Vol. 6, No. 1, 2015,99-103
ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 22 January 2015
www.researchmathsci.org
99
International Journal of
An Inventory Model with Three Rates of Production and
Time Dependent Deterioration Rate for Quadratic
Demand Rate
P.K.Lakshmidevi1 and M.Maragatham2
1
Department of Mathematics, Saranathan College of Engineering, Trichy – 12, India e-mail: sudhalakshmidevi28@gmail.com
Corresponding Author
2
PG and Research Department of Mathematics, Periyar E.V.R College Trichy – 23, India. e-mail:maraguevr@yahoo.co.in
Received 15 October 2014; accepted 29 November 2014
Abstract. An inventory model with three different rates of production and quadratic demand rate is considered. The shortages are allowed and deterioration rate is time dependent. The objective is to determine the optimal total cost and the optimal time schedule of the plan for the proposed model. To illustrate the results of this model, numerical example is presented.
Keywords: Inventory, quadratic demand, time dependent deterioration AMS Mathematics Subject Classification (2010): 90B05
1. Introduction
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payments for n-cycles. Yu [8] discussed a ordering policy for two – phase deteriorating items with changing deterioration rate. A production inventory model with two rates of production, backorders and variable production cycle is analyzed by Bhowmick and Samauta. The optimal values of inventory levels are derived using Hessian matrix. In this paper, a continuous production control inventory model with three rates of production and time dependent deterioration rate for quadratic demand rate is developed. The production is started at one rate and after some time it may be switched over to another rate is possible to real life situations. Such a situation is desirable in the sense that starting at low rate of production, a large quantum stock of manufactured items at the initial stage is avoided, leading to reduction in the holding cost, the new production rate is used. For demand, the quadratic function in time and time dependent deterioration rate is considered.
2. Notations and assumption Notations
Iଵ - The inventory level at time ݐଵ Iଶ - The inventory level at time ݐଶ Iଷ - The inventory level at time ݐସ ൣݐଷ,ݐସ൧ – The shortage period T – The time length of the plan ܫ(ݐ) – The inventory level at time t ଵ - The production rate in [0,ݐଵ] ଶ - The production rate in [ݐଵ, ݐଶ] ଷ - The production rate in [ݐସ, ܶ]
h – The holding cost per unit per unit time d – The deteriorating cost per unit per unit time s – The shortage cost per unit per unit time p – The production cost per unit per unit time TC- The total cost per unit time
Assumptions
1. The demand rate ݂(ݐ) = ܽ + ܾݐ + ܿݐଶ, ܿ > 0, ܾ > ܽ > 0 2. The deterioration rate ߠ(ݐ) = ߙݐ, ߙ > 0
3. The production rates ଷ> ଶ> ଵ 4. The shortages are allowed.
5. There is no replacement or repair of deteriorated items during the cycle under consideration.
6. The deterioration is instantaneous.
7. The time for allowing shortages is same as back order time
3. Model formulation
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durationሾ0, ݐଵሿ, the production is at the rate ଵ and the consumption by demand. At the time duration ሾݐଵ, ݐଶሿ, the production is at the rate ଶ and the consumption by demand. At the time duration ሾݐଶ, ݐଷሿ, there is no production, but only consumption by demand. At the time duration ሾݐଷ, ݐସሿ, the shortages is allowed. ሾݐସܶሿisthe duration of time to backlog at production rate ଷ. The cycle then repeats itself after time T. The deterioration is instantaneous with the rate ߙݐ in the duration ሾ0, ݐଷሿ. The model is represented by figure 1.
Inventory level
0 ࢚ ࢚ ࢚ ࢚ T Time
Figure 1: A production inventory model
The change of inventory level can be described as follows ௗூ(௧)
ௗ௧ + ߠ(ݐ)ܫ(ݐ) = ଵ− ݂(ݐ), 0 < ݐ < ݐଵ (1) ௗூ(௧)
ௗ௧ + ߠ(ݐ)ܫ(ݐ) = ଶ− ݂(ݐ), ݐଵ< ݐ < ݐଶ (2) ௗூ(௧)
ௗ௧ + ߠ(ݐ)ܫ(ݐ) = −݂(ݐ), ݐଶ< ݐ < ݐଷ (3) ௗூ(௧)
ௗ௧ = −݂(ݐ), ݐଷ< ݐ < ݐସ (4) ௗூ(௧)
ௗ௧ = ଷ− ݂(ݐ), ݐସ< ݐ < ܶ (5) With boundary conditionsܫ(0) = 0, ܫ(ݐଵ) = ܫଵ, ܫ(ݐଶ) = ܫଶ, ܫ(ݐଷ) = 0, ܫ(ݐସ) = ܫଷ, ܫ(ܶ) = 0 (6) The solution of the above equations are given by
I(t) =
ە ۖ ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۖ
ۓ eିಉ౪మమ ቂp
ଵt − at −ୠ୲ మ
ଶ −
ୡ୲య
ଷ +
୮భ୲య
−
ୟ୲య
−
ୠ୲ర
଼ −
ୡ୲ఱ
ଵቃ , 0 < t < tଵ
eିಉ౪మమ pଶ(t − tଵ) − a(t − tଵ) − ୠ
ଶ(tଶ− tଵଶ) − ୡ
ଷ(tଷ− tଵଷ) + pଶα(tଷ− tଵଷ) − ୟ
(tଷ− tଵଷ) −ୠ଼(tସ− t
ଵସ) −ୡଵ(tହ− tଵହ) + Iଵeಉ౪భమమ
, tଵ< t < tଶ
eିಉ౪మమ −a(t − tଶ) −ୠ
ଶ(tଶ− tଶଶ) − ୡ
ଷ(tଷ− tଶଷ) − ୟ
(tଷ− tଶଷ) − ୠ
଼(tସ− tଶସ) − ୡ
ଵ(tହ− tଶହ) + Iଶe ಉ౪మమ
మ ൨ , tଶ< t < tଷ −a(t − tଷ) −ୠଶ(tଶ− tଷଶ) −ୡଷ(tଷ− tଷଷ),tଷ< t < tସ
pଷ(t − tସ) − a(t − tସ) −ୠଶ(tଶ− tସଶ) −ଷୡ(tଷ− tସଷ) + Iଷ, tସ< t < T
(7)
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ܫଶ= ݁ିഀమమమ ቂܽ(ݐଷ− ݐଶ) +ଶ(ݐଷଶ− ݐଶଶ) +ଷ(ݐଷଷ− ݐଶଷ) +ఈ (ݐଷଷ− ݐଶଷ) + ఈ
଼ (ݐଷସ− ݐଶସ) +ఈଵ(ݐଷହ− ݐଶହ)ቃ (9)
By considering the continuity at ݐଵ, Iଵ= eି
α౪భమ
మ ቂpଵtଵ− atଵ−ୠ୲భమ
ଶ − ୡ୲భయ ଷ + ୮భα୲భయ − ୟα୲భయ − ୠα୲భర ଼ − ୡα୲భఱ
ଵ ቃ (10)
Therefore, after considering the continuity points, the solution becomes ܫ(ݐ) = ە ۖ ۖ ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۖ ۖ ۓ ݁ିഀమమ ቈଵݐ − ܽݐ −ܾݐଶ 2 −ܿݐ
ଷ 3 +ଵߙݐ
ଷ 6 −ܽߙݐ
ଷ 6 −ܾߙݐ
ସ 8 −ܿߙݐ
ହ
10 , 0 < ݐ < ݐଵ
݁ିഀమమ ۏ ێ ێ ۍଵݐଵ+ ଶ(ݐ − ݐଵ) − ܽݐ −ܾݐ ଶ 2 −ܿݐ
ଷ 3 +ଵߙݐ
ଷ
6 + ଶߙ(ݐଷ− ݐଵଷ) −ܽߙݐ ଷ 6 −ܾߙݐ8 −ସ ܿߙݐ10ହ ےۑ
ۑ ې
, ݐଵ< ݐ < ݐଶ
݁ିഀమమ −ܽ(ݐ − ݐଷ) −ܾ
2 (ݐଶ− ݐଷଶ) −ܿ3 (ݐଷ− ݐଷଷ) −ܽߙ6 (ݐଷ− ݐଷଷ) −ܾߙ8 (ݐସ− ݐଷସ) −ܿߙ10 (ݐହ− ݐଷହ)൨, ݐଶ< ݐ < ݐଷ −ܽ(ݐ − ݐଷ) −ܾ2 (ݐଶ− ݐଷଶ) −ܿ
3 (ݐଷ− ݐଷଷ),ݐଷ< ݐ < ݐସ ଷ(ݐ − ݐସ) − ܽ(ݐ − ݐଷ) −ܾ2 (ݐଶ− ݐଷଶ) −ܿ3 (ݐଷ− ݐଷଷ) + ܫଷ,ݐସ< ݐ < ܶ
(11)
Total inventory carried over the period ሾ0, ܶሿ = ܫ(ݐ)݀ݐ௧భ + ܫ(ݐ)݀ݐ௧௧మ
భ + ܫ(ݐ)݀ݐ
௧య
௧మ (12)
Total number of deteriorating items in ሾ0, ݐଷሿ = ଵݐଵ+ ଶ(ݐଶ− ݐଵ) − ݂(ݐ)݀ݐ௧య
= ଵݐଵ+ ଶ(ݐଶ− ݐଵ) − ܽݐଷ−ଶଵܾݐଷଶ−ଵଷܿݐଷଷ (13) Total shortage occurred in ሾݐଷ, ܶሿ = − ܫ(ݐ)݀ݐ௧௧ర
య − ܫ(ݐ)݀ݐ
்
௧ర (14)
Total number of units produced in ሾ0, ܶሿ = ଵݐଵ+ ଶ(ݐଶ− ݐଵ) + ଷ(ܶ − ݐସ) (15) Total cost ܶܥ = ܪܥ + ܦܥ − ܵܥ + ܲܥ
= ℎ න ܫ(ݐ)݀ݐ ௧భ + න ܫ(ݐ)݀ݐ ௧మ ௧భ + න ܫ(ݐ)݀ݐ ௧య ௧మ + ݀ ଵݐଵ+ ଶ(ݐଶ− ݐଵ) − න ݂(ݐ)݀ݐ ௧య −ݏ ቂ− ܫ(ݐ)݀ݐ௧ర ௧య − ܫ(ݐ)݀ݐ ்
௧ర ቃ + ሾଵݐଵ+ ଶ(ݐଶ− ݐଵ) + ଷ(ܶ − ݐସ)ሿ (16)
The time for allowing shortages is same as back order time (ie), ݐସ− ݐଷ= ܶ − ݐସ
Therefore ݐଷ= 2 ∗ ݐସ− ܶ (17) The necessary conditions for ܶܥ(ݐଵ, ݐଶ, ݐସ, ܶ) to be minimum are
డ் డ௧భ = 0,
డ் డ௧మ = 0,
డ் డ௧ర = 0,
డ்
డ் = 0 (18) Solving these equations , we get the optimal values ݐଵ∗, ݐଶ∗, ݐସ∗, ܶ∗ which minimize total cost provided they satisfy the following sufficient condition
103 =
ۉ ۈ ۈ ۈ ۈ ۇ
డమ்
డ௧భమ
డమ்
డ௧మడ௧భ
డమ்
డ௧రడ௧భ
డమ்
డ்డ௧భ
డమ்
డ௧భడ௧మ
డమ்
డ௧మమ
డమ்
డ௧రడ௧మ
డమ்
డ்డ௧మ
డమ்
డ௧భడ௧ర
డమ்
డ௧మడ௧ర
డమ்
డ௧రమ
డమ்
డ்డ௧ర
డమ்
డ௧భడ்
డమ்
డ௧మడ்
డమ்
డ௧రడ்
డమ்
డ்మ ی
ۋ ۋ ۋ ۋ ۊ
is positive definite. (19)
If the solutions obtained from equations (18) do not satisfy the sufficient condition (19), then no feasible solution will be optimal for the set of parameter values taken to solve equations (18) is considered. Such a situation will imply that the parameter values are inconsistent and there is some error in their estimation.
3.1. Numercial example
Let ܽ = 0.1, ܾ = 0.2, ܿ = 0.003, ߙ = 0.02, ଵ= 2, ଶ= 4, ଷ= 6, ℎܿ = ܴݏ. 2,
݀ܿ = ܴݏ3, ݏܿ = ܴݏ6, ܿ = ܴݏ10 Using MATLAB program the following are calculated, Then optimal values of ݐଵ= 1.3496, ݐଶ= 3.0748, ݐଷ= 3.1419, ݐସ= 3.5117, ܶ = 3.8815.
The total cost ܶܥ = ܴݏ167.7252. The production quantity = 11.8188.
4. Conclusion
A continuous production inventory model for time dependent deteriorating items with shortages in which three different rates of production and quadratic demand rate is considered. The case of change of production is very useful in practical situations. By starting at a low rate of production, a large quantum stock of manufactured item, at the initial stage is avoided, leading to reduction in the holding cost. The variation in production rate provides a way resulting consumer satisfaction and earning potential profit.The total cost of the system and the optimal values for ݐଵ, ݐଶ, ݐସ, ܶ is derived for quadratic demand rate and time dependent deterioration rate.
REFERENCES
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3. J.Bhowmick and G.P.Samauta, A deterministic inventory model of deteriorating items with two rates of production, shortages and variable production cycle, ISRN Applied Mathematics,(2011) 1-16.
4. R.Kasthuri, P.Vasanthi, S.Ranganayaki and C.V.Seshaiah, Multi-Item inventory model involving three constraints : A Karush-Kuhn-Tucker conditions approach, American Journal of Operations Research, 1 (2011) 155-159.
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6. M.Maragatham and P.K.,Lakshmidevi, An inventory model for non instantaneous deteriorating items under conditions of permissible delay in payments for n-cycles, International Journal of Fuzzy Mathematical Archive, 2 (2013) 49-57.
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