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Economic production quantity (EPQ) research has typically focused on the cost of production processes, but has not employed accurate calculation to assess factors influencing ordering costs, because one of their assumptions is the raw materials that are product timely. However, the production and transport process of raw materials are influencing factors and increase the holding cost of raw materials, either by incre asing or reducing the total relevant cost. [1] combined [2]’s concept of holding cost of raw materials and [3]’s two-level trade credit and limited storage capacity model to develop innovative and detailed EPQ model that considers the holding cost of non-deteriorating raw materials to closer to the real world. However, some raw materials have deteriorated should be considered. Therefore, this research extends [1]’s model to consider the holding cost of deteriorating raw materials. Four theorems for determining the optimal cycle time and the total relevant cost were developed using cost minimization. Finally, sensitivity analyses are used to find out the effects of the parameters to determine the ordering policies.

[

[

1) In [

2) In [

Trade credit stimulates retailers to purchase larger quantities of goods, as well as more storage capacity in which to store those goods. [

[

As mentioned above, we found there is lack about the holding cost of deteriorating raw materials in the total relevant cost. Moreover, [

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Note: Column N + t for [

complete inventory model by incorporating the holding cost of non-deteriorating raw materials with two-level trade credit and limited storage capacity. Therefore, this research extends [

Q the order size.

P the production rate.

D the demand rate.

A the ordering cost.

T the cycle time.

ρ = 1 − D P > 0.

L max the storage maximum.

I m ( t ) the inventory function for raw materials.

θ the deterioration rate, 0 ≤ θ < 1.

s the unit selling price per item.

c the unit purchasing price per item.

h_{m} the unit holding cost per item for raw materials in a raw materials warehouse.

h_{o} the unit holding cost per item for product in an owned warehouse.

h_{r} the unit holding cost per item for product in a rented warehouse.

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

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

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

M the manufacturer’s trade credit period offered by the supplier.

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

W the storage capacity of an owned warehouse.

t w i the point in time when the inventory level increases to W when the production period is W P − D .

t w d the point in time when the inventory level decreases to W when the production cease period is T − W D .

t w d − t w i the time of rented warehouse is

{ 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 the 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) Backlogging is not allowed.

5) A single item is considered.

6) Time period is infinite.

7) Replenishment rate is infinite.

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

9) Storage capacity of raw materials warehouse is unlimited.

10) If the order quantity is larger than the manufacturer’s OW (owned warehouse) storage capacity, then the manufacturer will rent an RW (rented warehouse) with unlimited storage capacity. When demand occurs, it is first replenished from the RW which has storage that exceeds the items. The RW takes first in last out (FILO), and products in the OW or RW will not deteriorate.

11) During the period 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, keeps his or 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 have to pay any interest payable.

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

13) The manufacturer can accumulate revenue and earn interest after his or her customer pays the amount of the purchasing cost to the manufacturer until the end of the trade credit period offered by the supplier. In other words, the manufacturer can accumulate revenue and earn interest during the period from N to M with rate I e under the condition of trade credit.

14) The manufacturer keeps the profit for use in other activities.

The model considers three stages of a supply chain system. It assumes that the supplier prepares the deteriorating raw materials for production, and the deteriorating raw materials are expected to decrease by the inventory function I m ( t ) with the deterioration rate θ (from time 0 to t s ). The quantity of products is expected to increase with time to the maximum inventory level (from 0 to t s ); the products are sold on demand at the same time. After production stops (at time t s ), the products are sold only on demand until the quantity reaches zero (at time T), as shown in

The annual total relevant cost consists of the following element.

As shown in

d I m ( t ) d t + θ I m ( t ) = − P , 0 ≤ t ≤ t s . (1)

By using the boundary condition I m ( t s ) = 0 , we obtain

I m ( t ) = P θ ( e θ ( t s − t ) − 1 ) , 0 ≤ t ≤ t s . (2)

We will then set the cycle time T and the optimal quantity Q.

( P − D ) t s − D ( T − t s ) = 0,

t s = D P T . (3)

Q = I m ( 0 ) = P θ ( e θ D P T − 1 ) . (4)

Annual ordering cost is

A T . (5)

Annual purchasing cost is

c × Q × 1 T = c P θ T ( e θ D P T − 1 ) . (6)

Annual holding cost is

1) As shown in

h m × ∫ 0 t s I m ( t ) d t × 1 T = h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] . (7)

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 is

h o × T × L max 2 × 1 T = D T h o ρ 2 . (8)

b) W ≤ D T ρ , as shown in

Annual holding cost in owned warehouse is

h o × [ ( t w d − t w i ) + T ] W 2 × 1 T = W h o − W 2 h o 2 D T ρ . (9)

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 is

0. (10)

b) W ≤ D T ρ , as shown in

Annual holding cost in rented warehouse is

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 . (11)

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

1) 0 < T ≤ M − N .

Annual interest payable is

0. (12)

2) M − N ≤ T ≤ M .

Annual interest payable is

0. (13)

3) M ≤ T ≤ P M D , as shown in

Annual interest payable is

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

4) M ≤ P M D ≤ T , as shown in

Annual interest payable is

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 . (15)

Five cases to occur in annual interest earned.

1) 0 < T ≤ N and T ≤ M − N , as shown in

Annual interest earned is

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 . (16)

2) 0 < T ≤ N and M − N ≤ T , as shown in

Annual interest earned is

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

3) N ≤ T ≤ M and T ≤ M − N , as shown in

Annual interest earned is

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 . (18)

4) N ≤ T ≤ M and M − N ≤ T , as shown in

Annual interest earned is

s I e × [ ( M − N ) × D ( M − N ) 2 ] × 1 T = s I e D ( M − N ) 2 2 T . (19)

5) N ≤ M ≤ T , as shown in

Annual interest earned is

s I e × [ ( M − N ) × D ( M − N ) 2 ] × 1 T = s I e D ( M − N ) 2 2 T . (20)

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)-(20), the 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 ( 21 a ) T R C 2 ( T ) , if W D ρ ≤ T < M − N ( 21 b ) T R C 3 ( T ) , if M − N ≤ T < M ( 21 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 21 d ) T R C 5 ( T ) , if P M D ≤ T (21e)

where

T R C 1 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + D T h o ρ 2 − s I e D ( 2 M − 2 N − T ) 2 , (22)

T R C 2 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + 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 , (23)

T R C 3 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + 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 , (24)

T R C 4 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + 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 ( M − N ) 2 2 T − s I e D ( M − N ) 2 2 T , (25)

T R C 5 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + 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 . (26)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of

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 ) , and T R C 4 ( P M D ) = T R C 5 ( P M D ) .

Case 2. M − N ≤ W D ρ < M .

According to Equations (1)-(20), the 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 ( 27 a ) T R C 6 ( T ) , if M − N ≤ T < W D ρ ( 27 b ) T R C 3 ( T ) , if W D ρ ≤ T < M ( 27 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 27 d ) T R C 5 ( T ) , if P M D ≤ T (25e)

where

T R C 6 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + D T h o ρ 2 − s I e D ( M − N ) 2 2 T . (28)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of

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 ) , and T R C 4 ( P M D ) = T R C 5 ( P M D ) .

Case 3. M ≤ W D ρ < P M D .

According to Equations (1)-(20), the 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 ( 29 a ) T R C 6 ( T ) , if M − N ≤ T < M ( 29 b ) T R C 7 ( T ) , if M ≤ T < W D ρ ( 29 c ) T R C 4 ( T ) , if W D ρ ≤ T < P M D ( 29 d ) T R C 5 ( T ) , if P M D ≤ T (29e)

where

T R C 7 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + D T h o ρ 2 + c I p D ( T − M ) 2 2 T − s I e D ( M − N ) 2 2 T . (30)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of

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 ) , and T R C 4 ( P M D ) = T R C 5 ( P M D ) .

Case 4. P M D ≤ W D ρ .

According to Equations (1)-(20), the 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 ( 31 a ) T R C 6 ( T ) , if M − N ≤ T < M ( 31 b ) T R C 7 ( T ) , if M ≤ T < P M D ( 31 c ) T R C 8 ( T ) , if P M D ≤ T < W D ρ ( 31 d ) T R C 5 ( T ) , if W D ρ ≤ T (31e)

where

T R C 8 ( T ) = A T + c P θ T ( e θ D P T − 1 ) + h m P θ T [ 1 θ ( e θ D P T − 1 ) − D P T ] + 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 . (32)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of

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 ) , and T R C 8 ( W D ρ ) = T R C 5 ( W D ρ ) .

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

Equations (22)-(26), (28), (30), and (32) yield the first order and second-order derivatives as follows.

T R C ′ 1 ( T ) = 1 T 2 { − A − ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + D ( h o ρ + s I e ) 2 T 2 } , (33)

T R C ″ 1 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] } , (34)

T R C ′ 2 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + W 2 ( h o − h r ) D ρ + D ( h r ρ + s I e ) T 2 } , (35)

T R C ″ 2 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ } , (36)

T R C ′ 3 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 + D h r ρ T 2 } , (37)

T R C ″ 3 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ − s I e D ( M − N ) 2 } , (38)

T R C ′ 4 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e D ( M − N ) 2 + D ( h r ρ + c I p ) T 2 } , (39)

T R C ″ 4 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ + c I p D M 2 − s I e D ( M − N ) 2 } , (40)

T R C ′ 5 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + W 2 ( h o − h r ) D ρ + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ρ ( h r + c I p ) T 2 } , (41)

T R C ″ 5 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 } , (42)

T R C ′ 6 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + s I e D ( M − N ) 2 + D h o ρ T 2 } , (43)

T R C ″ 6 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] − s I e D ( M − N ) 2 } , (44)

T R C ′ 7 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] − c I p D M 2 + s I e D ( M − N ) 2 + D ( h o ρ + c I p ) T 2 } , (45)

T R C ″ 7 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + c I p D M 2 − s I e D ( M − N ) 2 } , (46)

T R C ′ 8 ( T ) = 1 2 T 2 { − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ρ ( h o + c I p ) T 2 } , (47)

and

T R C ″ 8 ( T ) = 1 T 3 { 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 } . (48)

Let

G 1 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] , (49)

G 2 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ , (50)

G 3 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ − s I e D ( M − N ) 2 , (51)

G 4 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ + c I p D M 2 − s I e D ( M − N ) 2 , (52)

G 5 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + W 2 ( h r − h o ) D ρ − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 , (53)

G 6 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] − s I e D ( M − N ) 2 , (54)

G 7 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] + c I p D M 2 − s I e D ( M − N ) 2 , (55)

and

G 8 = 2 A + ( c + h m θ ) [ 2 P θ ( e θ D P T − 1 ) − 2 D T e θ D P T + θ D 2 P T 2 e θ D P T ] − c I p ( P − D ) M 2 − s I e D ( M − N ) 2 . (56)

Equations (49)-(56) imply

G 4 > G 3 > G 5 > G 8 , (57)

G 4 > G 7 > G 6 > G 8 , (58)

and

G 2 > G 1 > G 6 > G 8 . (59)

Equations (33)-(48) reveal the following results.

Lemma 1. T R C ′ i ( T ) is increasing on T > 0 if G i > 0 for all i = 1 ∼ 8 . That is, T R C i ( T ) is convex on T > 0 if G i > 0 .

T R C ′ i ( T ) = { < 0 , if 0 < T < T i * ( 60 a ) = 0 , if T = T i * ( 60 b ) > 0 , if T i * < T < ∞ (60c)

Equations (60a)-(60c) imply that T R C i ( T ) is decreasing on ( 0, T i * ] and increasing on [ T i * , ∞ ) for all i = 1 ∼ 8 . Solving optimal cycle T i * ( T ) ( i = 1 ∼ 8 ) by T R C ′ i ( T ) = 0 ( i = 1 ∼ 8 ) .

Case 1. W D ρ < M − N .

Equations (33), (35), (37), (39), and (41) yield

T R C ′ 1 ( W D ρ ) = T R C ′ 2 ( W D ρ ) = Δ 12 2 ( W D ρ ) 2 , (61)

T R C ′ 2 ( M − N ) = T R C ′ 3 ( M − N ) = Δ 23 2 ( M − N ) 2 , (62)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (63)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (64)

where

Δ 12 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] + D ( h o ρ + s I e ) ( W D ρ ) 2 , (65)

Δ 23 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( M − N ) − 1 ) − D ( M − N ) e θ D P ( M − N ) ] + W 2 ( h o − h r ) D ρ + D ( h r ρ + s I e ) ( M − N ) 2 , (66)

Δ 34 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P M − 1 ) − D M e θ D P M ] + W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 + D h r ρ M 2 , (67)

Δ 45 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( P M D ) − 1 ) − D ( P M D ) e θ D P ( P M D ) ] + W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e D ( M − N ) 2 + D ( h r ρ + c I p ) ( P M D ) 2 . (68)

Equations (65)-(68) imply

Δ 12 < Δ 23 < Δ 34 < Δ 45 . (69)

Case 2. M − N ≤ W D ρ < M .

Equations (33), (37), (39), (41), and (43) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (70)

T R C ′ 6 ( W D ρ ) = T R C ′ 3 ( W D ρ ) = Δ 63 2 ( W D ρ ) 2 , (71)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (72)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (73)

where

Δ 16 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( M − N ) − 1 ) − D ( M − N ) e θ D P ( M − N ) ] + D ( h o ρ + s I e ) ( M − N ) 2 , (74)

Δ 63 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] + s I e D ( M − N ) 2 + D h o ρ ( W D ρ ) 2 . (75)

Equations (67), (68), (74), and (75) imply

Δ 16 ≤ Δ 63 < Δ 34 < Δ 45 . (76)

Case 3. M ≤ W D ρ < P M D .

Equations (33), (39), (41), (43), and (45) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (77)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (78)

T R C ′ 7 ( W D ρ ) = T R C ′ 4 ( W D ρ ) = Δ 74 2 ( W D ρ ) 2 , (79)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (80)

where

Δ 67 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P M − 1 ) − D M e θ D P M ] + s I e D ( M − N ) 2 + D h o ρ M 2 , (81)

Δ 74 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] − c I p D M 2 + s I e D ( M − N ) 2 + D ( h o ρ + c I p ) ( W D ρ ) 2 . (82)

Equations (68), (74), (81), and (82) imply

Δ 16 ≤ Δ 67 ≤ Δ 74 < Δ 45 . (83)

Case 4. P M D ≤ W D ρ .

Equations (33), (41), (43), (45), and (47) yield

T R C ′ 1 ( M − N ) = T R C ′ 6 ( M − N ) = Δ 16 2 ( M − N ) 2 , (84)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (85)

T R C ′ 7 ( P M D ) = T R C ′ 8 ( P M D ) = Δ 78 2 ( P M D ) 2 , (86)

T R C ′ 8 ( W D ρ ) = T R C ′ 5 ( W D ρ ) = Δ 45 2 ( W D ρ ) 2 , (87)

where

Δ 78 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( P M D ) − 1 ) − D ( P M D ) e θ D P ( P M D ) ] − c I p D M 2 + s I e D ( M − N ) 2 + D ( h o ρ + c I p ) ( P M D ) 2 , (88)

Δ 85 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ρ ( h o + c I p ) ( W D ρ ) 2 . (89)

Equations (74), (81), (88), and (89) imply

Δ 16 ≤ Δ 67 ≤ Δ 78 ≤ Δ 85 . (90)

Based on the above arguments, the following results holds.

Lemma 2.

1) If Δ 12 ≤ 0 , then

a) G 1 > 0 and G 2 > 0 ,

b) T 1 * and T 2 * exist,

c) T R C 1 ( T ) and T R C 2 ( T ) are convex on T > 0 .

2) If Δ 16 ≤ 0 , then

a) G 1 > 0 and G 6 > 0 ,

b) T 1 * and T 6 * exist,

c) T R C 1 ( T ) and T R C 6 ( T ) are convex on T > 0 .

3) If Δ 34 ≤ 0 , then

a) G 3 > 0 and G 4 > 0 ,

b) T 3 * and T 4 * exist,

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

4) If Δ 85 ≤ 0 , then

a) G 5 > 0 and G 8 > 0 ,

b) T 5 * and T 8 * exist,

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

5) If Δ 45 ≤ 0 , then

a) G 4 > 0 and G 5 > 0 ,

b) T 4 * and T 5 * exist,

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

6) If Δ 78 ≤ 0 , then

a) G 7 > 0 and G 8 > 0 ,

b) T 7 * and T 8 * exist,

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

7) If Δ 74 ≤ 0 , then

a) G 4 > 0 and G 7 > 0 ,

b) T 4 * and T 7 * exist,

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

8) If Δ 67 ≤ 0 , then

a) G 6 > 0 and G 7 > 0 ,

b) T 6 * and T 7 * exist,

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

Proof. 1. (a) If Δ 12 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] + D ( h o ρ + s I e ) ( W D ρ ) 2 . (91)

Equation (91) implies

G 1 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + h o ρ + s I e ] > 0. (92)

G 2 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + h r ρ + s I e ] > 0. (93)

Equations (59), (92), and (93) demonstrate G 2 > G 1 > 0 .

b) Lemma 1 implies that T 1 * and T 2 * exist.

c) Equations (34), (36), and lemma 1 imply that T R C 1 ( T ) and T R C 2 ( T ) are convex on T > 0 .

2. a) If Δ 16 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( M − N ) − 1 ) − D ( M − N ) e θ D P ( M − N ) ] + D ( h o ρ + s I e ) ( M − N ) 2 . (94)

Equation (94) implies

G 1 ≥ D ( M − N ) 2 [ ( c + h m θ ) θ D P e θ D P ( M − N ) + h o ρ + s I e ] > 0. (95)

G 6 ≥ D ( M − N ) 2 [ ( c + h m θ ) θ D P e θ D P ( M − N ) + h o ρ ] > 0. (96)

Equations (59), (95), and (96) demonstrate G 1 > G 6 > 0 .

b) Lemma 1 implies that T 1 * and T 6 * exist.

c) Equations (34), (44), and lemma 1 imply that T R C 1 ( T ) and T R C 6 ( T ) are convex on T > 0 .

3. a) If Δ 34 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P M − 1 ) − D M e θ D P M ] + W 2 ( h o − h r ) D ρ + s I e D ( M − N ) 2 + D h r ρ M 2 . (97)

Equation (97) implies

G 3 ≥ D M 2 [ ( c + h m θ ) θ D P e θ D P M + h r ρ ] > 0. (98)

G 4 ≥ D M 2 [ ( c + h m θ ) θ D P e θ D P M + h r ρ + c I p ] > 0. (99)

Equations (57), (98), and (99) demonstrate G 4 > G 3 > 0 .

b) Lemma 1 implies that T 3 * and T 4 * exist.

c) Equations (38), (40), and lemma 1 imply that T R C 3 ( T ) and T R C 4 ( T ) are convex on T > 0 .

4. a) If Δ 85 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] + c I p ( P − D ) M 2 + s I e D ( M − N ) 2 + D ρ ( h o + c I p ) ( W D ρ ) 2 . (100)

Equation (100) implies

G 5 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + ρ ( h o + c I p ) ] + W 2 ( h r − h o ) D ρ > 0. (101)

G 8 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + ρ ( h o + c I p ) ] > 0. (102)

Equations (57), (101), and (102) demonstrate G 5 > G 8 > 0 .

b) Lemma 1 implies that T 5 * and T 8 * exist.

c) Equations (42), (48), and lemma 1 imply that T R C 5 ( T ) and T R C 8 ( T ) are convex on T > 0 .

5. a) If Δ 45 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( P M D ) − 1 ) − D ( P M D ) e θ D P ( P M D ) ] + W 2 ( h o − h r ) D ρ − c I p D M 2 + s I e D ( M − N ) 2 + D ( h r ρ + c I p ) ( P M D ) 2 . (103)

Equation (103) implies

G 4 ≥ D ( P M D ) 2 [ ( c + h m θ ) θ D P e θ D P ( P M D ) + h r ρ + c I p ] > 0. (104)

G 5 ≥ D ( P M D ) 2 [ ( c + h m θ ) θ D P e θ D P ( P M D ) + ρ ( h r + c I p ) ] > 0. (105)

Equations (57), (104), and (105) demonstrate G 4 > G 5 > 0 .

b) Lemma 1 implies that T 4 * and T 5 * exist.

c) Equations (40), (42), and lemma 1 imply that T R C 4 ( T ) and T R C 5 ( T ) are convex on T > 0 .

6. a) If Δ 78 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( P M D ) − 1 ) − D ( P M D ) e θ D P ( P M D ) ] − c I p D M 2 + s I e D ( M − N ) 2 + D ( h o ρ + c I p ) ( P M D ) 2 . (106)

Equation (106) implies

G 7 ≥ D ( P M D ) 2 [ ( c + h m θ ) θ D P e θ D P ( P M D ) + h o ρ + c I p ] > 0. (107)

G 8 ≥ D ( P M D ) 2 [ ( c + h m θ ) θ D P e θ D P ( P M D ) + ρ ( h o + c I p ) ] > 0. (108)

Equations (58), (107), and (108) demonstrate G 7 > G 8 > 0 .

b) Lemma 1 implies that T 7 * and T 8 * exist.

c) Equations (46), (48), and lemma 1 imply that T R C 7 ( T ) and T R C 8 ( T ) are convex on T > 0 .

7. a) If Δ 74 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P ( W D ρ ) − 1 ) − D ( W D ρ ) e θ D P ( W D ρ ) ] − c I p D M 2 + s I e D ( M − N ) 2 + D ( h o ρ + c I p ) ( W D ρ ) 2 . (109)

Equation (109) implies

G 4 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + ( h o ρ + c I p ) ] + W 2 ( h r − h o ) D ρ > 0. (110)

G 7 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + ( h o ρ + c I p ) ] > 0. (111)

Equations (58), (110), and (111) demonstrate G 4 > G 7 > 0 .

b) Lemma 1 implies that T 4 * and T 7 * exist.

c) Equations (40), (46), and lemma 1 imply that T R C 4 ( T ) and T R C 7 ( T ) are convex on T > 0 .

8. a) If Δ 67 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P M − 1 ) − D M e θ D P M ] + s I e D ( M − N ) 2 + D h o ρ M 2 . (112)

Equation (112) implies

G 6 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + h o ρ ] > 0. (113)

G 7 ≥ D ( W D ρ ) 2 [ ( c + h m θ ) θ D P e θ D P ( W D ρ ) + ( h o ρ + c I p ) ] > 0. (114)

Equations (58), (113), and (114) demonstrate G 7 > G 6 > 0 .

b) Lemma 1 implies that T 6 * and T 7 * exist.

c) Equations (44), (46), and lemma 1 imply that T R C 6 ( T ) and T R C 7 ( 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 .

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

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

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

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

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

Proof. 1) If 0 < Δ 12 , then 0 < Δ 12 < Δ 23 < Δ 34 < Δ 45 . So, lemmas 1, 2, and Equations (60a)-(60c) imply

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

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

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

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

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

Since T R C ( T ) is continuous on T > 0 , Equations (21a)-(21e) and 1.1-1.5 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 * ) .

2) If Δ 12 ≤ 0 < Δ 23 , then Δ 12 ≤ 0 < Δ 23 < Δ 34 < Δ 45 . So, lemmas 1, 2, and Equations (60a)-(60c) imply

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

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

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

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

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

Since T R C ( T ) is continuous on T > 0 , Equations (21a)-(21e) and 2.1-2.5 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 * ) .

3) If Δ 23 ≤ 0 < Δ 34 , then Δ 12 < Δ 23 ≤ 0 < Δ 34 < Δ 45 . So, lemmas 1, 2, and Equations (60a)-(60c) imply

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

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

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

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

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

Since T R C ( T ) is continuous on T > 0 , Equations (21a)-(21e) and 3.1-3.5 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 * ) .

4) If Δ 34 ≤ 0 < Δ 45 , then Δ 12 < Δ 23 < Δ 34 ≤ 0 < Δ 45 . So, lemmas 1, 2, and Equations (60a)-(60c) imply

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

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

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

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

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

Since T R C ( T ) is continuous on T > 0 , Equations (21a)-(21e) and 4a-4e 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 * ) .

5) If Δ 45 ≤ 0 , then Δ 12 < Δ 23 < Δ 34 < Δ 45 ≤ 0 . So, lemmas 1, 2, and Equations (60a)-(60c) imply

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

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

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

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

e) 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 (21a)-(21e) and 5.1-5.5 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 argument above arguments, we have completed the proof of theorem 1. □

Applying lemmas 1, 2, and Equations (27a)-(27e), the following results hold.

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

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

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

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

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

5) 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 (29a)-(29e), the following results hold.

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

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

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

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

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

5) 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 (31a)-(31e), the following results hold.

Theorem 4. Suppose P M D ≤ W D ρ .

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

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

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

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

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

To find out the critical parameters in this research, [

From the computational outcomes, we can determine T * and T R C ( T * ) from the sensitivity analyses for for this research, [

According to

1) this research model

a) Positive & Major: the ordering cost A.

b) Positive & Minor: the unit holding cost per item for product in a rented warehouse h_{r}.

c) Negative & Minor: the unit selling price per item s, the unit holding cost per item for raw materials in a raw materials warehouse h_{m}, the unit holding cost per item for product in an owned warehouse h_{o}, the interest rate payable I_{p}, and the interest rate earned I_{e}.

d) Negative & Major: the unit purchasing price per item c and the deterioration rate θ.

2) [

a) Positive & Major: the ordering cost A.

b) Positive & Minor: none.

c) Negative & Minor: the unit selling price per item s, the unit purchasing price per item c, the unit holding cost per item for raw materials in a raw materials warehouse h_{m}, the unit holding cost per item for product in an owned warehouse h_{o}, the unit holding cost per item for product in a rented warehouse h_{r}, the interest rate payable I_{p}, and the interest rate earned I_{e}.

Parameters | +/− | this research | [ | [ |
---|---|---|---|---|

A | −50% | 0.323796413 | 0.336873359 | 0.353809894 |

−25% | 0.356136518 | 0.370559305 | 0.389189423 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.404310371 | 0.420751426 | 0.441904987 | |

+50% | 0.426353536 | 0.443723500 | 0.466031997 | |

s | −50% | 0.389714048 | 0.405541903 | 0.425930795 |

−25% | 0.385376100 | 0.401021954 | 0.421183603 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.376549626 | 0.391825666 | 0.411524966 | |

+50% | 0.372057574 | 0.387145613 | 0.406609619 | |

c | −50% | 0.406384273 | 0.417864966 | 0.447136879 |

−25% | 0.391824025 | 0.405680770 | 0.429428145 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.372601550 | 0.389210740 | 0.406358844 | |

+50% | 0.365912858 | 0.383377581 | 0.398410010 | |

h_{m} | −50% | 0.389613329 | 0.406049944 | |

−25% | 0.385227856 | 0.401164096 | ||

0% | 0.380988522 | 0.396450476 | ||

+25% | 0.376887341 | 0.391899198 | ||

+50% | 0.372916925 | 0.387501152 | ||

h_{o} | −50% | 0.391628615 | 0.407536772 | 0.428025957 |

−25% | 0.386345341 | 0.402031840 | 0.422244261 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.375555056 | 0.390789406 | 0.410436607 | |

+50% | 0.370041463 | 0.385045114 | 0.404403516 | |

h_{r} | −50% | 0.382462299 | 0.401566710 | 0.426988203 |

−25% | 0.381654883 | 0.398746393 | 0.421093248 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.380429265 | 0.394545054 | 0.412530499 | |

+50% | 0.379953137 | 0.392938258 | 0.409322090 | |

I_{p} | −50% | 0.396194694 | 0.417864966 | 0.447136879 |

−25% | 0.387619621 | 0.405680770 | 0.429428145 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.375705573 | 0.389210740 | 0.406358844 | |

+50% | 0.371396477 | 0.383377581 | 0.398410010 | |

I_{e} | −50% | 0.389714048 | 0.405541903 | 0.425930795 |

−25% | 0.385376100 | 0.401021954 | 0.421183603 | |

0% | 0.380988522 | 0.396450476 | 0.416382291 | |

+25% | 0.376549626 | 0.391825666 | 0.411524966 | |

+50% | 0.372057574 | 0.387145613 | 0.406609619 | |

θ | −50% | 0.388544380 | ||

−25% | 0.384723975 | |||

0% | 0.380988522 | |||

+25% | 0.377335518 | |||

+50% | 0.373762524 |

Parameters | +/− | this research | [ | [ |
---|---|---|---|---|

A | −50% | 35267.01504 | 34992.21730 | 1613.304009 |

−25% | 36021.43281 | 35691.12590 | 2266.223520 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 37278.79577 | 36855.97352 | 3354.422706 | |

+50% | 37907.47725 | 37438.39734 | 3898.522299 | |

s | −50% | 36880.93796 | 36487.38963 | 3010.092101 |

−25% | 36765.52613 | 36380.46967 | 2910.207607 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 36534.70246 | 36166.62974 | 2710.438620 | |

+50% | 36419.29063 | 36059.70978 | 2610.554126 | |

c | −50% | 19908.61022 | 19695.93987 | 2687.620477 |

−25% | 28279.36227 | 27984.74479 | 2748.971795 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 45020.86635 | 44562.35461 | 2871.674431 | |

+50% | 53391.61838 | 52851.15952 | 2933.025750 | |

h_{m} | −50% | 36414.30792 | 36021.07272 | |

−25% | 36532.21111 | 36147.31121 | ||

0% | 36650.11429 | 36273.54970 | ||

+25% | 36768.01747 | 36399.78820 | ||

+50% | 36885.92065 | 36526.02669 | ||

h_{o} | −50% | 36332.28614 | 35934.95967 | 2454.531969 |

−25% | 36491.20021 | 36104.25469 | 2632.427541 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 36809.02837 | 36442.84472 | 2988.218684 | |

+50% | 36967.94245 | 36612.13974 | 3166.114256 | |

h_{r} | −50% | 36646.73504 | 36256.45940 | 2772.085535 |

−25% | 36648.42467 | 36265.00455 | 2791.204324 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 36651.80392 | 36282.09484 | 2829.441902 | |

+50% | 36653.49354 | 36290.63999 | 2848.560691 | |

I_{p} | −50% | 36610.72995 | 36195.93987 | 2687.620477 |

−25% | 36630.42212 | 36234.74479 | 2748.971795 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 36669.80647 | 36312.35461 | 2871.674431 | |

+50% | 36689.49864 | 36351.15952 | 2933.025750 | |

I_{e} | −50% | 36880.93796 | 36487.38963 | 3010.092101 |

−25% | 36765.52613 | 36380.46967 | 2910.207607 | |

0% | 36650.11429 | 36273.54970 | 2810.323113 | |

+25% | 36534.70246 | 36166.62974 | 2710.438620 | |

+50% | 36419.29063 | 36059.70978 | 2610.554126 | |

θ | −50% | 36445.25871 | ||

−25% | 36547.47934 | |||

0% | 36650.11429 | |||

+25% | 36753.16317 | |||

+50% | 36856.63035 |

d) Negative & Major: none.

3) [

a) Positive & Major: the ordering cost A.

b) Positive & Minor: none.

c) Negative & Minor: the unit selling price per item s, the unit holding cost per item for product in an owned warehouse h_{o}, the unit holding cost per item for product in a rented warehouse h_{r}, and the interest rate earned I_{e}.

d) Negative & Major: the unit purchasing price per item c and the interest rate payable I_{p}.

Therefore, when making decisions on the order cycle time, variables with a relatively large influence must be considered as priority, while those with a small

influence can be processed later.

On the other hand, it is seen that the variables impact the annual total relevant cost T R C ( T * ) for this research, [

1) this research model

a) Positive & Major: the unit purchasing price per item c.

b) Positive & Minor: the ordering cost A, the unit holding cost per item for raw materials in a raw materials warehouse h_{m}, the unit holding cost per item for product in an owned warehouse h_{o}, and the interest rate payable I_{p}.

c) Negative & Minor: the unit holding cost per item for product in a rented warehouse h_{r}, the interest rate earned I_{e}, and the deterioration rate θ.

d) Negative & Major: none.

2) [

a) Positive & Major: the unit purchasing price per item c.

b) Positive & Minor: the ordering cost A, the unit holding cost per item for raw materials in a raw materials warehouse h_{m}, the unit holding cost per item for product in an owned warehouse h_{o}, the unit holding cost per item for product in a rented warehouse h_{r}, and the interest rate payable I_{p}.

c) Negative & Minor: the unit selling price per item s and the interest rate earned I_{e}.

d) Negative & Major: none.

3) [

a) Positive & Major: the ordering cost A and the unit holding cost per item for product in an owned warehouse h_{o}.

b) Positive & Minor: the unit purchasing price per item c, the unit holding cost per item for product in a rented warehouse h_{r}, and the interest rate payable I_{p}.

c) Negative & Minor: the interest rate earned I_{e}.

d) Negative & Major: the unit selling price per item s.

Therefore, when making decisions on the annual total relevant cost, variables with a relatively large influence can be considered as priority, while those with a small influence can be processed later.

We can organize the relative parameters impact to T * and T R C ( T * ) for this research, [

One of traditional EPQ model’s assumptions is that the raw materials required for production are timely, so that the holding cost of raw materials will be ignored. [

We reach the following conclusions in management practice after the sensitivity analyses:

1) When making decisions on the order cycle time T * under limited resources, it gives priority order to the ordering cost A and the unit purchasing price per item c.

Impact | this research | [ | [ |
---|---|---|---|

Positive & Major | A | A | A |

Positive & Minor | h_{r} | ||

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

Negative & Major | c, θ | c, I_{p} |

Impact | this research | [ | [ |
---|---|---|---|

Positive & Major | c | c | A, h_{o} |

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

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

Negative & Major | s |

2) When making decisions on the annual total relevant cost T R C ( T * ) under limited resources, it only considers the unit purchasing price per item c.

This research provides more precise decisions for practical business decisions. Although adding the holding cost of raw materials increases the complexity of the model, but it’s useful and contributes to the field of industrial management.

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

Yen, G.-F., Lin, S.-D. and Lee, A.-K. (2019) EPQ Inventory Model for Deteriorating Raw Materials with Two-Level Trade Credit and Limited Storage Capacity under Alternate Due Date of Payment. Open Access Library Journal, 6: e5795. https://doi.org/10.4236/oalib.1105795