## Simulation of Economic Production Quantity Model for Deteriorating Item

**Uttam Kumar Khedlekar**^{1,}, **Diwakar Shukla**^{1}

^{1}Department of Mathematics and Statistics Dr. Hari Singh Gour University Sagar, M. P., India

### Abstract

A manufacturing business may be affected due to disruption produced either in production system or in raw material supply. The disruptions in production system occur due to appearance of uncertainty and unplanned events. For example, skilled labor problem, machine failure problem, system repair problem etc. Inventory managers need to monitor the raw material supply and anticipate the shortage before the system gets disrupted. The deterioration among produced items is also a worry factor for managers. In this paper, we suggest a production inventory model subject to condition of occurrence of disruption at input level and deterioration at output level. A comparative approach has been adopted between models with and without disruption. The main focus is on specific type disruption like sudden reduction in supply of raw material. Its effect has been examined in comparative approach. Parametric simulation is used to generate graphs and useful suggestions are made for inventory managers.

### At a glance: Figures

**Keywords:** disrupted production system, inventory control, shortage, deterioration

*American Journal of Modeling and Optimization*, 2013 1 (3),
pp 25-30.

DOI: 10.12691/ajmo-1-3-1

Received December 14, 2012; Revised May 26, 2013; Accepted May 27, 2013

**Copyright**© 2014 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Khedlekar, Uttam Kumar, and Diwakar Shukla. "Simulation of Economic Production Quantity Model for Deteriorating Item."
*American Journal of Modeling and Optimization*1.3 (2013): 25-30.

- Khedlekar, U. K. , & Shukla, D. (2013). Simulation of Economic Production Quantity Model for Deteriorating Item.
*American Journal of Modeling and Optimization*,*1*(3), 25-30.

- Khedlekar, Uttam Kumar, and Diwakar Shukla. "Simulation of Economic Production Quantity Model for Deteriorating Item."
*American Journal of Modeling and Optimization*1, no. 3 (2013): 25-30.

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### 1. Introduction

There are many reasons that a production system gets disrupted like machine breakdown, unexpected events or emergency crises. An oil-drilling company may be disrupted due to failure in electricity supply, failure of drilling machines whereas an oil refining company faces problems of uneven crude oil supply, non-availability of other raw materials or earthquake and strike etc. It is assumed that a process is in control at the beginning of a production run. Suddenly the supply of raw material reduction by a constant factor, in the whole time cycle, production rate reduced. The modified production run ends at the same time as was before disruption. The change in optimal parameters before and after disruption has been examined in this paper.

Lin and Kroll (2006) one pioneer to solve the production problem under an imperfect production system subject to random breakdowns. Liao (2007) established an EPQ model by giving permission to delay in payment for the buyer to manufacturers.

A single vendor and multi-buyer inventory policy for a deteriorating item was due to Yang and Wee (2002). Teng and Chang (2005) presented an economic production quantity model for deteriorating items when the demand rate depends on not only on-display stock, but also on the selling price per unit of an item which may influence by economic policy, political scenario or agriculture productivity or both get affected. A similar approach has been followed by Hou and Lin (2006) on the deterministic economic order quantity model by taking into account the inflation and the time value of money for the deteriorating items with price and stock-dependent selling rate. By dividing the demand rate into multiple segments, Shukla and Khedlekar (2010*a*) have introduced three-component demand rate for the newly launched deteriorating item.

Joglekar (2003) used a linear demand function with price sensitiveness and allowed retailers to use a continuous increasing price strategy in an inventory cycle. He derived the retailer’s optimal profit by ignoring all the inventory costs. His results are restricted to growing market only, neither for stable market nor for a declining market. Expenditure sources like ordering cost, safety features, lead time and numbers of lots are the integral parts of decision making.

A number of structural properties of the inventory system are studied analytically by Samanta and Roy (2004) by determination of production cycle time and backlog for deteriorating item, which follows an exponential distribution. Qi *et al.* (2004) analyzed the supply chain-coordination with demand disruption in a deterministic scenario. Giri *et al.* (1996) who computed the optimal policy of an EOQ model with dynamic costs. The model they proposed is very basic though, since they have considered the very special case where the holding and ordering costs are linear functions of time. The other shortcoming of that paper is that the deterioration rate is also a linear function of time, and the algorithm they proposed in order to solve the problem is only valid as long as the demand rate is a linear function of time.

A central policy presented by Benjaafar EIHafsi (2006) specify a single product assemble-to-order system for my components, an end–product to serve and customer classes and problem solved as a Markov decision process and characterize the structure of an optimal policy.

We refer some useful contribution to reader Shukla and Khedlekar (2012), Lo (2007) and Khedlekar and Shukla (2013), Shukla *et al*. (2009, 2010*b*, 2010*c*). He *et al.* (2010) obtained optimal production time to facilitate the manufacturer sell the item in multiple markets by considering constant demand rate, but they do not readjust the production system. Due to above contribution we incorporate the deteriorating factor with constant demand and adjust the disrupted production system with shortages and when it occurs an optimal time of placing an order is obtained along with order quantity from the spot market.

### 2. Model Description

Suppose a deteriorating item, manufactured by a manufacturer, sold to customers. It has demand rate production rate *p>**µ* in each cycle. Since *p - µ>0 *so* *inventory accumulates at manufacture’s level and stops production at time *T*_{p} due to excess stock of that item. Now inventory reduces constantly until entire stuck vanishes at time *H** *(*T*_{p}<*H*), it is normal phenomenon for any production cum storage system.

Now assume that disruption occurs in this process at time *T*_{d} (*T*_{d }<* T*_{p}) due to assignable causes (like strike, lake of raw material) and due to random causes (like failure of power, fatique) etc. We consider the disruption in production as sudden cut off in the supply of raw material by a fixed amount say. This reduces *p** *into *p+* ( may be positive or negative) at time* T*_{d}.

Now this cut off remain maintained constantly throughout the production cycle time *H*. the inventory curve (Figure 2) bears a shift in shape due to this region of disruption. The time of maximum inventory shifts to (>*T*_{p}) and maximum level of inventory reduces accordingly so as to finish up the production cycle at same time H (or *H*). This disruption may minor or major. In case of minor level, the inventory vanishes at time *H, *but in case of major level, the stock condition (or supply) may be lesser to the demand rate *µ* and so the inventory reduces to zero before the time *H*.

### 3. Production System Without Disruption

To compare the model output first, management optimizes the production system run without disruption with production rate *p* (per unit time) stopped at production time *T*_{p} and there after till time *H,* inventory depicted due to demand rate *µ* and deterioration rate *θ *of items (see Figure 1). The presentations in differential equations for two periods [0, *T*_{p}] and [*T*_{p}, *H*] satisfy throughout its domain.

**Fig**

**ure**

**1**

**.**Production System without disruption

**Fig**

**ure**

**2**

**.**Production System after disruption

,boundary condition

(1) |

,boundary condition

(2) |

On solving Equation (1) and Equation (2) with boundary conditions we get

(3) |

(4) |

As per Figure 1 inventory level *I*_{1}(*t*) and* I*_{2}(*t*) are equal at time *T*_{p}.

*i.e*. *I*_{1}(*T*_{p}) =* I*_{2}(*T*_{p}) yields

(5) |

If *θ <<*1, then

(6) |

**Proposition 1**. If* θ <<*1 then *T*_{p} is in increasing in *θ.*

By Equation (6) one can write

(7) |

This proved the** **result_{*}

As *θ* increases optimal production time* T*_{p} increases that is more product required to producing. One can conclude that to keep low deterioration is effective way to keep lower cost of production of items.

### 4. Production System with Disruption

As previous section production rate remains unchanged but in practice production system is always disruption due to unplanned and thus we consider the production system little changed by and disruption time is *T*_{d}* *. If, then production rate decreases and, if then production rate increases.

**Proposition 2.**

then manufacturer still satisfies the demand even production system has been disrupted, otherwise If then production system unable to satisfy demand that is there will be shortages due to production disruption.

**Proof:** Suppose the production system disrupted at time *T*_{d} as (see Figure 2) and there after the production rate will be thus presentations of two differential equations for intervals [0, *T*_{d}] and [*T*_{d}, *H*] are

(8) |

(9) |

with boundary condition

On solving Equation (9) with boundary condition we get

(10) |

If this means production system satisfy the demand of items.

That isthen still satisfy the demand.

If this means production system does not satisfy the demand of items.

That is then there will be shortages in the system.

This proved the lemma_{*}

Again if then we find optimal production time (with disruption) * *such that at time* H* entire stock will be sold-out and inventory level will be zero.

If there will be shortages in the system and in this situation we will find the optimum time *T*_{r}* *of placing the order and respective order quantity *Q*_{r}.

**Proposition 3. **If then production time with disruption is obtained by.

**Proof:** If

or that is on hand inventory is .

Therefore we will find out the optimal time(see Figure 2) when we stopped the production after disruption in such a manner that stock remains zero at time *H*. the presentations of two differential equations for intervals [*T*_{d ,}*T*_{p}^{d}] and [*T*_{p}^{d}*, H*] are

(11) |

Boundary condition

, boundary condition

(12) |

On solving (11) with boundary condition we get

(13) |

(14) |

Using condition

(15) |

If

By Equation (10),

(16) |

and thus by Equation (14), order quantity will be

(17) |

This proved the result_{*}

**4.1. An Application with Simulation**

For application we assumed a particular case when * p** *= 350, *µ *= 300, *Δ**p** *= -100, *θ *= 0.01, *H *= 15 and* **T*_{d}_{ }=1, on applying then we get *I*_{2}*(H)>0 *and thus by Equation (6) and Equation (15)* *gives* **T*_{p}*=*0.14,* *= 13.71 days. We simulate the application on same data in which other parameters are invariant.

**Fig**

**ure**

**3**

**.**(Effect of Td on )

**Fig**

**ure**

**4**

**.**(Effect of H on )

**Fig**

**ure**

**5**

**.**

**(Effect of θ on)**

**Fig**

**ure**

**6**

**.**(Comparison between and

*T*

_{p}with and without disruption w.r.t.

*θ*)

**4.2. Discussion**

Time horizon *H* linearly increases the time *T*_{p} and both (see Table 1). Production timeafter disruption is linearly decreases as disruption *T*_{d} decreases (see Figure 3) that is system get disrupted later is in favor of management. Also the production time is not longer if it gets disrupted later. If deterioration increase thenis in linearly increases (see Figure 4), the same result followed by *T*_{p }with respect to deteriorations (see Figure 5). There is a shift of stop time of production before and after disruption (see Figure 6), and both are highly sensitive on deterioration.

### 5. Conclusion

The deterioration factor affects negatively when disruption is present in the production system. In beginning, if production rate is higher than demand rate, then inventory managers are benefitted. It provides a little accumulation of stock to the managers. But, due to specific type of disruption, the reserve inventory helps to manage the market only up to a shorter period. It is interesting fact that when disruption appears its early occurrence may be dangerous but later occurrence is manageable.

Production model with disruption is quite different from the production model without disruption. The stop time for production after disruption is always greater than the stop time of production time without disruptions. If system get disrupted later it goes to favour of production system so, management should delay the disruptions as well as possible.

If production disrupted time is longer, then it is difficult to manage and it need to order more quantity from the spot market. The demand parameter highly affected the policy. Thus, the performance of any production system will be robust to demand variations and model uncertainty. The proposed model can be further extended by considering the more realistic assumption like time dependent production along with time dependent demand even production system gets disrupted, and deterioration may follow a probability distribution function.

### Acknowledgement

We are thankful to referee for his very helpful comments and suggestions for the overall improvement of the paper.

### References

[1] | Benjaafar, S., EIHafsi, M., “Production and inventory control of a single product assemble-to-order system with multiple customer classes”, Management Science, 52(12). 1896-1912. (2006). | ||

In article | CrossRef | ||

[2] | Giri, B.C., “Goswami, A., and Chaudhuri, K.S., An EOQ model for deteriorating items with time varying demand and costs”, Journal of the Operational Research Society, 47(11). 1398-1405. (1996). | ||

In article | |||

[3] | He, Y., Wang, S.Y., and Lai, K.K., “An optimal production-inventory model for deteriorating items with multiple-market demand”, European Journal of Operational Research, 203(3). 593-600. (2010). | ||

In article | CrossRef | ||

[4] | Hou, K.L., and Lin, L.C., “An EOQ model for deteriorating items with price-and stock-dependent selling rates under inflation and value of money”, International Journal of System Science, 37(15). 1131-1139. (2006). | ||

In article | CrossRef | ||

[5] | Joglekar, P., “Optimal price and order quantity strategies for the reseller of a product with price sensitive demand”, Proceeding Academic Information Management Sciences, 7(1). 13-19. (2003). | ||

In article | |||

[6] | Khedlekar, U.K., and Shukla, D., “Dynamic inventory model with logarithmic demand”, Opsearch, 50(1). 1-13. (2013). | ||

In article | CrossRef | ||

[7] | Khedlekar, U.K., Shukla, D., and “Managerial efficiency with disrupted production system” International Journal of Operations Research, 9(3). 141-150. (2012). | ||

In article | |||

[8] | Liao, J.J., “On an EPQ model for deteriorating items under permissible delay in payments”. Applied Mathematical Modeling, 31(3). 393-403. (2007). | ||

In article | CrossRef | ||

[9] | Lin, G.C., and Kroll, D.E., “Economic lot sizing for an imperfect production system subject to random breakdowns”, Engineering Optimization, 38(1). 73-92. (2006). | ||

In article | CrossRef | ||

[10] | Lo, M.C., “Decision supports system for the integrated inventory model with general distribution demand”, Information Technology Journal, 6(7). 1069-1074. (2007). | ||

In article | CrossRef | ||

[11] | Qi, X., Bard, J.F., & Yu, G., “Supply chain coordination with demand disruptions”, Omega, 32. 301-312. (2004). | ||

In article | CrossRef | ||

[12] | Samanta, G.P., and Roy, A., “A production inventory model with deteriorating items and shortages”, Yugoslav Journal of Operations Research, 14(2). 219-230. (2004). | ||

In article | CrossRef | ||

[13] | Shukla, D. and Khedlekar, U.K. (2010a). An order level inventory model with three-Component demand rate (TCDR) for newly launched deteriorating item. International Journal of Operations Research (IJOR-Taiwan), 7(2): 61-70. | ||

In article | |||

[14] | Shukla, D., Chandel, R.P.S., Khedlekar, U.K., and Agrawal, R.K., “Multi-items inventory model with time varying holding cost and variable deterioration”, Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering & Medicine, 1(8). 223-227. (2010b). | ||

In article | |||

[15] | Shukla, D., Khedlekar, U.K., Chandel, R.P.S., and Bhagwat, S., “Simulation of inventory policy for product with price and time dependent demand for deteriorating item”, International Journal of Modeling, Simulation, and Scientific Computing, 3(1). 1-30. (2010c). | ||

In article | |||

[16] | Shukla, D., Khedlekar, U.K., and Bhupendra, “An inventory model with three warehouses”, Indian Journal of Mathematics and Mathematical Science, 5(1). 39-46. (2009). | ||

In article | |||

[17] | Teng, J.T., and Chang, C.T., “Economic production quantity model for deteriorating items with price and stock dependent demand”, Computer & Operations Research, 32(2). 297-308. (2005). | ||

In article | CrossRef | ||

[18] | Yang, P.C., and Wee, H.M., “A single-vendor and multiple-buyers production-inventory policy for a deteriorating item”, European Journal of Operational Research, 143(3). 570-581. (2002). | ||

In article | CrossRef | ||