OPSEARCH DOI 10.1007/s12597-013-0122-9 APPLICATION ARTICLE
Multi-item production inventory model with fuzzy rough coefficients via geometric programming approach
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D. K. Jana · K. Maity · T. K. Roy
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Accepted: 2 February 2013 © Operational Research Society of India 2013
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Abstract Inventory Management and production planning are essential tasks for every company in the industry. Therefore, the development of a large set of Economic Order Quantity (EOQ) models is needed. In this paper, a Fuzzy Rough (Fu-Ro) multi-item Economic Production Quantity (EPQ) model is developed with constant demand, infinite production rate having flexibility and reliability consideration in production process and demand dependent unit production cost and shortages under the restrictions on storage area. Some of the inventory parameters are considered as trapezoidal fuzzy rough numbers and the model is formulated following cost minimization principle. Then by geometric programming (GP) technique we solve the problem. The model is illustrated through numerical example.
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Keywords Trapezoidal fuzzy rough number · Production · Flexibility · Reliability · Geometric programming
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D. K. Jana () Department of Applied Science, Haldia Institute of Technology, Haldia, Purba Midna Pur, 721657, West Bengal, India e-mail:
[email protected] K. Maity Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, Bhupatinagar, Purba Medinipur, 721425, West Bengal, India e-mail: kalipada
[email protected] T. K. Roy Department of Mathematics, Bengal Engineering and Science University, Shibpur Howrah, 711103, West Bengal, India e-mail: roy t
[email protected]
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1 Introduction
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In the early 20th century, the first models for the combined optimization of the batchproduction and inventory level problem were derived from the basic Economic Order Quantity (EOQ) model. Before this, mathematical methods had started emerging to optimize the size of inventory and orders and since then, there has been an increasing number of contributions, which have improved and extended the basic model in many ways. One of these extensions allows for a finite production rate. The different EOQmodels are most often used in a continuous-review setting and it is assumed that the inventory can be monitored every moment in time. It is imperative to acknowledge the importance of production aspects in supply chain management, especially in processbased industries. For these applications, it is important to find solutions that allow the production to be efficient while keeping inventory low (Bjork and Carlsson, [3], Bjork [4]). This tradeoff problem is found in many supply chains. A specific application that inspired the author to conduct this research is found in the paper industry supply chains in the Nordic countries. These supply chains consist of a few, large paper producing companies and quite many distributors that operate independently from the producers. Typically a large paper machine is producing several products in large quantities. There are often substantial uncertainties found in these supply chains and these cannot be captured by probabilistic measures (Islam and Roy [15]). This may result in a situation, where management wants to increase batch sizes in order to produce to stock. There are also other reasons for the uncertainty in the batch sizes. All these uncertainties will lead to uncertain cycle time. Therefore, an EPQmodel that models a production process, with several different products, is developed in this paper. Uncertainty is common in real life problems such as randomness, fuzziness and roughness. In 1965, Zadeh [26] first have introduced the idea of fuzzy set theory, fuzzy set theory has been well developed and applied in a wide variety of real problems. Possibility theory was also proposed by Zadeh [27] and developed by many researchers such as Dubois and Prade [6]. However, in a decision-making process, we may face a hybrid uncertain environment where fuzziness and roughness exist at the same time. In such cases, a fuzzy rough variable is a useful tool. Fuzziness and roughness play an important role among types of uncertainty problems. The concept of fuzzy rough sets introduced by Dubois and Prade [7, 8] play a key role dealing with the two types of uncertainty simultaneously. Dubois and Prade [9] first studied the fuzzification problem of rough sets. Fuzzy programming and rough programming have been proposed for decisions under uncertainty environment. In these programming models, roughness and fuzziness are considered as separate aspects. Several researchers have considered the issue of combining fuzziness and roughness in a general framework for the study of fuzzy rough sets. Morsi and Yakout [13] have depoled an axiomatics for fuzzy rough sets. At present, the fuzzy rough set has been applied to practical problems. Furthermore, the fuzzy rough set theory was used to extract fuzzy decision rules from fuzzy information systems. In 2008, Xu and Zhao [23] also developed a class of fuzzy rough expected value multi-objective decision making model and
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its application to inventory problems. Maity [18] has also developed Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem. Next, Shi et al. [21] have developed a probability maximization model based on rough approximation and its application to the inventory problem. Recently, Xu and Zhao [25], Mondals et at. [20] have considered fuzzy rough inventory models, but none has considered the fuzzy rough multi-item production inventory problem via GP. For the inventory problem, the classical inventory decision-making models are considered a single-item. However, single-item inventories seldom occur whereas multi-item inventories are common in real-life situations. Several researchers (cf. Lee and You [16], Taleizadeh et al. [22], Balkhi and Foul [2], Hartley [12]) discussed multi-item classical inventory models under resource constraints. In reality, the parameters involved in solving inventory problems may be uncertain and fuzzy rough in nature. For example, production costs of an item may be dependent upon the total quantity to be produced. But in the inventory system the amount produced within a scheduling period may be uncertain and range within an interval arising from specific requirements, such as local conditions and customer demand. In such situations, the fuzzy rough theory can be used for the formulation of inventory models. Therefore, there is strong motivation for further research in the area, where we consider fuzzy rough constrained in multi-item production inventory problems. Geometric Programming(GP) is a class amongst nonlinear programming techniques. Due to certain advantages over the other optimization procedures, GP has been effectively used to solve many real-life decision-making problems with posynomial/ signomial objective function and constraints. Duffin et al. [10] first showed that the GP technique could be used with some advantages for optimization problems of a particular type. Later Hariri and Abou-el-ata [11], Abou-el-ata and Kotb [1], Leung [17], Mandal et al. [19] solved some inventory problems using GP method. Here, we propose to apply the GP technique to the crisp and fuzzy-rough multi-item production inventory problems. The proposed model considers a multi-item economic production quantity (EPQ) inventory model with constant demand, infinite production rate having flexibility and reliability consideration in production process and demand dependent unit production cost and shortages under the restrictions on storage area. Some of the inventory parameters are considered as trapezoidal fuzzy rough numbers and model is formulated following cost minimization principle. Then by geometric programming (GP) and GRG techniques, we have solved the proposed models.
2 Necessary knowledge about fuzzy rough In this section, we will state some basic concepts, theorems and lemmas on fuzzy rough theory by Xu and Zhou [24] These results are crucial for the remainder of this paper.
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Definition 1 In Xu and Zhou [24] proposed some definitions and discussed some important properties of fuzzy rough variables. Let U be a universe, and X a set representing a concept. Then its lower approximation is defined by X = {x ∈ U | R −1 (x) ⊂ X}
(1)
and the upper approximation is defined by X=
(2)
R(x)
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x∈X
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Definition 2 The collection of all sets having the same lower and upper approximations is called a rough set, denoted by (X, X).The figure of a rough set is depicted in Fig. 1.
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Example 1 Let ξ focus on the continuous set in the one dimension real space R. There are still some vague sets which cannot be directly fixed and need to be described by the rough approximation. For example, set R be the universe, a similarity relation!is defined as a ¯ b if and only if |a − b| ≤ 10. We have that for the set [20, 50], its lower approximation [20,50] = [30,40] and its upper approximation [20, 50] = [10, 60]. Then the upper and lower approximation of the set [20,50] make up a rough set ([30, 40], [10, 60]) which is the collection of all sets having the same lower approximation [30, 40] and upper approximation [10,60].
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Definition 3 A fuzzy rough variable ξ is a fuzzy variable with uncertain parameter ρ ∈ X, where X is approximated by (X, X) according to the similarity relation R, namely, X ⊆ X ⊆ X. For convenience, we usually denote ρ (X, X)R expressing that ρ is in some set A which is approximated by (X, X) according to the similarity relation R, namely, X ⊆ A ⊆ X.
Fig. 1 A rough set
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Example 2 Let’s consider the LR fuzzy variable ξ with the following membership function, ⎧ ρ−x ⎪ ⎪ L if ρ − α < x < ρ ⎪ ⎪ ⎨ α μξ (x) = 1 if x = ρ ⎪ ⎪ x − ρ ⎪ ⎪ if ρ < x < ρ + β ⎩L β
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If ρ ([1, 2], [0, 3])R , then ξ is called a fuzzy rough variable.
3 Single-objective Fu Ro model
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Let us consider the following single-objective decision making model with fuzzy rough coefficients:
(3)
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⎧ ⎨ Max f
(x, ξ ) gr (x, ξ ) ≤ 0, r = 1, 2, ..., p ⎩ s.t x∈X
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where x is a n-dimensional decision vector, ξ = (ξ1 , ξ2 , ξ3 , ..., ξn ) is a Fu-Ro vector, f (x, ξ ) are objective functions, i = 1, 2, ..., m. Because of the existence of Fu-Ro vector ξ , problem (3) is not well-defined. That is, the meaning of maximizing f (x, ξ ) is not clear and constraints gr (x, ξ ) ≤ 0, r = 1, 2, ..., p do not define a deterministic feasible set.
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3.1 Equivalent crisp model for single objective problem with Fu Ro parameters:
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For the single-objective model (3) with Fu-Ro parameters, we cannot deal with it directly, we should use some tools to make it have mathematical meaning, we then can solve it. In this subsection, we employ the expected value operator to transform the fuzzy rough model into Fu-Ro EVM i.e. crisp model. Based on the definition of the expected value of fuzzy rough events f (x, ξ ) and gr (x, ξ ), the general model for Fu-Ro EVM is proposed as follows, ⎧ ⎪ ⎪ Max E f (x, ξ ) ⎪ ⎪ ⎨ ⎧ ⎨ E g (x, ξ ) ≤ 0, r = 1, 2, ..., p ⎪ r ⎪ s.t ⎪ ⎪ ⎩ ⎩ x∈X
(4)
where x is n-dimensional decision vector and ξ is n-dimensional fuzzy rough variable.
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Lemma 1 Assume that ξ and η are the introduction of variables with finite expected values. Then for any real numbers a and b, we have E[aξ + bη] = aE[ξ ] + bE[η]
(5)
Proof The proof of the Lemma is in reference Xu and Zhou [24].
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Theorem 1 If trapezoidal fuzzy rough numbers c˜¯ij are defined as c˜¯ij (λ) = (cij 1 , cij 2 , cij 3 , cij 4 ) with cij t ([cij t 2 , cij t 3 ], [cij t 1, cij t 4 ]), for i = 1, 2, , m, j = 1, 2, , n, t = 1, 2, 3, 4, x = (x1 , x2 , ..., xm ), 0 ≤ cij t 1 ≤ cij t 2 < cij t 3 ≤ cij t 4 . then E[c˜¯1T x],E[c˜¯2T x],...,E[c˜¯nT x] is equivalent to 1
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c1j t k xj , c2j t k xj , ..., cnj t k xj , 16 16 16 4
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j =1 t =1 k=1
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j =1 t =1 k=1
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(6)
j =1 t =1 k=1
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Proof The proof of the theorem is in reference Xu and Zhou [24].
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Theorem 2 If trapezoidal fuzzy rough numbers a˜¯ rj , b˜¯ r defined as follows, a˜¯ rj (λ) = (a rj 1 , a rj 2 , a rj 3 , arj 4 ) with a rj t ([arj t 2, arj t 3], [arj t 1, arj t 4]), b˜¯r (λ) = (br1 , br2 , br3 , br4 ) with brt ([brt 2 , brj t 3], [brt 1, brt 4]), for r = 1, 2, , p, j = 1, 2, ..., n, t = 1, 2, 3, 4, 0 ≤ art 1 ≤ art 2 < art 3 ≤ art 4, 0 ≤ brt 1 ≤ brt 2 < brt 3 ≤ brt 4 . T x] ≤ E[b˜¯ ], r = 1, 2, ..., p is equivalent to then E[a˜¯ rj rj 1
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arj t k xj ≤ brt k , r = 1, 2, ..., p 16 16 4
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Proof The proof of the theorem is in reference Xu and Zhou [24].
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4 Notations and assumptions The following notations and assumptions are used in developing the model. 4.1 Notations Inventory system involves n items and for i t h (i = 1, 2, ..., n) item, (i) (ii) (iii) (iv)
Di c˜¯0i c1i c˜¯2i
= Demand per unit time (a decision variable) = Unit production cost which is fuzzy rough in nature, = Holding cost per unit per unit time, = Fuzzy rough shortage cost per unit per unit time,
(7)
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c3i = Set-up cost Qi = Order level (a decision variable) Si = Shortage level (a decision variable) Wai = Crisp, Fuzzy rough area required to store one unit of item ri = Production process reliability (a decision variable) Ti = Cycle length T1i = Duration of time so long inventory level is positive in a cycle, λkj i = Dual variables λ = Vector of λkj i ’s W0 = Crisp, Fuzzy rough maximum allowable storage space I0 = Crisp, Fuzzy rough maximum investment in inventory S0 = Crisp, Fuzzy rough maximum allowable total shortage cost qi (t) = The inventory level of the i-th item at time t, T C = Total average cost, E[T C] = Expected total average cost
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(v) (vi) (vii) (viii) (ix) (xii) (x) (xi) (xi) (xii) (xiii) (xiv) (xv) (xvi) (xvii)
4.2 Assumptions
For i-th (i = 1, 2, ..., n) item the following assumptions are made.
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(iv)
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(iii)
Production rate is infinite i.e., production is instantaneous with zero lead time. Shortages are allowed and fully backlogged with a restriction on total shortage S0 The unit production cost c0i is a power function of demand i.e., c0i = −β c0di Di i , βi (> 1), c0di (> 0) are constants (Cheng [5]). Total cost of interest and depreciation per production cycle is inversely related to set-up cost and directly related to process reliability (Cheng [5]) according to the following equation
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(i) (ii)
−b
c
f (c3i , ri ) = ari c3i ri ri ri
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where ari , bri , cri are positive constants. Here the process reliability level ri (0 ≤ ri ≤ 1) means that of all the items produced in a production run only ri percent are of acceptable quality that can be used to meet demand.
5 Mathematical formulation In this model the inventory level gradually decreases only to meet the demand of the customers. Therefore, if qi (t) is the inventory level of the i-th(i = 1, 2, ..., n) item at time t, we have dqi (t) = −Di , dt
0 ≤ t ≤ Ti
(8)
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with the boundary conditions ⎧ r Q − Si at ⎪ ⎨ i i at qi (t) = 0 ⎪ ⎩ −Si at
t =0 t = T1i
(9)
t = Ti
5.1 Crisp model
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Solving Eq. (8) using Eq. (9), the crisp problem for minimizing the average cost subject to total space, total inventory investment cost and total shortage cost constraints is the following: n
1 1−β 2 c0di Di i ri−1 + c1di ri−1 Q−1 i (ri Qi −Si ) 2
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Min T C(D, Q, c3 , S, r) =
i=1
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1 −1 −1 2 + c3i Di ri−1 Q−1 i + c2di ri Qi Si 2 −b c −1 + ari c3i ri ri ri Di Q−1 i
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subject to
(10)
k1i Ri ≤ 1
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n
−βi
k2i Di
Qi ≤ 1
i=1 n
k3i Di−1 Si2 ≤ 1
i=1
Ri−1 ri Qi − Ri−1 Si ≤ 1 ri ≤ 1, i = 1, 2, ...., n,
where, c1di = c21i , c2di = c22i , Ri = ri Qi − Si , k1i = wWai0 , k2i = cI0di , k3i = 0 The above expressions of cost and constraints are of signomial form.
c2di 2S0 .
5.2 Fuzzy rough model Here, we consider the space constraint, total inventory investment cost constraint and total shortage cost constraint as fuzzy rough constraints. Then the constraints
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as well as the problem (10) can be expressed in fuzzy rough environment as follows. Min T ˜¯C(D, Q, c3 , S, r, R)=
subject to
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1 1−β −1 −1 2 c˜¯0di Di i ri−1 + c1i ri−1 Q−1 i Ri +c3i Di ri Qi 2 i=1 1 −bri cri −1 −1 2 + c˜¯2di ri−1 Q−1 S +a c r D Q (11) ri i i i i i 3i 2 n
w˜¯ ai Ri ≤ W˜¯ 0
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i=1 n
−β c˜¯0di Di i Qi ≤ Iˆ0
i=1
i=1
2
c˜¯2di Di−1 Si2 ≤ S˜¯0
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Ri−1 ri Qi − Ri−1 Si ≤ 1 ri ≤ 1, i = 1, 2, ...., n.
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Where w˜¯ ai ,W˜¯ 0 , c¯˜0di , c¯˜2di , S˜0 , I˜0 are trapezoidal fuzzy rough numbers, then by Lemma 1 and Theorems 1 & 2 the above Fu-Ro model is converted to equivalent crisp problem to:
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5.2.1 Equivalent crisp problem of fuzzy rough model
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1 1−β 2 Min E T ˜¯C(D, Q, c3 , S, r, R) = E c˜¯0di Di i ri−1 + c1i ri−1 Q−1 i Ri 2
subject to
i=1
1 ˜ −1 −1 2 +c3i Di ri−1 Q−1 i + E[c¯2di ]ri Qi Si 2 −bri cri −1 −1 +ari c3i ri Di Qi (12) n
i=1 n
E(w˜¯ ai Ri ) ≤ E(W˜¯ 0 ) −βi
E[c˜¯0di ]Di
Qi ≤ E[I˜¯0 ]
i=1 n
1 i=1
2
E[c˜¯2di ]Di−1 Si2 ≤ E[S˜¯0 ]
Ri−1 ri Qi − Ri−1 Si ≤ 1 ri ≤ 1, i = 1, 2, ...., n,
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Where ˜¯ is denoted the trapezoidal fuzzy rough numbers. Where trapezoidal fuzzy rough numbers c˜¯0di , c˜¯2di , S˜¯ 0 , I˜¯0 are defined as follows E[c˜¯0di ] =
1
c0dij t k 16 n
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j =1 t =1 k=1
c˜¯0di = (c0dij 1 , c0dij 2 , c0dij 3 , c0dij 4 ), c0dij t ([c0dij t 2, c0dij t 3], [c0dij t 1, c0dij t 4]) E[c˜¯2di ] =
1
c2dij t k 16 n
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j =1 t =1 k=1
E[S˜¯0 ] =
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c˜¯2di = (c2dij 1 , c2dij 2 , c2dij 3 , c2dij 4 ), c2dij t ([c2dij t 2, c2dij t 3], [c2dij t 1, c2dij t 4]) 1
S0j t k 16 n
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j =1 t =1 k=1
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S˜¯0 = (S 0j 1 , S 0j 2 , S 0j 3 , S 0j 4 ), S 0j t ([S0j t 2, S0j t 3], [S0j t 1, S0j t 4]) n 4 4 1
E[I˜¯0 ] = S0j t k 16 j =1 t =1 k=1
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I˜¯0 = (S 0j 1 , I 0j 2 , I 0j 3 , I 0j 4 ), I 0j t ([I0j t 2 , I0j t 3], [I0j t 1, I0j t 4]), i = 1, 2, ..., n, t = 1, 2, 3, 4 where, i = 1, 2, ...., n, α(0 < α < 1) is a predefined probability level, wavy
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bar ¯ represents fuzzy rough and E[] represents fuzzy rough expectation of fuzzy rough variable. Following Eqs. 6 and 7, the problem (12) reduces to If we substitute cf 0di = E[c˜¯0di ], cf 2di = E[c˜¯2di ], Sf 0 = E[S˜¯0 ], If 0 = E[I˜¯0 ], E[c˜¯0di ] = c˜¯f 0di , 12 E[c˜¯2di ] = c˜¯f 2di , kf 1i =
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the above problem reduces into Min ET C(D,Q,c3 ,S,r,R)=
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subject to
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i=1 n
i=1 n
E[w˜¯ ai ] , kf 2i E[W˜¯ 0 ]
=
E[c˜¯0di ] , kf 3i E[I˜¯0 ]
=
E[c˜¯2di ] , 2E[S˜¯ 0 ]
in
1−βi −1 1 −1 −1 2 ri + c1di ri−1 Q−1 i Ri +c3i Di ri Qi
cf 0di Di
2
1 −bri cri −1 −1 −1 2 −1 + cf 2di ri Qi Si + ari c3i ri Di Qi (13) 2 kf 1i Ri ≤ 1 −βi
kf 2i Di
Qi ≤ 1
kf 3i Di−1 Si2 ≤ 1
i=1
Ri−1 ri Qi − Ri−1 Si ≤ 1 ri ≤1, (i = 1, 2, ...., n),
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Therefore, we are to solve the problems given by Eq. 13 by GP method. As all the terms of the problem (13) are of posynomial type, we can solve them using the GP technique.
6 Solutions of the models by geometric programming technique 6.1 Crisp model
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Corresponding dual problem of the problem (10) is ⎧ n ⎨ 5 3 dmi e0 λ0mi kj i ej λj ii e(3+i) λ(3+i)1i Max V (λ) = ⎩ λ0mi λj ii λ(3+i)1i i=1 m=1 j =i e(3+i) −λ(3+i)2i × (14) λ(3+i)2i where d1i = c1i , d2i = c22i , d3i = 1, d4i = c24i , d5i = ai , e0 =
n 5
λ0mi , ej =
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λj ii ,
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j = 1, 2, 3, i = 1, 2, ..., n, and subject to normality and orthogonality conditions: λ0mi = 1
i=1 m=1
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(1 − βi )λ01i + λ03i + λ05i − βi λ2ii − λ3ii = 0 −λ01i − λ02i − λ03i − λ04i + (ci − 1)λ05i + λ(3+i)1i + λ(3+n+i)ii = 0 −λ02i − λ03i − λ04i − λ05i + λ2ii + λ(3+i)1i = 0 2λ04i + 2λ3ii − λ(3+i)2i = 0 2λ02i + λ1ii − λ(3+i)1i + λ(3+i)2i = 0 λ03i − bi λ05i = 0
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The required non-negative conditions are: e3i = λ(3+i)1i − λ(3+i)2i > 0; λ0mi > 0, m = 1, 2, .., 5; λj ii > 0, j = 1, 2, 3; λ(3+i)ki > 0, k = 1, 2; and λ(3+n+i)ii > 0, c4i wi k1i = A , k2i = cI1i0 , k3i = 2S , i = 1, 2, ..., n. 0 0 The system of dual equations in Eq. 14 are (6n + 1) linear equations in 11n variables, which can be written as AT W = B, where A is the coefficient matrix of λkj i ’s of order (6n + 1) × 11n, W = [λkj i ]T and B are column vectors having 11n and (6n + 1) elements, respectively. We can write (6n + 1) linear equations in (6n + 1) unknowns in terms of d(= 5n − 1) independent variables. If A1 , is the coefficient matrix (non-singular) of (6n + 1) dependent variables and W1 = [λkj i ]T and W2 = [u1 , u2 , , ..., ud ]T are the column vectors of (6n + 1) dependent and (5n − 1) independent variables, respectively, we have W1 = [A1 ]−1 B − [A1 ]−1 A2 W2 where A2 is the coefficient matrix of (5n − 1) independent variables in A.
(15)
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Now, we write Z(λ) = logV (λ). The values of ui which determine the optimal values of the dual variables satisfying the normality and orthogonality conditions are given by fi ≡
∂Z(λ) = 0, i = 1, 2, ..., d. ∂ui
(16)
One can obtain the optimal values of the decision variables using Duffin et al. [10] theorem of geometric programming on an objective function using optimal dual variables.
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6.2 Equivalent crisp problem of fuzzy-rough model
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The corresponding dual problem of the problem (13) is ⎧ n ⎨ 5 3 dmi e0 λ0mi kfj i ej λj ii e(3+i) λ(3+i)1i Max V (λ) = ⎩ λ0mi λj ii λ(3+i)1i j =i i=1 m=1 e(3+i) −λ(3+i)2i × (17) λ(3+i)2i
n
d3i = 1, d4i =
cf 4i 2 ,
d5i = ai , e0 =
5 n
λ0mi ,
i=1 m=1
λj ii , j = 1, 2, 3, i = 1, 2, ..., n, and subject to normality and orthogonality
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ej =
c2i 2 ,
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where d1i = cf 1i , d2i =
i=1 n 5
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conditions: λ0mi = 1
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(1 − βi )λ01i + λ03i + λ05i − βi λ2ii − λ3ii = 0 −λ01i − λ02i − λ03i − λ04i + (ci − 1)λ05i + λ(3+i)1i + λ(3+n+i)ii = 0 −λ02i − λ03i − λ04i − λ05i + λ2ii + λ(3+i)1i = 0 s2λ04i + 2λ3ii − λ(3+i)2i = 0 2λ02i + λ1ii − λ(3+i)1i + λ(3+i)2i = 0 λ03i − bi λ05i = 0
The required non-negative conditions are: e3i = λ(3+i)1i − λ(3+i)2i > 0; λ0mi > 0, m = 1, 2, .., 5; λj ii > 0, j = 1, 2, 3; λ(3+i)ki > 0; k = 1, 2 and λ(3+n+i)ii > 0; with i = 1, 2, .., n., Here, the system of dual equations in (26) are (6n + 1) linear equations in 11n variables. The degree of difficulty in this case also is (5n − 1). Applying the same procedure described earlier, we obtain optimal dual variables, decision variables and objective value. We can write (6n + 1) linear equations in (6n + 1) unknowns in terms of d(= 5n − 1) independent variables. If A1 , is the coefficient matrix (non-singular) of (6n + 1) dependent variables and W1 = [λkj i ]T and W2 = [u1 , u2 , , ..., ud ]T are
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the column vectors of (6n + 1) dependent and (5n − 1) independent variables, respectively, we have W1 = [A1 ]−1 B − [A1 ]−1 A2 W2
(18)
where A2 is the coefficient matrix of (5n − 1) independent variables in A. Now, we write Z(λ) = logV (λ). The values of ui which determine the optimal values of the dual variables satisfying the normality and orthogonality conditions are given by fi ≡
∂Z(λ) = 0, i = 1, 2, ..., d. ∂ui
(19)
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One can obtain the optimal values of the decision variables using Duffin et al. [10] theorem of geometric programming on an objective function using optimal dual variables.
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6.3 Numerical illustrations
The models are illustrated for two items, i.e., for n = 2. 6.3.1 Crisp model
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Crisp model is illustrated using the following example. In this example parametric values for two items are given in Table 1. Other parametric values are: A0 = $325 sq.ft, I0 = $223, S0 = $34. The optimum values of the dual variables are:
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λ∗ ≡ (λ∗011 , λ∗021, λ∗031, λ∗041, λ∗051 , λ∗012, λ∗022, λ∗032, λ∗042 , λ∗052, λ∗111, λ∗122, λ∗211, λ∗222, λ∗311, λ∗322 , λ∗411, λ∗421, λ∗512, λ∗522 , λ∗611, λ∗722) = (0.2155549, 0.2797219, 0.1824117, 0.0097511, 0.0331726, 0.1028633, 0.1213298 0.0439604, 0.0052205, 0.0158322, 0.0369242, 0.03809517, 0.3029099, 0.1259343, 0.0040239, 0.0073654, 0.4311154, 0.1752083, 0.0022454, 0.0873042, 0.0045193, 0.0481934).
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Form the above input data, the dual problem (14) of crisp model (10) is solved using both GP and GRG (gradient search method) techniques and are presented in Table 2. It is clear from Table 2, that for assumed parametric values the GP method gives better results than the GRG method for the crisp model under consideration. Table 1 The input data for crisp model Item
c1i
c2i
c4i
ai
βi
wi
bri
cri
1
821
2.5
3.45
1234
1.52
8.21
1.45
6.12
2
934
3.0
3.46
1267
1.60
9.25
1.51
7.13
OPSEARCH Table 2 The optimum results for crisp model Method
Item
Di∗
Q∗i
Si∗
∗ c3i
ri∗
Cost($)
GRG
1
102.53
52.08
31.13
16.70
0.9359
610.56
2
94.46
54.54
36.85
23.41
0.9741
1
108.19
57.54
24.74
19.87
0.92417
2
101.81
61.45
34.39
22.78
0.9691
GP
598.85
PY
6.3.2 Equivalent crisp problem of fuzzy rough model
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In this example parametric values for two items for crisp data are given in Table 1 and remaining trapezoidal fuzzy rough numbers are depicted in Table 3. Input data of different crisp parameters for case-I and case-II are given in Table 1, remaining trapezoidal fuzzy rough numbers are depicted in Table 3. Optimal values of objective and decision variables are given in Table 4. The optimum values of the dual variables are: ∗ λ ≡ λ∗011 , λ∗021, λ∗031, λ∗041 , λ∗051, λ∗012, λ∗022, λ∗032, λ∗042 , λ∗052, λ∗111, λ∗122, λ∗211, λ∗222 , λ∗311, λ∗322, λ∗411, λ∗421 , λ∗512, λ∗522, λ∗611, λ∗722 = (0.1245632, 0.24518679, 0.12457642, 0.0054127, 0.03412547, 0.2415784, 0.14278954, 0.2694677, 0.0001243, 0.0303796, 0.1326454, 0.1056981, 0.3051823, 0.4035968, 0.00021457, 0.00541274, 0.2145784, 0.0214574, 0.34157, 0.000241257, 0.00214571, 0.0001241).
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In this case dual problem (17) of equivalent Fu-Ro problem (13) is solved by GP and results are presented in Table 4. In this case (17) is directly solved using GRG also and results are presented in Table 4. It is clear from Table 4, that for assumed parametric values the results obtained through both the GP method and GA technique are almost same. But GP can be applicable if the model can be transformed into an equivalent posynomial/ signomial problem. We find minimum cost in both crisp and fuzzy rough inventory system, it is seen that the results in fuzzy rough environments are better. Table 3 The input data for fuzzy rough variables S˜¯0
I tem
c˜¯1i
˜¯ c2i
1
(ζ1 − 20, ζ1 − 10, ζ1 + 10, ζ1 + 20) ζ1 = ([800, 820], [790, 860])
(γ1 − 3, γ1 − 2, γ1 + 2, γ1 + 3)
(ν1 − 5, ν1 − 3, ν1 + 3, ν1 + 5)
γ1 = ([25, 30], [24, 37])
ν1 = ([30, 45], [25, 47])
2
(ζ2 − 20, ζ2 − 10, ζ2 + 10, ζ1 + 20)
(γ2 − 3, γ2 − 2, γ2 + 2, γ2 + 3)
(ν2 − 5, ν2 − 3, ν2 + 3, ν2 + 5)
I tem
ζ2 = ([850, 930], [840, 960]) w¯˜ ai
γ2 = ([26, 31], [23, 33]) A˜¯0
ν2 = ([30, 42], [27, 60]) I˜¯0
1 2
(ρ1 − 2, ρ1 − 1, ρ1 + 1, ρ1 + 2)
(τ1 − 8, τ1 − 9, τ1 + 8, τ1 + 9)
(χ1 − 9, χ1 − 7, χ1 + 7, χ1 + 9)
ρ1 = ([7.5, 8.4], [6.4, 8.8])
τ1 = ([290, 310], [260, 350])
χ1 = ([196, 210], [180, 236])
(ρ2 − 2, ρ2 − 1, ρ2 + 1, ρ2 + 2)
(τ2 − 9, τ2 − 8τ2 + 8, τ2 + 9)
(χ2 − 9, χ2 − 7, χ2 + 9, χ2 + 7)
ρ2 = ([6.5, 9.5], [6, 10])
τ2 = ([295, 325], [280, 348])
ν2 = ([185, 210], [175, 260])
OPSEARCH Table 4 The optimum results using GP and FuRo-GA Method
Item
Di∗
Q∗i
Si∗
GP
1
146.10
56.21
31.85
16.52
0.6985
2
104.56
55.64
35.42
21.65
0.95412
1
141.00
59.63
11.72
31.19
0.96
2
116.78
56.29
17.61
33.46
0.89
GRG
∗ c3i
ri∗
Z
585.89 585.97
PY
7 Conclusion
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Here for the first time a multi-item EPQ model is developed in fuzzy-rough environment and solved using GP specially when some parameters are trapezoidal fuzzy rough numbers. A methodology is proposed to deal with imprecise constraints involving fuzzy rough parameters. Different approaches are followed to solve the models using different examples. It is found that GRG can be used successfully to solve this type of problems in different complicated situations. The methodology presented here is quite general and can be applied to solve optimization problems in different fields of science and technology. In this paper we develop an economic production inventory model under shortages with flexibility and reliability considerations. The constraints, considered are fuzzy rough. This procedure might be extended to two warehouse inventory models, Supply chain model, etc.
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Acknowledgment Second author Dr. Kalipada Maity thanks Minor research Project (PSW- 092, 12/13) under UGC, Govt. of India.
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