A single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints

doi:10.1016/j.ejor.2007.04.009

European Journal of Operational Research

Volume 188, Issue 1, 1 July 2008, Pages 121-139

https://doi.org/10.1016/j.ejor.2007.04.009 Get rights and content

Abstract

This paper develops a mathematical model for a single period multi-product manufacturing system of stochastically imperfect items with continuous stochastic demand under budget and shortage constraints. After calculating expected profit in general form in terms of density functions of the demand and percentage of imperfectness, particular expressions for those density functions are considered. Here the constraints are of three types: (i) both are stochastic, (ii) one stochastic and other one imprecise (fuzzy) and (iii) both imprecise. The stochastic constraints have been represented by chance constraints and fuzzy constraints in the form of possibility/necessity constraints. Stochastic and fuzzy constraints are transformed to equivalent deterministic ones using ‘here and now’ approach and fuzzy relations respectively. The deterministic problems are solved using a non-linear optimization technique-Generalized Reduced Gradient Method. The model is illustrated through numerical examples. Sensitivity analyses on profit functions due to different permitted ‘aspiration’ and ‘confidence’ levels are presented.

Introduction

In most of the classical economic order quantity (EOQ) inventory models developed since 1915, the demand is considered to be deterministic - constant or time-dependent or stock dependent. In some realistic situations like newsboy problem etc., demand is uncertain in stochastic sense.

Hadley and Whitin [20] first extended the classical EOQ inventory model to the stochastic model. Traditional stochastic models have been discussed primarily with the demand and lead time uncertainties [cf. Nahmias [37] and Silver [42]]. Contini [7] developed an algorithm for stochastic goal programming in which the random variables are normally distributed with known means and variances. In this model the stochastic problem has been transformed into an equivalent deterministic quadratic programming problem, where the objective functions consisted of maximizing the probability of a vector of goals lying in the confidence region of predefined size. Fitzsimmoms and Sullivan [15] developed an algorithm using probabilistic goals based on the concept of chance constraints due to Charnes and Cooper [5] where the goals can be stated in terms of probability of satisfying the aspiration levels. Teghem et al. [43] and Leclercq [30] presented interactive methods in stochastic programming. Shah [41] discussed a probabilistic time scheduling model for an exponential decaying inventory when delays in payment are permissible. Ben-Daya and Raouf [2], Kalpakam and Sapan [25], [26] studied perishable inventory models with stochastic lead-time. Karmarkar [27] derived a lot size model with lead time and established the relationships between lot sizes and lead times for batch manufacturing shops with queues. The models discussed above are EOQ models that deal with instantaneous inventory replenishment policy only.

An economic production quantity (EPQ) model deals with an inventory -cum production system in which procurement of inventory occurs through production within the cycle itself. Till date several research papers have been developed on EPQ models having constant or dynamic or stock dependent demand with/without shortages. In some models unit production costs are taken as functions of production rate. Khouja [28], Khouja and Mehrej [29] solved EPQ model considering constant demand and no shortage, taking the production rate as a decision variable. Hong et al. [21] developed an EPQ model considering linearly time varying demand and constant production rate. Goswami and Chaudhuri [17] developed such a model considering shortages. Zhou [49], [50] considered EPQ models over a finite time horizon with a linear trend in demand and shortages. He considered different production rate in different production cycle to fulfill the increasing demand. Giri and Chaudhuri [16] discussed a production-inventory model considering linearly time dependent demand and production rate dependent unit production cost with fully backlogged shortages. Aliyu et.al [1] developed an EPQ system with shortages and back order. Sana et al [40] developed an EPQ model for deteriorating items with trended demand and shortages. Maity and Maiti [34] extended the research in considering the EPQ model with inventory dependent demand under inflation and discounting.

The models discussed above did not consider the defectiveness in the produced quantities. In reality all the produced units can not be of good quality in a production system. Defective items as a result of imperfect production process were initially considered by Proteous [38] and later by several researchers such as Goyal and Cardenas-Barron [18], Ben-Daya [3], Goyal et al. [19]. Salama and Jaber [39] considered imperfect quality items in stochastic environment.

In general, the EPQ models are formulated under crisp resource constraints. In real life, it may not be so. For example, at the beginning of a business, it may be launched with some capital. But during the period of business, it may happen that to meet the unexpected increased demand, the production rate may have to be stepped up and in doing so, the organization would have to invest some more capital. This augmented amount is normally fuzzy in nature for a new company, for which past data are not available. Regarding an old company, it is possible to have the past data for the variation in the budget and the said may be represented by a probability distribution. Hence the resource constraints become stochastic or imprecise in nature. During the last few years, there are some EPQ models [cf. Chang [4], Chen and Hsieh [6], Hsieh [23], Hsieh and Chen [22], Lee and Yao [31], Tsourveloudis et al. [44], Yao and Wu [47]] formulated in fuzzy environment in the literature. Earlier investigations normally took one parameter or a resource as fuzzy and solved using fuzzy set theory and extension principle. They did not consider fuzzy and stochastic parameters and/or both fuzzy and stochastic constraints in a single model. In these models, resource constraints were not imposed in possibility/necessity sense. Such a complex model has been investigated in this paper.

The chance constrained programming (CCP) technique is one which is used to solve problems involving chance constraints i.e., constraints having random parameters. The CCP was originally developed by Charnes and Cooper [5] and has, in recent years, been extended in several directions for various applications. Liu and Iwamura [32] solved chance constraint programming with fuzzy parameters.

Analogous to chance constraint programming with stochastic parameters, in a fuzzy environment, it is assumed that some constraints with a least possibility are satisfied. Again some constraints may be satisfied with some predefined necessity also [cf. Dubois and Prade [10], [11]]. These possibility and necessity resource constraints may be imposed as per the demand of the situation. Zadeh [48], Dubois and Prade [8], [9], [13], [14], Wang and Hwang [46] introduced the necessity and possibility constraints which are very relevant to the real life decision making problems and presented the process of defuzzification for these constraints. Liu and Iwamura [32], [33] have extended these ideas to linear/non-linear programming problems. Inuiguchi et al. [24] discussed production-planning problem under possibility constraint. Maity and Maiti [35] also developed an optimal control model under possibility and necessity constraints. In these papers the resource constraints in different contexts such as job scheduling, portfolio selections etc. were introduced in possibility and necessity senses. Following the above authors, in this paper we have imposed possibility and/or necessity constraints along with a stochastic resource constraint on an EPQ model. Here foundation and derivation of possibility and necessity measures have been established more explicitly with the help of Lemma 1, Lemma 2.

In this paper, we consider a multi-product manufacturing system producing stochastically imperfect items for stochastic demand under different types (stochastic/possibilistic/necessity) of budgetary and limited shortages constraints. The stochastic constraints are converted into crisp ones using ‘here and now’ method [cf. Mohon [36]] for chance constraints and imprecise constraints are difuzzified following Dubois and Prade [12] and Liu and Iwamura [32], [33]. The model is of single period for each item and the cycle lengths for different items are constants but different for different items. The total demand for a cycle is considered as stochastic. The rates of defective units are also stochastic. The problems are formulated with different combinations of different types of constraints and their equivalent deterministic ones are solved using a non-linear gradient-based optimization technique – Generalized Reduced Gradient Method. The model is illustrated numerically. For different ‘aspiration’ and/or ‘confidence’ levels of constraints, variation in profit has been presented. The focus of this paper is two-fold. A multi-item production model has been derived with stochastic imperfect production and stochastic demand over a single period. In this paper, fuzzy resource constraints in the form of possibility and necessity constraints have been derived and for the first time, these constraints along with traditional chance constraints have been imposed in a stochastic inventory model.

This paper is organized as follows: 1. Introduction; 2. Assumptions and notations; 3. Mathematical model; 4. Constraints; 5. Chance Constraint Programming Technique; 6. Possibility/Necessity Programming Technique; 7. Different cases; 8. Numerical Examples; 9. Discussion; 10. Sensitivity Analysis; 11. Conclusion; References.

Section snippets

Assumptions and notations

Assumptions

(i)
The inventory system is an imperfect production system and involves multiple items.
(ii)
This is a single period inventory model.
(iii)
Production rate is finite and constant.
(iv)
Total demand over the period of cycle is stochastic and uniform over time.
(v)
Percentage of imperfectness is stochastic.
(vi)
Shortages are permitted and fully back-logged.
(vii)
Screening costs for all items are same.

The inventory system involves n items and for ith item $(i = 1, 2, \dots, n)$ following Notations are used:

(i)
A_i’s, B_i’s and R_i’s are constants in the

Mathematical model

Case-I:
When shortages do not occur:
The inventory level q_i(t) governed by the differential equations, i = 1,2, … , n. $\begin{matrix} \frac{d q_{i} (t)}{d t} = (1 - e_{i}) P_{i} - \frac{x_{i}}{T_{i}}, & 0 ⩽ t ⩽ t_{1 i} \\ = - \frac{x_{i}}{T_{i}}, & t_{1 i} ⩽ t ⩽ T_{i} \end{matrix}\}$ with q_i(0) = 0.
Using initial conditions, we have the solution of (1) as $q_{i} (t) = \{\begin{matrix} (P_{ii} - \frac{x_{i}}{T_{i}}) t, & 0 ⩽ t ⩽ t_{1 i}, \\ P_{ii} t_{1 i} - \frac{x_{i}}{T_{i}} t, & t_{1 i} ⩽ t ⩽ T_{i}, \end{matrix}$ where P_ii = (1 − e_i)P_i [by notation (xii)].
Since shortages do not occur, we must have q_i(T_i) ⩾ 0. $\Rightarrow x_{i} ⩽ P_{ii} t_{1 i} .$ Now by notations Q_i = P_it_1i and Q_ii = P_iit_1i.
Expected holding cost for non-defective units of ith item, using (2), (3), is given by $C_{1 i} \int_{0}^{1} [\int]$

Probabilistic constraints

(i)
When limitations on total production cost and screening cost is probabilistic then $Prob [\sum_{i = 1}^{n} (C_{p i} + s_{c}) Q_{i} ⩽ B] ⩾ r_{1}, 0 < r_{1} < 1 .$
(ii)
When limitation on shortages becomes probabilistic, the constraint becomes $Prob [\sum_{i = 1}^{n} C_{s i} Q_{s i} ⩽ S_{M}] ⩾ r_{2}, 0 < r_{2} < 1 .$

Fuzzy (Possibility/necessity) constraints

If ${\tilde{C}}_{p i}$ , ${\tilde{C}}_{s i}$ and $\tilde{B}$ , ${\tilde{S}}_{M}$ are imprecise in nature, then the above constraints are of the following form: $(i) \sum_{i = 1}^{n} ({\tilde{C}}_{p i} + s_{c}) Q_{i} ⩽ \tilde{B}, (ii) \sum_{i = 1}^{n} {\tilde{C}}_{s i} Q_{s i} ⩽ {\tilde{S}}_{M} .$ (Here wavy bar ‘∼’ denotes fuzzyfication of the parameters).

Hence the problem (16) is reduced to $\begin{matrix} Maximize EAP (Q_{1}, Q_{2}, \dots, Q_{n}) \\ s.t. \\ \sum_{i = 1}^{n} ({\tilde{C}}_{p i} + s \end{matrix}$

Chance constrained programming technique

A stochastic non-linear programming problem with some linear chance constraints can be expressed as $Max Z (x_{1}, x_{2}, \dots, x_{n})$ $\begin{matrix} s.t. \\ Prob [\sum_{j = 1}^{n} a_{ij} x_{j} ⩽ b_{i}] ⩾ r_{i}, \end{matrix}$ $x_{j} ⩾ 0, j = 1, 2, \dots, n, r_{i} \in (0, 1), i = 1, 2, \dots, k,$ where a_ij, b_i are normal random variables and r_i are specified probabilities. For simplicity, we assume that the decision variables x_j are deterministic. Here we consider the case: only b_i’s are normally distributed random variables with known means and variances.

Let E(b_i) and Var(b_i) denote the mean and variance of the

Possibility/necessity in fuzzy environment

Any fuzzy subset $\tilde{a}$ of $R$ (where $R$ represents a set of real numbers) with membership function $μ_{\tilde{a}} (x) : R \to [0, 1]$ is called a fuzzy number. Let $\tilde{a}$ and $\tilde{b}$ be two fuzzy quantities with membership functions $μ_{\tilde{a}} (x)$ and $μ_{\tilde{b}} (x)$ , respectively. Then according to Dubois and Prade [9], Liu and Iwamura [32], [33] $Pos (\tilde{a} * \tilde{b}) = \sup {\min (μ_{\tilde{a}} (x), μ_{\tilde{b}} (y)), x, y \in R, x * y},$ $Nes (\tilde{a} * \tilde{b}) = \inf {\max (1 - μ_{\tilde{a}} (x), μ_{\tilde{b}} (y)), x, y \in R, x * y},$ where the abbreviation Pos represents possibility, Nes represents necessity and ∗ is any of the relations <,

Both constraints are stochastic

Model-7.1.1:
In this case our problem is (Ref. article 4a..) $\begin{matrix} Max EAP (Q_{1}, Q_{2}, \dots, Q_{n}) \\ s.t. \\ Prob [\sum_{i = 1}^{n} (C_{p i} + s_{c}) Q_{i} ⩽ B] ⩾ r_{1}, \\ Prob [\sum_{i = 1}^{n} C_{s i} Q_{s i} ⩽ S_{M}] ⩾ r_{2}, \\ Q_{i} ⩾ 0, 0 < r_{1} < 1, 0 < r_{2} < 1 . \end{matrix}$

Hence according to Eq. (23) the optimizing problem (30) is restated as:

Max EAP (Q_{1}, Q_{2}, \dots, Q_{n})

\begin{matrix} s.t. \\ \sum_{i = 1}^{n} (C_{p i} + s_{c}) Q_{i} - E (B) - \sqrt{Var (B)} Φ (r_{1}) ⩽ 0, \end{matrix}

\sum_{i = 1}^{n} C_{s i} Q_{s i} - E (S_{M}) - \sqrt{Var (S_{M})} Φ (r_{2}) ⩽ 0,

Q_{i} ⩾ 0, Q_{s i} ⩾ 0, i = 1, 2, \dots, n, 0 < r_{1} < 1, 0 < r_{2} < 1 .

One constraint is stochastic and other one is fuzzy

In this case there are four sub cases. We consider the model under the different cases of the constraints.

Model-7.2.1
Budget constraint as stochastic and shortage constraint

Numerical illustration

Now, the above models are illustrated numerically.

The common input parameters, which are used in each of the above models, are given in Table 1.

The other data for the respective models 7.1.1, 7.2.1, 7.2.2, 7.2.3 and 7.2.4 are given below:

Model-7.1.1:
C_p1 = 2, C_p2 = 3, E(B) = 510, σ(B) = 10, Φ(r₁) = −1.645,
C_s1 = 3.7, C_s2 = 5.3, E(S_M) = 45, σ (S_M) = 3, Φ(r₂) = −1.645.
Model-7.2.1:
C_p1 = 2, C_p2 = 3, E(B) = 510, σ(B) = 10, Φ(r₁) = −1.645,
${\tilde{C}}_{s 1} = (3.5, 3.7, 3.9)$ , ${\tilde{C}}_{s 2} = (4.9, 5.3, 5.5)$ , ${\tilde{S}}_{M} = (37, 40, 43)$ , η₂ = 0.05.
Model-7.2.2:
C_p1 = 2, C_p2 = 3, E(B) = 510, σ(B) = 10, Φ(r₁) = −1.645,
${\tilde{C}}_{s 1} = (3.5$

Discussion

The results of the stochastic model with different types of constraints have been tabled in Table 2. Here, the constraints are both stochastic or one stochastic and other one being fuzzy or both fuzzy. From the results, it is observed that budget constraint plays the vital role. Keeping shortage constraint as stochastic, profit is 240.02$ (cf. Table 2; model 7.2.3) when the budget constraint is of necessity type. But, it goes up to 287.95$ (cf. Table 2; model 7.2.4) with possibility type budget

Sensitivity analysis

Sensitivity analyses are performed for maximization of the objective function with respect to different ‘confidence level’ of fuzzy constraints. The study has been done only for the models 7.2.2, 7.3.2 and 7.3.4. For other models, sensitivity analyses can be done in similar way.

The changes of the objective function EAP in model 7.2.2 due to changes in Φ(r₁) and η₂ are represented through the 3-D graph in Picture 1. In model 7.2.2 the base values taken for r₁(Φ(r₁)) and η₂ are 0.95(−1.645) and

Conclusion

In this paper, we propose an extension to economic production lot size model for imperfect items in which the production rate is assumed to be finite and demand rate is stochastic (continuous) under uncertain budget and shortage constraints. Here we also consider that the percentage of defective items is stochastic and the natures of uncertainty in the constraints are stochastic and/or fuzzy.

In the stochastic/fuzzy-possibility/necessity type non-linear programming problem, we first convert the

References (50)

M. Ben-Daya
The economic production lot-sizing problem with imperfect production process and imperfect maintenance
International Journal of Production Economics
(2002)
D. Dubois et al.
Ranking fuzzy numbers in the setting of Possibility theory
Information Sciences
(1983)
D. Dubois et al.
The mean value of a fuzzy number
Fuzzy sets and systems
(1987)
S.K. Goyal et al.
Note on economic production quantity model for items with imperfect quality a practical approach
International Journal of Production Economics
(2002)
M. Inuiguchi et al.
The usefulness of possibility programming in production planning problems
International Journal of Production Economics
(1994)
S. Kalpakam et al.
(S-1, S) perishable system with stochastic lead time
Mathematics and Computer Modeling
(1995)
M. Khouja
The economic production lot size model under volume flexibility
Journal of the Computer and Operations Research
(1995)
B. Liu et al.
Chance constraint Programming with fuzzy parameters
Fuzzy Sets and Systems
(1998)
K. Maity et al.
Possibility and necessity constraints and their defuzzification – A multi-item production-inventory scenario via optimal control theory
European Journal of Operational Research
(2007)
S. Sana et al.
A production-inventory model for a deteriorating item with trended demand and shortages
European Journal of Operational Research
(2004)

N.H. Shah

Probabilistic time-scheduling model for an exponentially decaying inventory when delays in payments are permissible

International Journal of Production Economics

(1993)

J. Teghem et al.

Strange: An interactive method for multi-objective linear programming under uncertainty

European Journal of Operational Research

(1986)

N.C. Tsourveloudis et al.

Fuzzy work-in-process inventory control of unreliable manufacturing systems

Information Sciences

(2000)

J. Wang et al.

Fuzzy decision modeling for supply chain management

Fuzzy Sets and Systems

(2005)

J.T. Wang et al.

A fuzzy set approach for R&D portfolio selection using a real options valuation model

Omega-International Journal of Management Science

(2007)

L.A. Zadeh

Fuzzy sets as a basis for a theory of possibility

Fuzzy Sets and Systems

(1978)

M.D.S. Aliyu et al.

Multi-item-multi-plant inventory control of production systems with shortages/backorders

International Journal of Systems Science

(1999)

M. Ben-Daya et al.

On the constrained multi-item single period inventory problem

International Journal of Production Management

(1993)

S.C. Chang

Fuzzy production inventory for fuzzy product quantity with triangular fuzzy number

Fuzzy Sets and Systems

(1999)

A. Charnes et al.

Chance constrained programming

Management Science

(1959)

S.H. Chen, C.H. Hsieh, Optimization of fuzzy production inventory model under fuzzy parameters, in: Proceedings of the...

B. Contini

A stochastic approach to goal programming

Operations Research

(1978)

D. Dubois, H. Prade, System of linear fuzzy constraints, Purdue University Electrical Engineering Technique Report...

D. Dubois et al.

Fuzzy Sets and Systems: Theory and Applications

(1980)

D. Dubois et al.

Possibility Theory

(1988)

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Production, Manufacturing and LogisticsA single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints

Abstract

Introduction

Section snippets

Assumptions and notations

Mathematical model

Probabilistic constraints

Fuzzy (Possibility/necessity) constraints

Chance constrained programming technique

Possibility/necessity in fuzzy environment

Both constraints are stochastic

One constraint is stochastic and other one is fuzzy

Numerical illustration

Discussion

Sensitivity analysis

Conclusion

International Journal of Production Economics

Information Sciences

Fuzzy sets and systems

International Journal of Production Economics

International Journal of Production Economics

Mathematics and Computer Modeling

Journal of the Computer and Operations Research

Fuzzy Sets and Systems

European Journal of Operational Research

European Journal of Operational Research

International Journal of Production Economics

European Journal of Operational Research

Information Sciences

Fuzzy Sets and Systems

Omega-International Journal of Management Science

Fuzzy Sets and Systems

Multi-item-multi-plant inventory control of production systems with shortages/backorders

International Journal of Systems Science

On the constrained multi-item single period inventory problem

International Journal of Production Management

Fuzzy production inventory for fuzzy product quantity with triangular fuzzy number

Fuzzy Sets and Systems

Chance constrained programming

Management Science

A stochastic approach to goal programming

Operations Research

Fuzzy Sets and Systems: Theory and Applications

Possibility Theory

Production, Manufacturing and Logistics
A single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints