B0370709

American Journal of Engineering Research (AJER)

2014

American Journal of Engineering Research (AJER) e-ISSN : 2320-0847 p-ISSN : 2320-0936 Volume-03, Issue-07, pp-07-09 www.ajer.org Research Paper

Open Access

Random Supply, Costant Lead Times and Quadratic Backorder Costs. For Inventory Model (M, T) Dr. Martin Omorodion Joseph Ayo Babalola University Ikeji – Arakeji, Nigeria ABOUT THE AUTHOR BS.c (Bristol) Economics with Statistics MS.c (Wales) Statistics MS.c (Birmingham) Computer Science Ph.D Operational Research A.I.A Associate of the Institute of Actuaries Abstract: - This paper considers the inventory costs for the (M.T) model in which the backorder costs is quadratic, supply is continuous and lead time is constant.

Use is made of series 1: Inventory Model (M.T) with quadratic costs and continuous lead times.

The inventory cost for fixed M (maximum re-order level) and constant lead times is averaged over the states of M. Supply is assumed to follow a gamma distribution. The inventory costs for the model, random supply, constants lead time and quadratic costs are thereby derived.

Keywords: - Gamma distribution, normal distribution, maximum reorder level, quadratic backorder costs and constant lead times.

I. INTRODUCTION In inventory costs for fixed M and constant lead times obtained in series 1 is averaged over the states of M. M is assured to be a gamma variate. Demand during the lead time is normally distributed.

The inventory costs for fixed M (maximum re-order level) when the cost of a backorder was quadratic function of the length of time excluding the cost of stockouts dependent only on the number of stockouts equation 14 of series 1.

II. LITERATURE REVIEW Zipkin (2006) threats both fixed and random lead times and examines both stationary and limiting distributions under different assumptions.

Bartsimas (1999) in his paper “probabilistic service level guarantee in make-to-stock, considered both linear and quadratic inventory costs and backorder costs�.

Pritibhushan (2008) in his paper. „A note in Bernoulli demand inventory model presents a simple-item, continuous monitoring inventory model with probabilistic demand for the item and probabilistic lead time of order replacementâ€&#x;.

Hadley and Within (1972) extensively developed the inventory model (M.T) for constant lead times and linear backorder costs.

Random Supply Constant lead time and quadratic backorder costs. Series 2 Since the supply is a random variable the maximum re-order cover M would vary. Similarly M follows a gamma distribution. T becomes the only control parameter. The probability density function of M, u(M) � M =

esp ÎźM M đ?‘Łâˆ’1 Îźđ?‘Ł đ?‘Ł

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đ?‘Ł>0

đ?‘€>0

(1)

Page 7

American Journal of Engineering Research (AJER)

2014

Rc + S DT b1 hc + hc M − DL − + (G5 M, T + L − G5 M, L + b2 + T 2 T T b3 (G6 M, T + L − (G6 M, L + G 𝑀, T + L − G7 M, L T 7 DT 𝑀 − 𝐷𝑇 Where G5 M, T = σ2 T g M − − M − DT F 2 σ T 𝜎2𝑇 Multiplying by U(M) σ2 T esp − μM Mv−1 μv 𝑀 − 𝐷𝑇 σ2 T esp−μM μv 𝑀 − 𝐷𝑇 U M G5 M, T 𝑔 − 𝐹 2 𝑣 𝑣 σ T σ2 T ∞ Let G15 T = ∪ 𝑢 𝑀 𝐺5 𝑀, 𝑇 𝑑𝑀 Substituting for U M G5 M, T integrating and simplifying we have C=

G15

𝑣

2

𝑣−2 −𝑧 𝑣+1 2

− 𝑧=1 𝑖=0

G6 M, T = M−DT σ2 T

𝑣−1−1 𝑖

𝑖=0

𝑣−1 𝑖

𝐷𝑇 𝑧=1

𝑣!

2 𝜇 𝑣+1−𝑧 !

2

𝐷 − 𝜇𝜎 𝑇

𝜎2𝑇 2

𝑣−1−𝑖

𝑣−𝑧 2

𝑣

2

μ 𝜇 𝜎 𝑇 + 𝑒𝑠𝑝 − 𝐷𝜇𝑇 𝑣 2

F

𝑣−1

σ2 T 𝑣 𝜇2 𝜎 2 𝑇 T = 𝜇 𝑒𝑠𝑝 − 𝐷𝜇𝑇 𝑣 2

𝑖=0

𝑣+1−2+𝑧 𝑖

𝑣−𝑧−𝑖 𝑖

𝐷𝑇 − 𝜇𝜎 2 𝑇

2

𝐷𝑇 − 𝜇𝜎 𝑇

𝑣−𝑧−2𝑖

𝑖

𝜎2𝑇 2

𝑖

𝑖

𝜎2𝑇 2

𝑣+1−𝑧−2𝑖

(2)

σ4 𝐷𝑇 2 σ2 M 𝑀2 + + − 𝑇𝑀𝑉 + 4D3 2 2D2 2𝐷 σ2 T

+

2

σ2

M

σ4

D

T − D2 − D g M −

σ2 T

− 4D 3 𝑒𝑠𝑝

2𝐷𝑀 𝜎2

𝐹

𝑀+𝐷𝑇

(3)

𝜎 2𝑇

Multiplying by U (M) σ4 DT2 σ2 Mv +1 esp − μv μv v U M G6 M, T = + + − T + M + 4D3 2 2D2 2D 𝑣 2 v v M − DT σ2 T σ M esp − μM μ 𝑀 − 𝐷𝑇 + T− Mv−1 − 𝑔 2 2D D 𝑣 σ2 T σ2 T −

esp −μM μv M v −1 4D 3 𝑣

𝑒𝑠𝑝

∞ u 0

2𝐷𝑀 𝜎2

𝐹

𝑀−𝐷𝑇

(4)

σ2 T

Let G16 T = M G5 M, T dM Substituting for U(M) G2 (M, T) integrating and simplifying we have 𝑣−𝑧 𝜇 𝑣 𝑒𝑠𝑝 𝜇 2 𝜎 2 𝑇−𝐷𝜇𝑇 𝜎4 𝐷𝑇 2 𝑣−1 ! 𝑣−𝑧−𝑖 2 𝑣 𝐺16 𝑇 = + 𝑧=1 𝑖=0 𝜇 2 𝑣−2 ! 𝑣 4𝐷 3 2 2 𝑣+2 𝑣+2−𝑧 𝑖 2 𝑣+1 ! 𝑣+2+𝑧+𝑖 2 2 2 𝑣+2−𝑧−2𝑖 𝜎 𝑇 𝐷𝑇 − 𝜇𝜎 𝑇 + 𝑧=1 𝜇 2 𝑣+2−𝑧 ! 𝑧=0 2 𝑖 𝑣+1 (𝑣+1−𝑧)/2

𝑧=1

𝑖=0

𝑣!

2 𝜇 𝑣+1−𝑧

𝑣 𝜎 𝑇 𝜇 𝑒𝑠𝑝 + 2 2

𝑣+1−𝑧−𝑖 𝑖

𝜇2 𝜎 2 𝑇 2 − 𝐷𝜇𝑇 𝑣

𝑣 2

𝑖=0 𝑣

𝑣−2 𝑖

2

𝜎 𝑇− 2 𝐷 2

𝐷𝑇 − 𝜇𝜎 2 𝑇

𝑣−2𝑖

𝑣−2 2

𝜎 𝑇 2

−

𝜎 2𝑇

𝑖

2 2

𝐷𝑇 − 𝜇𝜎 2 𝑇

𝑣+1−𝑧−2𝑖

𝜎 −𝑇 2𝐷2

𝑣−1 2

𝑍=0

𝑣−1−𝑖 𝑖

𝜇𝑣 𝑒𝑠𝑝

𝑖

(5)

2

𝐷𝑇 − 𝜇𝜎 𝑇

𝑣−1−2𝑖

𝑖

𝜎2𝑇 2

2𝑖

−

1 𝐷

𝜇2 𝜎 2 𝑇 − 𝐷𝜇𝑇 2 𝑣 4𝐷3 𝑖

(𝑣 − 1)! 𝜎2𝑇 2 𝑣−𝑧−2𝑖 𝐷𝑇 − 𝜇𝜎 𝑇 6 2 2𝐷 2 𝑧=1 𝑖=0 𝜇 − 2 𝑣−2 𝜎 Substituting T for L 𝐺7 𝑀, 𝐷 𝑀3 𝜎 2 𝑀2 𝜎 4 𝑀2 𝜎6 𝜎 2 𝑇 2 𝑇 2 𝑀 𝑇 3 𝑇𝑀2 𝑀 − 𝐷𝑇 =− − 3− + + − + − − 2 𝐹 4 5 6 2 𝐷 3𝐷 2𝐷 2𝐷 4𝐷 2𝐷 𝐷 3 𝐷 𝜎2𝑇

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American Journal of Engineering Research (AJER) 1

+ +

đ?œŽ2đ?‘‡ đ?œŽ6 4đ??ˇ 6

đ?‘€ − đ??ˇđ?‘‡

đ?‘”

−

đ?œŽ2 đ?‘‡ 2đ??ˇđ?‘€

đ?‘’đ?‘ đ?‘?

2 đ?œŽ 2 đ?‘€đ?‘‡ 2 đ?œŽ 2 đ?‘‡ 3 đ?œŽ 2 đ?‘€đ?‘‡ đ?œŽ 4 đ?‘€đ?‘‡ đ?œŽ 4 đ?‘‡ 2 8đ?œŽ 6 đ?‘‡ − + + + + 5 3 đ??ˇ2 3đ??ˇ 3đ??ˇ3 2đ??ˇ4 6đ??ˇ3 đ??ˇ

đ?‘€+đ??ˇđ?‘‡

đ??š

đ?œŽ2

2014

(7)

đ?œŽ 2đ?‘‡

Multiplying by U(M) and simplifying đ??ş7 đ?‘€, đ?‘‡ đ?‘’đ?‘ đ?‘? −đ?œ‡đ?‘€ đ?œ‡đ?‘Ł đ?‘€2+đ?‘Ł đ?œŽ2 đ?‘‡ đ?œŽ4 đ?‘‡2 1+đ?‘Ł đ?‘Ł đ?‘ˆ đ?‘€ = = + đ?‘€ − + đ?‘€ + đ??ˇ đ?‘Ł 3đ??ˇ3 2đ??ˇ4 đ??ˇ2 2đ??ˇ5 đ??ˇ 6 2 2 6 6 đ?‘Łâˆ’1 đ?‘Ł đ?œŽ đ?œŽ đ?‘‡ 8đ?œŽ đ?‘‡ đ?œŽ 2đ??ˇđ?‘€ đ?‘’đ?‘ đ?‘? −đ?œ‡đ?‘€ đ?‘€ đ?œ‡ đ?‘€ + đ??ˇđ?‘‡ +đ?‘€đ?‘Ł+1 − + 5 đ?‘€ đ?‘Łâˆ’1 + 6 đ?‘’đ?‘ đ?‘? đ??š 4đ??ˇ6 6đ??ˇ3 đ??ˇ 4đ??ˇ đ?œŽ2 đ?‘Ł đ?œŽ2 đ?‘‡ Let G17 (T)=U(M)G7 (M,T)dM Then substituting for U(M) G7 (M,L+T) Integrating and simplifying đ??ş17

đ?œ‡đ?‘Ł đ?œ‡2 đ?œŽ 2 đ?‘‡ đ?‘‡ = đ?‘’đ?‘ đ?‘? − đ??ˇđ?œ‡đ?‘‡ đ?‘Ł 2

2

đ??ˇđ?‘‡ − đ?œ‡đ?œŽ đ?‘‡

đ??ˇđ?‘‡ − đ?œ‡đ?œŽ 2 đ?‘‡ đ??ˇđ?‘‡ − đ?œ‡đ?œŽ 2 đ?‘‡ đ?‘Ł/2 đ?‘–=0

đ?‘Łâˆ’1 đ?‘–

đ??ˇđ?‘‡ − đ?œŽ 2 đ?‘‡

đ?‘Ł+2−đ?‘§ −2đ?‘–

đ?‘–

2

đ?‘–

đ?œŽ2đ?‘‡ 2

đ??ˇđ?‘‡ − đ?œ‡đ?œŽ 2 đ?‘‡ đ?‘Łâˆ’2đ?‘–

đ?œŽ 2đ?‘‡ 2

+

đ?œŽ đ?‘‡ 2

đ?‘Ł+1−đ?‘§ −2đ?‘–

đ?‘Łâˆ’đ?‘§âˆ’2đ?‘–

đ?œŽ2đ?‘‡ 2

đ?‘–

đ?‘Ł+3 (đ?‘Łâˆ’đ?‘§+3)/2

�=1

đ?œŽ4 đ?‘‡2 + 2đ??ˇ5 đ??ˇ 6

+

đ?‘–

1 3đ??ˇ3

2

(đ?‘Ł + đ?‘§)! đ?œ‡đ?‘§ đ?‘Ł + 3 − đ?‘§

đ?‘–=0 đ?‘Ł+1−2 đ?‘Ł+1 2

�=1 �=0 2

đ?‘Ł! đ?œ‡đ?‘§ đ?‘Ł + 1 − đ?‘§ đ?‘Ł

3

đ?œŽ đ?œŽ đ?‘‡ đ?‘‡ + + 6 2 4đ??ˇ 2đ??ˇ 3

đ?‘Łâˆ’2 2

�=1 �=0

đ?œ‡đ?‘Ł + đ?‘’đ?‘ đ?‘? đ?œ‡2 đ?œŽ 2 đ?‘‡ − đ??ˇđ?œ‡đ?‘‡ đ?‘Ł

đ?‘Łâˆ’2đ?‘–

đ?œŽ 2đ?‘‡ 2

đ?œŽ6

+ 4đ??ˇ 6 đ?‘’đ?‘ đ?‘?

đ?‘–

+ đ?œ‡ 2đ?œŽ 2đ?‘‡ 2

đ?œŽ 2đ?‘‡3 3đ??ˇ

+

− đ??ˇđ?œ‡đ?‘‡

đ?œŽ 4đ?‘‡2 6đ??ˇ 3

+

� �=1

8

đ?‘Ł+3−đ?‘§âˆ’đ?‘– đ?‘– đ?‘Ł+1−đ?‘§âˆ’đ?‘– đ?‘–

đ?‘Łâˆ’1 ! đ?‘Łâˆ’đ?‘§âˆ’đ?‘– đ?‘– đ?‘Łâˆ’đ?‘§

đ?œ‡đ?‘§

− 8đ?œŽ 6 đ?‘‡ đ??ˇ5

đ?‘Łâˆ’2 2

đ?‘–=0

2 đ?œŽ2 đ?‘‡2 đ?œŽ2 đ?‘‡ đ?œŽ4 đ?‘‡ + 3+ 4 3 đ??ˇ2 3đ??ˇ 2đ??ˇ đ?‘Łâˆ’1 2

đ?‘Łâˆ’đ?‘–−1 đ?‘– (đ?‘Łâˆ’1)! đ?‘Łâˆ’đ?‘§âˆ’đ?‘– 2đ??ˇ đ?‘§ đ?‘– đ?œ‡ − 2 đ?‘Łâˆ’đ?‘§ ! đ?‘–=0

(9)

đ?œŽ

2đ?‘– đ?œŽ 2đ?‘‡ đ?‘Łâˆ’đ?‘§âˆ’đ?‘– đ??ˇđ?‘‡ − đ?œ‡đ?œŽ 2 đ?‘‡ đ?‘Łâˆ’đ?‘§âˆ’2đ?‘– 2 đ?‘– Hence averaging the inventory costs for fixed M over the states of M over the states of M we obtain the inventory cost for variable M Rc +đ?‘† â„Žđ?‘?đ?‘Ł đ??ˇđ?‘‡ đ?‘? đ?‘? +â„Žđ?‘? b C= + − â„Žđ?‘? đ??ˇđ??ż + + 1 đ??ş15 đ?‘‡ + đ??ż − đ??ş15 đ??ż + 2 − G16 L + 3 G17 T + L − G17 L đ?‘‡

đ?œ‡

2

�

�

G16 T + L − G16 L

T

(10)

III. IMPACT OF THE STUDY The study will enable industries or organizations having thousands of items in their warehouses located in various locations of the world and items supplied by different manufacturers with accurately use realistic lead times in arriving at their inventory cost.

In many of such cases lead times are not constant.

Expressing lead time as continuous gives a more realistic estimate of inventory costs.

REFERENCES [1] [2] [3] [4]

[5]

Bertimas D (1999), Probabilistic Service Level Guarantee in make-to-stock, citeseerx.ist.psu.edu Hadley and Whitin (1972), Analysis of Inventory Systems, John Hiby and Sons Inc. N.Y. Sinha Pritibhushan (2008), A note on Bernonlli Demand Inventory Model, Decision Vol. 35, January – June.

Singha D, Tripathi R.P and Mishra T (2013), “Inventory model with non-instantaneous receipt and the exponential demand rate under trade credits, International Journal of Modern Mathematical Science (3), 154-165.

Zipkin Paul (2006), Stochastic lead times in continuous-time inventory models, Naval Research Logistic, 21 Nov. 2006.

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