Inventory models for breakable items with stock dependent demand and imprecise constraints

Mathematical and Computer Modelling 52 (2010) 1771–1782

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Inventory models for breakable items with stock dependent demand and imprecise constraints Anirban Saha a,∗ , Arindam Roy b , Samarjit Kar c , Manoranjan Maiti d a

School of Applied Sciences, Haldia Institute of Technology, Hatiberia, Purba-Medinipur, WB, Pin-721 657, India

b

Department of Computer Science, Contai P.K. College, Contai, Purba-Medinipur, WB, India

c

Department of Mathematics, National Institute of Technology, Durgapur, WB, Pin-713 209, India

d

Department of Applied Mathematics, Vidyasagar University, Midnapore, WB, Pin-721 102, India

article

info

Article history: Received 3 August 2009 Received in revised form 2 July 2010 Accepted 5 July 2010 Keywords: Inventory Multi-item EOQ Breakable unit Possibility Necessity

abstract This paper develops multi-item Economic Order Quantity (EOQ) inventory models for breakable units with stock dependent demand under imprecise constraints. The units are damaged due to the accumulated stress of the stocked items kept in stacked form and the damaged function, i.e. rate of breakability per unit time may be linear or non-linear function of current stock level. Here shortages are not allowed. Both the crisp and fuzzy models have been formulated as profit maximization problems with crisp/imprecise space and budget constraints and solved by using a gradient based non-linear programming technique, Generalised Reduced Gradient (GRG) Method. The models are illustrated with a numerical example and some sensitivity analyses have been presented. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction In the present competitive market, the inventory/stock is decoratively displayed through electronic media to attract the customers and to push the sale. Levin et al. [1], Schary and Becker [2] and Wolfe [3] established the impact of product availability for stimulating demand. Many researchers considered the demand function as stock dependent polynomial functions, i.e. demand depends on on-hand inventory. In the case of a polynomial demand function, demand goes to zero as inventory becomes nil. This is an impractical proposition. It is the fact that demand rate will go down as the displayed inventory level decreases. But, there may be some dedicated customers accustomed to a particular brand, who will wait for that product whether it is available or not. So the demand rate is stock dependent until the inventory reduces to a certain given level (which depends on the product and the place of business), after which the demand rate becomes constant. The classical inventory model developed by Harris in 1915 demands the specific requirements of deterministic cost and demand without deterioration of the items in stock. Gradually, the concept of deterioration/damage in inventory system caught up the mind of inventory researchers. In most of the real life inventory problems, the effect of deterioration/damage is a natural phenomenon. In classical inventory models of deteriorating items, it is assumed that goods deteriorate physically with the progress of time during their normal storage. A large number of research papers had been developed in inventory management by several researchers, viz. Deb and Chaudhuri [4,5], Chung and Ting [6], Goyal and Gunasekaran [7], Hariya and Benkherrouf [8] and others who have considered damage function as either a constant or a function of time t. But Mondal and Maiti [9] first considered the inventory of the items made of glass, china clay, ceramic, etc. which are normally stored

∗

Corresponding author. Tel.: +91 3224 253061; fax: +91 3224 252800/253062. E-mail addresses: [email protected], [email protected] (A. Saha), [email protected] (A. Roy), [email protected] (S. Kar), [email protected] (M. Maiti). 0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.07.004

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in heaped stock and are damaged due to the accumulated stress and the amount of damage by the stress varies with the size of stock. Hence for the above materials, damage function is a function of on-hand stock q(t ). Again, for these breakable materials damage function may be linear or non-linear in q(t ), the exact form of which can be obtained from the stress–strain relations for the materials. But, the finding of market research—‘‘More Display, More Demands’’ puts a retailer of the items of glass, china clay, ceramic etc. in a conflicting situation. He/she is tempted to go for a large heaped stock in order to create higher demand for the item by display and then takes the advantage of more consumption by the people. But as a result, he/she simultaneously invites more damage to his/her units as damageability increases with the increase of piled stock. Hence, in addition to the usual contradiction embedded in the holding and setup costs of an inventory system, the retailer faces the above contradiction and tries to take an optimum decision for the maximum profit. Here, in this paper, some inventory models in these conflicting situations are formulated. Till now, algorithms have been developed for solving the inventory problems when inventory parameters like total available space and total budget allocation for replenishment etc. are precisely known. But in real life situation, these parameters may be uncertain in non-stochastic sense. In the competitive market, it is not possible to do the business with predefined fixed budgetary capital. Initially, a decision maker (DM) may start with an amount, but at a later stage, to meet the sudden increase of demand or to avail the sudden fall in the price of the commodity, he/she is forced to augment some more capital as per demand of the situation. Hence, in this case, budgetary allocation is imprecise. Similar may be the case for total available space. Earlier, Mondal and Maiti [9,10] formulated a single item inventory model for a breakable unit with stock dependent demand and they assumed that the demand was linear or non-linear function of current stock level as well as the breakable unit B(q) depended on the current stock level. The dependency was both linear and non-linear, i.e. in general B(q) = aqγ , 0 ≤ γ ≤ 1, 0 ≤ a ≤ 1. But the model was solved for γ = 1, 21 only, not for general values of γ in crisp environment. They did not considered also the multi-items. Later Maiti and Maiti [11–13] developed some production–inventory/inventory models for breakable items in crisp environment. Though the general form of breakability function was considered, they did not formulate the problems with imprecise constraints of total available space and total budget. Here, in this paper, some multi-item inventory models with stock dependent demand under crisp/imprecise space and budget constraints are formulated for breakable items which get damaged due to the accumulated stress of heaped stocks during storage. Although the accumulated stress is a function of on-hand inventory and time, for simplicity, it has been assumed that the number of damaged items depend only on the current stock level. This dependency may be both linear and non-linear. Here shortages are not allowed. Also, in this paper, we consider that the demand depends on current stock but becomes constant when the stock falls below a certain level (i.e. Q1i for ith item). Similarly, we consider the breakability of the items up to a certain stock level (Q0i for ith item) and thereafter, there is no breaking of the items when the stock falls below the level Q0i . Both the crisp and fuzzy models have been formulated as average profit maximization problems with crisp and imprecise space and budget constraints respectively and finally the models are solved by using a gradient based non-linear programming technique (i.e. GRG) for general values of γ . In the case of fuzzy model, the problem is converted to a chance constrained programming problem using possibility/necessity measure of fuzzy event. The models are illustrated with numerical example and some sensitivity analyses on expected profit have been presented. 2. Chance constraint programming problem under fuzzy environment with possibility and necessity measures If A˜ and B˜ be two fuzzy subsets of real numbers < with membership functions µA˜ and µB˜ respectively, then taking degree of uncertainty as the semantics of fuzzy number, according to Liu and Iwamura [14], Dubois and Prade [15,16] and Zimmermann [17]:





Pos A˜ ∗ B˜ = sup min µA˜ (x), µB˜ (y) , x, y ∈ <, x ∗ y ,








where the abbreviation ‘Pos’ represents possibility measure and ∗ is any one of the relations >, <, =, ≤, ≥. On the other hand necessity measure of the fuzzy event A˜ ∗ B˜ is a dual of possibility measure. The grade of necessity of an event is the grade of impossibility of the opposite event and is defined as









Nes A˜ ∗ B˜ = 1 − Pos A˜ ∗ B˜ ,

˜ where the abbreviation ‘Nes’ represents necessity measure and A˜ ∗ B˜ represents complement of the event A˜ ∗ B. If the objective function does not contain any fuzzy parameter, then the basic technique of chance constrained programming in a fuzzy environment is to convert the possibility/necessity based constraints to their respective deterministic equivalents according to predetermined confidence levels. Following lemmas are used to solve the inventory model in this paper. Lemma 1. If a˜ = (a1 , a2 , a3 , a4 ) be a Trapezoidal Fuzzy Number (TrFN) with a1 > 0 and b be a crisp number, then Pos a˜ ≥ b ≥ α




iff

a4 − b a4 − a3

≥ α.

A. Saha et al. / Mathematical and Computer Modelling 52 (2010) 1771–1782

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Proof. It is clear from Fig. 1a that

 1  
 a4 − b Pos a˜ ≥ b =  a − a3   4

for a3 ≥ b for a3 ≤ b ≤ a4 otherwise.

0

Fig. 1a. Ranking a TrFN with a crisp number.

Since 0 ≤ α ≤ 1, therefore Pos a˜ ≥ b ≥ α iff




a 4 −b a4 −a3

≥ α.



Lemma 2. If a˜ = (a1 , a2 , a3 , a4 ) be a TrFN with 0 < a1 and b be a crisp number, then Nes a˜ ≥ b ≥ α




iff

b − a1 a2 − a1

≤ (1 − α) .

Proof. We have Nes a˜ ≥ b ≥ α




 ⇒ 1 − Pos(˜a < b) ≥ α ⇒ Pos(˜a < b) ≤ (1 − α) .

Fig. 1b. Ranking a TrFN with a crisp number.

From Fig. 1b it is clear that Pos a˜ < b =




b − a1 a2 − a1

≤ (1 − α) .

Therefore, Nes a˜ ≥ b ≥ α iff




b−a1 a2 −a1

≤ (1 − α).



3. Assumptions and notations A deterministic multi-item inventory model for breakable items with infinite rate of replenishment and stock dependent demand is developed under the following assumptions and notations (for ith item, i = 1, 2, 3, . . . , m): 1. 2. 3. 4.

The demand Di (qi ) is linear function and stock dependent. Breakable function Bi (qi ) is linear or non-linear function of current stock level. si be the selling price per unit item. CHi be the holding cost per unit item per unit time.

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Fig. 2. Graphical representation of proposed inventory system.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

CBi be the breaking cost per unit item. Cpi be the purchasing cost per unit item. C3i be the setup cost per replenishment. S1i be the required space per unit item. Ti is the duration of a complete cycle. Q0i and Q1i are inventory levels at time t0i and t1i respectively. Qi be the initial stock, i.e. replenishment size. qi (t ) be the on-hand inventory at time t. Shortages are not allowed. Lead time is negligible. L be the total available space. R be the total budget. Replenishment rate is infinite, but replenishment size is finite. The time horizon of the inventory system is infinite.

4. Mathematical model 4.1. Crisp model In our proposed inventory model, we consider the demand (which is stock dependent) and breakable functions for the ith item (i = 1, 2, 3, . . . , m) as follows: Di (qi ) =

d0i + d1i (qi − Q1i ), d0i , 0 ≤ qi ≤ Q1i



Q1i ≤ qi

(1)

and Bi (qi ) =



b(qi − Q0i )γ , Q0i ≤ qi 0, 0 ≤ qi ≤ Q0i

(2)

where qi = Q1i be the stock at time t = t1i and qi = Q0i be the stock at time t = t0i . Q0i is the amount of stock up to which the breakability of the items is considered and also we consider that Q0i lies in between Qi and Q1i , i.e. Q1i ≤ Q0i ≤ Qi . We assume that Qi be the initial stock and inventory reduces to zero at time Ti (see Fig. 2). Therefore, the differential equation for the model is dqi dt

= −Di (qi ) − Bi (qi );

i = 1, 2, 3, . . . , m

(3)

i.e.

( −d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ , Q0i ≤ qi = −d0i − d1i (qi − Q1i ), Q1i ≤ qi ≤ Q0i dt −d0i , 0 ≤ qi ≤ Q1i

dqi

(4)

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with

Z

Q0i

1

t0i =

−d0i − d1i (qi − Q1i ) − bi (qi − Q0i )γ

Qi

Z

Q1i

1

t1i =

−d0i − d1i (qi − Q1i )

Q0i 0

Z

1

Ti =

−d0i

Q1i

dqi

(5)

dqi

(6)

dqi .

(7)

The total breaking cost, for the ith item, is CBi θi (Qi ), where total number of breakable items over the period (0, T ) b(qi − Q0i )γ dqi

Q0i

Z

θi (Qi ) =

−d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Qi

.

(8)

Similarly, the total inventory holding cost, for the ith item, is CHi Gi (Qi ), where Q0i

Z

Gi (Qi ) =

−d0i − d1i (qi − Q1i ) − b(qi − Q0i

Qi

Q1i

Z

qi dqi

)γ

+ Q0i

qi dqi

−d0i − d1i (qi − Q1i )

Z

0

+ Q1i

qi dqi

(9)

−d0i

and the total selling price, for the ith item, is si ψi (Qi ), where

ψi (Qi ) =

Q0i

Z

Qi

[d0i + d1i (qi − Q1i )]dqi + −d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Z

Q1i Q0i

[d0i + d1i (qi − Q1i )]dqi + −d0i − d1i (qi − Q1i )

Z

0 Q1i

d0i dqi

−d0i

.

(10)

Therefore the average profit for the ith item is Zi (Qi ) = [Selling Price − (Setup Cost + Purchase Cost + Holding Cost + Breaking Cost)]/Total Time

  1  si ψi (Qi ) − C3i + Cpi Qi + CHi Gi (Qi ) + CBi θi (Qi ) .

=

(11)

Ti

Model 1: Summing the average profit for all items, the problem is: Maximize Z Z =

m X

Zi (Qi ) =

m X 1 

Ti i =1 i=1 subject to the constraints m X

si ψi (Qi ) − C3i + Cpi Qi + CHi Gi (Qi ) + CBi θi (Qi )





S1i Qi ≤ L

i =1 m

X

Cpi Qi ≤ R,

i =1

where L and R are total available space and budget respectively and Qi > Q0i > Q1i ; 0 < t0i < t1i < Ti with

Z

Q0i

1

t0i =

−d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Qi

Z

Q1i

1

t1i =

−d0i − d1i (qi − Q1i )

Q0i

Z

0

Ti = Q1i

1

−d0i

dqi

dqi

dqi .

Now, we considered the different forms of breakability function. Model 1a: Damage function is linear. Let Bi (qi ) =

b(qi − Q0i ), Q0i ≤ qi 0, 0 ≤ qi ≤ Q0i .



(In Eq. (2), it is assumed that γ = 1.)

(12)

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Using this form for Bi (qi ), we have

Qi (d1i + b) − (bQ0i + d1i Q1i − d0i ) t0i = log d1i + b Q0i (d1i + b) − (bQ0i + d1i Q1i − d0i ) Qi (d1i + b) − (bQ0i + d1i Q1i − d0i ) Q0i d1i − (d1i Q1i − d0i ) 1 1 log log t1i = + d1i + b Q0i (d1i + b) − (bQ0i + d1i Q1i − d0i ) d1i Q1i d1i − (d1i Q1i − d0i ) Qi (d1i + b) − (bQ0i + d1i Q1i − d0i ) 1 Q1i + 1 log Q0i d1i − (d1i Q1i − d0i ) + log Ti = d0i d1i + b Q0i (d1i + b) − (bQ0i + d1i Q1i − d0i ) d1i Q1i d1i − (d1i Q1i − d0i )  
1 d1i (bQ0i + d1i Q1i − d0i ) ψi (Qi ) = d0i Ti − d1i Q1i t1i + t0i + e−(d1i +b)t0i − 1 (d1i + b) d1i + b  


1 d1i Qi −(d1i +b)t0i − 1 + (d1i Q1i − d0i ) (t1i − t0i ) + − e ed1i (t0i −t1i ) − 1 − Q0i ed1i (t0i −t1i ) − 1 d1i + b d1i  
(bQ0i + d1i Q1i − d0i ) 1 Gi (Qi ) = t0i + e−(d1i +b)t0i − 1 (d1i + b) d1i + b  
(d1i Q1i − d0i )
Qi 1 −(d1i +b)t0i d1i (t0i −t1i ) − e −1 + (t1i − t0i ) + e −1 d1i + b d1i d1i  
Q0i d1i (t0i −t1i ) d0i 2 (Ti − t1i2 ) − e − 1 + (Q1i + d0i t1i )(Ti − t1i ) − d1i 2  

b(bQ0i + d1i Q1i − d0i ) 1 bQi θi (Qi ) = t0i + e−(d1i +b)t0i − 1 − e−(d1i +b)t0i − 1 − bt0i Q0i . (d1i + b) d1i + b d1i + b 1

With the above expressions of Gi (Qi ), ψi (Qi ), θi (Qi ), t0i , t1i and Ti , we derive the expression for profit function Zi (Qi ) from the expression (11). Model 1b: Damage function is non-linear. Let Bi (qi ) =



b(qi − Q0i )γ , Q0i ≤ qi 0, 0 ≤ qi ≤ Q0i .

(13)

(In Eq. (2), it is assumed that 0 < γ < 1.) With the above expression, we have

Z

Q0i

1

t0i =

−d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Qi

Z

Q1i

1

t1i =

−d0i − d1i (qi − Q1i )

Q0i

Z

0

1

Ti =

Q1i −d0i Z Q0i Gi (Qi ) = Qi

ψi (Qi ) =

Z

θi (Qi ) =

Z

Q0i Qi Q0i

Qi

dqi

dqi

dqi

+

−d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Q0i

[d0i + d1i (qi − Q1i )]dqi + −d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ b(qi − Q0i )γ dqi

−d0i − d1i (qi − Q1i ) − b(qi − Q0i )γ

Q1i

Z

qi dqi

Z

Q1i Q0i

qi dqi

−d0i − d1i (qi − Q1i )

Z

0

+ Q1i

[d0i + d1i (qi − Q1i )]dqi + −d0i − d1i (qi − Q1i )

Z

qi dqi

−d0i 0 Q1i

d0i dqi

−d0i

.

With the above expressions of Gi (Qi ), ψi (Qi ), θi (Qi ), t0i , t1i and Ti , we derive the expression for profit function Zi (Qi ) from the expression (11). In this model, we perform the integrations using Trapezoidal Rule of Numerical Integration for γ = 41 , 12 , 34 . (Please see Appendix(b).) 4.2. Fuzzy model Here in this model, we assume that the total budget and total available space are imprecise and both are represented by Trapezoidal Fuzzy Number (TrFN), i.e. R˜ = (R1 , R2 , R3 , R4 ) and L˜ = (L1 , L2 , L3 , L4 ) respectively.

A. Saha et al. / Mathematical and Computer Modelling 52 (2010) 1771–1782

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Model 2: Let the imprecise inventory model be defined as: Maximize Z Z =

m X

m X 1 

Zi (Qi ) =

Ti i =1 i=1 subject to the constraints m X

si ψi (Qi ) − C3i + Cpi Qi + CHi Gi (Qi ) + CBi θi (Qi )





S1i Qi ≤ L˜

i =1 m

X

Cpi Qi ≤ R˜ ,

i =1

where L˜ and R˜ are imprecise total available space and total budget respectively and Qi > Q0i > Q1i ; 0 < t0i < t1i < Ti with the expression of t0i , t1i and Ti same as in crisp model. Model 2a: When decision maker prefers to optimize the objective function w. r. to optimistic resource constraints, the above problem reduces to Maximize Z

) m X ˜ S1i Qi ≥ α1 pos L ≥ i=1 ( ) m X Cpi Qi ≥ α2 , pos R˜ ≥ (

subject to

i =1

L4 −

m P

S1i Qi

i =1

i.e.

≥ α1

L4 − L3

R4 −

m P

Cpi Qi

i=1

≥ α2 ,

R4 − R3

where α1 and α2 are confidence levels. Model 2b: On the other hand, when decision maker desires to optimize the objective function w. r. to pessimistic resource constraints, the above problem reduces to Maximize Z

( subject to

nes L˜ ≥

m X

)

( ≥ α3 ;

S1i Qi

nes R˜ ≥

i=1

( i.e.

pos L˜ <

m X

) S1i Qi

pos R˜ <

i=1 m P

i.e.

L2 − L1

≥ α4

Cpi Qi

X

) Cpi Qi

< (1 − α4 ) ,

i =1 m P

S1i Qi − L1

i=1

)

i=1 m

( < (1 − α 3 ) ;

m X

Cpi Qi − R1

i =1

≤ (1 − α3 ) ;

≤ (1 − α4 ) ,

R2 − R1

where α3 and α4 are confidence levels. 5. Numerical results 5.1. Crisp model To illustrate the models numerically, we assume the following values for the inventory parameters: s1 = Rs. 8,

s2 = Rs. 7.5,

CB1 = Rs. 5,

CB2 = Rs. 4.5,

d11 = 0.25,

d02 = 20,

S11 = 2,

S12 = 3,

C31 = Rs. 100, CH1 = Re. 0.7, d12 = 0.22,

L = 600,

C32 = Rs. 110,

Cp1 = Rs. 5,

CH2 = Re. 0.6,

Q01 = 70,

Q11 = 40,

b = 0.05, Q02 = 75,

Cp2 = Rs. 4.5, d01 = 25, Q12 = 45,

R = 1200.

Table 1 shows the optimum solutions of the crisp models for linear and different non-linear forms of damage function, i.e. for γ = 1 and other different values.

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Table 1 Optimum results for crisp models with crisp resource constraints and different breakabilities.

γ

∗ t01

∗ t11

T1∗

∗ t02

∗ t12

T2∗

Q1∗

Q2∗

Z∗

1

1.229 1.621

2.278 2.670

3.878 4.270

1.423 1.124

2.719 2.421

4.969 4.671

118.281 135.935

121.146 109.376

40.380 43.881

1.780

2.830

4.430

0.988

2.284

4.534

143.228

104.514

45.821

1.824

2.873

4.473

0.952

2.249

4.499

145.122

103.252

46.649

3 4 1 2 1 4

Table 2 (Particular cases). Optimum results for crisp model with crisp resource constraints with constant breakabilities (i.e. γ = 0) and without breakabilities (i.e. b = 0). Cases ↓

∗ t01

∗ t11

T1∗

∗ t02

∗ t12

T2∗

Q1∗

Q2∗

Z∗

Constant breakabilities (i.e. γ = 0) Without breakabilities (i.e. b = 0)

1.832 1.824

2.881 2.874

4.482 4.474

0.948 0.954

2.244 2.251

4.494 4.501

145.4 145.132

103.067 103.245

46.996 47.253

5.2. Fuzzy model For fuzzy model the imprecise parameters L˜ and R˜ are TrFN as follows: L˜ = (565, 590, 610, 635) and R˜ = (1165, 1190, 1210, 1235) and all other parameters are same as in crisp model. The maximum returns for optimistic and pessimistic resource constraints from Fuzzy Models have been calculated for the confidence level α1 = α2 = 0.9 and α3 = α4 = 0.09 respectively and the results are displayed in Table 3. Table 3 Optimum results for fuzzy models with optimistic (and pessimistic) resource constraints and different breakabilities.

γ

∗ t01

∗ t11

T1∗

∗ t02

∗ t12

T2∗

Q1∗

Q2∗

Z∗

1

1.263 (1.193) 1.676 (1.566) 1.842 (1.720) 1.858 (1.762)

2.313 (2.242) 2.725 (2.616) 2.891 (2.769) 2.908 (2.812)

3.913 (3.842) 4.325 (4.216) 4.491 (4.369) 4.508 (4.412)

1.501 (1.346) 1.191 (1.058) 1.050 (0.926) 1.043 (0.891)

2.797 (2.642) 2.487 (2.355) 2.346 (2.223) 2.339 (2.187)

5.047 (4.892) 4.737 (4.605) 4.596 (4.473) 4.589 (4.437)

119.929 (116.620) 138.684 (133.230) 146.365 (140.156) 146.875 (142.005)

124.214 (118.170) 111.711 (107.096) 106.590 (102.480) 106.250 (101.247)

40.446 (40.287) 44.245 (43.504) 46.316 (45.316) 47.190 (46.098)

3 4 1 2 1 4

Table 4 (Particular cases). Optimum results for fuzzy models with optimistic (and pessimistic) resource constraints with constant breakabilities (i.e. γ = 0) and without breakabilities (i.e. b = 0). Cases ↓

∗ t01

∗ t11

T1∗

∗ t02

∗ t12

T2∗

Q1∗

Q2∗

Z∗

Constant breakabilities (i.e. γ = 0) Without breakabilities (i.e. b = 0)

1.861 (1.771) 1.863 (1.769)

2.91 (2.82) 2.912 (2.818)

4.51 (4.42) 4.512 (4.418)

1.044 (0.886) 1.046 (0.891)

2.34 (2.182) 2.342 (2.188)

4.59 (4.432) 4.592 (4.438)

146.875 (142.289) 146.875 (142.097)

106.25 (101.057) 106.25 (101.185)

47.552 (46.429) 47.828 (46.689)

Note: The values without brackets represent the results for optimistic resource constraints and the values within brackets represent the results for pessimistic resource constraints.

6. Discussion The results in Tables 1–4 are as per the expectation. In Tables 1 and 3, average profit increases as breakability decreases. It is to be noticed that in Table 1, changes in the profit decreases as breakability decreases. Actually, for γ = 12 and γ = 14 , the breakable amounts are so small that it does not affect the profit much and for this reason, profits in these cases are all most equal to the profit for the model with constant breakability (γ = 0) presented in Table 2. Also Table 2 depicts that profit without breakability is more than the case with breakability, though the difference is not much. This small difference is due to the small value of ‘b’, the constant coefficient of breakability. As ‘possibility’ means the satisfaction of constraints in optimistic sense and ‘necessity’ is in pessimistic sense, the amounts of profits with possibility constraints are higher than the corresponding profits with necessity constraints for all values of γ . This behaviour is depicted in Tables 3 and 4. In all these cases also, the profits for the models with small breakability parameter and without breakability are almost same due to small value of ‘b’. 7. Sensitivity analysis For the above models, three types of sensitivity analyses are performed. First, for the crisp model, the effect of changes in the value of the coefficient of breakable function b on the maximum average profit while holding the other parameters

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remain unchanged. Second, for the crisp model also, the effect of changes in the value of the stock Q01 (when Q02 is fixed) on the maximum average profit while holding the other parameters remain unchanged and next third for the fuzzy model, the effect of changes in the confidence level on the optimum returns of the average profit for optimistic and pessimistic constraints. 7.1. The effect of changes in the value of the coefficient of breakable function b on the maximum average profit (Table 5 and Fig. 3) In the crisp model, when the coefficient of breakable function b is increased by 20%, the maximum average profit goes down approximately by 2.94%, 1.47%, 0.62%, 0.26% for γ = 1, γ = 43 , γ = 12 , γ = 14 respectively. Similarly, when the coefficient of breakable function b is decreased by 20%, the maximum average profit goes up approximately by 3.04%, 1.49%, 0.62%, 0.26% for γ = 1, γ = 34 , γ = 12 , γ = 14 respectively. The maximum average profits for different values of b in

γ = 1, γ = 34 , γ =

1 2

and γ =

1 4

are as follows in Table 5 and Fig. 3.

Table 5 Sensitivity analysis on maximum average profit w.r.t. breakability coefficient.

γ 1 3 4 1 2 1 4

b 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

45.682 46.565

44.239 45.878

42.889 45.202

41.608 44.536

40.380 43.881

39.194 43.236

38.172 42.601

37.348 41.975

36.668 41.356

46.973

46.684

46.395

46.107

45.821

45.535

45.250

44.966

44.682

47.141

47.018

46.895

46.772

46.649

46.527

46.404

46.282

46.159

Fig. 3. Graphical representation of Table 5.

The graphical representation (Fig. 3) of Table 5 shows that the effect of changes in the value of the coefficient of breakable function b on the maximum average profit is much greater for γ = 1 than others. Also, it is clear from Fig. 3 that changes in the value of the coefficient of breakable function b has not so much effect on the maximum average profit for γ = 14 though for all 4 cases increase in the value of b shows the decrease of profit as well as decrease of the value of b shows the increase of profit. This behaviour of profit function w.r.t. breakability is normal and as per expectation. 7.2. The effect of changes in the value of the stock Q01 (when Q02 is fixed) on the maximum average profit (Table 6 and Fig. 4) In the crisp model, when the stock Q01 , keeping Q02 fixed, is increased by 10%, the maximum average profit goes up approximately by 2.67%, 1.04%, 0.36%, 0.12% for γ = 1, γ = 43 , γ = 21 , γ = 14 respectively. Similarly, when the stock Q01 , keeping Q02 fixed, is decreased by 10%, the maximum average profit goes down approximately by 3.04%, 1.16%, 0.39%, 0.12% for γ = 1, γ = 34 , γ = 12 , γ = 14 respectively. The maximum average profits for different values of b in

γ = 1, γ = 34 , γ =

1 and γ = 14 are as follows in Table 6 and Fig. 4. 2 The graphical representation (Fig. 4) of Table 6 shows that the effect of changes in the value of the stock Q01 , keeping Q02 fixed, the maximum average profit is much greater for γ = 1 than others. Also, it is clear from Fig. 4 that changes in the

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Table 6 Sensitivity analysis on maximum average profit w.r.t. q01 (when Q02 is fixed).

γ

Q01 42

49

56

63

70

77

84

91

98

3 4

34.354 41.531

36.165 42.202

37.755 42.815

39.152 43.374

40.380 43.881

41.458 44.339

42.404 44.749

43.230 45.116

43.948 45.440

1 2

45.037

45.252

45.454

45.644

45.821

45.985

46.136

46.276

46.403

1 4

46.408

46.471

46.532

46.592

46.649

46.704

46.757

46.807

46.854

1

Fig. 4. Graphical representation of Table 6.

value of the coefficient of breakable function b has not so much effect on the maximum average profit for γ = 14 , because this is, in graph, almost a straight line though for all 4 cases increase in the value of Q01 shows the increase of profit as well as decrease of the value of Q01 shows the decrease of profit. 7.3. The effect of changes in the confidence level α on the optimum return of the average profit for optimistic and pessimistic constraints (Figs. 5 and 6) The graphical representation for the Fuzzy Model presents the maximum average profit vs. possibility and maximum average profit vs. necessity for γ = 12 in Figs. 5 and 6 respectively. In both cases, as expected, profit decreases with the increase of confidence levels.

Fig. 5. Profit vs. confidence level in possibility.

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Fig. 6. Profit vs. confidence level in necessity.

8. Concluding remarks This paper may be useful for the research in marketing and operations managements. As, according to marketing norms, demand varies with the displayed quantity, inventory management of breakable items has been considered here incorporating the effect of displayed inventory on demand. Again, as items made of glass, ceramics, china clay, mud etc. are of special type and get damaged due to heaped stock, the damage function is assumed accordingly. In this paper, a multi-item inventory model is developed for breakable items with both linear and non-linear damaged function and stock dependent demand. As the methods used here are quite general, this approach can be used for the other inventory models with price discount—All Unit Discount (AUD), Incremental Quantity Discount (IQD) or Combination of both, allowing shortages, etc. Appendix (a) For a realistic inventory model, at any time t, the number of consumed units must be greater than the number of damaged/breakable units, otherwise the purpose of inventory will be defeated. Hence, we should have Di (qi ) > Bi (qi ) i.e. d0i + d1i (qi − Q1i ) > b(qi − Q0i )γ . Now it is clear that γ cannot be negative, because if γ is negative then decrease of current stock level creates more damage of items, but this is impossible. Again, it is clear from the above expression that the inequality holds if γ ≤ 1. So, 0 ≤ γ ≤ 1. (b) In Model 1b the integrations are solved by using the Trapezoidal rule of Numerical Integrations by taking 4 intervals, i.e. 5 ordinates. Rb Let us consider, we have to find out a f (x) dx, then applying the rule we get b

Z

f (x) dx = a

( b − a) 8



 

f (a) + f (b) + 2 f

3a + b 4



 +f

a+b 2



 +f

a + 3b



4

and taking this formula we evaluate all of the integrals in Model 1b. References [1] R.I. Levin, C.P. Mclaughlin, R.P. Lamone, J.F. Kottas, Production/Operations Management: Contemporary Policy for Managing Operating System, McGraw-Hill, New York, 1972, p. 373. [2] A. Schary, R.C. Becker, Distribution and final demand. The influence of availability, Mississippi Valley Journal of Business and Economics 8 (1972) 17–26. [3] H.B. Wolef, A model for control of style merchandise, Industrial Management Review 9 (1968) 69–82. [4] M. Deb, K.S. Chaudhuri, An EOQ model for items with finite rate of production and variable rate of deterioration, Opsearch 23 (1986) 175–181. [5] M. Deb, K.S. Chaudhuri, A note on the heuristic for replenishment of trended inventories considering shortages, Journal of Operational Research Society 38 (1987) 459–563. [6] K.J. Chung, P.S. Ting, A heuristic for replenishment of deteriorating items with linear trend in demand, Journal of Operational Research Society 44 (12) (1993) 1235–1241. [7] S.K. Goyal, Gunasekaran, An integrated production–inventory marketing model for deteriorating items, Computers & Industrial Engineering 28 (1995) 755–762.

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[8] M.A. Hariya, L. Benkherrouf, Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand, European Journal of Operational Research 79 (1994) 123–137. [9] M. Mandal, M. Maiti, Inventory model for damageable items with stock dependents demand and shortages, Opsearch 34 (3) (1997) 155–166. [10] M. Mandal, M. Maiti, Inventory of damageable items with variable replenishment rate, stock dependent demand and some units in hand, Applied Mathematical Modelling 23 (1999) 799–807. [11] M. Maiti, M. Maiti, Inventory of damageable items with variable replenishment and unit production cost via simulated annealing method, Computers & Industrial Engineering 49 (2005) 432–448. [12] M. Maiti, M. Maiti, Production policy for damageable items with variable cost function in an imperfect production process via genetic algorithm, Mathematical and Computer Modelling 42 (2005) 977–990. [13] M. Maiti, M. Maiti, Multi-item shelf-space allocation of breakable items via genetic algorithm, Journal of Applied Mathematics and Computing 20 (1–2) (2006) 327–343. [14] B. Liu, K. Iwamura, A note on chance constrained programming with fuzzy co-efficients, Fuzzy Sets and System 100 (1998) 229–233. [15] D. Dubois, H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information Sciences 30 (1983) 183–224. [16] D. Dubois, H. Prade, Fuzzy Sets and System—Theory and Applications, Academic, New York, 1980. [17] H.J. Zimmermann, Fuzzy, in: Fuzzy Sets and System—Theory and Applications, second ed., Kluwer, Boston, 1991.