^{1}

^{2}

^{1}

The traditional economic production quantity (EPQ) model assumes that raw materials are supplied timely. But during the production and transport process of raw materials could change the holding cost of raw materials, therefore, they should be considered in 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 consider the holding cost of non-deteriorating raw materials. It’s closer to the real world. However, in reality, most of the raw materials are deteriorating; it also needs to be considered. Therefore, this research extends [1]’s model to consider the holding cost of deteriorating raw materials. We use cost minimization to develop the total relevant cost and determine the optimal cycle time by four theorems. Finally, we use sensitivity analyses to investigate the effects of the parameters on ordering policies.

The economic order quantity (EOQ) model [

[

Recently, [

As mentioned above, studies that consider the impact of the holding cost of deteriorating raw materials in the total relevant cost are limited or nonexistent. Moreover, [

The mathematical model is developed based on the following.

Author | Model | N − t | LSC | NRM | DRM | |
---|---|---|---|---|---|---|

[ | EOQ | V | ||||

[ | EPQ | V | V | |||

[ | EOQ | V | V | |||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EOQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | V | |||

[ | EOQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | V | |||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EOQ | V | ||||

[ | EPQ | V | V | |||

[ | EPQ | V | ||||

[ | EOQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EPQ | V | ||||

[ | EOQ | V | ||||

[ | EOQ | V | V | V | ||

[ | EPQ | V | V | V | ||

This research | EPQ | V | V | V |

Note: Column N − t for [

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 the manufacturer at time t ∈ [ 0, T ] , then the customer will have a trade credit period N − t and make the payment at time N.

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 ≤ N .

Annual interest payable is

0. (12)

2) 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)

Three cases to occur in annual interest earned.

1) 0 < T ≤ N , as shown in

Annual interest earned is

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

2) N ≤ T ≤ M , as shown in

Annual interest earned is

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

3) N ≤ M ≤ T , as shown in

Annual interest earned is

s I e × [ ( D N + D M ) ( M − N ) 2 ] × 1 T = s I e D ( M 2 − N 2 ) 2 T . (18)

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 < N ,

2) 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 < N .

According to Equations (1)-(18), 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 ( 19 a ) T R C 2 ( T ) , if W D ρ ≤ T < N ( 19 b ) T R C 3 ( T ) , if N ≤ T < M ( 19 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 19 d ) T R C 5 ( T ) , if P M D ≤ T (19e)

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 ( M − N ) , (20)

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 ( M − N ) , (21)

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 ( 2 M T − N 2 − T 2 ) 2 T , (22)

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 ( T − M ) 2 2 T − s I e D ( M 2 − N 2 ) 2 T , (23)

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 2 − N 2 ) 2 T . (24)

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 ( N ) = T R C 3 ( 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. N ≤ W D ρ < M .

According to Equations (1)-(18), the total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < N ( 25 a ) T R C 6 ( T ) , if N ≤ T < W D ρ ( 25 b ) T R C 3 ( T ) , if W D ρ ≤ T < M ( 25 c ) T R C 4 ( T ) , if M ≤ T < P M D ( 25 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 ( 2 M T − N 2 − T 2 ) 2 T . (26)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of T R C 1 ( N ) = T R C 6 ( 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)-(18), the total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < N ( 27 a ) T R C 6 ( T ) , if N ≤ T < M ( 27 b ) T R C 7 ( T ) , if M ≤ T < W D ρ ( 27 c ) T R C 4 ( T ) , if W D ρ ≤ T < P M D ( 27 d ) T R C 5 ( T ) , if P M D ≤ T (27e)

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 2 − N 2 ) 2 T . (28)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of T R C 1 ( N ) = T R C 6 ( 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)-(18), the total relevant cost T R C ( T ) can be expressed by

T R C ( T ) = { T R C 1 ( T ) , if 0 < T < N ( 29 a ) T R C 6 ( T ) , if N ≤ T < M ( 29 b ) T R C 7 ( T ) , if M ≤ T < P M D ( 29 c ) T R C 8 ( T ) , if P M D ≤ T < W D ρ ( 29 d ) T R C 5 ( T ) , if W D ρ ≤ T (29e)

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 2 − N 2 ) 2 T . (30)

T R C ( T ) is continuous at T, T ∈ [ 0, ∞ ) because of T R C 1 ( N ) = T R C 6 ( 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 (20)-(24), (26), (28), and (30) 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 2 T 2 } , (31)

T R C ′ 1 ( T ) = 1 T 3 { 2 A + 2 ( c + h m θ ) [ P θ ( e θ D P T − 1 ) − D T e θ D P T ] + θ D 2 P T 2 e θ D P T } , (32)

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 T 2 } , (33)

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 } , (34)

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 N 2 + D ( h r ρ + s I e ) T 2 } , (35)

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 N 2 } , (36)

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 + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + s I e ) T 2 } , (37)

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 + D M 2 ( c I p − s I e ) + s I e D N 2 } , (36)

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 + D M 2 ( s I e − c I p ) − s I e D N 2 + c I p P M 2 + D ρ ( h r + c I p ) T 2 } , (39)

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 + D M 2 ( c I p − s I e ) + s I e D N 2 − c I p P M 2 } , (40)

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 N 2 + D ( h r ρ + s I e ) T 2 } , (41)

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 N 2 } , (42)

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 ] + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + c I p ) T 2 } , (43)

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 ] + D M 2 ( c I p − s I e ) + s I e D N 2 } , (44)

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 ] + D M 2 ( s I e − c I p ) − s I e D N 2 + c I p P M 2 + D ρ ( h o + c I p ) T 2 } , (45)

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 ] + D M 2 ( c I p − s I e ) + s I e D N 2 − c I p P M 2 } . (46)

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 ] , (47)

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 , (48)

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 N 2 , (49)

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 + D M 2 ( c I p − s I e ) + s I e D N 2 , (50)

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 + D M 2 ( c I p − s I e ) + s I e D N 2 − c I p P M 2 , (51)

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 N 2 , (52)

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 ] + D M 2 ( c I p − s I e ) + s I e D N 2 , (53)

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 ] + D M 2 ( c I p − s I e ) + s I e D N 2 − c I p P M 2 . (54)

Equations (47)-(54) imply

G 4 > G 5 > G 8 , (55)

G 4 > G 7 > G 8 , (56)

G 3 > G 2 > G 1 , (57)

and

G 3 > G 6 > G 1 . (58)

Equations (31)-(46) 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 * ( 59 a ) = 0 , if T = T i * ( 59 b ) > 0 , if T i * < T < ∞ (59c)

Equations (59a)-(59c) 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 < N .

Equations (31), (33), (35), (37), and (39) yield

T R C ′ 1 ( W D ρ ) = T R C ′ 2 ( W D ρ ) = Δ 12 2 ( W D ρ ) 2 , (60)

T R C ′ 2 ( N ) = T R C ′ 3 ( N ) = Δ 23 2 N 2 , (61)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (62)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (63)

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 ( W D ρ ) 2 , (64)

Δ 23 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P N − 1 ) − D N e θ D P N ] + W 2 ( h o − h r ) D ρ + D h r ρ N 2 , (65)

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 N 2 + D ( h r ρ + s I e ) M 2 , (66)

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 + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + s I e ) ( P M D ) 2 . (67)

Equations (64)-(67) imply

Δ 12 < Δ 23 < Δ 34 < Δ 45 . (68)

Case 2. N ≤ W D ρ < M .

Equations (31), (35), (37), (39), and (41) yield

T R C ′ 1 ( N ) = T R C ′ 6 ( N ) = Δ 16 2 N 2 , (69)

T R C ′ 6 ( W D ρ ) = T R C ′ 3 ( W D ρ ) = Δ 63 2 ( W D ρ ) 2 , (70)

T R C ′ 3 ( M ) = T R C ′ 4 ( M ) = Δ 34 2 M 2 , (71)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (72)

where

Δ 16 = − 2 A − 2 ( c + h m θ ) [ P θ ( e θ D P N − 1 ) − D N e θ D P N ] + D h o ρ N 2 , (73)

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 N 2 + D ( h r ρ + s I e ) ( W D ρ ) 2 . (74)

Equations (66), (67), (73), and (74) imply

Δ 16 ≤ Δ 63 < Δ 34 < Δ 45 . (75)

Case 3. M ≤ W D ρ < P M D .

Equations (31), (37), (39), (41), and (43) yield

T R C ′ 1 ( N ) = T R C ′ 6 ( N ) = Δ 16 2 N 2 , (76)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (77)

T R C ′ 7 ( W D ρ ) = T R C ′ 4 ( W D ρ ) = Δ 74 2 ( W D ρ ) 2 , (78)

T R C ′ 4 ( P M D ) = T R C ′ 5 ( P M D ) = Δ 45 2 ( P M D ) 2 , (79)

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 N 2 + D ( h r ρ + s I e ) M 2 , (80)

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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + c I p ) ( W D ρ ) 2 . (81)

Equations (67), (73), (79), and (80) imply

Δ 16 ≤ Δ 67 ≤ Δ 74 < Δ 45 . (82)

Case 4. P M D ≤ W D ρ .

Equations (31), (39), (41), (43), and (45) yield

T R C ′ 1 ( N ) = T R C ′ 6 ( N ) = Δ 16 2 N 2 , (83)

T R C ′ 6 ( M ) = T R C ′ 7 ( M ) = Δ 67 2 M 2 , (84)

T R C ′ 7 ( P M D ) = T R C ′ 8 ( P M D ) = Δ 78 2 ( P M D ) 2 , (85)

T R C ′ 8 ( W D ρ ) = T R C ′ 5 ( W D ρ ) = Δ 45 2 ( W D ρ ) 2 , (86)

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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + c I p ) ( P M D ) 2 , (87)

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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + c I p P M 2 + D ρ ( h o + c I p ) ( W D ρ ) 2 . (88)

Equations (73), (80), (87), and (88) imply

Δ 16 ≤ Δ 67 ≤ Δ 78 ≤ Δ 85 . (89)

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 23 ≤ 0 , then

a) G 2 > 0 and G 3 > 0 ,

b) T 2 * and T 3 * exist,

c) T R C 2 ( T ) and T R C 3 ( T ) are convex on T > 0 .

3) 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 .

4) If Δ 63 ≤ 0 , then

a) G 3 > 0 and G 6 > 0 ,

b) T 3 * and T 6 * exist,

c) T R C 3 ( T ) and T R C 6 ( T ) are convex on T > 0 .

5) 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 .

6) 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 .

7) 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 .

8) 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 .

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 ( W D ρ ) 2 . (90)

Equation (90) 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 ] > 0. (91)

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

Equations (54), (91), and (92) demonstrate G 2 > G 1 > 0 .

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

c) Equations (32), (34), 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 23 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P N − 1 ) − D N e θ D P N ] + W 2 ( h o − h r ) D ρ + D h r ρ N 2 . (93)

Equation (93) implies

G 2 ≥ D N 2 [ ( c + h m θ ) θ D P e θ D P N + h r ρ ] > 0. (94)

G 3 ≥ D N 2 [ ( c + h m θ ) θ D P e θ D P N + ( h r ρ + s I e ) ] > 0. (95)

Equations (54), (94), and (95) demonstrate G 3 > G 2 > 0 .

b) Lemma 1 implies that T 2 * and T 3 * exist.

c) Equations (34), (36), and lemma 1 imply that T R C 2 ( T ) and T R C 3 ( T ) are convex on T > 0 .

3. a) If Δ 16 ≤ 0 , then

2 A ≥ − 2 ( c + h m θ ) [ P θ ( e θ D P N − 1 ) − D N e θ D P N ] + D h o ρ N 2 . (96)

Equation (96) implies

G 1 ≥ D N 2 [ ( c + h m θ ) θ D P e θ D P N + h o ρ ] > 0. (97)

G 6 ≥ D N 2 [ ( c + h m θ ) θ D P e θ D P N + ( h o ρ + s I e ) ] > 0. (98)

Equations (55), (97), and (98) demonstrate G 6 > G 1 > 0 .

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

c) Equations (32), (42), and lemma 1 imply that T R C 1 ( T ) and T R C 6 ( T ) are convex on T > 0 .

4. a) If Δ 63 ≤ 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 ) ] − s I e D N 2 + D ( h r ρ + s I e ) ( W D ρ ) 2 . (99)

Equation (99) implies

G 3 ≥ 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 ) ] + W 2 ( h r − h o ) D ρ > 0. (100)

G 6 ≥ 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. (101)

Equations (55), (100), and (101) demonstrate G 3 > G 6 > 0 .

b) Lemma 1 implies that T 3 * and T 6 * exist.

c) Equations (36), (42), and lemma 1 imply that T R C 3 ( T ) and T R C 6 ( T ) are convex on T > 0 .

5. 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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + c I p P M 2 + D ρ ( h o + c I p ) ( W D ρ ) 2 . (102)

Equation (102) 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. (103)

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. (104)

Equations (51), (103), and (104) demonstrate G 5 > G 8 > 0 .

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

c) Equations (40), (46), and lemma 1 imply that T R C 5 ( T ) and T R C 8 ( T ) are convex on T > 0 .

6. 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 + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + s I e ) ( P M D ) 2 . (105)

Equation (105) 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. (106)

G 5 ≥ 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 ) ] − c I p P M 2 > 0. (107)

Equations (51), (106), and (107) demonstrate G 4 > G 5 > 0 .

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

c) Equations (38), (40), and lemma 1 imply that T R C 4 ( T ) and T R C 5 ( T ) are convex on T > 0 .

7. 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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + c I p ) ( P M D ) 2 . (108)

Equation (108) 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. (109)

G 8 ≥ 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 ) ] − c I p P M 2 > 0. (110)

Equations (52), (109), and (110) demonstrate G 7 > G 8 > 0 .

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

c) Equations (44), (46), and lemma 1 imply that T R C 7 ( T ) and T R C 8 ( T ) are convex on T > 0 .

8. 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 ) ] + D M 2 ( s I e − c I p ) − s I e D N 2 + D ( h r ρ + c I p ) ( W D ρ ) 2 . (111)

Equation (111) 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. (112)

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. (113)

Equations (52), (112), and (113) demonstrate G 4 > G 7 > 0 .

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

c) Equations (38), (44), and lemma 1 imply that T R C 4 ( 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 < 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 (59a)-(59c) 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 , N ] .

c) T R C 3 ( T ) is increasing on [ 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 (19a)-(19e) 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 (59a)-(59c) 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 * , N ] .

c) T R C 3 ( T ) is increasing on [ 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 (19a)-(19e) 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 (59a)-(59c) 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 , N ] .

c) T R C 3 ( T ) is decreasing on [ 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 (19a)-(19e) 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 (59a)-(59c) 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 , N ] .

c) T R C 3 ( T ) is decreasing on [ 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 (19a)-(19e) and 4.1-4.5 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 (59a)-(59c) 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 , N ] .

c) T R C 3 ( T ) is decreasing on [ 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 (19a)-(19e) 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 (25a)-(25e), the following results hold.

Theorem 2. Suppose 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 (27a)-(27e), 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 (29a)-(29e), 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 * .

We executes the sensitivity analyses for this research, [

With the computational outcomes, we can compare for T * and T R C ( T * ) 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 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 deterioration rate θ.

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

2) [

a) Positive & Major: the ordering cost A, the unit holding cost per item for product in a rented warehouse h_{r}, and the interest rate earned I_{e}.

b) Positive & Minor: none.

c) Negative & Minor: the unit purchasing price per item c and the unit holding cost per item for raw materials in a raw materials warehouse h_{m}.

d) Negative & Major: the unit selling price per item s, the unit holding cost per item for product in an owned warehouse h_{o}, and the interest rate payable I_{p}.

3) [

a) Positive & Major: the ordering cost A and the interest rate earned I_{e}.

b) Positive & Minor: the interest rate payable I_{p}.

c) Negative & Minor: the unit purchasing price per item c, the unit holding

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

A | −50% | 0.270233815 | 0.101786596 | 0.327853393 |

−25% | 0.308263707 | 0.205770647 | 0.365752394 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.362872495 | 0.309262520 | 0.421410042 | |

+50% | 0.387290961 | 0.349705848 | 0.446645529 | |

s | −50% | 0.320429903 | 0.36875616 | 0.415415749 |

−25% | 0.353084192 | 0.316762278 | 0.405123971 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.319433195 | 0.194027125 | 0.383713181 | |

+50% | 0.301196267 | 0.079376429 | 0.372546631 | |

c | −50% | 0.313929929 | 0.34787516 | 0.418378299 |

−25% | 0.341413098 | 0.267593486 | 0.404628808 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.329239493 | 0.259153569 | 0.386870646 | |

+50% | 0.330211234 | 0.256525704 | 0.380796006 | |

h_{m} | −50% | 0.297988701 | 0.344291552 | |

−25% | 0.340421870 | 0.266881021 | ||

0% | 0.336680821 | 0.262664107 | ||

+25% | 0.333061397 | 0.258640968 | ||

+50% | 0.329557115 | 0.254797204 | ||

h_{o} | −50% | 0.348680773 | 0.297528747 | 0.406832431 |

−25% | 0.342733499 | 0.279860234 | 0.400745084 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.330516996 | 0.245821301 | 0.388284193 | |

+50% | 0.324235566 | 0.229212566 | 0.381901310 | |

h_{r} | −50% | 0.328130944 | 0.181953853 | 0.399872524 |

−25% | 0.332846661 | 0.225942070 | 0.396914129 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.339860159 | 0.294847498 | 0.392651524 | |

+50% | 0.342539502 | 0.323848207 | 0.391065153 | |

I_{p} | −50% | 0.339166974 | 0.450722838 | 0.380709435 |

−25% | 0.337755094 | 0.348933273 | 0.388867371 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.335835953 | 0.178927888 | 0.398768059 | |

+50% | 0.335154093 | 0.068271068 | 0.401999000 | |

I_{e} | −50% | 0.368756160 | 0.221426362 | 0.440417015 |

−25% | 0.353084192 | 0.242921867 | 0.418119460 | |

0% | 0.336680821 | 0.262664107 | 0.394563833 | |

+25% | 0.319433195 | 0.281022833 | 0.369509613 | |

+50% | 0.301196267 | 0.298253637 | 0.342628209 | |

θ | −50% | 0.343327907 | ||

−25% | 0.339965886 | |||

0% | 0.336680821 | |||

+25% | 0.333470273 | |||

+50% | 0.330331881 |

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

A | −50% | 33,763.13964 | 33,437.31541 | 534.510096 |

−25% | 34,674.33045 | 34,339.88184 | 1218.313736 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 36,192.98180 | 35,844.15922 | 2357.986469 | |

+50% | 36,952.30747 | 36,596.29791 | 2927.822836 | |

s | −50% | 36,371.40551 | 36,020.89415 | 2491.884636 |

−25% | 35,902.53082 | 35,556.45734 | 2140.017370 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 34,964.78145 | 34,627.58372 | 1436.282838 | |

+50% | 34,495.90676 | 34,163.14691 | 1084.415571 | |

c | −50% | 18,766.54316 | 18,591.89054 | 1697.209763 |

−25% | 27,100.09964 | 26,841.95553 | 1742.679933 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 43,767.21261 | 43,342.08552 | 1833.620272 | |

+50% | 52,100.76910 | 51,592.15051 | 1879.090442 | |

h_{m} | −50% | 35,238.69360 | 34,896.51322 | |

−25% | 35,336.17486 | 34,994.26687 | ||

0% | 35,433.65613 | 35,092.02053 | ||

+25% | 35,531.13739 | 35,189.77418 | ||

+50% | 35,628.61866 | 35,287.52784 | ||

h_{o} | −50% | 35,174.46516 | 34,829.60381 | 1443.910425 |

−25% | 35,304.06065 | 34,960.81217 | 1616.030264 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 35,563.25162 | 35,223.22888 | 1960.269941 | |

+50% | 35,692.84710 | 35,354.43724 | 2132.389781 | |

h_{r} | −50% | 35,418.99651 | 35,079.46804 | 1765.193270 |

−25% | 35,426.32632 | 35,085.74428 | 1776.671687 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 35,440.98594 | 35,098.29677 | 1799.628520 | |

+50% | 35,448.31575 | 35,104.57302 | 1811.106937 | |

I_{p} | −50% | 35,433.65390 | 35,091.89054 | 1697.209763 |

−25% | 35,433.65501 | 35,091.95553 | 1742.679933 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 35,433.65725 | 35,092.08552 | 1833.620272 | |

+50% | 35,433.65837 | 35,092.15051 | 1879.090442 | |

I_{e} | −50% | 36,371.40551 | 36,020.89415 | 2491.884636 |

−25% | 35,902.53082 | 35,556.45734 | 2140.017370 | |

0% | 35,433.65613 | 35,092.02053 | 1788.150103 | |

+25% | 34,964.78145 | 34,627.58372 | 1436.282838 | |

+50% | 34,495.90676 | 34,163.14691 | 1084.415571 | |

θ | −50% | 35,264.66951 | ||

−25% | 35,349.02050 | |||

0% | 35,433.65613 | |||

+25% | 35,518.56949 | |||

+50% | 35,603.77148 |

cost per item for product in an owned warehouse h_{o}, and the unit holding cost per item for product in a rented warehouse h_{r}.

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

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}, the unit holding cost per item for product in a rented warehouse h_{r}, the interest rate payable I_{p}, and the deterioration rate θ.

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

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, the unit purchasing price per item c, the unit holding cost per item for product in an owned warehouse h_{o}, and the interest rate payable I_{p}.

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

c) Negative & Minor: none.

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

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, [

From the basic EPQ model introduced by [

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

Positive & Major | A | A, h_{r}, I_{e} | A, I_{e} |

Positive & Minor | h_{r} | I_{p} | |

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

Negative & Major | s, c, I_{e} | s, h_{o}, I_{p} | s |

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

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

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

Negative & Minor | s, I_{e} | s, I_{e} | |

Negative & Major | s, I_{e} |

two-level trade credit and limited storage capacity, it’s all assumed that the raw materials required for production are timely. And [

After the sensitivity analyses, we reach the following conclusions in practical management:

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

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.

Even though adding the holding cost of raw materials increases the complexity of the model, but it approximates real-world situations and provides more precise decisions for practical business 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) Optimal Eco- nomic Production Quantity Policies Consi- dering the Holding Cost of Deteriorating Raw Materials under Two-Level Trade Credit and Limited Storage Capacity. Open Access Library Journal, 6: e5794. https://doi.org/10.4236/oalib.1105794