Behret and Kahraman 2011).

### 8.1.1 Techniques Known and Used for Controlling Inventories in Mexico

In 2007 the corporation Corporate Resources Management (CRM) conducted a study on the Mexican industry about techniques known and used for controlling the system of inventories. The results were published in Campos (2010), where corporations of various economic sectors and different sizes participated (39% large, 37% medium, and the remaining small and micro), obtaining the following results.

Regarding the best known techniques, the results were Point of Reorder 92.7%, Economic Order (EOQ) 87.2%, Material Requirements Planning (MRP) 78.0%, Maximum and Minimum 75.6%, Periodical Reviews 67.5%, and System Kanban 50%.

In regard to the most used, the study concluded that 90.2% preferred Point of Reorder for the planning of inventories, a very simple technique which is better for materials that show a constant demand, a very unusual characteristic in today’s markets, and its result generates excess and lack of materials; next was the technique of Economic Order (EOQ) with 76.9%, then Maximum and Minimum with 65.9%, and, lastly, Periodical Reviews with 37.5%.

On the other hand, only 58.5% has used the MRP technique, which requires a certain control over the management of planning variables (forecast, delivery time, and lot size), as well as applying a historical statistic of data, to be updating information.

According to this study, the best known and used in Mexico are Point of Reorder and Economic Order, simple determining techniques of low maintenance with characteristics that are rarely found in the real world, which do not guarantee good service to the corporation.

The low use of MRP (Material Requirements Planning) technique continues to draw attention, even though the corporations that use it, generally the variables that supply the said system (such as the case of inventory of security, forecast, or lotting), are not properly calculated by a total lack of knowledge of the various models and techniques of inventories administration; likewise, demand behaviors are not properly analyzed, and it is assumed as known and constant. Inventory levels are also not updated regularly.

Other important aspect discussed in these article is the little or nonexistent frequency with which it provides maintenance and updates to the dates and variables of the inventories system; for this action to be considered, there is an additional cost in Mexican companies, a delicate aspect in the administration of the same. Consequently, the safety inventories and the quantity of materials to be ordered are generally not adequate.

Besides, in this study (Campos 2010), the frequency of use was also determined; and about this concept, two perspectives were analyzed: the first one refers to the percentage of the controlled parts for the technique and second, the percentage of total sum of all inventories.

The first perspective, Point of Reorder, controls the largest volume of materials, while the second, Periodical Reviews, controls the least amount of parts.

As far as the sum of inventories is concerned, the study showed that MRP controls from 10% to 25% of the units, which in total value equals 40–60%, which indicates that MRP is used to control products that most impact costs.

Consequently, the inventories models which will be used in this job are MRP and EOQ, being the most utilized in the Mexican industry. The former is applied to control products that impact costs the most, and the second to control the largest volume of materials. The strategy of using fuzzy sets to determine demand was also proposed, which is one of the main problems that Mexican companies have when managing their systems of inventories, which generally are estimated based on subjective experience and judgments, since it is considered an additional cost to provide maintenance and updates to the data and system variables. So these models will be presented and resolved considering the demand as a fuzzy number, allowing for the incorporation the experience and empirical knowledge that we have to determine estimation and behavior.

### 8.1.2 Data from Inventory Records, Problems with Uncertainty and EOQ and MRP Models

Inventory control is an important field in the supply chain. An adequate regulation of the same may significantly improve company profit. In 1913, the Economic Order Quantity (EOQ) equation was introduced by Harris (1990). Since then, a great number of academic works have been published which describe numerous basic model variables EOQ. For a review, see Brahimi et al. (2006).

The EOQ model tends to have few parameters, and all data entries to the model are assumed to be known, so the amount of the order which minimizes the total cost function may easily be determined. This model also assumes that demand characteristics and delivery time are known with certainty, which renders having a simple and direct mathematical structure to model demand during delivery time. Nevertheless, the lack of culture in place to gather data and maintain history of the same makes operation of this model difficult.

In the same fashion as in traditional literature about inventories, the fuzzy Economical Order Quantity (EOQ) model has provided a genesis in the evolution of fuzzy models of inventories. According to Guiffrida, Alfred L. Kent State (2010), six works have discussed the basic EOQ model, under different parameter schemes. The following table shows this review, where C indicates the model with non-fuzzy parameter/attribute and F denotes the model with fuzzy parameter/attribute.

The model MRP has received much attention, and there is plenty of literature about it (Orlicky 1975). Nevertheless, the classic solution procedures that have been used do not optimize production decisions. Having as the objective to count with optimum solutions to minimize costs, the MRP problem has been studied through mathematical programming, considering the problem with a determinist structure, which allows for having a more manageable model.

MRP has been used primarily when there is a dependency between items’ demands; the use of this model of control of inventories allows for the reduction of their levels and may predict the material requirements for the horizon of planning (Table 8.1).
Table 8.1

Fuzzy models for EOQ

Model

Parameter of entry quantity to order

Order cost

Maintenance cost

Annual demand

Park (1987)

C

C

Vujosevic (1996)

C

C

Lee and Yao (1999)

F

C

Yao et al.(2000)

F

F

Yao and Chiang (2003)

C

F

Wang et al. (2007)

C

C

Source: Alfred L. Guiffrida, Kent State University. Cap 8. Fuzzy, Inventory Models

However, its implementation requires precise and complete information about the demand of materials and production planning. Having this information implies that companies manage control and registry of activities, which is the reason why few Mexican companies use this model.

For the fuzzy model of Material Requirements Planning (MRP), only a few research works have been accomplished. In Lee and Wu (2006), the application of Fuzzy Sets Theory is to the problem of lot size in an MRP system of one stage. In Mula et al. (2008), is developed a new lineal programming model, labeled NNRPD, for short-term production planning in a MRP manufacturing environment with capacity restrictions, multi-product, multi-level, and multi-period. In Vasant (2004), a curve is used as a belonging function for the selection of a mixture of products in a chocolate factory, where the information available is imprecise or fuzzy. In Mula et al. (2007), a model of optimization is formulated which takes into account the uncertainty that exists in the demand in the market, as well as pending orders (delayed), the concept of possibility programming is used. Such focus allows for the modeling of ambiguity of the market’s demand, information of costs, etc., which could be present in the production planning systems. In Pendharkar (1997), the fuzzy dynamic programming is used to solve a problem of inventories with production programming, where the linguistic states, such as “stock must be zero at the end of the planning horizon” and “reduce, as much as possible, production capacity,” are used to describe administratively the fuzzy aspirations for the inventory and the reduction in production capacity, of a possible market pullout.

Then, in both models, it is required to have several parameters for its performance; in this work, demand is considered as the primary parameter to be known. Therefore, it is necessary to have with information to determine it or with a strategy by which it can be estimated.

A strategy to consider for the inventories control systems with this problem is the theory of fuzzy sets, which was introduced by Zadeh (1975) and can be applied to model the demand behavior more realistically and use empirical and subjective knowledge of the administration.

## 8.2 Economic Order Quantity (EOQ) Model with Fuzzy Demand, Without Production or Deficit

The Economic Order Quantity models, as already mentioned, represent in Mexico a considerable percentage among the most widely used, still present in companies, particularly micro, small, and medium. The basic EOQ calculates the reposition of order size for an inventory system; for a determined item, see Soodong and Noble (2000). The average CP(Q) cost depends on the amount of Q orders that are done to cover the demand and the number of units stored. In the following figure, we can observe how CP(Q) is convex function. And in Q* the values of costs for maintaining inventory and for ordering are the same (Fig. 8.1). Fig. 8.1Graphic of the average cost in an EOQ model. (Source: Hillier and Lieberman. Mac Graw Hill. Quantitative methods for administration, Cap. 11)
Assumed in the model are the following: The demand for items is a known and positive constant. The delivery time is zero, meaning that any time frame done is received immediately. And the involved costs are also known. The behavior of the policy of inventories for an Economic Order Quantity model, without production or deficit, is presented in the following figure (Fig. 8.2). Fig. 8.2Model behavior with no production or deficit. (Source: Hillier and Lieberman. Mac Graw Hill. Quantitative methods for administration, Cap. 11)
The diagram presented in Fig. 8.3 was the result of the analysis conducted with the solution of several examples when using different EOQ models and different fuzzy numbers; in the determination of demand, see Flores Brito (2010). The application of proposal stages, for the incorporation of demand, as a fuzzy number in the Economic Order, allows for the solution systematically, bringing the results in a simple way and facilitating its analysis. Fig. 8.3Proposed stages to solve the model of EOQ, with the con demand as a fuzzy number. (Source: Prepared by the authors)
Stages of the solving process:
1. (a)

According to the data that you have in the problem and at the behavior of the storage level, the model to be applied is decided.

2. (b)

The definition of demand as a fuzzy number implies that knowledge is within certain information values. For example, if demand is defined as a triangular fuzzy number, it means its value can be found within three data (minimum, the most likely, and the maximum). The definition of a fuzzy number can be determined, through the membership function, or for its graphic representation. It is necessary to get values for the α-cuts; at this point, it is necessary to determine the “step length,” which means the increase that α − cuts will have, where / can be found between 0 and 1. The value of the increase will imply the precision of the calculations.

3. (c)

It is recommended to do the calculations separately, this is to say, for the equations which determine the optimum policy, first making the calculations that do not include the part of the fuzziness; once these values are obtained, the fuzziness is incorporated. For this point we will have many values for each variable, according to the step length of the α − cuts that have been decided on; this is due to the fact that calculations have been made for each value of the demand related to the α − cuts.

4. (d)

When calculating each variable, a fuzzy number will be obtained, which originated from the incorporation of the demand as a fuzzy number; in this way if the demand has been defined as a triangular fuzzy number (TFN), the results for the other variables will originate a TFN as well. It is recommended to make graphs for each variable, as this allows for results verification.

5. (e)

Once the calculation has been done, using the demand as a fuzzy number (FN), it is necessary to carry out an approximation to the obtained FN; the difference between the calculations and approximations is that the calculations are obtained using the equations and the demand as a FN, while the approximations are obtained using only the values which resulted from the calculations and which define the FN, calculating the α-cuts; that is to say, now each variable is considered as an FN. The aforementioned allows for a definite result for certain values; if the demand is a trapezoidal fuzzy number (TrFN), then they will be obtained for the other variables TrFN. Differences between the calculated values and the obtained approximations can be observed in the graphs.

6. (f)

When we have the graphs where we observe the calculated values and the approximations for each one of the variables, we can get to the end of the process if the differences between one curve and another one are not representatives; otherwise, it is recommended to make an analysis of the deviations’ curves.

Following is an example of the Economic Order model without production and without deficit, considering the unknown demand, which is solved as a diffuse EOQ model, following the previous stages.

Be the following data: d=?, units/year; k = 25 um/order; and C = 6.25 um/unit. Where h is 20% of the purchase cost as per the company’s policy, therefore h = (20) (6.25) = 1.25 um*unit/year. To determine the optimum amount per order that should be used to minimize the costs, establish what is the average cost associated with the optimum order amount, specifying how many orders would the company making per year.

For this case it is considered that the demand is not clearly known, which happens in various real situations. It is admitted that the demand is a triangular fuzzy number (stage 2) in the following manner:
$$\tilde{d}=\left(a,b,c\right)$$
$$\tilde{d}=\left(5800,8500,13700\right)$$
Its membership function is defined as:
$${\mu}_d(x)=\Big\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}x<5800\\ {}\frac{x-5800}{2700},& \mathrm{if}\ 5800\le x\le 8500\\ {}\frac{-x-13700}{5200},& \mathrm{if}\ 8500\le x\le 13700\end{array}\\ {}0,\kern0.5em \mathrm{if}x>13700\end{array}}$$
(8.1)
And can be written for the α − cuts
$${d}_{\propto }=\left[5800+\left(8500-5800\right)\alpha, 13700-\left(13700-8500\right)\alpha \right],\alpha \in \left[0,1\right]$$
(8.2)
Now the calculations for the α − cuts are carried out, with an increment of 0.1, from 0 to 1 (Table 8.2).
Table 8.2

Calculations of the α − cuts

α

dα

0

5800

13700

0.1

6070

13180

0.2

6340

12660

0.3

6610

12140

0.4

6880

11620

0.5

7150

11100

0.6

7420

10580

0.7

7690

10060

0.8

7960

9540

0.9

8230

9020

1

8500

8500

The way to do the calculations consists in solving the part that is not fuzzy and incorporating the fuzzy part after (stage 3); it is important to mention that at the end a new fuzzy number for each variable will be obtained.

For example, to determine the required amount (Q), the area which involves the ordering cost will be solved (k) and the cost to maintain in inventory (h). On the other hand, the fuzzy number d will be calculated, this last calculation will be carried out for the α − cuts, and precision will depend on the person doing the calculations. In this case, the increment of α is 0.1.

For each determined variable (stage 4), the graph from the obtained fuzzy number is presented, considering the demand as TFN. Subsequently an approximation is presented, which means taking into account the three real numbers that define the number and make the ∝ − cuts, as opposed to obtaining the values by calculating with fuzzy number d.

Determining the amount of the order is obtained by incorporating the fuzzy part and the non-fuzzy part.
$$Q=\sqrt{\frac{2k}{h}}\cdot \sqrt{5800+(2700)\alpha, 13700-(5200)\alpha }$$
(8.3)
In Table 8.3, the obtained results are shown for the order size.
Table 8.3

Triangular calculations and approximates of Q

α

Q

$$\tilde{Q}\left[482+(101)\alpha, 740-(157)\alpha \right]$$

0

482

740

482

740

0.1

493

726

492

725

0.2

504

712

502

709

0.3

514

697

512

693

0.4

525

682

522

677

0.5

535

666

532

662

0.6

545

651

543

646

0.7

555

634

553

630

0.8

564

618

563

615

0.9

574

601

573

599

1

583

583

583

583

From Table 8.3, it can be observed, in the first column, the values for α with a step length of 0.1; in columns 2 and 3, the results of the application of the formula Q are presented, considering the demand as a fuzzy number; and columns 4 and 5 show the results of the triangular approximations (using the limits obtained from the calculations of columns 2 and 3); this can be represented as follows.

The fuzzy number is obtained, whose triangular approximation (stage 5) is:
$$\tilde{Q}=\left(482,583,740\right)$$

In the graphs, it is observed that the difference between the Q value obtained from the calculations, using the TFN of the demand, and the value of the triangle approximation is minimum, as shown in the following figure.

In the same way that Q was obtained, the value for the following variables is calculated (continuation from stage 4); next the results from the calculations are presented and the corresponding graphs (Fig. 8.4). Fig. 8.4Graphic representation of Q. (Source: Prepared by the authors with data from Table 8.3)

The length of period (T) and the number of requests (N) are variables which are demand-driven; however, the value of N is obtained, using the value of Q previously calculated.

The obtained results are presented for the N variable in Table 8.4.
Table 8.4

Calculation and triangular approximations of $$\tilde{N}$$

α

$$N=\frac{d_{\alpha }}{Q}$$

$$\tilde{N}=\left[12+(3)\alpha, 19-(4)\alpha \right]$$

0

12

19

12

19

0.1

12

18

12

18

0.2

13

18

13

18

0.3

13

17

13

17

0.4

13

17

13

17

0.5

13

17

13

17

0.6

14

16

14

16

0.7

14

16

14

16

0.8

14

15

14

15

0.9

14

15

14

15

1

15

15

15

15

Table 8.5

Calculations and triangular approximations of T

α

$$T=\sqrt{\frac{2k}{h}}\cdot \frac{1}{\sqrt{d_{\alpha }}}$$

$$\tilde{T}=\left[0.054+(0.014)\alpha, 0.083-(0.015)\alpha \right]$$

0

0.08305

0.05403

0.054

0.083

0.1

0.08118

0.05509

0.055

0.082

0.2

0.07943

0.05621

0.057

0.080

0.3

0.07779

0.05740

0.058

0.079

0.4

0.07625

0.05867

0.060

0.077

0.5

0.07480

0.06003

0.061

0.076

0.6

0.07342

0.06149

0.063

0.074

0.7

0.07212

0.06306

0.064

0.073

0.8

0.07089

0.06475

0.066

0.071

0.9

0.06972

0.06659

0.067

0.070

1

0.06860

0.06860

0.069

0.069

The $$\tilde{N}$$ value has a triangular approximation same as$$\tilde{N}=\left(2,15,19\right)$$. Whereof, the graphic representation is Fig. 8.5.
Table 8.6

Calculations and triangular approximations of CP

α

$$\mathrm{CP}\left(\tilde{Q}\right)=\sqrt{2 hk}\cdot \sqrt{d_{\alpha }}+c{d}_{\alpha }$$

$$\overset{\sim }{\mathrm{CP}}(Q)=\left[\begin{array}{l}36852+(17002)\alpha, \\ {}86550-(32696)\alpha \end{array}\right]$$

0

36852

86550

36852

86550

0.1

38553

83283

38552

83281

0.2

40254

80015

40252

80011

0.3

41955

76746

41953

76741

0.4

43656

73477

43653

73472

0.5

45356

70208

45353

70202

0.6

47056

66938

47053

66932

0.7

48756

63668

48753

63663

0.8

50455

60397

50454

60393

0.9

52155

57126

52154

57124

1

53854

53854

53854

53854 Fig. 8.5Graphic representation of number of requests (N). (Source: Prepared by the authors with data from Table 8.4)

In the same way for T, it was necessary to calculate it without the fuzzy part and do the calculations after, including the fuzzy part. The results from the calculations for α − cuts are represented in the following table, followed by the results of the approximations of T as a fuzzy number.

The results are represented graphically in Fig. 8.5.

Finally, the average cost of the inventory, which also depends on the demand, is defined as follows as TFN.

The graphic representation is shown in the following figure (Fig. 8.6). Fig. 8.6Graphic representation of the length period (N). (Source: Prepared by the authors with data from Table 8.5)

### 8.2.1 Analysis of Results

The comparison between the data obtained and the fuzzy numbers determined by calculating approximations using α − cuts has been presented in the graphs and tables shown. It can be observed that the approximation is acceptable; therefore, it is considered necessary on this context to calculate general expressions for the deviation between the real curves and its approximations which, on the other hand, give rise to more complicated Eqs. (8.14); for this example, the result can be written as follows:
$$Q=\left(482,583,740\right)\hat{\mathrm{C}}\mathrm{P}(Q)=\left(602,729,925\right)$$
(8.4)
$$T=\left(0.054,0.0686,0.083\right)\kern0.5em \overset{\frown }{N}\kern0.5em =\left(12,15,19\right)$$
(8.5)
From the previous results, it can be said that the most probable value for Q is 583 units, while the smallest and greater values for the size of the request are 482 and 749, respectively. For CP (Q), T, and N, a similar description is given, as the calculations depend on the same triangular fuzzy number (TFN) (Fig. 8.7). Fig. 8.7Graphic representation of CP, which depends on Q. (Source: Prepared by the authors with data from Table 8.6)

## 8.3 MRP Model Considering the Demand of a Fuzzy Number

In 1947 George Dantzig developed the simplex algorithm to solve linear programming (LP) problems. This technique is applied to a variety of problems in the fields of industry, health, economy, transportation, etc. For this reason, linear programming is a well-studied area and one of the most used tools by companies.

The application of fuzzy sets in mathematical programming for the most part consists of transforming the classic theories in equivalent fuzzy models (see Kaufmann and Gil 1987). In practical situations, for a typical linear programming problem, it is not reasonable to demand that the objective function or the restrictions be specified in a precise way; in such situations, some type of fuzzy linear programming should be used.

Fuzzy or flexible linear programming (FLP) can be applied in different cases, for example, when the right side of the restrictions is a fuzzy number or when the technological coefficients are fuzzy numbers or when both previous cases are present. This work only focuses on analyzing the first case, due to the fact that the parameter to be determined is the demand and corresponds to the right side of a restriction. Such types of restrictions are called flexible restrictions.

A linear programming problem with flexible restrictions is defined as follows.

Maximize
$$z=\sum \limits_{j=1}^n{c}_j{x}_j$$
(8.6)
Subject to
$$\sum \limits_{j=1}^n{a}_{ij}{x}_j\le {B}_i,\kern0.875em i=1,2,3,\dots, m$$
(8.7)
$${x}_j\ge 0,\kern0.875em j=1,2,3,\dots, n$$
(8.8)
Considering that Bi is a fuzzy number of trapezoidal form (TrFN), with the following membership function:
$${B}_i=\Big\{\begin{array}{cc}1,& x\le {d}_i\\ {}\frac{-x+{d}_i+{p}_i}{p_i},& {d}_i\le x\le {d}_i+{p}_i\\ {}0,& x\ge {d}_i+{p}_i\end{array}}$$
(8.9)
where $$x\in \Re$$.
As can be seen in Fig. 8.8, the graphic form of the fuzzy number is linear and descendent of di to (di + pi)di. Fig. 8.8Fuzzy number for the FLP problem. (Source: Mula et al. (2007))

Once the type of fuzzy number that will be used to represent the parameter under uncertainty has been defined, to solve FLP, it is necessary to calculate the fuzzy set of optimal values; thus, it is necessary to calculate the superior limit (zu) and inferior (zl) for the objective function. The way to calculate these limits is found when solving a problem LP for each z, as follows:

LP problem to obtain zl

Maximize
$$z=\sum \limits_{j=1}^n{c}_j{x}_j$$
(8.10)
subject to
$$\sum \limits_{j=1}^n{a}_{ij}{x}_j\le {d}_i,\kern0.875em i=1,2,3,\dots, m$$
(8.11)
$${x}_j\ge 0,\kern0.875em j=1,2,3,\dots, n$$
(8.12)

LP problem to obtain zu

Maximize
$$z=\sum \limits_{j=1}^n{c}_j{x}_j$$
(8.13)
subject to
$$\sum \limits_{j=1}^n{a}_{ij}{x}_j\le {d}_i+{p}_i,\kern0.875em i=1,2,3,\dots, m$$
(8.14)
$${x}_j\ge 0,\kern0.875em j=1,2,3,\dots, n$$
(8.15)
It can be seen that each new LP considers the limits of the fuzzy number. The fuzzy set of optimal values is represented by G, which is defined as follows:
$$G(x)=\Big\{\begin{array}{cc}1,& {z}_u\le \mathrm{CX}\\ {}\frac{\mathrm{CX}-{z}_l}{z_u-{\mathrm{z}}_l},& {z}_l<\mathrm{CX}<{z}_u\\ {}0,& \mathrm{CX}\le {z}_l\end{array}}$$
(8.16)

Introducing a new variable, λ where λ ∈ [0,1], we have the following LP problem (see Zimmermann 1993).

Maximize
$$\lambda$$
(8.17)
subject to
$$\lambda \le \frac{\mathrm{CX}-{z}_l}{z_u-{z}_l}$$
(8.18)
$$\lambda {p}_i+\sum \limits_{j=1}^n{a}_{ij}{x}_j\le {d}_i+{p}_i,\kern0.875em i=1,2,3,\dots, m$$
(8.19)
$$\lambda {x}_j\ge 0,\kern0.875em j=1,2,3,\dots, n$$
(8.20)
where λ represents the maximum grade of membership, inside the fuzzy set of optimal values (G(x)), value which varies between zl and zu (see Martínez Fonseca 2001).

Reordering the objective function, we obtain the following:

Maximize
$$\lambda$$
(8.21)
subject to
$$\lambda \left({z}_u-{z}_l\right)-\mathrm{CX}\le {z}_l$$
(8.22)
$$\lambda {p}_i+\sum \limits_{j=1}^n{a}_{ij}{x}_j\le {d}_i+{p}_i,\kern0.875em i=1,2,3,\dots, m$$
(8.23)

An MRP model is presented next, considering the flexible restrictions, that is, as a fuzzy linear programming model. Thereafter, a numeric example is presented, which allows for an analysis of the results, considering the demand of a number under uncertainty. To solve it, the material studied in this section will be applied.

The following model was proposed in (Kaufmann and Gil 1987). However, for the purpose of this work, some modifications have been done. As observed, the model handles capacity and demand restrictions; in demand restrictions, the restriction is balance between what we have, what is owed, and what is required.

For capacity restriction, the model includes overtime costs and idle time; however, such restriction may not contemplate such terms, if the company that utilizes it does not cover such costs or if it is not required to analyze them in the system.

Therefore, capacity restriction will be defined only according to the time required to produce the item or items and by the produced quantity in a period, which cannot go over the available capacity in such period.

The model’s formulation is as follows:

Minimize
$$z=\sum \limits_{j=1}^I\sum \limits_{t=1}^T\left({\mathrm{cp}}_i{p}_{it}+{\mathrm{ci}}_i{\mathrm{INVT}}_{it}+{\mathrm{crd}}_i{\mathrm{Rd}}_{it}\right)+\sum \limits_{r=1}^R\sum \limits_{t=1}^T\left({\mathrm{ctoc}}_{rt}{\mathrm{Toc}}_{rt}+{\mathrm{ctex}}_{rt}{\mathrm{Tex}}_{rt}\right)$$
(8.24)
subject to
$${\mathrm{INVT}}_{i,t-1}-{\mathrm{Rd}}_{i,t-1}+{p}_{i,t-T{S}_i}+{\mathrm{RP}}_{i,t}-{\mathrm{INVT}}_{i,t}+{\mathrm{Rd}}_{i,t}={d}_{i,t};\kern0.5em \forall i,\forall t$$
(8.25)
$${\mathrm{Rd}}_{i,T}=0;\kern0.5em \forall i$$
(8.26)
$${p}_{it},{\mathrm{INVT}}_{i,t},{\mathrm{Rd}}_{i,t},{\mathrm{Toc}}_{rt},{\mathrm{Tex}}_{rt}\ge 0;\kern0.5em \forall i,\forall r,\forall t$$
(8.27)

The objective function of this model includes, as first part, the minimization of the sum of the costs, such as costs for producing the produced quantity, inventory costs for material quantity found in inventory, and the costs for missing material for each item.

In the second part of the function, the sum of the costs originated by overtime and idle time is minimized by the resources.

The first restriction is to cover the demand, which is a balance between what comes in, what goes out, and what is withheld of the product (i), during a period of time (t).

The second restriction is necessary for when the company has a limited capacity, as it includes the capacity to produce the resources, in a given case that the company includes overtime policies, are considered into this restriction.

The third restriction for each product (i) allows for the delays to be covered in the last period and, lastly, the non-negative restrictions.

Considering a fuzzy number does exist which represents the uncertainty and as a consequence, the demand’s behavior is as follows.
$${B}_i(x)=\Big\{\begin{array}{cc}1,& \mathrm{if}x\le {d}_i\\ {}\frac{-x+{d}_i+{p}_i}{p_i},& \mathrm{if}{d}_i\le x\le {d}_i+{p}_i\\ {}0,& \mathrm{if}x\ge {d}_i+{p}_i\end{array}}$$
(8.28)

The model is defined as shown below, where we can see that the objective function and the demand restriction are modified; the other restrictions are defined in the same way.

Maximize
$$\lambda$$
(8.29)
Subject to
$$\begin{array}{l}\lambda \left({z}_u-{z}_i\right)-\sum \limits_{j=1}^I\sum \limits_{t=1}^T\left({\mathrm{cp}}_i{p}_{it}+{\mathrm{ci}}_i{\mathrm{INVT}}_{it}+{\mathrm{crd}}_i{\mathrm{Rd}}_{it}\right)\\ {}\kern4.75em +\sum \limits_{r=1}^R\sum \limits_{t=1}^T\left({\mathrm{ctoc}}_{rt}{\mathrm{Toc}}_{rt}+{\mathrm{ctex}}_{rt}{\mathrm{Tex}}_{rt}\right)\le -{z}_i\end{array}}$$
(8.30)
$$\begin{array}{l}\lambda \left({d}_{(it)u}-{d}_{(it)l}\right)+{\mathrm{INVT}}_{i\left(t-1\right)}-{\mathrm{Rd}}_{i\left(t-1\right)}+{p}_{it}-{\mathrm{TS}}_i+{\mathrm{RP}}_{it}-{\mathrm{INVT}}_{it}+{\mathrm{Rd}}_{it}\\ {}\kern3.25em \le {d}_{it};\kern0.5em \forall i,\forall t\end{array}}$$
(8.31)
$$\sum \limits_{i=1}^I{\mathrm{AR}}_{ir}{P}_{it}+{\mathrm{Toc}}_{rt}-{\mathrm{Tex}}_{rt}={\mathrm{CAP}}_{rt},\kern0.5em \forall r,\forall t$$
(8.32)
$${\mathrm{Rd}}_{iT}=0;\kern0.5em \forall i$$
(8.33)
$${p}_{it},{\mathrm{INVT}}_{it},{\mathrm{Rd}}_{it},{\mathrm{Toc}}_{rt},{\mathrm{Tex}}_{rt}\ge 0;\kern0.875em \forall i,\forall r,\forall t$$
(8.34)

Following is an example of the MRP model, considering the unknown demand and without capacity restrictions; the problem is solved as a fuzzy MRP model. The modifications that were done in the objective function and the restrictions that cover the demand are explained, and finally the obtained result is presented, considering the demand as a fuzzy number. Due to the extension of the modified code, such code is not presented. The software utilized to solve it was LINGO 6.0 (LINGO/PC s.f.).

The problem considers 11 items, of which only one is the final product; to build such item, it is necessary to mix and assemble with the scrap items. In the following figure, the list of materials is introduced (Fig. 8.9). Fig. 8.9Materials to produce a table. (Source: Mula et al. (2007))
The demand for the final product for the following four periods is shown in Table 8.7.
Table 8.7

Demand for the final product for the four periods

Period

Final product

1

100

2

160

3

160

4

240

Source: Own elaboration

The data presented in this problem is estimated data. For example, production capacity restrictions are not considered; what is considered is to cover the demand and to fulfill product production due to other subproducts.

In Table 8.8, costs of production, storage, and scraps for each product are presented, as well as the initial inventory. The example does not consider material delivery for scheduled receiving.
Table 8.8

Costs and initial inventory of the products

Product

Cost to produce $Cost to store$

Cost for scraps $Initial inventory (units) 1 100 4 2 0 2 40 5 8 0 3 30 5 2 0 4 2 0.3 5 250 5 10 5 5 10 6 10 3 7 10 7 120 6.48 58 15 8 130 6.48 58 15 9 50 2.22 2 5 10 70 2.22 0 10 11 100 2.22 25 5 With this data, the model can be formulated to solve it with LINGO. The number of periods modeled are T = 1, 2, 3, 4; the number of products is 11, for which I = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. The objective function seeks to minimize the sum of the costs, for which this is defined as follows: (cost to produce the product i times the production of i) + (cost for inventory of i times the inventory of i) + (cost for scraps of i times scraps of i) for everything i and for everything t. The variables are defined for all the products and for all the periods, for example, p11 means the production of the product i = 1 in the period t = 1, variable inv34 means the inventory of the product i = 3 in the period t = 4, and so on. As observed in the previous model, restrictions were added which allow for the determination of the demand of the products, which depend of the final product. For each product i, four restrictions were added to cover the demand, of t = 1...4. The characteristics of the model are that there are a total of 131 variables and 57 restrictions. The model was solved in 50 iterations, resulting in a value in the objective function of$ 951,824.00 (Mexican pesos).

### 8.3.1 Demand as a Fuzzy Number

The use of the demand as a fuzzy number means that modifications need to be made, so it is necessary to add a new variable and to modify the restrictions of the demand. For this, the model for each one of the limits needs to be solved.

Obtain the differences and solve the model again. First, it is necessary to define the demand as a fuzzy number; in such case, it is defined as a trapezoidal fuzzy number. Then the objective function’s limits are calculated, to alter define a model which includes both limits in the restrictions.

For the period t = 1, the demand behaves as shown in the following figure (Fig. 8.10). Fig. 8.10Behavior of the demand for period t = 1. (Source: Elaborated by the authors)
With membership function, where x ∈ ℜ
$${B}_i(x)=\Big\{\begin{array}{cc}1,& \mathrm{if}x\le 100\\ {}\frac{-x+250}{150},& \mathrm{if}100\le x\le 250\\ {}0,& \mathrm{if}x\ge 250\end{array}}$$
(8.35)
For the following periods, the data are shown in Table 8.9.
Table 8.9

Data per period

Period

di

di + pi

2

160

280

3

160

280

4

240

360

The following modification is made on one of the restrictions.
$$i=1$$
(8.36)
$${\mathrm{inv}}_{10}=0,\kern0.5em {\mathrm{rd}}_{10}=0,\kern0.5em {\mathrm{inv}}_{14}=0$$
(8.37)
$${X}^{f\ast }(150)+\left({\mathrm{inv}}_{10}-{\mathrm{rd}}_{10}+{p}_{11}-{\mathrm{inv}}_{11}\right)\le {d}_{11}$$
(8.38)
$${X}^{f\ast }(120)+\left({\mathrm{inv}}_{11}-{\mathrm{rd}}_{11}+{p}_{12}-{\mathrm{inv}}_{12}\right)\le {d}_{12}$$
(8.39)
$${X}^{f\ast }(120)+\left({\mathrm{inv}}_{12}-{\mathrm{rd}}_{12}+{p}_{13}-{\mathrm{inv}}_{13}\right)\le {d}_{13}$$
(8.40)
$${X}^{f\ast }(120)+\left({\mathrm{inv}}_{13}-{\mathrm{rd}}_{13}+{p}_{14}-{\mathrm{inv}}_{14}\right)\le {d}_{14}$$
(8.41)

As observed in the previous restrictions, the value found in parenthesis is the result of the difference between the limits of the fuzzy number. Letter x is equal to the Greek letter λ (new variable). The value of the right is the superior limit.

For the following restrictions, the same procedure is carried out. It is necessary to consider that the demand of the other products depends on the demand of product 1; thus, in the model, the value in parenthesis is defined by the inferior demand and the demand with the growth.

The objective function changes to a function of maximizing the new variable, and the previous changes, using the zu and zl.

The modified model has the following characteristics, 132 variables and 59 restrictions, and was solved in 54 iterations. The value of the objective function is \$ 940,077.54 (Mexican pesos).

By comparing the results, it can be observed that a better solution was obtained, by making the restrictions more flexible.

## 8.4 Conclusions

From the analysis of this work, it can be concluded that in the Mexican companies, the use of models for inventory administration is a problem, because they generally don’t have databases to administer such models and they see this as an additional expense for the company. They use the experience and empirical knowledge of the ones in charge of inventory administration to determine any necessary parameter for decision-making, specifically to estimate the demand, and generally the security inventories and the amount of requested materials are excessive.

As established in this study, a simple tool, of easy application and not so costly, which allows for the solution of this problem of Mexican companies, is the fuzzy sets. With such tool it is possible to determine the demand for inventory systems, even when there is no statistical information, but incorporating the experience and knowledge of the administration as empirical information.

Since it is very important to determine the demand and its behavior in the inventory control, using fuzzy sets to estimate it proves to be a very promising alternative for the Mexican industry.

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## Authors and Affiliations

• Ricardo Aceves-García
• 1
Email author
• Zaida E. Alarcón-Bernal
• 2
1. 1.Department of Systems, Faculty of EngineeringNational Autonomous University of MexicoMexico CityMexico
2. 2.Department of Biomedical Systems, Faculty of EngineeringNational Autonomous University of MexicoMexico CityMexico