Economic Ordering Policy for Eggs Items with Deterioration and Breakability under Constant Production in the Republic Of Yemen

Abdullah.M.A.Alshami1, Aniket.A.Muley2

1Thamer university, Thamer, Yemen

Educational Faculty, Mathematics department

2Swami Ramanand Teerth Marathwada University, Nanded, India

431606 Maharashtra, India

Mathematical School of Sciences Statistics

[email protected] , [email protected]

ABSTRACT

This is paper developed the policy of EOQ in an inventory control system under the effect of the time value of money inflation rate on inventory at store and production consideration. The proposed an economic order quantity model to manage deteriorating and break-ability rates for items which have both risks as break-ability and deterioration in stock by obtaining the optimal quantity, optimal total cost of lot size if break-ability risk or deteriorating risk is occurring once of them of both of them. The finite horizon planning, the demand, production rate and deterioration break-ability rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first; here solution procedure is proposed to derive the total cost with respect to the order quantity, the number of replenishment during the planning horizon with two cases for optimal economic order quantity based on the. Finally represented the EOQ versus the deterioration and break-ability rates for the real data of eggs in the stock of sample industry which is selected randomly form eggs industry in the Republic of Yemen. Sensitivity analysis for the proposed data was suggested the accepted range of the net constant rate of inflation to achieved optimal total cost under economic order quantity, discussing the goodness of proposed model under several values of deterioration, break-ability rates and representing the net constant discount rate of inflation. Rates and the representing the net constant discount rate of inflation.

Keywords: Inventory-Deterioration-Breakability- Production Rate -Total Cost Function (TC)

Introduction

There are researches represented models considered the deterioration rate or breakability rate in the model as Abdullah, Najeeb (2018) proposed a model to determine optimal size for breakable item with broken function for broken items with Weibull demand. Abdullah, Aniket developed the model for inventory with non-zero level. Ate Allah, Mohammed (2014) assumed the back ordering and financial for deteriorating items. Giri, Dohi (2005) represented format as the exact formulation of the stochastic EMQ model for an unreliable production system. Chung, Liao (2006) proposed a model to obtain optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. Covert, Philip (1973) optimizes the EOQ by the represented model for items with Weibull distribution deterioration. Elasayed, Teresi (1983) developed the model for EOQ by analysis of inventory systems with deteriorating items. Goyal, Giri (2001) used the technique of trends in the modeling of deteriorating inventory. Hill (1999) developed the optimal production and shipment policy for single-vendor single-buyer integrated production inventory problem. Lee, Hsu (2009) developed a two-warehouse production model for deteriorating inventory items with time-dependent demands. Lin, Kroll (2006) developed economic lot sizing for an imperfect production system subject to random breakdowns. Maity, Maity, Mondal, Maiti (2007) applied Chebyshev approximation for solving the optimal production inventory problem of deteriorating multi item. Swenseth, Godfrey (2002) developed incorporating transportation costs into inventory replenishment decisions. Wagner, Whithin (1958) developed a dynamic version of the economic lot size model. Weiss, Rosenthal (2002) developed the model for optimal ordering policies when anticipating a disruption in supply or demand. Yang, Wee (2002) developed models for single-vendor and multiple-buyers production-inventory policy for a deteriorating item. In general, deterioration is defined as the decay, damage, spoilage, evaporation, and obsolescence of stored items and it results in decreasing usefulness, breakability is defined as the broken items cannot use it under any circumstances the items, in this case, may have deterioration rate or breakability or both at the same time. So that the management and holding of inventories of breakable and deteriorating items becomes an important problem for inventory managers. In real-world problem, there are many items having deterioration and breakability characteristics at same time as the eggs and blub and each items packed in can made from glass as medicines and milk which are packed in glass, some fruits as watermelon and nosier, olive, olive oil, these types of items require advanced model to manage the items have close validity with minimum risk of deterioration breakability characteristics for industry to determine the optimal economic ordering quantity, optimal total cost of per unit in each those items. The proposed model concerned among all of the items especially eggs items as a sample of area study in the Republic of Yemen. All above-mentioned issues (deteriorating items, breakability items, inflation and time value of money) are separately regarded in some inventory models.

2. Material and methods

2.2. Assumptions and notions

2.2.1. Assumptions

In this paper, the mathematical model is developed with the following assumptions

The planning horizon is finite.

Single item inventory control.

Demand and deterioration and breakability rates are constant.

Production rate is constant.

Deterioration and breakability occur as soon as the items are received into inventory.

There is no replacement or repair of breakable items during the period under consideration.

The shortage is not allowed.

The inflation rate is constant and the time value of money is considered.

The lead time is zero.

The inventory level at the end of the planning horizon will be zero.

The cost factors are deterministic.

The number of replenishment is restricted to one integer.

The total relevant cost consists of fixed ordering, purchasing, holding interest payable, interest earned from sales revenue during the permissible period.

The last order is only being placed to satisfy the balance in the stock of the last period.

2.3. Notation

C_P = The present value of holding cost during the first replenishment cycle.

Q = The order quantity in the replenishment.

TC_A = The total fixed ordering cost (0, b).

TC_h = The total holding cost (0, b)

TC_P = The total purchasing cost (0, b).

TC_s = The total shortage cost (0, b).

TC = The total relevant cost (0, b).

2.4. Parameters

The mathematical model is representing the following parameters

A = The fixed ordering cost per replenishment, $\order.

C = The unit purchasing price at time zero, $\order.

C (t) = The unit purchasing price at time t, C(t)=Ce^(-RT).

D = The constant demand rate per unit time.

p = The production rate per cycle.

b = The length of the finite planning horizon.

i = The constant inflation rate.

I(t) = The inventory level at time t.

I_C = The interest charged per $per year by the supplier.

r = The discount rate representing the time value of money.

R=r-1 , representing the net constant discount rate of inflation.

T = The length of each replenishment cycle.

T_j = The total time that elapsed up to, including interest charges.

? t?_j = The time at which the inventory level in the j^th

V = The unit selling price at time t.

C_S=The shortage cost pre-unit time ,$\unit\unit time

C_S (t)=The shortage cost pre-unit time ,$\unit\unit time at time t ,C_s (t)=C_s e^(-RT).

V(t) = The selling price per unit at time t, V(t)= Ve^(-RT)

? = The constant deterioration rate, units/unit time.

? = The constant breakability rate, units/unit time.

3. Mathematical model

Let is the inventory level at any time t, , Depletion due to demand and deterioration, breakability will occur simultaneously. The first order differential equation that describes the instantaneous state of over the open interval (0, b) is given by.

(dI(t))/dt+?I(t)+?I(t)=-(P-D),0?t?t_1,0???1,0???1, P>D (1)

Fig.1Graphical representation of the inventory control diagram

(dI(t))/dt=-(P-D),t_1?t?T, Where for equation of number

I(t)=? I?_0 (t) ?_t^(t_1)??(P-D) e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t)

I(t)=? I?_0 (t) ?_t^(t_1)??(P-D)e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t) (2)

According to Eq.2 the maximum inventory quantity at the begin each period is given as

Q= (P-D)/(?+?) (e^((?+?) t_1 )-1),t_1=Fb/N (3)

3.1. Fixed ordering cost

We assumed the number of replenishment is N so that the fixed ordering cost over the planning horizon under the inflation consideration is:

TC_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^(-(N+1)RT)-1)/(e^(-RT)-1)),T=b/N (4)

Then

TC_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^((-(N+1)Rb)/N)-1)/(e^((-Rb)/N)-1)),T=b/N (5)

3.2. Holding cost excluding interest cost

We find the average inventory quantity to obtain holding cost

I ?=?_0^(t_1)??I(t)dt=?_0^(t_1)??(P-D)/(?+?)(e^(?+?)(t_1-t) dt= (P-D)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1) ?? (6)

By using Eq. 7 we have obtained holding cost is as follows

TC_h=?_(j=0)^(N-1)??I_h C_j I ?=?_(j=0)^(N-1)??I_h Ce^(-R_j T)=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?)?? t_1-1) (7)

Since T= b/N then equation number (8) it will be as

TC_h=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)(e^(-Rb)-1)/(e^(-Rb/N)-1) (8)

3.3. Purchasing cost

According to fig.1 of inventory level the purchasing cost of j^th cycle is calculated as

CP_j=C_j I_m=C_j (P-D)/(?+?)(e^((?+?) t_1 )-1) (9)

The total purchasing cost over the planning horizon can be obtained as

Special case for total purchasing cost when

TC_P=?_(j=0)^(N-1)??CP_j=C(P-D)/(?+?)(e^((?+?) t_1 )-1)(? (e^(-Rb)-1)/(e^(-Rb/N)-1)) (10)

3.4. Shortage cost

The shortage cost which is calculated as

?TC?_s=?_0^(N-1)??C_s e^(-jRT) C_j=? (C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)(e^(-Rb)-1)/(e^(- Rb/N)-1) (11)

TC=TC_A+TC_h+TC_P+?TC?_s

Then,

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)+ ?((C(P-D)+)/(?+?) (e^((?+?) t_1 )-1)[email protected]((P-D)I_h)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1))(e^(-Rb)-1)/(e^(- Rb/N)-1) (12)

3.5. Economic order quantity

3.5. 1.Economic order quantity

To find EOQ by minimizing the total cost function as the following

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)

+(C(P-D)(e^((?+?) t_1 )-1))/(?+?)+((P-D)I_h)/((??+?)?^2 ) (e^(((?+?) t_1)/N)-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)(e^(-Rb)-1)/(e^((-Rb)/N)-1) (13)

Since t_1=Fb/N

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)+ ?(C(P-D)/(?+?) (e^((?+?) t_1 )-1)[email protected]((P-D)I_h)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1))(e^(-Rb)-1)/(e^(- Rb/N)-1) (14)

By using Taylor’s series about origin for the exponential functions to obtain exactly solution as follows:

e^(-(N+1)Rb/N)-1,e^(-Rb/N)-1,e^(-Rb)-1,e^(- Rb/N)-1,

The e Eq. (14) became

TC=AF/t_1 +1 +(C(P-D)(e^((?+?) t_1 )-1))/(?+?)+(I_h (P-D)(e^((?+?) t_1 )-(?+?) t_1-1))/(?+?)^2 +(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)Fb/t_1 (15)

By taking derivate the Eq.14 With respect to t_1 to find out the minimum value of total cost function then

dTC/(dt_1 )=((-AFb)/?t_1?^2 +C(P-D)(e^((?+?) t_1 ) )+((P-D) I_h (e^((?+?) t_1 )-(?+?) ))/((?+?) )+C_s (P-D)(e^((?+?) t_1 ) ) )Fb/t_1 -Fb/?t_1?^2 (C(P-D)(e^((?+?) t_1 )-1))/(?+?)+(I_h (P-D)(e^((?+?) t_1 )-(?+?) t_1-1))/(?+?)^2 +(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)=0

?t_1?^* =±?((A+(C+C_s)(P-D))/((P-D)(C+C_s)(?+?)+I_h))

The feasible solution for the first runtime is positive case.

?t_1?^* =?((A+(C+C_s)(P-D))/((P-D)(C+C_s)(?+?)+I_h)) (16)

To test the type of critical point at second derivative for the total cost with respect to t_1 as

(d^2 TC)/(d?t_1?^2 ) |_(t_1=?t_1?^* )=2FAb/?t_1?^3 +(2Fb(C+C_s)(P-D)))/?t_1?^3 ;0

Since P ; D, Q;0

So that the total cost function has a minimum value at the point ? t?_1

The period of the first time run.

?Q_1?^* =(P-D)/(?+?) (e^((?+?) ?t_1?^* )-1) (17)

4. Sensitivity analysis

Fig.2.Inventory control of eggs

Example1

D=300,I_h=0.05$ ,C=,20$,A=15$,C_s=0.005 $\unit,P=800,?=1 ,t_1=12 days , N=24, F = 0.789, R= 0.5, b=365 day

Table1. The sensitivity analysis

? ? Q^* Q TC*(Q*) ?t_1?^*

0.0000225 0.0000448 9915.7911 4400 203393.5426 19.81836

0.0075 0.0075 4552.3296 4100 92080.8026 8.5343237

0.04 0.04 2359.0167 3800 47454.2715 4.0028581

0.045 0.045 2252.9162 3499.2 45308.1381 3.7823426

0.05 0.05 2162.8269 3199.2 43486.7701 3.594668

0.1 0.1 1673.6968 2899.2 33612.1610 2.5625572

0.2 0.2 1338.1209 2599.2 26851.0379 1.8194713

0.3 0.3 1201.1942 2298 24095.0121 1.4876419

0.4 0.4 1128.0852 1998 22623.8373 1.2892261

0.5 0.5 1084.7874 1698 21752.4128 1.1535975

0.6 0.6 1058.224 1398 21217.4926 1.0533772

0.7 0.7 1042.1732 450 20893.8896 0.9754307

0.8 0.8 1033.2573 4550 20713.6659 0.9125673

0.9 0.9 1029.4688 4248 20636.4539 0.8604761

0.99 0.99 1029.3798 3948 20633.6609 0.8205007

Fig.3. Graphical representation optimal first runtime VS deterioration rate Fig.4. . Graphical representation optimal first runtime VS break-ability rate

Fig.5. Graphical representation optimal actual quantity and EOQ VS break-ability rate Fig.6. Graphical representation optimal first runtime VS EOQ

Fig.7. Graphical representation actual total cost and optimal total cost Vs optimal first runtime Fig.8. Graphical representation actual total cost and optimal total cost Vs EOQ

Fig.9. Graphical representation the difference between the actual total cost and optimal total cost Vs optimal first runtime

5. Conclusion

In this paper, the deterioration, breakability represented in the proposed model at the same time to reduce the total cost of eggs items and assumed the daily demand is constant, developed the mathematical model by the above assumptions and applies it in real life-based in the collected data. The output of the mathematical model is minimizing the total cost of eggs items and obtaining the economic order quantity for several breakable rate values which is including the calculated deterioration, breakable rates for eggs items. The model is fitted with any values for the deterioration, breakability rates as percentage if and only if the percentage for deteriorating rate lies in the range 99 %, the percentage for breakability rate lies in the range 99% at same time the obtained economic order quantity is satisfied the obtaining optimal total cost. The economic order quantity was decreasing when the deterioration, breakability rates increased so that the total cost of eggs is decreasing because the order economic quantity decreased. The difference between the total costs under assumptions of the mathematical model is decreasing where optimal first runtime was decreasing as the fig. (9). The output of above figures the inventory model showed the optimal total cost, economic order quantity for the collected data, first cycle with 0.0000225 as deterioration rate, 0.0000448 as breakability rate obtained in maximum value with risk level as percentages of deterioration, breakability rates lie between 0%, 99% respectively, which it is satisfied the optimal total cost whereas it would be at minimum value when risk was increasing which is satisfied the optimal total cost, when the percentages of each deterioration, breakability are less respectively. The impression about the output of the proposed model for the eggs items is the committed policy of imported industry of eggs at the Republic of Yemen .the benefit is with risk level as percentage of deterioration is less percentage of breakability is less of the inventory of eggs in stock with 50% as representing the net discount rate of inflation. The optimal quantity was decreasing when optimal of first time run decreased.

6. References

Abdullah.M.A.Alshami, Najjeb.N.M.Maglas. (2018). An Inventory Model for Determining Optimal Tiles

size the Republic of Yemen with Weibull Distribution Demand . IJAR 16, 680–688.

Alshami, Abdullah, Muley, Aniket. (2018). Economic ordering policy for deteriorating and breakable item with non-zero level of inventory. Journal Of Emerging Technologies And Innovative Research. 5. 1-7.

Abad.P.L. (2001). Optimal price and order size for a reseller under partial back-ordering. Computers and Operations Research28, 53–65.

Chung. K.J, L. J. (2006). The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. International Journal of Production Economics100, 116–130.

Covert. R. P, P. C. (1973). An EOQ model for items with Weibull distribution deterioration. AIIE Transactions5, 323–326.

Elasayed.E.A, T. (1983). Analysis of inventory systems with deteriorating items. InternationalJournal of Production Research 21 , 449–460.

Giri.B. C, D. (2005). The exact formulation of the stochastic EMQ model for an unreliable production system. Journal of the Operational Research Society 56, 563–575.

Goyal.S.K, B. C. (2001). Recent trends in the modelling of deteriorating inventory. European Journal of Operational Research 134, 1–16.

Hill.R.Ml. (1999). The optimal production and shipment policy for single-vendor single-buyer integrated production–inventory problem. International Journal of Production Research3, 2463–2475.

Lee C.C, H. S. (2009). A two-warehouse production model for deteriorating inventory items with time- dependent demands. European Journal of Operational Research 194 , 700–710.

Lin.G.C, K. (2006). Economic lot sizing for an imperfect production system subject to random breakdowns. Engineering Optimization 38, 73–92.

Maity. A, M. K. (2007). Chebyshev approximation for solving the optimal production inventory problem of deteriorating multi-item. Mathematical and Computer Modelling45, 149–161.

Swenseth, S. G. (2002). Incorporating transportation costs into inventory replenishment decisions . International Journal of Production Economics 77, 113–130.Economic Ordering Policy for Eggs Items with Deterioration and Breakability under Constant Production in the Republic Of Yemen

Abdullah.M.A.Alshami1, Aniket.A.Muley2

1Thamer university, Thamer, Yemen

Educational Faculty, Mathematics department

2Swami Ramanand Teerth Marathwada University, Nanded, India

431606 Maharashtra, India

Mathematical School of Sciences Statistics

[email protected] , [email protected]

ABSTRACT

This is paper developed the policy of EOQ in an inventory control system under the effect of the time value of money inflation rate on inventory at store and production consideration. The proposed an economic order quantity model to manage deteriorating and break-ability rates for items which have both risks as break-ability and deterioration in stock by obtaining the optimal quantity, optimal total cost of lot size if break-ability risk or deteriorating risk is occurring once of them of both of them. The finite horizon planning, the demand, production rate and deterioration break-ability rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first; here solution procedure is proposed to derive the total cost with respect to the order quantity, the number of replenishment during the planning horizon with two cases for optimal economic order quantity based on the. Finally represented the EOQ versus the deterioration and break-ability rates for the real data of eggs in the stock of sample industry which is selected randomly form eggs industry in the Republic of Yemen. Sensitivity analysis for the proposed data was suggested the accepted range of the net constant rate of inflation to achieved optimal total cost under economic order quantity, discussing the goodness of proposed model under several values of deterioration, break-ability rates and representing the net constant discount rate of inflation. Rates and the representing the net constant discount rate of inflation.

Keywords: Inventory-Deterioration-Breakability- Production Rate -Total Cost Function (TC)

Introduction

There are researches represented models considered the deterioration rate or breakability rate in the model as Abdullah, Najeeb (2018) proposed a model to determine optimal size for breakable item with broken function for broken items with Weibull demand. Abdullah, Aniket developed the model for inventory with non-zero level. Ate Allah, Mohammed (2014) assumed the back ordering and financial for deteriorating items. Giri, Dohi (2005) represented format as the exact formulation of the stochastic EMQ model for an unreliable production system. Chung, Liao (2006) proposed a model to obtain optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. Covert, Philip (1973) optimizes the EOQ by the represented model for items with Weibull distribution deterioration. Elasayed, Teresi (1983) developed the model for EOQ by analysis of inventory systems with deteriorating items. Goyal, Giri (2001) used the technique of trends in the modeling of deteriorating inventory. Hill (1999) developed the optimal production and shipment policy for single-vendor single-buyer integrated production inventory problem. Lee, Hsu (2009) developed a two-warehouse production model for deteriorating inventory items with time-dependent demands. Lin, Kroll (2006) developed economic lot sizing for an imperfect production system subject to random breakdowns. Maity, Maity, Mondal, Maiti (2007) applied Chebyshev approximation for solving the optimal production inventory problem of deteriorating multi item. Swenseth, Godfrey (2002) developed incorporating transportation costs into inventory replenishment decisions. Wagner, Whithin (1958) developed a dynamic version of the economic lot size model. Weiss, Rosenthal (2002) developed the model for optimal ordering policies when anticipating a disruption in supply or demand. Yang, Wee (2002) developed models for single-vendor and multiple-buyers production-inventory policy for a deteriorating item. In general, deterioration is defined as the decay, damage, spoilage, evaporation, and obsolescence of stored items and it results in decreasing usefulness, breakability is defined as the broken items cannot use it under any circumstances the items, in this case, may have deterioration rate or breakability or both at the same time. So that the management and holding of inventories of breakable and deteriorating items becomes an important problem for inventory managers. In real-world problem, there are many items having deterioration and breakability characteristics at same time as the eggs and blub and each items packed in can made from glass as medicines and milk which are packed in glass, some fruits as watermelon and nosier, olive, olive oil, these types of items require advanced model to manage the items have close validity with minimum risk of deterioration breakability characteristics for industry to determine the optimal economic ordering quantity, optimal total cost of per unit in each those items. The proposed model concerned among all of the items especially eggs items as a sample of area study in the Republic of Yemen. All above-mentioned issues (deteriorating items, breakability items, inflation and time value of money) are separately regarded in some inventory models.

2. Material and methods

2.2. Assumptions and notions

2.2.1. Assumptions

In this paper, the mathematical model is developed with the following assumptions

The planning horizon is finite.

Single item inventory control.

Demand and deterioration and breakability rates are constant.

Production rate is constant.

Deterioration and breakability occur as soon as the items are received into inventory.

There is no replacement or repair of breakable items during the period under consideration.

The shortage is not allowed.

The inflation rate is constant and the time value of money is considered.

The lead time is zero.

The inventory level at the end of the planning horizon will be zero.

The cost factors are deterministic.

The number of replenishment is restricted to one integer.

The total relevant cost consists of fixed ordering, purchasing, holding interest payable, interest earned from sales revenue during the permissible period.

The last order is only being placed to satisfy the balance in the stock of the last period.

2.3. Notation

C_P = The present value of holding cost during the first replenishment cycle.

Q = The order quantity in the replenishment.

TC_A = The total fixed ordering cost (0, b).

TC_h = The total holding cost (0, b)

TC_P = The total purchasing cost (0, b).

TC_s = The total shortage cost (0, b).

TC = The total relevant cost (0, b).

2.4. Parameters

The mathematical model is representing the following parameters

A = The fixed ordering cost per replenishment, $\order.

C = The unit purchasing price at time zero, $\order.

C (t) = The unit purchasing price at time t, C(t)=Ce^(-RT).

D = The constant demand rate per unit time.

p = The production rate per cycle.

b = The length of the finite planning horizon.

i = The constant inflation rate.

I(t) = The inventory level at time t.

I_C = The interest charged per $per year by the supplier.

r = The discount rate representing the time value of money.

R=r-1 , representing the net constant discount rate of inflation.

T = The length of each replenishment cycle.

T_j = The total time that elapsed up to, including interest charges.

? t?_j = The time at which the inventory level in the j^th

V = The unit selling price at time t.

C_S=The shortage cost pre-unit time ,$\unit\unit time

C_S (t)=The shortage cost pre-unit time ,$\unit\unit time at time t ,C_s (t)=C_s e^(-RT).

V(t) = The selling price per unit at time t, V(t)= Ve^(-RT)

? = The constant deterioration rate, units/unit time.

? = The constant breakability rate, units/unit time.

3. Mathematical model

Let is the inventory level at any time t, , Depletion due to demand and deterioration, breakability will occur simultaneously. The first order differential equation that describes the instantaneous state of over the open interval (0, b) is given by.

(dI(t))/dt+?I(t)+?I(t)=-(P-D),0?t?t_1,0???1,0???1, P;D (1)

Fig.1Graphical representation of the inventory control diagram

(dI(t))/dt=-(P-D),t_1?t?T, Where for equation of number

I(t)=? I?_0 (t) ?_t^(t_1)??(P-D) e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t)

I(t)=? I?_0 (t) ?_t^(t_1)??(P-D)e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t) (2)

According to Eq.2 the maximum inventory quantity at the begin each period is given as

Q= (P-D)/(?+?) (e^((?+?) t_1 )-1),t_1=Fb/N (3)

3.1. Fixed ordering cost

We assumed the number of replenishment is N so that the fixed ordering cost over the planning horizon under the inflation consideration is:

TC_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^(-(N+1)RT)-1)/(e^(-RT)-1)),T=b/N (4)

Then

TC_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^((-(N+1)Rb)/N)-1)/(e^((-Rb)/N)-1)),T=b/N (5)

3.2. Holding cost excluding interest cost

We find the average inventory quantity to obtain holding cost

I ?=?_0^(t_1)??I(t)dt=?_0^(t_1)??(P-D)/(?+?)(e^(?+?)(t_1-t) dt= (P-D)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1) ?? (6)

By using Eq. 7 we have obtained holding cost is as follows

TC_h=?_(j=0)^(N-1)??I_h C_j I ?=?_(j=0)^(N-1)??I_h Ce^(-R_j T)=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?)?? t_1-1) (7)

Since T= b/N then equation number (8) it will be as

TC_h=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)(e^(-Rb)-1)/(e^(-Rb/N)-1) (8)

3.3. Purchasing cost

According to fig.1 of inventory level the purchasing cost of j^th cycle is calculated as

CP_j=C_j I_m=C_j (P-D)/(?+?)(e^((?+?) t_1 )-1) (9)

The total purchasing cost over the planning horizon can be obtained as

Special case for total purchasing cost when

TC_P=?_(j=0)^(N-1)??CP_j=C(P-D)/(?+?)(e^((?+?) t_1 )-1)(? (e^(-Rb)-1)/(e^(-Rb/N)-1)) (10)

3.4. Shortage cost

The shortage cost which is calculated as

?TC?_s=?_0^(N-1)??C_s e^(-jRT) C_j=? (C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)(e^(-Rb)-1)/(e^(- Rb/N)-1) (11)

TC=TC_A+TC_h+TC_P+?TC?_s

Then,

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)+ ?((C(P-D)+)/(?+?) (e^((?+?) t_1 )-1)[email protected]((P-D)I_h)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1))(e^(-Rb)-1)/(e^(- Rb/N)-1) (12)

3.5. Economic order quantity

3.5. 1.Economic order quantity

To find EOQ by minimizing the total cost function as the following

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)

+(C(P-D)(e^((?+?) t_1 )-1))/(?+?)+((P-D)I_h)/((??+?)?^2 ) (e^(((?+?) t_1)/N)-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)(e^(-Rb)-1)/(e^((-Rb)/N)-1) (13)

Since t_1=Fb/N

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)+ ?(C(P-D)/(?+?) (e^((?+?) t_1 )-1)[email protected]((P-D)I_h)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)+(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1))(e^(-Rb)-1)/(e^(- Rb/N)-1) (14)

By using Taylor’s series about origin for the exponential functions to obtain exactly solution as follows:

e^(-(N+1)Rb/N)-1,e^(-Rb/N)-1,e^(-Rb)-1,e^(- Rb/N)-1,

The e Eq. (14) became

TC=AF/t_1 +1 +(C(P-D)(e^((?+?) t_1 )-1))/(?+?)+(I_h (P-D)(e^((?+?) t_1 )-(?+?) t_1-1))/(?+?)^2 +(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)Fb/t_1 (15)

By taking derivate the Eq.14 With respect to t_1 to find out the minimum value of total cost function then

dTC/(dt_1 )=((-AFb)/?t_1?^2 +C(P-D)(e^((?+?) t_1 ) )+((P-D) I_h (e^((?+?) t_1 )-(?+?) ))/((?+?) )+C_s (P-D)(e^((?+?) t_1 ) ) )Fb/t_1 -Fb/?t_1?^2 (C(P-D)(e^((?+?) t_1 )-1))/(?+?)+(I_h (P-D)(e^((?+?) t_1 )-(?+?) t_1-1))/(?+?)^2 +(C_s (P-D))/(?+?)(e^((?+?) t_1 )-1)=0

?t_1?^* =±?((A+(C+C_s)(P-D))/((P-D)(C+C_s)(?+?)+I_h))

The feasible solution for the first runtime is positive case.

?t_1?^* =?((A+(C+C_s)(P-D))/((P-D)(C+C_s)(?+?)+I_h)) (16)

To test the type of critical point at second derivative for the total cost with respect to t_1 as

(d^2 TC)/(d?t_1?^2 ) |_(t_1=?t_1?^* )=2FAb/?t_1?^3 +(2Fb(C+C_s)(P-D)))/?t_1?^3 >0

Since P > D, Q>0

So that the total cost function has a minimum value at the point ? t?_1

The period of the first time run.

?Q_1?^* =(P-D)/(?+?) (e^((?+?) ?t_1?^* )-1) (17)

4. Sensitivity analysis

Fig.2.Inventory control of eggs

Example1

D=300,I_h=0.05$ ,C=,20$,A=15$,C_s=0.005 $\unit,P=800,?=1 ,t_1=12 days , N=24, F = 0.789, R= 0.5, b=365 day

Table1. The sensitivity analysis

? ? Q^* Q TC*(Q*) ?t_1?^*

0.0000225 0.0000448 9915.7911 4400 203393.5426 19.81836

0.0075 0.0075 4552.3296 4100 92080.8026 8.5343237

0.04 0.04 2359.0167 3800 47454.2715 4.0028581

0.045 0.045 2252.9162 3499.2 45308.1381 3.7823426

0.05 0.05 2162.8269 3199.2 43486.7701 3.594668

0.1 0.1 1673.6968 2899.2 33612.1610 2.5625572

0.2 0.2 1338.1209 2599.2 26851.0379 1.8194713

0.3 0.3 1201.1942 2298 24095.0121 1.4876419

0.4 0.4 1128.0852 1998 22623.8373 1.2892261

0.5 0.5 1084.7874 1698 21752.4128 1.1535975

0.6 0.6 1058.224 1398 21217.4926 1.0533772

0.7 0.7 1042.1732 450 20893.8896 0.9754307

0.8 0.8 1033.2573 4550 20713.6659 0.9125673

0.9 0.9 1029.4688 4248 20636.4539 0.8604761

0.99 0.99 1029.3798 3948 20633.6609 0.8205007

Fig.3. Graphical representation optimal first runtime VS deterioration rate Fig.4. . Graphical representation optimal first runtime VS break-ability rate

Fig.5. Graphical representation optimal actual quantity and EOQ VS break-ability rate Fig.6. Graphical representation optimal first runtime VS EOQ

Fig.7. Graphical representation actual total cost and optimal total cost Vs optimal first runtime Fig.8. Graphical representation actual total cost and optimal total cost Vs EOQ

Fig.9. Graphical representation the difference between the actual total cost and optimal total cost Vs optimal first runtime

5. Conclusion

In this paper, the deterioration, breakability represented in the proposed model at the same time to reduce the total cost of eggs items and assumed the daily demand is constant, developed the mathematical model by the above assumptions and applies it in real life-based in the collected data. The output of the mathematical model is minimizing the total cost of eggs items and obtaining the economic order quantity for several breakable rate values which is including the calculated deterioration, breakable rates for eggs items. The model is fitted with any values for the deterioration, breakability rates as percentage if and only if the percentage for deteriorating rate lies in the range 99 %, the percentage for breakability rate lies in the range 99% at same time the obtained economic order quantity is satisfied the obtaining optimal total cost. The economic order quantity was decreasing when the deterioration, breakability rates increased so that the total cost of eggs is decreasing because the order economic quantity decreased. The difference between the total costs under assumptions of the mathematical model is decreasing where optimal first runtime was decreasing as the fig. (9). The output of above figures the inventory model showed the optimal total cost, economic order quantity for the collected data, first cycle with 0.0000225 as deterioration rate, 0.0000448 as breakability rate obtained in maximum value with risk level as percentages of deterioration, breakability rates lie between 0%, 99% respectively, which it is satisfied the optimal total cost whereas it would be at minimum value when risk was increasing which is satisfied the optimal total cost, when the percentages of each deterioration, breakability are less respectively. The impression about the output of the proposed model for the eggs items is the committed policy of imported industry of eggs at the Republic of Yemen .the benefit is with risk level as percentage of deterioration is less percentage of breakability is less of the inventory of eggs in stock with 50% as representing the net discount rate of inflation. The optimal quantity was decreasing when optimal of first time run decreased.

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