International Journal of Pure and Applied Sciences and Technology

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1 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), pp International ournal of Pure and Applied Sciences and Tecnology ISSN Available online at Researc Paper Optimal Pricing and Production Sceduling Policies for an Inventory Model wit Stock Dependent Production and Weibull Decay Essey Kebede Mulune 1, * and K. Srinivasa Rao 2 1 Department of Statistics, Bair Dar University, P.O. Box-79, Bair Dar, Etiopia 2 Department of Statistics, Andra University, Visakapatnam , India * Corresponding autor, (esseykebede@gmail.com) (Received: ; Accepted: ) Abstract: Te classical Economic Order Quantity (EPQ) models consider tat te production rate is fixed and constant. But in many manufacturing and production processes te production rate is a function of stock on and. For tis sort of situations we develop and analyze an EPQ model wit te assumption tat te production rate is a function of te on-and inventory and demand is a function of selling price. It is furter assumed tat lifetime of te item is random and follows tree parameter Weibull distribution. Wit suitable cost considerations te total cost and profit rate functions are derived. By maximizing te profit rate function te optimal pricing and production sceduling policies are derived. Te sensitivity analysis of te model reveals tat te stock dependent nature of production rate is aving significant influence on te optimal production quantity and production up-time and tat te selling price dependent demand parameters tremendously influence te optimal values of te unit selling price and profit rate. Keywords: EPQ model, stock dependent production rate, selling price dependent demand, production sceduling. 1. Introduction Inventory models are matematical models wic elp a business firm to make optimal decisions as wen and ow muc to buy (or produce) so as to minimize its cost or maximize its profit. Since te first classical lot-size formula was developed in 1915, a wide variety of inventory models ave been developed and analyzed wit various assumptions to deal wit te real life situations. Te efficiency of an inventory model depends upon te suitable assumptions made on te constituent components of

2 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), te mode. Te constituent components of te model are (1) replenisment (2) demand pattern and (3) lifetime of te commodity. Te implicit assumption of te EOQ model is tat te items are obtained from te outside supplier. Given tis assumption, it is reasonable to assume tat te entire lot is delivered at an instant of time wit infinite rate of replenisment. However, in some oter places like production processes, manufacturing units, wareouses, te replenisment (production) is not instantaneous and may ave finite rate. In te study of non-instantaneous inventory models, economical production quantity (EPQ) model plays an important role. An EPQ model is an inventory model tat determines te quantity of product to be produced on a single facility so as to meet te demand over an infinite planning orizon. Te two most important decision variables in any EPQ model are te production run time and te optimal quantity to be produced. Decay or deterioration of pysical goods wile in stock is a common penomenon. Parmaceuticals, foods, vegetables and fruits are a few examples of suc items. Taking tis into consideration, deteriorating inventory models ave been widely studied in recent years. Gare and Scrader (1963) developed an economic order quantity model wit constant rate of decay. Covert and Pilip (1973) extended tis model considering a variable rate of deterioration by assuming a two-parameter Weibull distribution. Compreensive reviews of researc literature on deteriorating items are provided by Raafat (1991), Goyal and Giri (2001), Li, et al. (2010), Pentico and Drake (2011) and Bakker, et al. (2012). 2. Literature Review Acting as te driving force of te wole inventory system, demand is a key factor tat sould be taken into consideration in studying inventory systems. In classical inventory models te demand rate is assumed to be constant. In reality, it may vary wit time or wit price or wit te instantaneous level of inventory displayed in a supermarket. Manna, et al. (2007), Patra (2010), Teng, et al. (2011) studied economic order quantity models for deteriorating items wen demand is quadratic in time. Skouri, et al. (2009) developed an order level inventory model wit general ramp type demand rate and partial backlogging. Panda, et al. (2009), Panda and Saa (2010) and Manna and Ciang (2010) studied production inventory models aving time-dependent demand and finite rate of production wic is proportional to te demand rate. It as been observed tat for certain types of inventories, particularly consumer goods, eaps of stock will attract customers and te demand is a function of stock on-and. Due to tis fact, tis area of inventory teory researc as recently been receiving considerable attention. Mandal and Paujdar (1989), Giri, et al. (1996), Teng et al. (2005), Uma Maeswara Rao, et al (2010a), Yang, et al. (2010), Lee and Dye (2012) and oters ave developed inventory models were demand rate is a function of on-and inventory. Srinivasa Rao, et. al (2011) developed a production inventory system wit demand rate a function of production quantity. Dye, et al. (2007) developed a deterministic inventory model for deteriorating items were te demand rate is assumed to be continuous and differentiable function of price. Roy and Cauduri (2007) introduced an order-level inventory model for a deteriorating item, wit selling price dependent demand incorporating te concept of te special sale campaign by way of price reduction into te model. Maiti, et al. (2009) developed inventory model for an item in stocastic environment wit price-dependent demand over a finite time orizon considering probabilistic lead-time. Sana (2011) developed a finite time-orizon deterministic EOQ model were te rate of demand decreases quadratically wit selling price. Inventory models for deteriorating items aving multivariate demand functions were studied by several autors. Cang, et al. (2010) and Kanra, et al. (2010) considered selling price and stock level dependent demand rate and Uma Maeswara Rao, et al. (2010b) considered time and stock level

3 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), dependent demand rate. Sa and Pandey (2009) developed a model for deteriorating items were demand is a function of stock display and frequency of advertisement. But tese autors ave considered tat te replenisment is instantaneous or ave a fixed finite rate. Bunia and Maiti (1998) developed inventory model witout sortages for non deteriorating items by assuming production rate is linearly dependent on on-and inventory and demand rate on time. Mata and Goswami (2009) considered fuzzy replenisment. Sridevi et al. (2010) developed an inventory model for deteriorating items wit random replenisment. Recently, Mulune and Srinivasa Rao (2012) developed an inventory model for deteriorating items wit te assumption tat te production rate is dependent on stock on and and demand is a power function of time. But in some production units dealing wit food processing te production rate is dependent on stock on and and demand is a function of selling price. Hence, in tis paper an EPQ model for deteriorating items wit stock dependent production and selling price dependent demand is developed and analyzed. Using te differential equations te instantaneous state of inventory is derived. Wit suitable cost considerations te total cost function and profit rate function are derived. By maximizing te profit rate function te optimal selling price, optimal ordering quantity and optimal production scedule (te time to start production and time to stop production) are obtained. A numerical illustration is included to demonstrate te solution procedure of te model. Te sensitivity of te model wit respect to te parameters and te costs is also studied. Tis model includes some of te earlier models as particular cases for specific values of te parameters. Tis model is also extended to te case of witout sortages. 3. Notations and Assumptions 3.1 Notations set up cost inventory olding cost per unit per unit time sortage cost per unit per unit time per unit production cost (cost price) of te item selling price of te item ( ) rate of production at any time t () demand rate as a function of selling price ( ) inventory level at any time t (α, β, ) production quantity (total production) maximum inventory level maximum sortage level time point at wic production stops (production down time) time point at wic sortages begin (production up time) time point at wic production resumes production cycle time total production cost per unit time total revenue of te system per unit time profit rate function deterioration rate parameters

4 . Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), (τ, φ) (, θ) demand rate parameters production rate parameters 3.2 Assumptions Te following assumptions are made for developing te inventory model under study. i. Life time of te commodity is random and follows a tree parameter Weibull distribution wit probability density function ( )=( ) (),, >0, were, is te scale parameter, is te sape parameter and is te location parameter. ii. iii. Demand is a function of selling price and it is of te form ()=& ', were &>0, ' 0. If '=0 ten te demand rate will be constant. Production rate is a linear function of te on-and inventory, i.e. ) ( ), 0 ( )=(), 0,,- were, )>0, 0 1 and ( ) ( ) at any time wen replenisment takes place. If =0, ten it includes te finite rate of production. iv. Tere is no repair or replacement of deteriorated items. v. Te planning orizon is infinite. Eac cycle will ave lengt. vi. Te inventory olding cost per unit per unit time, te sortage cost per unit per unit time, te unit production cost per unit time and setup cost per cycle are fixed and known. 4. Inventory Model wit Sortages 4.1 Model Formulation Consider an inventory system for deteriorating items in wic te lifetime of te commodity is random and follows a tree parameter Weibull distribution were sortages are allowed and fully backlogged. In tis model te stock level for te item is initially zero. Ten production starts at time =0, and continues adding items to stock until te on-and inventory reaces its maximum level at time =. During te time (0,) demand is met from replenisment and te remaining will be accumulated in stock. At time = deterioration of te item starts and ence, in te interval (, ) stock is depleted by demand and deterioration wile production is continuously adding to it. At =, production is stopped and stock will be depleted by deterioration and demand until it reaces zero at time =. As demand is assumed to occur continuously, at tis point sortages begin to accumulate until te backlog reaces its maximum level of at =. At tis point production resumes meeting te current demand and clearing te backlog. Finally sortages will be cleared at time =. Ten te cycle will be repeated indefinitely. Tese types of production systems are common in food processing industries were production rate is directly proportional to te stock on and. Te scematic diagram representing te inventory system is sown in Figure 1.

5 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), ( ) 0 Fig. 1: Scematic diagram representing te inventory level of te system for te model wit sortages Te differential equations governing te system in te cycle time (0,T) are: 1( ) 1 1( ) 1 1( ) 1 1( ) 1 1( ) 1 =2) ( )3 2& '3, 0 (1) =2) ( )3 4( ) ( )5 2& '3, (2) = 4( ) ( )5 2& '3, (3) = 2& '3, (4) =) 2& '3, (5) wit boundary conditions, (0)=0, ( )=0, :;1 ()=0 Solving te differential equation (1) and using te boundary condition (0)=0, te instantaneous state of inventory at time t in 0 is obtained as ( )= () &') =1 >?, 0 (6) Solving te differential equations (2) (5) te instantaneous states of inventory at time t in te respective time intervals are: I( )=( ) &') 4>B() 5 CD 4>EB(E) 5 I I( )=(& ') () CD (E) 1F D (E) 1FG γ 1F =ϕ 1? G,γ ϕ (7)

6 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), I =(& ') () D (E) 1F, (8) K I( )= (& ')( ), (9) I( )= ( ) &')( ), (10) Using te equations (6) to (8), te total inventory for te respective time periods are obtained as follows. Te total inventory in 0 γ is D( )1 N = ( ) &') ϕ D(1 ϕ )1 N = ( ) &') = > 1? (11) Te total inventory in γ is DI( ) γ R W DD >(E)B4(E) () 51F1 P P γ γ 1 =() &') Q P =ϕ 1? V D 4>B() 5dt ϕ P O γ U Using Taylor series expansion of te exponential function, neglecting terms of iger order, integrating and simplifying we get DI( ) 1 =() &')XB( ) 1 2 (1 Z)( γ ) γ 1 6 ( γ ) 1 8 ( [ γ [ 2( ) B (2) ( ) B \Z ] α ( ) (B) ^ 2 (12) were, B= ϕ (ϕ 1) Similarly total inventory in te interval is I DI(t)1 I I =(& ')DD 4(E) () 5 1F1 K

7 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), =(& ')X_ ( ) B `_( ) 4( ) B ( ) B 5 ` 1 2 ( ) a ( ) B ( ) B 2 b (2) a( ) B ( ) B b α 2 a( ) (B) ( ) (B) b^ (13) Since ( ) is continuous at, equating (7) and (8) we get, I( )=(θ &') 4> B( ) 5 XB 1 2 ( ) B ^ 1 =(& ') ( ) X( ) a( ) B ( ) B b ^ (14) Te relationsip between and can be establised from te equality in equation (14). Also, te maximum inventory level, ( )= is obtained as = (& ') ( ) X( ) =( ) B ( ) B? ^ (15) Similarly, since ( ) is continuous at, evaluating ( ) at = and equating (9) and (10) and we get (& ')( )= () &')( ) (16) Tis equality can be used to establis te relationsip between and. Solving for we get, = (& ')( ) ) (17) And te maximum sortage level is, = 1 ) () &')(& ')( ) (18) Backlogged demand is c Z( )= D()1 I c = D(& ')1 I =(& ')( ) (19) Total Production in te cycle time (0,) is =D( )1 N D( )1 d D( )1 c

8 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), =)( ) D( )1 N D( )1 Terefore using equations (11), (12) and (17) we get =) (& ')( ) X () &') = > 1?^ (θ &')XB 1 2 (1 Z) [ α ( ) (B) 2 1 ( ) B \Z 1 2 ] 2 (2) ( ) B Z 1 2 γ (1 Z) 1 6 γ 1 8 γ [^ (20) Let (,,) be te total cost per unit time. Ten, (,,) is te sum of te setup cost per unit time, te production cost per unit time, te inventory olding cost per unit time and te sortage cost per unit time, i.e. (,,)= CD( )1 Tis implies, c CD( )1 D( )1 G I N c d D( )1 I D( )1 (,,)= ) _ (& ')( ) ` ) X () &') = > 1?^ ( ) (θ &')XB( ) 1 2 (1 Z)( γ ) 1 6 ( γ ) 1 8 ( [ γ [ 2( ) B (2) ( ) B \Z ] α ( ) (B) 2 ^ (& ')Xe ( ) B fe( ) 4( ) B ( ) B 5f 1 2 ( ) a ( ) B ( ) B b 2a( ) B ( ) B b (2) α 2 a( ) (B) ( ) (B) b^ 2) 2( ) (& ')() &') 3 (21) G

9 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Let (,,) be te total revenue per unit time. Ten, (,,)= D()1 N d d = D(& ')1 N =s(& ') Also let (,,) be te profit rate function. Ten, (,,)=(,,) (,,) Tis implies (,,)=s(& ') (,,) (22) Were (,,) is as defined in (21) above 4.2 Optimal Policies of te Model In tis section, we obtain te optimal pricing and ordering policies of te inventory model developed in section (4.1). Te problem is to find out te optimal values of, and s tat maximize te profit rate function (,,) over (0,). To obtain tese values we differentiate (,,) given in equation (22) wit respect to, and and equate tem to zero. Te condition for te solutions to be optimal is tat te determinant of te Hessian matrix is negative definite, i.e. i (,,) i i (,,) i i i (,,) i i = i (,,) i i i (,,) i i (,,) i i <0 i (,,) i i i (,,) i i i (,,) i Differentiating (,,) wit respect to and equating to zero we get ) ( ) (θ &')XB (1 Z) ( ) B (1 ) ( ) \Z 1 2 ] α ( ) B ^ (& ')Xe ( ) B fa( ) 1b ( ) B ( ) α ( ) B ^=0 (23)

10 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Differentiating (,,) wit respect to simplifying and equating to zero we get (& ') ) ( )(& ')() &') (& ')Xa1( ) b_( ) 4( ) B ( ) B 5 ` e ( ) B fa1 ( ) b ( ) ( ) B α 1 ( ) B^=0 (24) Differentiating (,,) wit respect to simplifying and equating to zero we get (& 2') '( ) '( ) 4 > 15 XB( ) 1 2 (1 Z)( γ ) 1 6 ( γ ) 1 8 ( [ γ [ 2( ) B ( ) B \Z ] α ( ) (B) 2 ^ ' X_ ( ) B a ( ) B ( ) B b 2a( ) B ( ) B b (2) ^ ' ( ) α a( ) (B) ( ) (B) b 2 (2) `_( ) 4( ) B ( ) B 5 ` 1 2 ( ) 2 X 2(& ') 1 ^=0 (25) ) Solving te equations (23), (24) and (25) simultaneously using numerical metods like, Newton Rapson metod one can obtain te optimal values of, and. Substituting tese optimal values of, and in to equations (17), (15), (18), (21) and (22) we obtain te optimal values for,,, and (,,) respectively. For eac set of optimal values, te determinant of te Hessian matrix is computed and verified for negative semi definiteness. 4.3 Numerical Illustration Consider te case of deriving an economic production quantity and production scedule for a food processing industry. Here te product is of a deteriorating type and as a random lifetime wic is assumed to follow a tree parameter Weibull distribution. Te following parameter values were suggested by marketing personnel in te unit. Te deterioration parameters α, β, and are estimated to vary over 0.02 to 0.08, 0.5 to 2.5 and 0.4 to 1 respectively. Stock dependent production rate parameters and θ vary over 0.55 to 0.7 and 70 to 100 respectively and selling price dependent demand rate parameters τ and ' vary over 45 to 60 and 0.7 to 1 respectively. Let te values for oter parameters be =5, =4, =3 and =75 all in appropriate units. Te cycle lengt is taken to be =6 units and te values of te parameters are varied to observe te trend in te optimal policies. Te optimal values of production downtime, production uptime, selling price, production quantity, and total profit are obtained and presented in Table 1.

11 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Table 1: Optimal solutions of te model wit sortages for different values of te parameters k l α β θ From Table 1 it is observed tat te optimal time values and and optimal production quantity ave positive relationsip wit te demand parameter φ, deterioration parameter, production parameter θ and unit production cost and negative relationsip wit te demand parameter τ, te deterioration parameters (α, β) and te unit olding cost. Production cost, olding cost and

12 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), production parameter θ ave significant influences on and. Also te stock dependent production rate parameter as significant positive relation wit. As te values of τ, α, β and are increasing, te optimal values of selling price and total profit are also increasing, wereas increasing te parameters ', and decrease te values of tese variables. Te demand parameters τ and φ ave considerable effect on and. 4.4 Sensitivity Analysis To study te effect of canges in te parameters on te optimal values of te decision variables of te model we perform a sensitivity analysis. Te following values of te parameters are considered for analysis: &=50, '=0.8, =0.02, =2, =0.8, )=90, =0.6, =4, =4 and =3. Sensitivity analysis is performed by canging te parameter values by -15%, -10%, -5%, 5%, 10% and 15%. First canging te value of one parameter at a time wile keeping all te rest at fixed values and ten canging te values of all te parameters simultaneously, te optimal values are computed. Te results are presented in Table 2. Te relationsips between parameters, costs and te optimal values are sown in Figure 2. From Table 2 it is observed tat, and are sensitive and tose of and are igly sensitive to te canges in te demand parameters τ and φ. A 15% decrease in te value of τ results in % and % decrease in and respectively and a 15% increase in τ results in % and % increase in te values of and respectively. Similarly a 15% decrease in φ results in % and % increase in te optimal values of and respectively. Te optimal production quantity is igly sensitive to te canges in te production rate parameters and θ. A 15% increase in and θ result in % and % increase in. Te optimal profit is moderately sensitive to θ but less sensitive to. Parameter Values &=50 '=0.8 =0.02 =2 =0.8 Table 2: Sensitivity Analysis of te model wit sortages Percentage Cange in te parameter Values Variable -15% -10% -5% 0% 5% 10% 15% =

13 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Table 2: Sensitivity Analysis of te model wit sortages Parameter Values )=90 =5 =4 2=3 All Parameters Percentage Cange in te parameter Values Variable -15% -10% -5% 0% 5% 10% 15% Unit production cost of te item as significant influence on and. For example a 15% increase in p increases by % and by %. Te optimal profit is also moderately sensitive to p. Te production downtime and production quantity are igly sensitive to te canges in te unit cost also. A 15% decrease in results in % increase in and % increase in. Canging all parameter values simultaneously will ave a significant effect on te optimal values of and but not to te oter decision variables. Variations in t 1 * % -10% -5% 0% 5% 10% 15% Percentage cange in parameters k ' α β θ p c2 Variations in t 3 * % -10% -5% 0% 5% 10% 15% Percentage cange in parameters k ' α β θ p c2 (a) (b)

14 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Variations in s* % -10% -5% 0% 5% 10% 15% Percentage cange in parameters (c) 600 k ' 550 α 500 β θ 350 p 300 c2 250 Variations in Q* -15% -10% -5% 0% 5% 10% 15% Percentage cange in parameters (d) k ' α β θ p c2 Variations in total cost % -10% -5% 0% 5% 10% 15% Percentage cange in parameters k ' α β θ p c2 (e) Fig. 2: Sensitivity analysis of system variables wit respect to canges in te parameters and cost for te model wit sortages 5. Inventory Model witout Sortages 5.1 Model Formulation Consider an inventory system for deteriorating items wen sortages are not allowed. Te scematic diagram representing te inventory system is sown in Figure 3. ( ) Fig. 3: Scematic diagram representing te inventory level of te system wit no sortages Te differential equations governing te system in te cycle time (0,) are: 1( ) 1 1( ) 1 0 =2) ( )3 2& '3, 0 (26) =2) ( )3 4( ) ( )5 2& '3, (27)

15 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), ( ) 1 = 4( ) ( )5 2& '3, (28) wit boundary conditions, (0)=0 and ()=0 Solving te differential equations (26), (27) and (28) we get ( )= () &') =1 >?, 0 (29) ( )=() &') 4>B() 5 XZ )B ( ^, 2 1 (30) ( )= (& ') () X a( )B ( ) B b ^, (31) were Z= > (> 1) Te total inventory in 0 is D( )1 N == ( ) &') 4 > 15 (32) Te total inventory in γ is D( ) 1 =( ) &')XZ 1 2 (1 Z) [ 1 ( ) B \Z 1 2 ] 2( ) B (2) ( ) (B) 2 Z 1 2 (Z 1) [^ (33) And te total inventory in is d D( )1 =(& ')D () d X ( )B ( ) B ^1 =(& ')X_ ( )B `_( ) 4( )B ( ) B 5 ` 2a( )B ( ) B b a( ) (B) ( ) (B) b (2) 2 ^ 1 2 ( ) a( )B ( ) B b (34) Te maximum inventory level, ( )= is

16 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), =() &') 4> B( ) 5 XZ 1 2 ( ) B ^ (35) 1 Production quantity Q in te cycle time of lengt T is obtained as =D( )1 N D( )1 Tis implies =) q 1 ( ) &')4> 15r () &')XZ 1 2 (1 Z) [ 2 (2) ( ) B 1 ( ) B \Z 1 2 ] ( ) (B) 2 Z 1 2 (Z 1) [^ (36) Let (,,)be te total cost per unit time. (,,) is te sum of setup cost per unit time, te production cost per unit time and inventory olding cost per unit time, i.e. (,,)= CD( )1 = ) D( )1 Tis implies (,,)= ) N N D( )1 D( )1 I D( )1 I G D( )1 X () &') a > 1b^ ( ) () &')XZ( ) 1 2 (1 Z)( ) 1 6 ( ) 1 8 ( [ [ 2( ) B (2) ( ) B 1 (& ') \Z 1 2 ] ( ) (B) 2 ^ X_ ( )B `._( ) 4( )B ( ) B 5 ` 1 2 ( ) a( )B ( ) B b 2a( )B ( ) B b (2). a( ) (B) ( ) (B) b 2 ^ (37)

17 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Let (,,) be te profit rate function. Ten (,,)=(& ') (,,) (38) were, (,,) is as defined in (37). 5.2 Optimal Policies of te Model In tis section, we derive te optimal policies of te inventory system developed in section (5.1). Te problem is to find out te optimal values of te production downtime, te cycle lengt and te unit selling price tat maximize te profit over te interval (0,). To obtain tese values we differentiate (,,) in equation (38) wit respect to, and and equate te resulting equations to zero. Te condition for optimality of te solution is tat te determinant of te Hessian matrix be negative definite. Tat is, i (,,) i i (,,) i i i (,,) i i = i (,,) i i i (,,) i i (,,) i i <0 i (,,) i i i (,,) i i i (,,) i Differentiating (,,) wit respect to and equating to zero we get ) ( ) () &')XZ (1 Z) ( ) B (1 ) ( ) \Z 1 2 ] ( ) B ^ (& ')Xe ( )B fa( ) 1b ( ) ( ) B ( ) B^=0 (39) Differentiating (,,) wit respect to and equating to zero we get ) X () &') a > 1b^ ( ) () &')XZ( ) 1 2 (1 Z)( ) 1 6 ( ) 1 8 ( [ [ 2( ) B (2) ( ) B 1 \Z 1 2 ] ( ) (B) 2 ^ (& ')X_1 ( )B `a1 ( ) b

18 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), e ( ) ( )B fe( ) 4( )B ( ) B 5f ( ) ( ) B 2a( )B ( ) B b 2( )B (2) 2 a( )(B) ( ) (B) b. ( )B^ (40) Differentiating (,,) wit respect to and equating to zero we get (& 2') '( ) s 1 a> 1bt '( ) XZ 1 2 (1 Z) [ 1 ( ) B \Z 1 2 ] 2 (2) ( ) B ( ) (B) 2 Z 1 2 (1 Z) [^ ' Xe ( )B fe( ) 4( )B ( ) B 5f 2a( )B ( ) B b (2) 2 a( )(B) ( ) (B) b 1 2 ( ) a( )B ( ) B b^=0 (41) Solving te equations (39), (40) and (41) simultaneously using numerical metods and verifying te determinant of Hessian matrix to be negative semi definite for concavity we get te optimal values for, and. 5.3 Numerical Illustration Consider te case of deriving an economic production quantity and oter optimal policies for a food processing industry. Te optimal values for,,, and for different values of te parameters and costs are determined and displayed in Table 3. From Table 3 it is observed tat increases wen τ, and increase and decreases wen all oter parameters increase. Te effect of τ and are relatively iger. On te oter and te cycle lengt increases wen te values of φ,, θ and increase. Te optimal selling price and optimal total profit increase if τ, α and β increase and decreases if ', and increase. Te optimal production quantity increases wen te values of τ,, θ and increase and decrease wen oter parameters increase. Te influence due to τ and θ is noteworty.

19 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Table 3: Optimal solutions of te model witout sortages for different values of te parameters k l α β θ Sensitivity Analysis Te extents to wic te optimal values are affected by te cange of te parameters is discussed in te sensitivity analysis of te model. Te following values of te parameters are considered for analysis: &=60, '=0.8, =0.02, =2, =0.8, )=50, =0.5, =4, and =7. Sensitivity analysis is performed by canging te parameter values by -15%, -10%, -5%, 5%, 10%

20 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), and 15%. First canging te value of one parameter at a time wile keeping all te rest at teir original values and ten canging te values of all te parameters simultaneously, te optimal values are computed. Te results are presented in Table 4. Te relationsips between parameters, costs and te optimal values are sown in Figure 4. From Table 4 we observe tat te production downtime is moderately sensitive to τ α, β and and less sensitive to oters. Te optimal cycle lengt is moderately sensitive to τ and θ and less sensitive to oters. Te optimal production quantity is igly sensitive to τ and θ and less sensitive to oters. Optimal selling price and optimal profit are igly sensitive to te demand rate parameters τ and '. For example, increasing τ by 15% results in % and % increase in te respective optimal values and. Tese optimal values are igly sensitive to te canges in te demand rate parameter φ also bot being negatively affected. Table 4: Sensitivity Analysis of te model witout sortages Parameter Values x =60 l=0.8 α=0.02 β =2 =0.8 =0.5 θ=50 =5 =7 Percentage Cange in te parameter Values Variable -15% -10% -5% 0% 5% 10% 15%

21 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), Table 4: Sensitivity Analysis of te model witout sortages Parameter Values =75 All Parameters Percentage Cange in te parameter Values Variable -15% -10% -5% 0% 5% 10% 15% Variations in t 1 * k ' α β θ p Vatiations in T* k ' α β θ p -15% -10% -5% 0% 5% 10% 15% -15% -10% -5% 0% 5% 10% 15% Percentage cange in parameters Percentage cange in parameters (a) (b) variations in s* % -10% -5% 0% 5% 10% 15% Percentage cange in parameters k ' α β θ p Variations in Q* % -10% -5% 0% 5% 10% 15% Percentage cange in parameters k ' α β θ p (c) (d)

22 Int.. Pure Appl. Sci. Tecnol., 17(1) (2013), (c) Fig. 4: Sensitivity analysis of system variables wit respect to te parameters and costs for te model witout sortages 6. Conclusion Production inventory models play a dominant role in production sceduling and resource allocation. In tis paper, two production inventory models (one allowing sortages in te inventory and te oter witout sortages) wit stock dependent production rate and selling price dependent demand rate aving tree parameter Weibull decay wit te objective of maximizing te total system profit, were developed. Te models were illustrated wit numerical examples and sensitivity analysis of te models wit respect to cost and parameters was also carried out. Different optimal selling prices, optimal order quantities, optimal time fractions of replenisment and optimal system profit were obtained for different coices of costs and parameters. Comparing te models wit and witout sortages for te same set of parameters it is observed tat te optimal selling price and optimal profit for te model witout sortages are greater tan te corresponding optimal values for tat of te model wit sortages, and te optimal production quantity for te model wit sortages is greater tan te corresponding optimal value for tat of te model witout sortages. It can be concluded from te numerical examples and sensitivity analysis tat te stock dependent nature of production rate is aving significant influence on te optimal production quantity and production up-time and te demand parameters tremendously influence te optimal values of te unit selling price and profit rate. Te models are developed by assuming tat te money value remains constant trougout te period of time. It is also possible to develop te EPQ models discussed in tis paper under inflation (time value of money). Te models developed for single item can also be extended to include multiple commodities. References Variations in total profit % -10% -5% 0% 5% 10% 15% Percentage cange in parameters [1] M. Bakker,. Riezebos and R.H. Teunter, Review of inventory systems wit deterioration since 2001, European ournal of Operational Researc, 221(2012), [2] A.K. Bunia and M. Maiti, Deterministic inventory model for deteriorating items wit finite rate of replenisment dependent on inventory level, Comput. Oper. Res, 25(11) (1998), [3] C.T. Cang, Y.. Cen, T.R. Tsai and S.. Wu, Inventory models wit stock and price dependent demand for deteriorating items based on limited self space, Yugoslav ournal of Operations Researc, 20(1) (2010), [4] R.P. Covert and G.C. Pilip, An EOQ model for items wit Weibull distribution deterioration, AIIE Transactions, 5(1973), [5] C.Y. Dye, T.P. Hsie and L.Y. Ouyang, Determining optimal selling price and lot size wit a varying rate of deterioration and exponential partial backlogging, European ournal of Operations Researc, 181(2) (2007), [6] P.M. Gare and G.F. Scrader, An inventory model for exponentially deteriorating items, ournal of Industrial Engineering, 14(1963), k ' α β θ p

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