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The traditional Economic Production Quantity (EPQ) model focused on production process, used the ordering cost that includes relevant costs during the pre-production process. But, the ordering cost compris es the holding of raw materials that would affected by other factors would increase the total relevant cost, it cannot simply use the ordering cost to cover all. Therefore, this paper presents a new inventory model by considering the holding of raw materials under conditions of two-level trade credit under alternate due date of payment and limited storage capacity. According to cost-minimization strategy it develops four theorems to characterize the optimal solutions. Finally, it executes the sensitivity analysis and investigates the effects of the parameters in the annual total relevant costs.

[

[

1) [

2) [

The trade credit stimulates retailer purchase more quantities, and the increasing demand would cause more storage capacity to store goods. [

[

Q: the order size

P: production rate

D: demand rate

A: ordering cost

T: the cycle time

ρ = 1 − D P > 0

L_{max}: storage maximum

s: unit selling price per item

c: unit purchasing price per item

h_{m}: unit holding cost per item for raw materials in raw materials warehouse

h_{o}: unit holding cost per item for product in owned warehouse

h_{r}: unit holding cost per item for product in rented warehouse

I_{p}: interest rate payable per $ unit time (year)

I_{e}: interest rate earned per $ unit time (year)

t_{s}: time in years at which production stops

M: the manufacturer’s trade credit offered by the supplier

N: the customer’s trade credit period offered by the manufacturer

W: storage capacity of owned warehouse

tw_{i}: the point in time when the inventory level increases to W during the production period

tw_{d}: the point in time when the inventory level decreases to W during the production cease period = T − W D

tw_{d} − tw_{i}: the time of rented warehouse

= { D T ρ − W P − D + D T ρ − W D , if D T ρ > W 0 , if D T ρ ≤ W

T R C ( T ) : the total relevant cost per unit time of model when T > 0

T ∗ : the optimal solution of T R C (T)

1) Demand rate D is known and constant.

2) Production rate P is known and constant, P > D .

3) Shortages are not allowed.

4) A single item is considered.

5) Time period is infinite.

6) h r ≥ h o ≥ h m , M ≥ N and s ≥ c .

7) The storage capacity of raw materials warehouse is unlimited.

8) If the order quantity is larger than manufacturer’s OW storage capacity, the manufacturer will rent and the RW storage capacity is unlimited. When the demand occurs, if first is replenished from the RW which storages those exceeding items. It takes first in last out (FILO).

9) During the time the account is not settled, generated sales revenue is deposited in and interest-bearing account.

a) When M ≤ T , the account is settled at T = M , the manufacturer pays off all units sold and keeps his/her profits, and starts paying for the higher interest payable on the items in stock with rate I p .

b) When T ≤ M , the account is settled at T = M and the manufacturer does not need to pay any interest payable.

10) If a customer buys an item from the manufacturer at time t ∈ [ 0, T ] , then the customer will have a trade credit period N and make the payment at time N + t .

11) The manufacturer can accumulate revenue and earn interest after his/her customer pays for the amount of purchasing cost to the manufacturer until the end of the trade credit period offered by the supplier. That is, the manufacturer can accumulate revenue and earn interest during the period N to M with rate I e under the condition of trade credit.

12) The manufacturer keeps the profit for the use of the other activities.

The model structure have three stages of a supply chain system, this paper suppose that the supplier provide raw materials to the manufacturer to produce, the quantity of raw materials is expected to decrease in time (from time 0 to t s ). On the other hand, the quantity of products is expected to increase in time up to the maximum inventory level (from 0 to t s ), also sold on demand. After production stop (at time t s ), the products are only sold on demand until quantity reaches zero (at time T), as shown in

The annual total relevant cost consists of the following element.

As shown in

( P − D ) t s − D ( T − t s ) = 0 ,

t s = D T P . (1)

0 − Q t s − 0 = − P , 0 ≤ t ≤ T .

Q = P t s = D T . (2)

L max = ( P − D ) × t s = D T ρ . (3)

Annual ordering cost

= A T . (4)

Annual purchasing cost

= c Q × 1 T = c D . (5)

Annual holding cost

1) As shown in

= h m × Q × t s 2 × 1 T = D 2 T h m 2 P . (6)

2) Two cases occur in annual holding costs of owned warehouse.

a) D T ρ ≤ W , as shown in

Annual holding cost in owned warehouse

= h o × T × L max 2 × 1 T = D T h o ρ 2 . (7)

b) D T ρ > W , as shown in

Annual holding cost in owned warehouse

= h o × [ ( t w d − t w i ) + T ] W 2 × 1 T = W h o − W 2 h o 2 D T ρ . (8)

3) Two cases occur in annual holding costs of rented warehouse.

a) D T ρ ≤ W , as shown in

Annual holding cost in rented warehouse

= 0. (9)

b) D T ρ > W , as shown in

Annual holding cost in rented warehouse

= h r × ( t w d − t w i ) × ( L max − W ) 2 × 1 T = h r ( D T ρ − W ) 2 2 D T ρ . (10)

Four cases to occur in costs of annual interest payable for the items kept in stock.

1) 0 ≤ T < M − N .

Annual interest payable

= 0. (11)

2) M − N ≤ T < M .

Annual interest payable

= 0. (12)

3) M ≤ T < P M D , as shown in

Annual interest payable

= c I p × ( ( T − M ) × D ( T − M ) 2 ) × 1 T = c I p D ( T − M ) 2 2 T . (13)

4) P M D ≤ T , as shown in

Annual interest payable

= c I p × ( T × D T ρ 2 − M × ( P − D ) M 2 ) × 1 T = c I p ρ ( D T 2 − P M 2 ) 2 T . (14)

Three cases to occur in annual interest earned.

1) 0 ≤ T < N , as shown in

Annual interest earned

= s I e × { [ ( T + N ) − N ] × D T 2 + [ M − ( T + N ) ] × D T } × 1 T = s I e D ( 2 M − 2 N − T ) 2 . (15)

2) N ≤ T < M , as shown in

Annual interest earned

= s I e × ( ( M − N ) × D ( M − N ) 2 ) × 1 T = s I e D ( M − N ) 2 2 T . (16)

3) M < T , as shown in

Annual interest earned

= s I e × ( ( M − N ) × D ( M − N ) 2 ) × 1 T = s I e D ( M − N ) 2 2 T . (17)

From the above arguments, the annual total relevant cost for the manufacturer can be expressed as T R C ( T ) = annual ordering cost + annual purchasing cost + annual holding cost + annual interest payable + annual interest earned.

Because storage capacity W = D T ρ , there are four cases arise:

1) W D ρ < M − N ,

2) M − N ≤ W D ρ < M ,

3) M ≤ W D ρ < P M D ,

4) P M D ≤ W D ρ .

Case 1 W D ρ < M − N .

According to Equations (1)-(17), the annual total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < W D ρ ( 18 a ) T R C 2 ( T ) , if W D ρ ≤ T < M − N ( 18 b ) T R C 3 ( T ) , if M − N ≤ T < M ( 18 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 18 d ) T R C 5 ( T ) , if P M D ≤ T (18e)

where

T R C 1 ( T ) = A T + c D + D 2 T h m 2 P + D T h o ρ 2 − s I e D ( 2 M − 2 N − T ) 2 , (19)

T R C 2 ( T ) = A T + c D + D 2 T h m 2 P + W h o − W 2 h o 2 D T ρ + h r ( D T ρ − W ) 2 2 D T ρ − s I e D ( 2 M − 2 N − T ) 2 , (20)

T R C 3 ( T ) = A T + c D + D 2 T h m 2 P + W h o − W 2 h o 2 D T ρ + h r ( D T ρ − W ) 2 2 D T ρ − s I e D ( M − N ) 2 2 T , (21)

T R C 4 ( T ) = A T + c D + D 2 T h m 2 P + W h o − W 2 h o 2 D T ρ + h r ( D T ρ − W ) 2 2 D T ρ + c I p D ( T − M ) 2 2 T − s I e D ( M − N ) 2 2 T , (22)

T R C 5 ( T ) = A T + c D + D 2 T h m 2 P + W h o − W 2 h o 2 D T ρ + h r ( D T ρ − W ) 2 2 D T ρ + c I p ρ ( D T 2 − P M 2 ) 2 T − s I e D ( M − N ) 2 2 T . (23)

Since T R C 1 ( W D ρ ) = T R C 2 ( W D ρ ) , T R C 2 ( M − N ) = T R C 3 ( M − N ) , T R C 3 ( M ) = T R C 4 ( M ) , T R C 4 ( P M D ) = T R C 5 ( P M D ) , T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) .

Case 2 M − N ≤ W D ρ < M .

According to Equations (1)-(17), the annual total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < M − N ( 24 a ) T R C 6 ( T ) , if M − N ≤ T < W D ρ ( 24 b ) T R C 3 ( T ) , if M ≤ T < P M D ( 24 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 24 d ) T R C 5 ( T ) , if P M D ≤ T (24e)

where

T R C 6 ( T ) = A T + c D + D 2 T h m 2 P + D T h o ρ 2 − s I e D ( M − N ) 2 2 T . (25)

Since T R C 1 ( M − N ) = T R C 6 ( M − N ) , T R C 6 ( W D ρ ) = T R C 3 ( W D ρ ) , T R C 3 ( M ) = T R C 4 ( M ) , T R C 4 ( P M D ) = T R C 5 ( P M D ) , T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) .

Case 3 M ≤ W D ρ < P M D .

According to Equations (1)-(17), the annual total relevant cost T R C ( T ) can be expressed b

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < M − N ( 26 a ) T R C 6 ( T ) , if M − N ≤ T < M ( 26 b ) T R C 7 ( T ) , if M ≤ T < W D ρ ( 26 c ) T R C 4 ( T ) , if W D ρ ≤ T < P M D ( 26 d ) T R C 5 ( T ) , if P M D ≤ T (26e)

where

T R C 7 ( T ) = A T + c D + D 2 T h m 2 P + D T h o ρ 2 + c I p D ( T − M ) 2 2 T − s I e D ( M − N ) 2 2 T . (27)

Since T R C 1 ( M − N ) = T R C 6 ( M − N ) , T R C 6 ( M ) = T R C 7 ( M ) , T R C 7 ( W D ρ ) = T R C 4 ( W D ρ ) , T R C 4 ( P M D ) = T R C 5 ( P M D ) , T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) .

Case 4 P M D ≤ W D ρ .

According to Equations (1)-(17), the annual total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < M − N ( 28 a ) T R C 6 ( T ) , if M − N ≤ T < M ( 28 b ) T R C 7 ( T ) , if M ≤ T < P M D ( 28 c ) T R C 8 ( T ) , if P M D ≤ T < W D ρ ( 28 d ) T R C 5 ( T ) , if W D ρ ≤ T (28e)

where

T R C 8 ( T ) = A T + c D + D 2 T h m 2 P + D T h o ρ 2 + c I p ρ ( D T 2 − P M 2 ) 2 T − s I e D ( M − N ) 2 2 T . (29)

Since T R C 1 ( M − N ) = T R C 6 ( M − N ) , T R C 6 ( M ) = T R C 7 ( M ) , T R C 7 ( P M D ) = T R C 8 ( P M D ) , T R C 8 ( W D ρ ) = T R C 5 ( W D ρ ) , T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) .

In summary, all T R C i ( T ) ( i = 1 ∼ 8 ) are defined on T > 0 .

Equations (19)-(23), (25), (27) and (29) yield the first order and second-order derivatives as follows.

T R C ′ 1 ( T ) = − A T 2 + D 2 ( D P h m + ρ h o + s I e ) , (30)

T R C ″ 1 ( T ) = 2 A T 3 > 0 , (31)

T R C ′ 2 ( T ) = − 2 A + W 2 ( h o − h r ) D ρ 2 T 2 + D 2 ( D P h m + ρ h r + s I e ) , (32)

T R C ″ 2 ( T ) = 2 A + W 2 ( h r − h o ) D ρ T 3 > 0 , (33)

T R C ′ 3 ( T ) = − 2 A + W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ h r ) , (34)

T R C ″ 3 ( T ) = 2 A + W 2 ( h r − h o ) D ρ − s I e D ( M − N ) 2 T 3 , (35)

T R C ′ 4 ( T ) = − 2 A + W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ h r + c I p ) , (36)

T R C ″ 4 ( T ) = 2 A + W 2 ( h r − h o ) D ρ + c I p D M 2 − s I e D ( M − N ) 2 T 3 , (37)

T R C ′ 5 ( T ) = − 2 A + W 2 ( h o − h r ) D ρ + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ ( h r + c I p ) ) , (38)

T R C ″ 5 ( T ) = 2 A + W 2 ( h r − h o ) D ρ − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 T 3 , (39)

T R C ′ 6 ( T ) = − 2 A + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ h o ) , (40)

T R C ″ 6 ( T ) = 2 A − s I e D ( M − N ) 2 T 3 , (41)

T R C ′ 7 ( T ) = − 2 A − c I p D M 2 + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ h o + c I p ) , (42)

T R C ″ 7 ( T ) = 2 A + c I p D M 2 − s I e D ( M − N ) 2 T 3 , (43)

T R C ′ 8 ( T ) = − 2 A + c p ( P − D ) M 2 + s I e D ( M − N ) 2 2 T 2 + D 2 ( D P h m + ρ ( h o + c I p ) ) , (44)

and

T R C ″ 8 ( T ) = 2 A − c p ( P − D ) M 2 − s I e D ( M − N ) 2 T 3 . (45)

Let

G 3 = 2 A + W 2 ( h r − h o ) D ρ − s I e D ( M − N ) 2 , (46)

G 4 = 2 A + W 2 ( h r − h o ) D ρ + c I p D M 2 − s I e D ( M − N ) 2 , (47)

G 5 = 2 A + W 2 ( h r − h o ) D ρ − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 , (48)

G 6 = 2 A − s I e D ( M − N ) 2 , (49)

G 7 = 2 A + c I p D M 2 − s I e D ( M − N ) 2 , (50)

and

G 8 = 2 A − c p ( P − D ) M 2 − s I e D ( M − N ) 2 . (51)

Equations (46)-(51) imply

G 4 > G 3 > G 5 > G 8 , (52)

and

G 4 > G 7 > G 6 > G 8 . (53)

Equations (30)-(45) reveal the following results.

Lemma 1

1) T R C i ( T ) is convex on T > 0 if i = 1 , 2 .

2) T R C i ( T ) is convex on T > 0 if G i > 0 . Otherwise T R C ′ i ( T ) is increasing on T > 0 for all i = 3 ∼ 8 .

Solving

T R C ′ i ( T ) = 0, i = 1 ∼ 8 (54)

then

T 1 * = 2 A D ( D P h m + ρ h o + s I e ) , (55)

T 2 * = 2 A + W 2 ( h r − h o ) D ρ D ( D P h m + ρ h r + s I e ) , (56)

T 3 * = 2 A + W 2 ( h r − h o ) D ρ − s I e D ( M − N ) 2 D ( D P h m + ρ h r ) , if G 3 > 0 , (57)

T 4 * = 2 A + W 2 ( h r − h o ) D ρ + c I p D M 2 − s I e D ( M − N ) 2 D ( D P h m + ρ h r + c I p ) , if G 4 > 0 , (58)

T 5 * = 2 A + W 2 ( h r − h o ) D ρ − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 D ( D P h m + ρ ( h r + c I p ) ) , if G 5 > 0 , (59)

T 6 * = 2 A − s I e D ( M − N ) 2 D ( D P h m + ρ h o ) , if G 6 > 0 , (60)

T 7 * = 2 A + c I p D M 2 − s I e D ( M − N ) 2 D ( D P h m + ρ h o + c I p ) , if G 7 > 0 , (61)

and

T 8 * = 2 A − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 D ( D P h m + ρ ( h o + c I p ) ) , if G 8 > 0 , (62)

are the respective solutions of Equation (54). Furthermore, if T i * exists, then T R C i ( T ) is convex on T > 0 and

T R C ′ i ( T ) = { < 0 if 0 < T < T i * ( 63 a ) = 0 if T = T i * ( 63 b ) > 0 if T i * < T (63c)

Equations (63a)-(63c) imply that T R C i ( T ) is decreasing on ( 0, T i * ] and increasing on [ T i * , ∞ ) for all i = 1 ∼ 8 .

Case 1 W D ρ < M − N .

Equations (30), (32), (34), (36) and (38) yield

T R C ′ 1 ( W D ρ ) = T R C ′ 2 ( W D ρ ) = Δ 12 2 ( W D ρ ) 2 , (64)

T R C ′ 2 ( M − N ) = T R C ′ 3 ( M − N ) = Δ 23 2 ( M − N ) 2 , (65)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (66)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (67)

where

Δ 12 = − 2 A + D ( W D ρ ) 2 ( D P h m + ρ h o + s I e ) , (68)

Δ 23 = − 2 A + W 2 ( h o − h r ) D ρ + D ( M − N ) 2 ( D P h m + ρ h r + s I e ) , (69)

Δ 34 = − 2 A + W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 + D M 2 ( D P h m + ρ h r ) , (70)

Δ 45 = − 2 A + W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e ( M − N ) 2 + D ( P M D ) 2 ( D P h m + ρ h r + c I p ) . (71)

Equations (68)-(71) imply

Δ 12 < Δ 23 < Δ 34 < Δ 45 . (72)

Case 2 M − N ≤ W D ρ < M .

Equations (30), (34), (36), (38) and (40) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (73)

T R C ′ 6 ( W D ρ ) = T R C ′ 3 ( W D ρ ) = Δ 63 2 ( W D ρ ) 2 , (74)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (75)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (76)

where

Δ 16 = − 2 A + D ( M − N ) 2 ( D P h m + ρ h o + s I e ) , (77)

Δ 63 = − 2 A + s I e D ( M − N ) 2 + D ( W D ρ ) 2 ( D P h m + ρ h o ) . (78)

Equations (70), (71), (77) and (78) imply

Δ 16 ≤ Δ 63 < Δ 34 < Δ 45 . (79)

Case 3 M ≤ W D ρ < P M D .

Equations (30), (36), (38), (40) and (42) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (80)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (81)

T R C ′ 7 ( W D ρ ) = T R C ′ 4 ( W D ρ ) = Δ 74 2 ( W D ρ ) 2 , (82)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (83)

where

Δ 67 = − 2 A + s I e D ( M − N ) 2 + D M 2 ( D P h m + ρ h o ) , (84)

Δ 74 = − 2 A − c I p D M 2 + s I e D ( M − N ) 2 + D ( W D ρ ) 2 ( D P h m + ρ h o + c I p ) . (85)

Equations (71), (77), (84 and (85) imply

Δ 16 ≤ Δ 67 ≤ Δ 74 < Δ 45 . (86)

Case 4 P M D ≤ W D ρ .

Equations (30), (38), (40), (42) and (44) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (87)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (88)

T R C ′ 7 ( P M D ) = T R C ′ 8 ( P M D ) = Δ 78 2 ( P M D ) 2 , (89)

T R C ′ 8 ( W D ρ ) = T R C ′ 5 ( W D ρ ) = Δ 45 2 ( W D ρ ) 2 , (90)

where

Δ 78 = − 2 A − c I p D M 2 + s I e D ( M − N ) 2 + D ( P M D ) 2 ( D P h m + ρ h o + c I p ) , (91)

Δ 85 = − 2 A + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ( W D ρ ) 2 ( D P h m + ρ ( h o + c I p ) ) . (92)

Equations (77), (84), (91) and (92) imply

Δ 16 ≤ Δ 67 ≤ Δ 78 ≤ Δ 85 . (93)

Based on the above arguments, the following results hold.

Lemma 2

A) If Δ 23 ≤ 0 , then

(a1) G 3 > 0 ,

(a2) T 3 * exists,

(a3) T R C 3 ( T ) is convex on T > 0 .

B) If Δ 34 ≤ 0 , then

(b1) G 3 > 0 and G 4 > 0 ,

(b2) T 3 * and T 4 * exist,

(b3) T R C 3 ( T ) and T R C 4 ( T ) are convex on T > 0 .

C) If Δ 45 ≤ 0 , then

(c1) G 4 > 0 and G 5 > 0 ,

(c2) T 4 * and T 5 * exist,

(c3) T R C 4 ( T ) and T R C 5 ( T ) are convex on T > 0 .

D) If Δ 85 ≤ 0 , then

(d1) G 5 > 0 and G 8 > 0 ,

(d2) T 5 * and T 8 * exist,

(d3) T R C 5 ( T ) and T R C 8 ( T ) are convex on T > 0 .

E) If Δ 16 ≤ 0 , then

(e1) G 6 > 0 ,

(e2) T 6 * exists,

(e3) T R C 6 ( T ) is convex on T > 0 .

F) If Δ 67 ≤ 0 , then

(f1) G 6 > 0 and G 7 > 0 ,

(f2) T 6 * and T 7 * exist,

(f3) T R C 6 ( T ) and T R C 7 ( T ) are convex on T > 0 .

G) If Δ 74 ≤ 0 , then

(g1) G 4 > 0 and G 7 > 0 ,

(g2) T 4 * and T 7 * exist,

(g3) T R C 4 ( T ) and T R C 7 ( T ) are convex on T > 0 .

H) If Δ 78 ≤ 0 , then

(h1) G 7 > 0 and G 8 > 0 ,

(h2) T 7 * and T 8 * exist,

(h3) T R C 7 ( T ) and T R C 8 ( T ) are convex on T > 0 .

Proof. A) (a1) If Δ 23 ≤ 0 , then

2 A ≥ W 2 ( h o − h r ) D ρ + D ( M − N ) 2 ( D P h m + ρ h r + s I e ) . (94)

Equation (94) implies

G 3 ≥ D ( M − N ) 2 ( D P h m + ρ h r ) > 0. (95)

(a2) Equation (57) and Lemma 1 imply that T 3 * exists.

(a3) Equation (35) and Lemma 1 imply that T R C 3 ( T ) is convex on T > 0 .

B) (b1) If Δ 34 ≤ 0 , then

2 A ≥ W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 + D M 2 ( D P h m + ρ h r ) . (96)

Equation (96) implies

G 3 ≥ D M 2 ( D P h m + ρ h r + c I p ) > 0. (97)

Equations (52) and (97) demonstrate G 4 > G 3 > 0 .

(b2) Equations (57), (58) and Lemma 1 imply that T 3 * and T 4 * exist.

(b3) Equations (35), (37) and Lemma 1 imply that T R C 3 ( T ) and T R C 4 ( T ) is convex on T > 0 .

C) (c1) If Δ 45 ≤ 0 , then

2 A ≥ W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e ( M − N ) 2 + D ( P M D ) 2 ( D P h m + ρ h r + c I p ) . (98)

Equation (98) implies

G 5 ≥ D ( P M D ) 2 ( D P h m + ρ ( h r + c I p ) ) > 0. (99)

Equations (52) and (99) demonstrate G 4 > G 5 > 0 .

(c2) Equations (58), (59) and Lemma 1 imply that T 4 * and T 5 * exist.

(c3) Equations (37), (39) and Lemma 1 imply that T R C 4 ( T ) and T R C 5 ( T ) are convex on T > 0 .

D) (d1) If Δ 85 ≤ 0 , then

2 A ≥ c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ( W D ρ ) 2 ( D P h m + ρ ( h o + c I p ) ) , (100)

Equation (100) implies

G 8 ≥ D ( W D ρ ) 2 ( D P h m + ρ ( h o + c I p ) ) > 0. (101)

Equations (52) and (101) demonstrate G 5 > G 8 > 0 .

(d2) Equations (59), (62) and Lemma 1 imply that T 5 * and T 8 * exist.

(d3) Equations (39), (45) and Lemma 1 imply that T R C 5 ( T ) and T R C 8 ( T ) are convex on T > 0 .

E) (e1) If Δ 16 ≤ 0 , then

2 A ≥ D ( M − N ) 2 ( D P h m + ρ h o + s I e ) . (102)

Equation (102) implies

G 6 ≥ D ( M − N ) 2 ( D P h m + ρ h o ) > 0. (103)

(e2) Equations (60) and Lemma 1 imply that T 6 * exists.

(e3) Equations (41) and Lemma 1 imply that T R C 6 ( T ) is convex on T > 0 .

F) (f1) If Δ 67 ≤ 0 , then

2 A ≥ D M 2 ( s I e − c I p ) − s I e D N 2 + D ( P M D ) 2 ( D P h m + ρ h o + c I p ) . (104)

Equation (104) implies

G 7 ≥ D ( P M D ) 2 ( D P h m + ρ h o + c I p ) > 0. (105)

Equations (53) and (105) demonstrate G 7 > G 6 > 0 .

(f2) Equations (60), (61) and Lemma 1 imply that T 6 * and T 7 * exist.

(f3) Equations (41), (43) and Lemma 1 imply that T R C 6 ( T ) and T R C 7 ( T ) are convex on T > 0 .

G) (g1) If Δ 74 ≤ 0 , then

2 A ≥ − c I p D M 2 + s I e D ( M − N ) 2 + D ( W D ρ ) 2 ( D P h m + ρ h o + c I p ) . (106)

Equation (106) implies

G 4 ≥ W 2 ( h r − h o ) D ρ + D ( W D ρ ) 2 ( D P h m + ρ h o + c I p ) > 0. (107)

Equations (53) and (107) demonstrate G 4 > G 7 > 0 .

(g2) Equations (58), (61) and Lemma 1 imply that T 4 * and T 7 * exist.

(g3) Equations (37), (43) and Lemma 1 imply that T R C 4 ( T ) and T R C 7 ( T ) are convex on T > 0 .

H) (h1) If Δ 78 ≤ 0 , then

2 A ≥ − c I p D M 2 + s I e D ( M − N ) 2 + D ( P M D ) 2 ( D P h m + ρ h r + c I p ) . (108)

Equation (108) implies

G 8 ≥ D ( P M D ) 2 ( D P h m + ρ ( h r + c I p ) ) > 0. (109)

Equations (53) and (109) demonstrate G 7 > G 8 > 0 .

(h2) Equations (61), (62) and Lemma 1 imply that T 7 * and T 8 * exist.

(h3) Equations (43), (45) and Lemma 1 imply that T R C 7 ( T ) and T R C 8 ( T ) are convex on T > 0 .

Incorporate the above arguments, we have completed the proof of Lemma 2. □

Theorem 1 Suppose W D ρ < M − N . Hence,

A) if 0 < Δ 12 , then T R C ( T * ) = T R C 1 ( T 1 * ) and T * = T 1 * .

B) if Δ 12 ≤ 0 < Δ 23 , then T R C ( T * ) = T R C 2 ( T 2 * ) and T * = T 2 * .

C) if Δ 23 ≤ 0 < Δ 34 , then T R C ( T * ) = T R C 3 ( T 3 * ) and T * = T 3 * .

D) if Δ 34 ≤ 0 < Δ 45 , then T R C ( T * ) = T R C 4 ( T 4 * ) and T * = T 4 * .

E) if Δ 45 ≤ 0 , then T R C ( T * ) = T R C 5 ( T 5 * ) and T * = T 5 * .

Proof. A) If 0 < Δ 12 , then 0 < Δ 12 < Δ 23 < Δ 34 < Δ 45 . So, Equations (63a)-(63c), lemma 1 and 2 imply

(a1) T R C 1 ( T ) is decreasing on ( 0, T 1 * ] and increasing on [ T 1 * , W D ρ ] .

(a2) T R C 2 ( T ) is increasing on [ W D ρ , M − N ] .

(a3) T R C 3 ( T ) is increasing on [ M − N , M ] .

(a4) T R C 4 ( T ) is increasing on [ M , P M D ] .

(a5) T R C 5 ( T ) is increasing on [ P M D , ∞ ) .

Since T R C ( T ) is continuous on T > 0 , Equations (18a)-(18e) and (a1)-(a5) reveal that T R C ( T ) is decreasing on ( 0, T 1 * ] and increasing on [ T 1 * , ∞ ) . Hence, T * = T 1 * and T R C ( T * ) = T R C 1 ( T 1 * ) .

B) If Δ 12 ≤ 0 < Δ 23 , then Δ 12 ≤ 0 < Δ 23 < Δ 34 < Δ 45 . So, Equations (63a)-(63c), lemma 1 and 2 imply

(b1) T R C 1 ( T ) is decreasing on [ 0, W D ρ ] .

(b2) T R C 2 ( T ) is decreasing on [ W D ρ , T 2 * ] and increasing on [ T 2 * , M − N ] .

(b3) T R C 3 ( T ) is increasing on [ M − N , M ] .

(b4) T R C 4 ( T ) is increasing on [ M , P M D ] .

(b5) T R C 5 ( T ) is increasing on [ P M D , ∞ ) .

Since T R C ( T ) is continuous on T > 0 , Equations (18a)-(18e) and (b1)-(b5) reveal that T R C ( T ) is decreasing on ( 0, T 2 * ] and increasing on [ T 2 * , ∞ ) . Hence, T * = T 2 * and T R C ( T * ) = T R C 2 ( T 2 * ) .

C) If Δ 23 ≤ 0 < Δ 34 , then Δ 12 < Δ 23 ≤ 0 < Δ 34 < Δ 45 . So, Equations (63a)-(63c), lemma 1 and 2 imply

(c1) T R C 1 ( T ) is decreasing on [ 0, W D ρ ] .

(c2) T R C 2 ( T ) is decreasing on [ W D ρ , M − N ] .

(c3) T R C 3 ( T ) is decreasing on [ M − N , T 3 * ] and increasing on [ T 3 * , M ] .

(c4) T R C 4 ( T ) is increasing on [ M , P M D ] .

(c5) T R C 5 ( T ) is increasing on [ P M D , ∞ ) .

Since T R C ( T ) is continuous on T > 0 , Equations (18a)-(18c) and (c1)-(c5) reveal that T R C ( T ) is decreasing on ( 0, T 3 * ] and increasing on [ T 3 * , ∞ ) . Hence, T * = T 3 * and T R C ( T * ) = T R C 3 ( T 3 * ) .

D) If Δ 34 ≤ 0 < Δ 45 , then Δ 12 < Δ 23 < Δ 34 ≤ 0 < Δ 45 . So, Equations (63a)-(63c), lemma 1 and 2 imply

(d1) T R C 1 ( T ) is decreasing on [ 0, W D ρ ] .

(d2) T R C 2 ( T ) is decreasing on [ W D ρ , M − N ] .

(d3) T R C 3 ( T ) is decreasing on [ M − N , M ] .

(d4) T R C 4 ( T ) is decreasing on [ M , T 4 * ] and increasing on [ T 4 * , P M D ] .

(d5) T R C 5 ( T ) is increasing on [ P M D , ∞ ) .

Since T R C ( T ) is continuous on T > 0 , Equations (18a)-(18e) and (d1)-(d5) reveal that T R C ( T ) is decreasing on ( 0, T 4 * ] and increasing on [ T 4 * , ∞ ) . Hence, T * = T 4 * and T R C ( T * ) = T R C 4 ( T 4 * ) .

E) If Δ 45 ≤ 0 , then Δ 12 < Δ 23 < Δ 34 < Δ 45 ≤ 0 . So, Equations (63a)-(63c), lemma 1 and 2 imply

(e1) T R C 1 ( T ) is decreasing on [ 0, W D ρ ] .

(e2) T R C 2 ( T ) is decreasing on [ W D ρ , M − N ] .

(e3) T R C 3 ( T ) is decreasing on [ M − N , M ] .

(e4) T R C 4 ( T ) is decreasing on [ M , T 4 * ] .

(e5) T R C 5 ( T ) is decreasing on [ P M D , T 5 * ] and increasing on [ T 5 * , ∞ ) .

Since T R C ( T ) is continuous on T > 0 , Equations (18a)-(18e) and (d1)-(d5) reveal that T R C ( T ) is decreasing on ( 0, T 5 * ] and increasing on [ T 5 * , ∞ ) . Hence, T * = T 5 * and T R C ( T * ) = T R C 5 ( T 5 * ) .

Incorporating all the above arguments, we have completed the proof of Theorem 1. □

Applying Lemmas 1, 2 and Equations (24a)-(24e), the following results hold.

Theorem 2 Suppose M − N ≤ W D ρ < M . Hence,

A) if 0 < Δ 16 , then T R C ( T * ) = T R C 1 ( T 1 * ) and T * = T 1 * .

B) if Δ 16 ≤ 0 < Δ 63 , then T R C ( T * ) = T R C 6 ( T 6 * ) and T * = T 6 * .

C) if Δ 63 ≤ 0 < Δ 34 , then T R C ( T * ) = T R C 3 ( T 3 * ) and T * = T 3 * .

D) if Δ 34 ≤ 0 < Δ 45 , then T R C ( T * ) = T R C 4 ( T 4 * ) and T * = T 4 * .

E) if Δ 45 ≤ 0 , then T R C ( T * ) = T R C 5 ( T 5 * ) and T * = T 5 * .

Applying Lemmas 1, 2 and Equations (26a)-(26e), the following results hold.

Theorem 3 Suppose M ≤ W D ρ < P M D . Hence,

A) if 0 < Δ 16 , then T R C ( T * ) = T R C 1 ( T 1 * ) and T * = T 1 * .

B) if Δ 16 ≤ 0 < Δ 67 , then T R C ( T * ) = T R C 6 ( T 6 * ) and T * = T 6 * .

C) if Δ 67 ≤ 0 < Δ 74 , then T R C ( T * ) = T R C 7 ( T 7 * ) and T * = T 7 * .

D) if Δ 74 ≤ 0 < Δ 45 , then T R C ( T * ) = T R C 4 ( T 4 * ) and T * = T 4 * .

E) if Δ 45 ≤ 0 , then T R C ( T * ) = T R C 5 ( T 5 * ) and T * = T 5 * .

Applying Lemmas 1, 2 and Equations(28a)-(28e), the following results hold.

Theorem 4 Suppose P M D ≤ W D ρ . Hence,

A) if 0 < Δ 16 , then T R C ( T * ) = T R C 1 ( T 1 * ) and T * = T 1 * .

B) if Δ 16 ≤ 0 < Δ 67 , then T R C ( T * ) = T R C 6 ( T 6 * ) and T * = T 6 * .

C) if Δ 67 ≤ 0 < Δ 78 , then T R C ( T * ) = T R C 7 ( T 7 * ) and T * = T 7 * .

D) if Δ 78 ≤ 0 < Δ 85 , then T R C ( T * ) = T R C 8 ( T 8 * ) and T * = T 8 * .

E) if Δ 85 ≤ 0 , then T R C ( T * ) = T R C 5 ( T 5 * ) and T * = T 5 * .

We execute the sensitivity analysis by Maple 18.00 to find out the unique solution T i * when T R C ′ i ( T ) = 0, i = 1 ∼ 8 .

2*Impact | T*, Q* | TRC(T) | ||
---|---|---|---|---|

This paper | [ | This paper | [ | |

Positive & Major | A | A | c | A |

Positive & Minor | - | - | A, h_{m}, h_{o}, h_{r}, I_{p} | c, h_{o}, h_{r}, I_{p} |

Negative & Minor | s, h_{m}, h_{o}, h_{r}, I_{e} | s, h_{o}, h_{r}, I_{e} | s, I_{e} | s, I_{e} |

Negative & Major | c, I_{p} | c, I_{p} | - | - |

Given the parameters P = 5000 units / year , D = 3500 units / year , A = $ 1200 / order , s = $ 30 / unit , c = $ 10 / unit , h m = $ 1 / unit / year , h o = $ 3 / unit / year , h r = $ 6 / unit / year , I p = $ 0.3 / year , I e = $ 0.1 / year , M = 100 days = 100 / 365 year , N = 50 days = 50 / 365 year , W = 400 units .

We increase/decrease 25% and 50% of parameters at the same time to execute the sensitivity analysis. Based on the computational results and compare with [

EPQ models are being widely used as a decision making tool in practice. Nearly a hundred years, scholars have focused on the production process, but omitted the importance of raw materials during the pre-production process. However, the related costs of raw materials will directly or indirectly affect the annual total relevant cost, thereby generating significant errors so that an overall consideration is needed.

Therefore, this paper presents a new inventory model applies raw materials in [

The authors declare no conflicts of interest regarding the publication of this paper.

Yen, G.-F., Lin, S.-D. and Lee, A.-K. (2018) EPQ Policies Considering the Holding Cost of Raw Materials with Two-Level Trade Credit under Alternate Due Date of Payment and Limited Storage Capacity. Open Access Library Journal, 5: e5041. https://doi.org/10.4236/oalib.1105041