Computational Study for Disrupted Production System with Time Dependent Demand

Computational Study for Disrupted Production System with Time Dependent Demand

Uttam Kumar Khedlekar^1*, Diwakar Shukla² and RPS Chandel³

1, 2 Department of Mathematics and Statistics, Dr. Harisingh Gour Central University, Sagar, M P, India, 470003

3Department of Mathematics, Government Vivekananda College Lakhnadon, M P, India Received 4 July 2012, revised 02 September 2013, accepted 26 January 2014

A production system often gets disrupted due to uncertain and un-planned events like labor problem, inadequate manufacturing, machine breakdown, power supply failure etc. and manufacturer fails to deliver the product to retailers well in time. This may reduce the reliability and faith of customers on the company/product. So, manufacturer needs to study the variation of demand and customer arrival pattern before the system gets disrupted and readjust the reproduction time according to disruption. A flexible managerial decision policy for disruption based production system is required at this juncture. The paper considers the same and incorporates variable demand with the constant production rate. We have solved the disruption problem analytically to determine the production period before and after disruptions. Both increasing and decreasing trend in production run has been studied for deteriorating item. A numerical example and simulation study appended for sensitivity analysis in order to find which parameter is responsible for significant changes in disrupted production system.

Keywords: Inventory,Production run, Disrupted production system, Replenishment time, Deterioration, Shortage.

AMS Subject Classification: 90 B 05, 90 B 30, 90 B 50.

Introduction

Control and maintenance of the production system have attracted attention of inventory managers.

Many reasons exist for disruptions in a production system like machine failure, supply chain disruption, unexpected events or unwanted crises. An oil drilling company may be disrupted due to electricity supply, failure of drilling machines whereas an oil refining company may face problem of crude oil supply, availability of other raw materials, earthquake strike.

The quantity received of products by the firm may differ from the ordered quantities, which also creates uncertainty in the system. The classic EOQ model does not include chances of disruption in supply.

Parlar and Berkin (1991) modeled for the economic ordered quantity under disruption in which demand was deterministic. Berk and Antonio (1994) showed the cost function in Parlar and Berkin (1991) was uncorrected and provided the better model. Lin and Kroll (2006) solved the production problem under an imperfect production system subject to random machine breakdowns. Ma et al. (2010) revisited the

same idea with assumption that after a period the production process may shift to an out-of-control state at random time, machine produces defective items, and could not be repaired. Mishra and Singh (2011) considered Weibull distribution for deterioration and analyzed the model in different situations.

Chen and Zhang (2010) with three-echelon supply chain system consisting of supplier, manufacturer and customers. Teng and Chang (2005) suggested an economic production quantity model for deteriorating items when the demand rate depends not only on-display stock, but also on the selling price per unit of item which may be influenced by economic policy, political scenario or agriculture productivity or both get affected. Further Qi et al. (2004) analyzed the supply chain-coordination under demand disruption in deterministic scenario. The supply shortages for managerial purpose were investigated due to Yang et al. (2005) and they obtained solution by greedy method. A number of structural properties of the inventory system analytically presented by Samanta and Roy (2004) by the determination of production cycle time and backlog for deteriorating item. Chandel and Khedlekar (2013) solved the

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*Author for correspondence Email: uvkkcm@yahoo.co.in

inventory problem by using two and three warehouse setups at different locations. Kumar and Sharma (2012a) formulated the replacement policy for perishable item by using queuing theory approach and presented an optimal policy.

A central policy presented by Benjaafar and EIHafsi (2006) contains a single product assemble-to-order system for components, an end–product to serve to customer classes Khedlekar and Shukla (2013) presented the dynamic pricing policy with logarithmic demand. Hendricks and Singhal (2005) studied the average abnormal stock returns of firm who experienced high penalty.

The supply chain disruption risk is also measured by Tomlin (2006) and presented a flexible solution for disruptions. Exponential demand for deteriorating product has been considered by Ouyang et al.

(2005). We refer some other useful contributions, due to Howick and Eden (2001), Shukla et al.

(2010 & 2012), Shukla and Khedlekar (2009, 2014), Khedlekar (2012), and He et al. (2010), Weiss and Rosenthal (1992), Kumar and Sharma (2012b). This paper is intended to formulate the model for inventory problem with disruption in production system after certain duration. The demand considered as linear which follows most of the consumable and fashionable items. We shall investigate the impact of demand variation on disruption factor. There may be other reasons for uncertainty but we consider only disruptions in production run.

Assumptions and Notations

Suppose deteriorating items are manufactured by a manufacturer and sold to customers. The demand arising from the market is linear at rate (a + bt), the production rate is constant p>a>0, in each cycle. Due to this inventory accumulates at rate (p - (a + bt)) till time horizon H. If production stops at the time (Tp) and thus there after inventory depicted due to the demand and deterioration rate. Because of uncertainty and unplanned events, production run disrupts at time Td < Tp. After disruption manufacturer needs to compute the production period with disruptionT_p^d. Therefore, some shortage may occur in the system before time H. Lead time is assumed negligible and deterioration rate is constant. Following notations bearing the concepts utilized in the discussion.

H : Time horizon.

p : Production rate.

θ : Rate of deterioration.

a : Initial demand of item.

b : Parameter of demand governing increasing (b > 0) or decreasing (b < 0) trend, b < a.

D(t) : Demand of items D(t)=(a + bt), where p > a > 0.

Tp : Production time without disruption.

Td : Production disruption time when system get disrupted.

pd

T ^: New production period with disruptions.

Tr : Time of placing the order when shortages occur in the system.

Qr : Order quantity (shortages) for placing the order when shortage occurs.

Ii(t) : On hand inventory at time t, for i = 1, 2, 3, 4.

Production Model without Disruption

To compare the proposed model first we design the model without disruption. Suppose management optimizes the production system may run without disruption with production rate p (per unit time). With normal production run manufacturer decides to stop it at time T and, no need to produce the items till time H. Inventory depicted due to demand rate (a bt ) and there is deterioration rate θ of items. The inventory depletion of normal production is shown in fig.1- I). The differential equations for two different periods [0, Tp] and [Tp, H] satisfy throughout the domain given in equation (1) and (2).



( )



) ( )

( ₁

1 t I t p a bt

dtI

d     _, 0tT_p,

boundary condition I₁

 

0 0 … (1) ),

( ) ( )

( ₂

2 t I t a bt

dtI

d    _Tp ≤_t≤_H,

boundary condition I₂

 

H 0 … (2) On solving equation (1) and (2) with boundary conditions and usingI₁(T_p) = I₂(T_p), we get







 ^



p

b H b a e b a

e ^Tp (p )  ^H( ²  ²  )



Or 











p

b H b a e b a

Tp 1log (p )  ^H( ² ²  )

 … (3)

Corollary 1

Production time without disruption is increasing order in linearly decreasing demand, constant and in linearly increasing demand.

i.e.

 

Tp b_0 ^

 

Tp b_0 ^

 

Tp _b0

So the production time without disruption is highly affected by demand variations.

Corollary 2

If



a^3b^3Hb



ape^H b^2, then Tp (for all b), increases over θ.

i.e.



^{3 3 2}



3 _p 0,

H p

T

e a b H b b ap

dT

d p e





  

 

   

  for



a^3b^3Hb



ape^H b^2 … (4)

Corollary 3

Rate of change of production time with respect to deterioration is lesser in constant demand than in linearly increasing demand.

i.e.

0 0

0  

 























b p b

p b

p

d dT d

dT d

dT







Note 1: As θ increases, the optimal production time Tp also increases which reveals that manufacturer has to produce more items when deterioration rate is high.

Proposed Production Model with Disruption In section 3, the production rate was constant but in real life practice the production system often disrupted due to unavoidable events and crises and thus we considered that after a duration the production system disrupted and little changed byΔPand disruption time is Td (see fig. 1-II).

IfΔP0, then the production rate decreases and, ifΔP0, then the production rate increases. So, in all we assume that disrupted production rate is



pΔp



0.

Lemma 1

If: ^{ p}



^{(pab e) H(p a bH b  )}



 

/



e^TdH 



,then manufacturing system satisfies the demand even production system get disrupted.

If  ^{p   p}



^{(pab e) H(p a bH  )b}



 

/ e^TdH  , then production system is unable to satisfy the demand and there will be shortage due to

disruption.

Proof

Suppose the production system disrupts at time Td (see fig. 1-II). Thereafter the production rate will

Δp

p . Now two differential equations in intervals [0, Td] and [Td, H] are

), ( ) ( )

( ₁

1 t I t p a bt

dtI

d     0≤_t≤_Td, at

boundary condition _I1

( )

_{0 =0} … (5) ),

( )

( )

( ₂

2 t I t p p a bt

dtI

d      T_d ≤t≤H … (6)

With boundary condition

1( )_d 2( )_d p a bT^d b2 ^Td p a b2

I T I T e ^

   

     

      

Solving equation (6) with boundary condition,

2( ) p ^{Td t} p a b2 ^t p p a bt b2

I t  e^{ }   e^  

       

     

If I₂

 

H  0, we say production system satisfies the linear demand of items.

It provides^{ p}



^{(pab e) H}



^{(p a bH  )b}

 

 

/ e^TdH 

Now manufacturer satisfies the demand.

If I₂

 

H  0, production system does not satisfy the linear demand of items.

It provides ^{( )   p p}



^{(pab e) H(p a bH  )b}



 

/ e^TdH

At this stage due to disruption shortages occurs in the system.

This proves the lemma.

Note 2: If I₂

( )

H ≥0, we can find optimal production time (with disruption)

 

T p^d such that at time H, the entire stock will be sold-out and inventory level will be zero.

Note 3:IfI₂

( )

H _<0, shortage occurs in the system and we can find optimum time Tr for placing the order of similar items from competitor and can find the respective order quantity Qr.

Lemma 2

If I₂

( )

H ≥₀, then the production time with disruption

 

Tp^d is obtained by

1log ( )

( )

d p

T H

d p a b p e e a b H b

T p p

 

    

 

        

    

… (7)

Proof

If I₂

( )

H ^≥₀

or ^{ p}



^{(pab e) H (p a bH b  )}



 

/ e^TdH  that is on-hand inventory I₂

 

H 0 exists. So, we can find out when to stop the production after disruption in such a manner that stock remains zero at time H. The optimal time T_p^d is in fig. 1-I. Two differential equations for intervals [Td ,Tpd] and [Tpd, H] are given in equations (8) and (9).

), ( )

( )

( ₂

2 t I t p p a bt

dtI

d      _Td ≤_t≤_Tp^d

… (8) With boundary conditions

1( )_d 2( )_d p a bT^d b2 ^Td p a b2

I T I T e^

   

      

      

), ( ) ( )

( ₃

3 t I t a bt

dtI

d    _Tp^d ≤_t≤_Hand boundary conditionI₃

 

H 0 … (9)

Solving (8) and (9) with boundary conditions and usingI₂(T_p^d)I₃(T_p^d), we get

) (

) (

p p

b H b a e e p b a

e ^Tdp p ^{Td H}

















 











 ^{ }



or

1log ( )

( )

d p

T H

d p a b p e e a b H b

T p p

 

    

 

        

    

… (10) This proves the lemma.

Corollary 4

If ^{ p}



^(p

^

^a

^

^{b e) H }

^

^{(p a bH  )b}



 

/  e

^TdH

 

_thenTp^dis increases over Td . by equation (10)

( ) 0,

p d

d

d T

T H

d

dT p e

dT p a b p e e a b H b



 



    

  

      

for



pb



a … (11)

So, the duration T_p^dincreases over Td.

Corollary 5

Production period with disruption is lower in for linearly decreasing demand compared to constant and increasing demand.

i.e.

 

T_p^d _b0 ^

 

T_p^d _b0^

 

T_p^d _b0

So, the production period with disruption highly depends on demand variations.

Lemma 3

If ^{( )   p p}



^{(pab e) H (p a bH b  )}



 

/ e^TdH  , manufacturer could not fulfill the demand and there will be shortage. Now replenishment time Tr and order quantity Qr are

a, p p T_r pT^d







  for 1



r



^Tr



^{H Tr}



r b H T e b e

Q ^{  }

 ^{  } ^{ }

 ₂ 1 … (12)

Proof

If I₂

( )

H <0, then production system does not fulfill the exponential demand,

Fig. 1—Production System

Or when ^{( )   p p}



^{(pab e) H(p a bH b  )}



 

/ e^TdH  , shortage occurs in the system. Suppose Tr and Qr (see fig. 2) are time of placing an order and amount of ordered quantities respectively.

Solving equation (8) and put I2(Tr)=0, we can find Tr with the help of following equation

2( )_r p p a bT^r b2 ^Tr p a b2 p ^{Td Tr} 0

I T e^ e^{ }

    



        

      

If  1_{, then}

a p p T_r pT^d







  … (13)

The differential equation for this situation is ),

( )

( )

( ₃

3 t I t p p a bt

dtI

d      T_r ≤t≤H_, boundary conditionI₃

 

H 0 … (14) Solving (14), we have

  ^{ }  

3( ) p p a 1 ^{H t} b ^t b2 1 ^{H t}

I t  e^{ }  H t e^  e^{ }

  

     

Then Q_r I T3

 

_r p p a



1 e^{H Tr}



   

  



_r



^Tr 2



1 ^{H Tr}



b H T e ^ b e^{ }

 ^  ^

    … (15)

This proves the lemma.

Application and Sensitive Analysis

For application we assume case when p = 350, a = 300, Δp = -100, θ = 0.022, H = 20 days and Td = 3days. On applying the proposed model we get I2(H) < 0 and by equations (3) to (15), we get Tp = 5.66383, T_p^d 6.691, Tr = 18 and Qr = 73.89 units.

We simulates model assuming the full operation limit p = 625 units per day. But at the beginning, the manufacturer could not attain 100% production and

disruption occurs in the production system. Model performance and sensitiveness for model parameters like capacity limit (p), demand (b) and time Td are shown in table 1.

As per table 1, demand parameter b highly affects the model output parameters. Little increment in b produces big gap between deman and production which could be managed by increment in production rate (∆p = 100 units per day). Combined effect of both the production time increases linearly and production period after disruption followed the same. Due to high demand of product, manufacturer manages by increasing the producction run and so manager has less on-hand inventory (I₃(H)) at the end of cycle time H (see fig 3). Similarly, on increasing T , the _p model output parameter related to on-hand inventory I3(H) decereases sharply.

Full capacity of production run (100%) assumes 625 units per day but the production capacity increases everyday from 56% to 100% (that is from 375 to 625 units). Although disruption occurs simultaneously and due to this there is downfall in production by 50 units per day. The combined effect reduces the production time Tp (see component of fig.4), and also reduces the production time after

Fig. 2—Production system after disruption, T_p^d =H

Fig. 3—T_pwith repect to b

Fig. 4—T_p^d,T_pand Tr with repect to p

disruption T_p^d(see component of fig. 4). Due to production period with disruption T_p^d occur earlier resulted that it required less quantity but later (see component of fig.4). This reveals that increasing the capacity of production run helps to manage the system often it disrupted.

According to table 1, production period with disruption (T_p^d) for constant demand (when b = 0) is lesser than the linearly increasing demand (when b > 0).

The same has been reported by corollary 5.

It reveals that high demand needed to produce more quantity even the system get disrupted and managers

has to take decision accordingly to the demand pattern. From table 2, the compound result illustrates by positive and negative value both of ∆p. On increasing the positive value of ∆p the replenishment time increases (see fig. 5) that is after long time we have to order for shortage. But for increasing negative values, we have to order earlier with less quantity and, after a certain time (25^th day) order quantity almost constant.

Case of Stochastic Demand

In whatever covered earlier, the demand function was D(t)= (a + bt), p > a > 0 and b is real, the demand function is deterministic. However, in real

Table 1—Sensitiveness with respect to capacity limit and other parameters

P Capacity Limit

(%)

b θ T_d ^ΔP T_p T_p^{d I2(H) Tr Qr}

350 56% 10 0.022 7 100 12.80 11.57 129 - -

350 56% 8 0.022 7 100 11.63 10.64 133 - -

350 56% 6 0.022 7 100 10.43 09.68 137 - -

350 56% 4 0.022 7 100 09.20 08.71 142 - -

375 60% 0 0.01 1 -50 12.11 13.71 -44 18 74

400 64% 0 0.01 1 -50 11.39 12.79 -42 16 50

425 68% 0 0.01 1 -50 10.76 11.99 -40 19 294

450 72% 0 0.01 1 -50 10.19 11.28 -38 20 488

475 76% 0 0.01 1 -50 09.68 10.65 -36 21 728

500 82% 0 0.01 1 -50 09.22 10.09 -35 22 1014

525 84% 0 0.01 1 -50 08.80 09.58 -33 23 1345

550 88% 0 0.01 1 -50 08.41 09.12 -31 24 1721

575 92% 0 0.01 1 -50 08.06 08.71 -29 25 2141

600 96% 0 0.01 1 -50 07.74 08.33 -27 26 2604

625 100% 0 0.01 1 -50 07.44 07.98 -25 27 3110

Table 2—Comparison with increasing / decreasing trend in production run

p a Td Δp Tp d

Tp ^{I2(H) Tr Qr}

350 100 4 -25 6.10 6.26 I2(H)<0 20.1 22

350 100 4 25 6.10 5.96 49.47 - -

350 100 4 -50 6.10 6.45 I2(H)<0 20.2 39

350 100 4 50 6.10 5.84 74.47 - -

350 100 4 -75 6.10 6.67 I2(H)<0 20.4 69

350 100 4 75 6.10 5.73 99.47 - -

350 100 4 -100 6.10 6.93 I2(H)<0 20.5 74

350 100 4 100 6.10 5.63 124.47 - -

350 100 4 -125 6.10 7.26 I2(H)<0 24 490

350 100 4 125 6.10 5.54 149.47 - -

350 100 4 -150 6.10 7.66 I2(H)<0 26 582

350 100 4 150 6.10 5.47 174.47 - -

350 100 4 -200 6.10 8.86 I2(H)<0 35 696

350 100 4 200 6.10 5.33 224.47 - -

life situation, often the demand function is of stochastic nature. Let us consider the following probability distribution for intial demand (a) and correspondingly the time period T_pand T_p^d have been computed in table 3, under assumption that production run increased by ∆p = 100 units at time

Td= 5 days.

Here ¹⁰ 1

1



^ 

 i

i i

P ,

Then Expected ( ) ¹⁰ 8.78

1









^

 i

i

i pi

p

p E T T p

T days

Expected ( ) ¹⁰ 7.96

1









^

 i

i i

d d

d E T T p

T_{p p pi} days

Then production period with disruption (T_p^d) is marginally reduced for stochastic demand than production period without disruptions. The unceratianity in demnad causes more problem even system get disrupted.

Conclusion, Recommendations and Future Research We have proposed a production-inventory model for deteriorating item with production disruption and analyzed the system under different situations. The suggested model helps to manufacturers to take decision while disruption occurs and they may reduce loss in term of revenue as well as in terms of reliability of product in order to safeguard the faith of customer for brand of the product. We have calculated the different time durations of the system and found that the impact of linear demand on disrupted production system differs from constant demand rate.

Further, it is observed that demand parameter highly affects the production policy in terms of production period before and after disruptions. Linearly decreasing demand is easy to manage rather than linearly increasing and constant. So, the performance of suggested disruption based production system is robust under demand variations and model uncertainties.

The combination of two strategies, one increasing and other decreasing, in a production run have shown difference among the production time before and after disruption and shortage. If demand rate has increasing trend then management needs to order more items from the spot market. Beside this, if the demand rate decreases, it needs to stop the production even earlier.

An increase in capacity planning is a good strategy for manufacturer when system gets disrupted.

Manufacturer has to make policy to keep balance between disruption and constant increment in capacity. If production rate increases (as ∆p positive) then the production period with disruption is shorter than earlier.

The proposed model may further be extended to the case of more realistic assumptions like stochastic demand, time dependent disrupted production process etc. One can consider time dependent deterioration rate and could extend the same in the fuzzy environment.

Acknowledgement

The authors are thankful to referee(s) for fruitful comments which helped to improve the quality of the manuscript.

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