View of Economic production quantity model with backorders and items with imperfect/perfect quality options

Journal of Applied Research and Technology

www.jart.icat.unam.mx

Journal of Applied Research and Technology 17 (2019) 250-257

Behrouz Afshar-Nadjafi ^a *, Hamed Pourbakhsh^b, Mohammad Mirhabibi^b, Hasan Khodaei^b, Babak Ghodami^b, Fahimeh Sadighi^b, Soheil Azizi^b

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

















𝑐^′

𝜋 𝜋̂

𝑡1= ^𝑏

𝑈−𝐷

𝑡2=^{𝑦 (1−}

𝐷 𝑈 )−𝑏

𝑈−𝐷

𝑡₃=^{𝑦 (1−}

𝐷 𝑈 – 𝑝)−𝑏

𝐷

𝑡₄= ^𝑏

𝐷

𝐼𝑚𝑎𝑥= 𝑦 (1 − ^𝐷

𝑈 ) – 𝑏

𝑇𝑃(𝑦, 𝑏) = 𝑠𝑦(1 − 𝑝) + 𝑣𝑦𝑝 − 𝑐𝑦 − 𝐾 − 𝑑𝑦 − 𝜋𝑏 − ^𝜋̂𝑏2

2𝐷(1−^𝐷

𝑈)−^{ℎ(𝑦(1−}

𝐷 𝑈)−𝑏)²

2(𝑈−𝐷) −^{ℎ(𝑦(1−}

𝐷

𝑈)−𝑏−𝑝𝑦)²

2𝐷

𝐸(𝑇) =^{(1−𝐸(𝑝))𝑦}

𝐷

𝐸𝑇𝑃𝑈(𝑦, 𝑏) = [𝑠𝐷(1 − 𝐸(𝑝)) + 𝑣𝐷𝐸(𝑝) − 𝑐𝐷 −^𝐾𝐷

𝑦 − 𝑑𝐷 − ^𝜋𝐷𝑏

𝑦 − ^𝜋̂𝑏2

2𝑦(1−^𝐷

𝑈)−^{ℎ𝐷(𝑦(1−}

𝐷 𝑈)−𝑏)² 2𝑦(𝑈−𝐷) − ^{ℎ(𝑦(1−}

𝐷

𝑈)−𝑏−𝐸(𝑝)𝑦)²

2𝑦 ] /(1 − 𝐸(𝑝))

𝜕𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏

𝑏^∗=^{[−𝜋𝐷+ℎ𝑦}

∗(1−𝐸(𝑝))](1−^𝐷_𝑈)

𝜋̂+ℎ

𝜕𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑦 = 0

𝑦^∗= √

2𝐷𝐾(𝜋̂+ℎ) (1−𝐷

𝑈)

−𝜋²𝐷²

2ℎ(𝜋̂+ℎ)((𝐸(𝑝)+0.5)+^𝐸2(𝑝) 2(1−𝐷 𝑈)

)−ℎ²(1−𝐸(𝑝))²

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏² = ^{−(𝜋̂+ℎ)}

𝑦(1−^𝐷_𝑈)(1−𝐸(𝑝))

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏𝜕𝑦 =^{𝜕2𝐸𝑇𝑃𝑈(𝑦,𝑏)}

𝜕𝑦𝜕𝑏 = ¹

𝑦²(1−𝐸(𝑝))[𝜋𝐷 +^{(𝜋̂+ℎ)𝑏}

(1−^𝐷 𝑈)]

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑦² = ⁻¹

𝑦³(1−𝐸(𝑝))[2𝐷(𝐾 + 𝜋𝑏) +^{(𝜋̂+ℎ)𝑏2}

(1−^𝐷

𝑈) ]



1



₂

𝛼1=^{𝜕2𝐸𝑇𝑃𝑈(𝑦,𝑏)}

𝜕𝑏² ≤ 0 ; ∀ 𝑏 , 𝑦

𝛼₂ = 𝑑𝑒𝑡∇²𝐸𝑇𝑃𝑈

(

𝑦, 𝑏

)

= 𝑑𝑒𝑡

[

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏²

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏𝜕𝑦

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑦𝜕𝑏

𝜕²𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑦²

]

=

(𝜋̂+ℎ) 𝑦⁴(1−^𝐷

𝑈)(1−𝐸(𝑝))²

[

2𝐷𝐾 −^𝜋

2𝐷²(1−^𝐷 𝑈)

𝜋̂+ℎ

]

; 𝑓𝑟𝑒𝑒 𝑖𝑛 𝑠𝑖𝑔𝑛



1



_{2 𝐸𝑇𝑃𝑈(𝑦, 𝑏)}

𝐸𝑇𝑃𝑈(𝑦, 𝑏)

𝜕𝐸𝑇𝑃𝑈(𝑦,𝑏)

𝜕𝑏 = 0 ^{𝜕𝐸𝑇𝑃𝑈(𝑦,𝑏)}

𝜕𝑦 = 0

𝑏^∗ 𝑦^∗

𝜋²𝐷²≤ min {^{2𝐷𝐾(𝜋̂+ℎ)}

(1−^𝐷

𝑈) , 2𝐷𝐾ℎ(1−𝐸(𝑝))² (2𝐸(𝑝)+1)(1−^𝐷_𝑈)+𝐸²(𝑝)}

𝐸𝑇𝑃𝑈(𝑦, 𝑏) 𝑏^∗ 𝑦^∗

𝜋²𝐷²≤^{2𝐷𝐾(𝜋̂+ℎ)}

(1−^𝐷

𝑈) 𝛼₂≥

0 𝐸𝑇𝑃𝑈(𝑦, 𝑏)

𝑏^∗≥ 0 𝜋²𝐷²≤ 2𝐷𝐾ℎ(1−𝐸(𝑝))²

(2𝐸(𝑝)+1)(1−^𝐷 𝑈)+𝐸²(𝑝)

𝑏^∗ 𝜋²𝐷²> 2𝐷𝐾ℎ(1−𝐸(𝑝))²

(2𝐸(𝑝)+1)(1−^𝐷_𝑈)+𝐸²(𝑝) 𝑏^∗< 0 (𝑦^∗, 𝑏^∗) 𝐸𝑇𝑃𝑈(𝑦, 𝑏) 𝑦 > 0

𝑏 ≥ 0 𝐸𝑇𝑃𝑈(𝑦, 𝑏) 𝑦 > 0

𝑏 ≥ 0 𝑏^∗= 0

𝜋 = 0

𝐸𝑇𝑃𝑈(𝑦, 𝑏) 𝑏^∗

𝑏^∗ 𝑦^∗

𝑏^∗=^ℎ𝑦

∗(1−𝐸(𝑝))(1−^𝐷_𝑈) 𝜋

̂+ℎ 𝑦^∗= √_2ℎ(𝜋̂+ℎ)((𝐸(𝑝)+0.5)(1−^{2𝐷𝐾(𝜋𝐷}_𝑈)+0.5𝐸^̂+ℎ)2(𝑝))−ℎ²(1−𝐸(𝑝))²(1−^𝐷_𝑈)

𝜋 = 0 𝐸(𝑝) = 0 𝑏^∗ 𝑦^∗

𝑏^∗= √^{2𝐷𝐾ℎ(1−}

𝐷 𝑈) 𝜋

̂(𝜋̂+ℎ)

𝑦^∗= √_ℎ(1−^{2𝐷𝐾𝐷}

𝑈)√^𝜋̂+ℎ

𝜋̂ (𝑐^′)

𝑐^′> 𝑐 (𝑝 = 0)

𝑇𝑃𝑈(𝑦, 𝑏) = (𝑠 − 𝑐^′)𝐷 −𝐷

𝑦(𝐾 + 𝜋𝑏) − ^𝜋̂𝑏

2+ℎ(𝑦(1−^𝐷_𝑈)−𝑏)² 2𝑦(1−^𝐷

𝑈)

𝑇𝑃𝑈(𝑦, 𝑏)

𝑏^∗∗=^[ℎ𝑦

∗−𝜋𝐷](1−^𝐷 𝑈) 𝜋̂+ℎ

𝑦^∗∗= √_ℎ(1−^2𝐷𝐾𝐷 𝑈)

(𝜋̂+ℎ) 𝜋

̂ −^𝜋2𝐷2

ℎ𝜋̂

𝑝%

𝑐′

𝑇𝑃𝑈(𝑦^∗∗, 𝑏^∗∗, 𝑐′) − 𝐸𝑇𝑃𝑈(𝑦^∗, 𝑏^∗) = (𝑠 − 𝑐^′)𝐷 − ^𝐷

𝑦^∗∗(𝐾 + 𝜋𝑏^∗∗) −^𝜋̂𝑏

∗∗2+ℎ(𝑦^∗∗(1−^𝐷_𝑈)−𝑏^∗∗)²

2𝑦^∗∗(1−^𝐷_𝑈) − 𝐸𝑇𝑃𝑈(𝑦^∗, 𝑏^∗)

𝑇𝑃𝑈(𝑦^∗∗, 𝑏^∗∗, 𝑐′) − 𝐸𝑇𝑃𝑈(𝑦^∗, 𝑏^∗) ≥ 0

𝑐^′≤ 𝑠 − 1

𝑦^∗∗(𝐾 + 𝜋𝑏^∗∗) −𝜋̂𝑏^∗∗2+ ℎ(𝑦^∗∗(1 −^𝐷_𝑈) − 𝑏^∗∗)² 2𝐷𝑦^∗∗(1 −𝐷

𝑈)

− ^{𝐸𝑇𝑃𝑈(𝑦∗,𝑏∗)}

𝐷

𝑝%

𝑈 =

12,000 𝐷 = 10,000 𝐾 = 450

𝜋̂ = 0.5 𝜋 = 1.2 ℎ =

75 𝑑 = 5 𝑠 = 220 𝑣 = 30

𝑐^′= 125 𝑐 = 100 𝐸(𝑝) = 6%

𝑦^∗= 1691.17 𝑏^∗= 236.71 𝐸𝑇𝑃𝑈(𝑦^∗, 𝑏^∗) = 1,010,030.28

𝑦^∗= 10241.09 𝑏^∗∗= 1669.06 𝑇𝑃𝑈(𝑦^∗∗, 𝑏^∗∗) = 947,165.47 𝑀𝑐 = 110

15%

𝑏^∗ 𝑦^∗

𝑀𝑐 = 110

𝑀𝑐 = 110

𝐸(𝑝)

𝑬(𝒑)

𝐸(𝑝)

𝐸(𝑝) 𝑀𝑐

𝐸(𝑝)

𝐸(𝑝)

𝑏^∗ 𝐸(𝑝) = 0

0 2,000 4,000 6,000 8,000 10,000 12,000

0 0.1 0.2 0.3

y* b*

𝐸(𝑝)

𝐸(𝑝) 𝐸(𝑝)

(𝑝) > 20%

𝐶′

𝐶′ 𝑀𝑐 = 110

800,000 850,000 900,000 950,000 1,000,000 1,050,000 1,100,000 1,150,000 1,200,000

0 0.1 0.2 0.3

ETPU TPU

𝑴𝒄 𝑬(𝒑).

𝑬(𝒑).

500,000 700,000 900,000 1,100,000 1,300,000

80 100 120 140 160

ETPU TPU

Omar, M., Zubir, M.B., & Moin, N.H. (2010). An alternative approach to analyze economic ordering quantity and economic production quantity inventory problems using the completing the square method. Computers and Industrial. Engineering, 59(2), 362-364.

Pentico, D.W., Drake, M.J., & Toews, C. (2009). The deterministic EPQ with partial backordering: a new approach. Omega, 37(3), 624-636.

𝑪^′.