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\title{Economic Lot Scheduling of Time Varying Demand with Stockout in a Jute Industry}
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\begin{document}

             \author[1]{Mehedi  Islam}

             \author[2]{Subrata  Talapatra}

             \author[3]{Ghazi Abu  Taher}

             \affil[1]{  }

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\date{\small \em Received: 10 December 2015 Accepted: 1 January 2016 Published: 15 January 2016}

\maketitle


\begin{abstract}
        


The economic lot scheduling problem (ELSP) creates challenge between lot sizing and sequencing. The ELSP?s primary goal is to minimize the total setup and holding expenditures of different products on a single machine. ELSP is a mathematical model. It deals with a company?s planning what to manufacture, when to manufacture and how much to manufacture. This paper deals with the Economic Lot Scheduling (ELS) of a Jute industry for time varying demand with Stock out. This model will help to understand the total production time and allocate individual time against each product. This also increases the cycle time for a given aggregate inventory. In reality, demands and capacities are varying with time. An aggregate plan is expected to give time varying capacities since the plan is to meet fluctuating demand. It is therefore necessary to model the more realistic situation where the demand and capacity vary each day. This model will provide a production schedule of a set of items in a single machine to minimizing the long run average holding and set up cost under the assumptions of time varying demand and production rates, allowing material stock out.

\end{abstract}


\keywords{economic lot scheduling problem (ELSP); inventory; time varying demand; multiproduct; lot-sizing and scheduling; sequence-dependent setups.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
he Economic Lot Scheduling Problem (ELSP) is assumed that the production facility in the incontrol state producing items of high quality. It finds the problem of production sequence, production times and idle time of several products. It will minimize the inventory and setup cost also. In this model, the items are produced and consumed simultaneously for a portion of the cycle time. The rate of consumption of items is varying throughout the month. The cost of production per unit is same irrespective of production lot size. Here stockout is permitted. It is assumed that the stockout will be satisfied from the units produced at a later date with a penalty. The items are not produced between the period, while the inventory consumes and the next cycle begins. Then another item might be produced. There must be a setup time between the two items. The total cycle length is T.\par
A particular product is produced at a rate of P, the demand of that product is D. Then, inventory will built up at a rate of P-D. Because the product consumes while production. The built up inventory will consume at a certain number of period, then cycles begins again. The operation of this model is shown in Fig. \hyperref[fig_0]{1}.\par
In the economic lot scheduling problem, it is not assumed that changeover times is sequence dependent. So, when the changeover times are sequence independent, then the economic lot scheduling problem essentially tries to minimize the total cost which is the sum of the ordering cost and carrying cost (Srinivasan, G., Quantitative Models in Operations   
\section[{Development of Disaggregation Method with Stockout}]{Development of Disaggregation Method with Stockout}\par
Three products are produced such as sacking, Hessian, CBC (Carpet Baking Clothe). The daily demand and the inventory of these products remain constant. The demand and inventory of the products are summarized form the last three years data as shown in Table \hyperref[tab_1]{1}. The capacity in each of the 3 days are 35, 37, 33 tons/day, respectively. Allocation time have to find for making the products.\par
The value of r represents the demand that can be met with the existing inventory. The production of jproduct has to be started before r j hours. The products are sorted according to increasing value of r. The order is found as Sacking-Hessian-CBC. The products will be produced in the said order. The process flow also depends upon the value of r. The maximum value of r is 0.666, from which the cycle time is counted considering the demand constant.\par
It is assumed that, the reasonable upper limit of the cycle time is (?? +     {\ref -------------------------------------------------(}   The maximum value of r is 6.65, from which the cycle time is counted, that is 1 day. Again, it is assumed that, the upper limit of the cycle time is  {\ref (}   {\ref --------------------------------------------------(}   {\ref ---------------(}  ----------------(5) ?? ?????? ? 0.857[ production time limit for CBC] --------------(  {\ref 6}) ?? ???? ? 6.5[ production time limit for Hessian] -------------(  {\ref 7})\par
?? ???? , ?? ?????? , ?? ???? , ?? ? 0 TORA software is used for solving the problem. The optimal solution of our problem is given by, ?? 1 = ?? ???? = 0 ? 0, ??\textbf{2} 
\section[{Result and Discussion}]{Result and Discussion}\par
This paper translates the ELS of a Jute industry for time varying demand with Stock out. At the beginning of every cycle, the existing inventories are worked out. The expected inventory at the end of each cycle has been calculated. These values are used to compute r j for finding the sequence of production. The order of production has changed in the two cycles because of the values of r j . Again the shortage of inventory in each cycle has also changed the order of production. Therefore, it is observed that, the order of production does not depend upon the values of r j , but depends on the values of stockout of inventories. The change of order satisfies all the constraints and factors. Finally the starting time of each product is calculated by LP. This is acceptable and the model provides flexibility in this regard.    
\section[{IV.}]{IV.}\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-3.png}
\caption{\label{fig_1}1 ??)}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-4.png}
\caption{\label{fig_2}??? = ?? 1 +}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{23}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Fig. 2 :Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{11}\includegraphics[]{image-7.png}
\caption{\label{fig_5}1 + 1 ??)}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-8.png}
\caption{\label{fig_6}1 )(}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{fig_7}}\end{figure}
     \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.34201183431952664\textwidth}P{0.10059171597633135\textwidth}P{0.06035502958579882\textwidth}P{0.09053254437869822\textwidth}P{0.2565088757396449\textwidth}}
\tabcellsep Sacking\tabcellsep CBC\tabcellsep Hessian\tabcellsep \\
Inventory (tons/day)\tabcellsep 10\tabcellsep 4\tabcellsep 7\tabcellsep \\
Demand (tons/day), D 1\tabcellsep 18\tabcellsep 6\tabcellsep 11\tabcellsep P 1 = 35 tons/day\\
D 2\tabcellsep 20\tabcellsep 7\tabcellsep 10\tabcellsep P 2 = 37 tons/day\\
D 3\tabcellsep 16\tabcellsep 9\tabcellsep 8\tabcellsep P 3 = 33 tons/day\\
r = Inventory/Demand\tabcellsep 0.555\tabcellsep 0.666\tabcellsep 0.636\tabcellsep \end{longtable} \par
 
\caption{\label{tab_1}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.1328125\textwidth}P{0.06640625\textwidth}P{0.0265625\textwidth}P{0.6242187499999999\textwidth}}
\tabcellsep \tabcellsep \tabcellsep Year 2016\\
\tabcellsep \tabcellsep \tabcellsep 15\\
\tabcellsep \tabcellsep \tabcellsep ( ) Volume XVI Issue I Version I\\
\tabcellsep \tabcellsep \tabcellsep of Researches in Engineering\\
\tabcellsep \tabcellsep \tabcellsep Global Journal\\
\tabcellsep Sacking\tabcellsep CBC\tabcellsep Hessian\\
Inventory (tons/day)\tabcellsep -50\tabcellsep 6\tabcellsep 65\end{longtable} \par
  {\small\itshape [Note: G © 2016 Global Journals Inc. (US)]} 
\caption{\label{tab_3}Table 2 :}\end{figure}
 			\footnote{Year 2016 G © 2016 Global Journals Inc. (US)} 		 		\backmatter   			 
\subsection[{Acknowledgements}]{Acknowledgements}\par
The author wishes his thanks to Managing Director of Khalishpur Jute Mill for his kind helped in this regard. 			  			  				\begin{bibitemlist}{1}
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\bibitem[Talapatra et al. (2014)]{b10}\label{b10} 	 		‘Implementation of Disaggregation Method in Economic Lot Scheduling of a Jute Industry under Constant Demand’.  		 			Subrata Talapatra 		,  		 			Abu Ghazi 		,  		 			Taher 		.  	 	 		\textit{International Conference on Mechanical, Industrial and Energy Engineering},  				 (Khulna-9203, BANGLADESH)  		2014. 26-27 December, 2014.  		 			Khulna University of Engineering \& Technology 		 	 
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\bibitem[Srinivasan]{b9}\label{b9} 	 		\textit{Quantitative Models in Operations and Supply Chain Management},  		 			G Srinivasan 		.  		 sbn: 978-81-203- 3981-1.  		 	 
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\bibitem[Gallego and Roundy ()]{b2}\label{b2} 	 		‘The extended economic lot scheduling problem’.  		 			G Gallego 		,  		 			R Roundy 		.  	 	 		\textit{Nav Res}  		1992. 39 p. .  	 
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\end{document}
