The values of safety factor optimization and coordination under random supply and demand.

Jian, Liu ; Weiming, Yi

ABSTRACT

The classic inventory management dominantly pays attention to the internal inventory control of business, and is neglectful of supply chain coordination and partnering. The local optimization approach results in impeded logistics, cost increase, lack of business and supply chain competitiveness. Supply chain coordination and partnering become significant means to improvement of supply chain performance, enhancement of business competitiveness. The paper introduces concept of effective inventory level, which is used to evaluate upstream shortage's impact on downstream inventory, models the inventory at warehouse and retailer under random lead time and demand, and makes the global optimization of safety factor to minimize channel inventory cost. It is shown that optimization and coordination of safety factor lead to inventory cost savings at two sites, especially under large lead time variability and stock value-adding rate. Meanwhile, authors present the coordination mechanism of global optimization of safety factor, i.e., cost-sharing contract, which makes both supplier and buyer benefiting from global optimization. Finally, this paper makes sensitivity analyses on value of global optimization.

INTRODUCTION

Inventories exist throughout the supply chain in various forms for various reasons. At any manufacturing point, distribution warehouses and retailers, they may exist as raw materials, work in progress, or finished goods. Ballou (1992) estimates that carrying these inventories can cost anywhere between 20 and 40% of their value per year. Inventory cost is a key factor that influences supply chain performance. Lee and Billington (1992) site several opportunities that exist in managing supply chain inventories. Among them are making coordinated decisions between the various echelons, incorporating sources of uncertainty, and designing proper supply chain performance measures. The central premise here is that the lowest inventories result when the entire supply chain is considered as a single system. Such coordinated decisions have produced spectacular results at Xerox (Stenross and Sweet, 1991), and at Hewlett Packard (Lee and Billington, 1995), which were able to reduce their respective inventory levels by over 25%.

Inventory systems are often subject to randomly changing environmental conditions that may affect the demand for the product, the supply, and the cost structure. The environment represents various important factors such as the randomly changing economic conditions, market conditions for new products or products that may be obsolete or any exogenous condition that may affect the demand as well as the supply and the cost parameters. Supply chain models in the literature that are related to uncertain environment mostly concentrate on the demand process, which may vary stochastically in a random environment. In most of the inventory models that involve uncertainties in the environment, the attention has been focused on the probabilistic modeling of the customer demand side. For example, large amount of literature have made qualitative and quantitative research on Bullwhip Effect of demand variability (Lee et al., 1997a, 1997b; Chen, 2000) and inventory collaboration under non-deterministic demand [Kefeng Xu and Yan Dong, 2000]. However, with economic globalization and intensification of competition, supply uncertainty increases apparently, which causes more and more important effect on supply chain performance. In previous literature, the uncertainty in the supply side has not received the amount of treatment it deserved. Up until the recent years supply uncertainty has received greater attention. In the literature, there is growing interest in models where an order that is placed may not be received due to uncertainty involved in the supply process. Parlar and Berkin (1991) propose an EOQ-type formulation where the supply is available or disrupted for random durations in the planning horizon. Parlar et al. (1995) consider a periodic review model with set-up costs using a Markovian supply availability structure in which the supply is either available or completely unavailable. They show the optimality of (S, s) policies where S depends on the supply state in the previous period. Gupta (1993), Parlar (1997) make extensions of the supplier reliability model incorporating random demand and multiple suppliers. Suleyman Ozekici and Mahmut Parla (1999) study infinite-horizon periodic review inventory models with unreliable suppliers where the demand, supply and cost parameters change with respect to a randomly changing environment. Bookbinder et al. (1999) probe into lead-time variability between successive stages in supply chain and effect of expedited orders on supply chain performance. Andersson and Marklund (2000) consider a model for decentralized inventory control in a two-level distribution system with one central warehouse and N non-identical retailers under random supply and demand. However, previous literatures assume that safety factor is given by the local optimization and have not studied safety factor's global optimization in supply chain. In addition, there is little literature on relationship between safety factor and supply uncertainty, demand uncertainty, inventory cost at different sites. Meanwhile, for making models tractable, some researches neglect variability in random delay caused by upstream shortages, and assume that shortage delay is constant. However, the approach impairs upstream shortage's influence on downstream inventory and reduces model effectiveness.

Our work differs from the previous ones in the sense that the effect of the upstream backorder viability on downstream inventory is considered and the global optimization of safety factor is made for reduction of supply chain inventory cost. Moreover, authors present cost-sharing contract, which ensures that both suppler and buyer benefit from the global optimization, and make sensitivity analyses on value of global optimization.

This paper is organized in the following way. In section 2, we pose all assumptions and notations. Section 3 describes two-level supply chain inventory model and section 4 approaches local and global optimization of safety factor. We put forward safety factor coordination mechanism in section 5. The numerical results are given in Section 6, which include model solution and sensitivity analyses. In section 7, we give some conclusions and point out some possible directions for future research.

MODEL ASSUMPTION AND NOTATION

The inventory system under consideration consists of one warehouse and one retailer. The replenishment lead-time for the warehouse and demand for the retailer are assumed to be normally distributed stochastic variables. Moreover, demands per period are independent. The transportation time between the warehouse and the retailer is assumed to be constant. When shortages occur at the warehouse, all demands from the retailers are fully backlogged and the backorders are filled according to a FIFO-policy. We consider a stationary base stock (order-up-to) control system in which each site reviews its inventory level at the beginning of each period and replenishes its inventory from the upstream site to bring the inventory level to the base stock level. For clarity in the remainder of this paper, we refer to the length of a review period as one period (R: =1), which is regarded as time unit. Let us introduce the following notation:

[mu] mean of demand during unit SW , SR order-up-to level at
 time for the and retailer warehouse retailer,
[sigma] standard deviation of BW backorders at warehouse,
 demand during unit time for hw, hr holding cost per
 the demand unit per time unit at
L lead time for warehouse with warehouse and retailer,
 mean [L.sub.0], standard pw, pr penalty cost per unit per
 deviation [[sigma].sub.L] time unit at warehouse
Y demand during lead time at and retailer,
 warehouse with Mean [Y.sub.0], ICW , ICR expected inventory
 standard deviation, cost at warehouse and
 [[sigma].sub.Y] retailer,
T transportation time from the TIC expected channel
 warehouse to the retailer, inventory cost.
DT demand during transportation SLW, SLR service levels at
 time for the retailer, warehouse and retailer.

MODEL DESCRIPTION AND ANALYSIS

Warehouse's Inventory Model

First of all, we analyze warehouse's demand process. Under stationary base stock policy, retailer's order per period is equal to downstream buyer's demand for it when the demands per period are independent and identically distributed (i.i.d.) random variables (Lee, 2000). Hence, demand process for warehouse is the same as one for retailer. Therefore, warehouse's demand during unit time is normally distributed with mean and variance 2. According to base stock policy, we obtain warehouse's expected inventory cost:

ICW = [h.sub.w] [[integral].sup.SW.sub.0] (SW - y)[f.sub.Y](y)dy + [P.sub.w] [[integral].sup.+[infinity].sub.SW](y - SW)[f.sub.Y](y)dy (1)

where fY (y) is probability density function for Y.

As demand during unit time for warehouse is normally distributed, demand Y during lead-time is normally distributed with mean [Y.sub.0], standard deviation [[sigma].sub.Y] :

[Y.sub.0] = [micro][L.sub.0], [[sigma].sub.Y] = [square root of [L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L]] (2)

Order-up-to level at warehouse is

SW = [micro][L.sub.0] + k [square root of [L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L]] (3)

where k denotes warehouse's safety factor.

Note that safety factor is the critical fraction of No stock-out. Safety factor k and service level (SLW) have the following relation:

k = [[PHI].sup.-1] (S L W), (4)

where service level (SLW) denotes probability of No stock-out and M(@) is standard normal cumulative distribution function. Substituting (3) into (1), we get (ref. Appendix):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where

G (k) = [[integral].sup.k.sub.-[infinity]] xf(x)dx < 0 (6)

f (x) is standard normal probability density function.

From expression (5), we know that warehouse's inventory cost increases in [micro], [sigma], [L.sub.0], [[sigma].sub.L], which means that warehouse can reduce inventory cost through partnering with external supplier to compress lead time or adding order frequency to decrease demand during unit time.

Retailer's Inventory Model

For exactly showing impact of shortages at warehouse on retailer inventory, we consider not only expected shortages but also shortages viability at warehouse. So we propose the heuristic concept, i.e., effective inventory level (EIL), which is defined as order-up-to level at retailer minus backorders at warehouse. EIL is expressed as

EIL = SR - [B.sub.W]. (7)

Due to warehouse shortages, demand during transportation time T is only covered by effective inventory level. So retailer's inventory cost is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [x.sub.+] = max(0, x), Z = [B.sub.W] + [D.sub.T], [f.sub.Z (z) is probability density function of random variable Z.

Note that demand [D.sub.T[ during transportation time is normally distributed. Moreover, generally speaking, [B.sub.W] is relatively small in comparison with [D.sub.Tz]. So we regarded Z as approximately normally distributed random variable with mean Z and standard deviation Z (ref. Appendix):

[[micro].sub.Z] = r(k)[square root of [L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L] + [T[micro]] (9)

[sigma].sub.Z] = [square root of U(k)[L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L] + [T[[sigma].sup.2]] (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

H (k) = [[integral].sup.k.sub.-[infinity]] [x.sup.2] [phi] (x)dx (12)

Retailer's order-up-to level is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Where l is safety factor at retailer. Substituting (13) into (8), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

From (14), we know that retailer's inventory cost increases in [micro], [sigma], [L.sub.0], [[sigma].sub.L]. Note that U(k) decreases in k. So retailer's inventory cost decreases in safety factor at warehouse. Therefore, inventory at retailer is affected by not only demand for itself but lead-time and safety factor at warehouse. Retailer's partnering with warehouse plays important role in reduction of its inventory cost.

POLICY OF SAFETY FACTOR OPTIMIZATION

Safety Factor Local Optimization

This section studies two kinds of safety factor optimization policy, i.e., local optimization and global optimization. We will compare the two policies to verify value of global optimization.

In the past, each business in supply chain makes the local optimization of safety factor to minimize its inventory cost, only attaches importance to its own inventory and doesn't pay attention to global performance in supply chain. Based on the local optimization policy, we use the first order condition for expression (5), (14) to get safety factors at warehouse and retailer under local optimization, denoted by [k.sup.+] , [l.sup.+], respectively:

[k.sup.*] = [[PHI].sup.-1][[p.sub.w/([p.sub.w] + [h.sub.w]) (15)

[l.sup.*] = [[PHI].sup.-1][[p.sub.r/([p.sub.r] + [h.sub.r]) (16)

From (4), (15), (16), service levels at warehouse and retailer are respectively:

[SLW.sup.*] = [p.sub.w]/([p.sub.w] + [h.sub.w]) (17)

[SLR.sup.*] = [p.sub.r]/([p.sub.r] + [h.sub.r]) (18)

Under local optimization policy, service level at each site increases in penalty cost and decreases in holding cost. Substituting (15) into (5) to get warehouse's inventory cost under local optimization:

[ICW.sup.*] = - G([k.sup.*])[h.sub.w]/1 - [PHI]([k.sup.*]) [square root of [L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L]] (19)

Substituting (16) into (14) to get retailer's inventory cost under local optimization:

[ICR.sup.*] = - G([l.sup.*])[h.sub.r]/1 - [PHI]([l.sup.*]) [square root of U[k.sub.*]([L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L]] + T[[sigma].sup.2 (19)

Channel inventory cost is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Because channel inventory cost [TIC.sup.+] is obtained by each site's local optimization of safety factor minimizing its inventory cost, in general, the optimal safety factors can't minimize channel inventory cost. This causes us to seek another safety factor optimization method, i.e., the supply-chain-oriented global optimization of safety factor.

Safety Factor Global Optimization

Since local optimization policy doesn't pay attention to supply chain coordination and partnering, it may result in high supply chain cost. With intensification of competition, competency of supply chain affects business' maintenance and development. Market competition has turned into one among supply chains. Improvement of supply performance becomes prevalent topic in business management. Therefore, it is necessary to study safety factor global optimization and coordination. The global optimization of safety factor is determination of suitable safety factor to minimize supply chain inventory cost, which is an effective approach to improvement of performance of entire supply chain.

From (5) and (14), we get supply chain inventory cost:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

We regard the warehouse and the retailer as a system. This system aims at minimizing system cost on condition that system service level, i.e., retailer's service level is ensured.

For convenience, we assume that system service levels toward external customers under local and global optimization policies are the same. So retailer's service level and safety factor under global optimization are respectively

[SLR.sup.*] = [SLR.sup.*] = [P.sub.r]/([P.sub.r] + [h.sub.r]), (23)

[l.sup.**] = [l.sup.*] = [[PHI].sup.-1]/[[P.sub.r]([P.sub.r] + [h.sub.r]). (24)

We want to minimize supply chain inventory cost, given system service level SLR. According to the first order condition, we have

[partial derivative]/[partial derivative]k (TIC) = 0. (25)

Subsisting l++ into (25), we get following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Solution of above equation, denoted by [k.sup.++], is optimal safety factor that minimizes channel inventory cost. However, we cannot get analytic expression of optimal safety factor [k.sup.++] from equation (26). Therefore, we only find numerical solution of the equation when other parameters' values are given. In section of number study, we will calculate values of optimal safety factor through computer program, and make analysis on relation between optimal safety factor and other parameters. Based on equation (26), channel inventory cost can be reduced through global optimization of safety factor when system service level doesn't change.

Note that G([l.sup.++])<0. From (26), we get:

[PHI]([k.sup.**]) > [p.sub.w]/([h.sub.w] + [p.sub.w]) = [PHI(k*).

so [k.sup.++] > [k.sup.+]. Therefore, Heightening safety factor at warehouse contributes to reduction of supply chain inventory cost while system service level is preserved.

SAFETY FACTOR COORDINATION MECHANISM

The global optimization of safety factor requires warehouse's heightening safety factor, which lead to increase of warehouse's inventory cost. For urging warehouse to participate in global optimization, retailer must make compensation for warehouse's cost increase. In other word, warehouse and retailer must make a cost-sharing contract so that both parties benefit from global optimization (Cachon et al., 2000; Moses et al, 2000). We assume that channel inventory cost is shared by fraction a, called as sharing factor. After sharing, inventory costs that warehouse and retailer really bear, denoted by [RICW.sup.++], [RICR.sup.++] respectively, are

[RICW.sup.**] = [aTIC.sup.**] (28)

[RICR.sup.**] = (1-a)[TIC.sup.**] (29)

For making both parties to participate in global optimization, sharing factor a must satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

From (28), (29), (30), we get sharing factor 's feasible interval:

(1 - [ICR.sup.*]/[TIC.sup.**], [ICW.sup.*]/[TIC.sup.**]) (31)

Sharing factor's feasible interval length is

d = [TIC.sup.*] - [TIC.sup.**]/[TIC.sup.**]) (32)

The expression (32) shows that sharing factor's feasible interval length increases in savings of channel inventory cost under global optimization. Therefore, the larger cost savings under global optimization is, cost-sharing contract has more choices.

The compensation that retailer pays warehouse is

t = [RICH.sup.**] - [ICR.sup.**] = (1-a)[TIC.sup.**] - [ICR.sup.**] (33)

After sharing, savings of inventory cost at warehouse and retailer are respectively

[DELTA]ICW = [ICW.sup.*] - [RICW.sup.**] = [ICW.sup.*] - [aTIC.sup.**] (34)

[DELTA]ICR = [ICR.sup.*] - [RICR.sup.**] = [ICR.sup.*] - (1-a)[TIC.sup.**] (35)

Based on cost-sharing contract, global optimization benefits both warehouse and retailer. So cost-sharing contract ensures that global optimization becomes a practicable means to reduction of inventory cost.

NUMERICAL STUDY

Value of Safety Factor Optimization

Through numerical study, this section reports the results that evaluate value of global optimization and its sensibility to related parameters.

Let

[micro]=12, [sigma]=3, [L.sub.0]=16, [[sigma].sub.L]=4, [h.sub.w]=1, [h.sub.r]=2, [p.sub.w]=3, [p.sub.r]=8, T=4. (36)

We use above model to get the results in table 1.

From table 1, warehouse's order-up-to level increases and retailer's one reduces after global optimization. In particular, channel inventory cost reduces through global optimization. The declining-rate of channel inventory cost is:

r = [TIC.sup.*] - [TIC.sup.**]/[TIC.sup.*] = 9.5% (37)

From (34), (35), savings of inventory cost at warehouse and retailer are: [DELTA]ICW = 5.6 [], [DELTA]ICR = 5.4. Inventory costs at warehouse and retailer reduce by 8.9% and 10.3%, respectively. This shows that the global optimization based on cost-sharing contract can reduce inventory cost at warehouse and retailer.

Effect of Lead-time and Demand Uncertainties

We set [sigma] = 1, 3; [[sigma].sub.L] = 2, 4, 6. Values of other parameters are the same as ones in expression (36). Note that safety factors at retailer and locally optimized safety factor at warehouse aren't affected by [sigma], [[sigma].sub.L]. So [1.sup.++] = [1.sup.+] = 0.84, [k.sup.+]=0.67. According to above model, we get the computational results in table 2. From table 2, declining-rate r of chain inventory cost increases in [[sigma].sub.L] and decreases in [sigma]. This means that the larger lead-time uncertainty is and the smaller demand uncertainty is, the larger values of safety factor's global optimization is. Moreover, the global optimization still has superiority over the local one under small demand uncertainty. For example, when [sigma] = 3, [[sigma].sub.L] = 2, channel inventory cost reduces by 6.9%, which is advantageous to improvement of business performance and competitiveness.

In addition, effect of lead-time variability on order-up-to level at warehouse and retailer is larger than that of demand variability. Therefore, control of lead-time variability is especially important to reduction channel inventory cost. This shows that warehouse must partner with its external supplier to reduce lead-time uncertainty.

Effect of Warehouse's Cost Parameters

Let [h.sub.w] = 0.5, 1; [p.sub.w] = 1, 3, 5. Similarly, values of other parameters are assumed to be the same as ones in (36). Safety factors at retailer are: [l.sup.++] = [l.sup.+] = 0.84. The computational results corresponding to different cost parameters at warehouse are shown in table 3.

From table 3, declining-rate r of channel inventory cost decreases in holding cost per unit [h.sub.w] and shortage penalty cost per unit [p.sub.w] at warehouse. Therefore, when cost parameters are fixed, the smaller cost parameters at warehouse are, the more effective global optimization is. In addition, order-up-to level at warehouse decreases in holding cost per unit at warehouse and increases in shortage penalty cost per unit at warehouse, whereas order-up-to level at retailer increases in holding cost per unit at warehouse and decreases in shortage penalty cost per unit at warehouse. This indicates that effect of cost parameters on order-up-to level at warehouse is contrary to that on order-up-to level at retailer.

Effect of Retailer's Cost Parameters

Let [h.sub.r] = 2, 3; [p.sub.r] = 3, 5, 7. Values of other parameters are the same as ones in expression (36). Since cost parameters, lead-time and demand standard deviation at warehouse are fixed, safety factor and order-up-to level under global optimization are: [k.sub.+] = 0.67, [SW.sup.+] =225. The computational results corresponding to different cost parameters at retailer are shown in table 4.

From table 4, declining-rate of channel inventory cost increases in holding cost per unit hr and shortage penalty cost per unit pr at retailer, which is contrary to that at warehouse. Hence, the larger ratio of retailer's cost parameters to warehouse's ones is, the larger value global optimization has. Note that cost parameters generally increase in value of stock. So global optimization is especially suitable to supply chain down that stock value-adding rate is comparative large. As postponement policy is increasingly applied to supply chain management practice, customizing activities are more and more close to end customers (Hoek, 1997). The distribution system will undertake more final processing and manufacturing activities, which make stock value-adding rate increasing. Therefore, under postponement policy, global optimization of safety factor is extremely contributive to reduction of distribution chain inventory cost. In addition, cost parameters at retailer affect order-up-to level. The larger cost parameters at retailer are, the higher order-up-to level at warehouse is. This means that warehouse not only want to control its cost parameters but also devote its energies to reducing cost parameters at retailer through the parties partnering.

CONCLUSION AND FUTURE RESEARCH

This study is mainly focused on values of safety factor optimization and coordination in two-level supply chain. We introduce definition of effective inventory level to incorporate upstream shortage's impact on downstream inventory into two-stage inventory model. Through modeling warehouse and retailer inventory, authors analyze relation between warehouse and retailer. From model analyses, it is known that safety factor, lead-time mean and variability affect inventory at retailer while demand at retailer impact inventory at warehouse. This reveals that supply chain partnering and coordination are key means to reduction of both parties inventory cost. In traditional inventory policy, determination of safety factor is business-oriented local optimization, which doesn't pay attention to supply chain coordination to rest in high channel inventory cost. Therefore, we develop the global optimization approach to selecting safety factor. The optimal safety factor can be obtained by solving the equation (26) through Mathematica program. In comparison with the local optimization, the global optimization has apparent advantages in reduction of channel inventory cost, especially under large supply uncertainty or high stock value-adding rate in supply chain. For realization of global optimization, we propose the cost-shaving contract to ensure that both warehouse and retailer benefit from global optimization. Length of feasible interval of sharing factor increases in declining-rate of channel inventory cost. This means that the more effective global optimization is, the more choices cost-sharing contract has. In a word, Safety factor optimization and coordination will be effective in reducing inventory cost for the system of buyer-supplier channel as a whole, even without changing any cost characteristics of the channel or service level at the end market. Global optimization provides business manager with more opportunities for improvement of business performance. Nowadays, with intense global competition and quick change of economic environment, supply and demand uncertainty increases rapidly. Therefore, safety factor coordination will play important role in supply chain inventory control. Authors believe that it will receive surely more and more attention.

Just like most research, our analysis has limits. For instance, this paper assumes that demand and lead-time are stationary. In future, we will relax this assumption and study safety factor coordination under non-stationary demand and lead-time. In addition, we will extend two-level series supply chain to multi-level network supply chain, and further explore value of safety factor optimization and coordination.

APPENDIX

Proof of Expression (5)

Proof. Let

Y - [micro][L.sub.0]/[square root of [L.sub.0][[sigma].sup.2] + [[micro].sup.2][[sigma].sup.2.sub.L]] (A.1)

Apparently, X follow standard normal distribution. Substituting (3) into (1) to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.2)

Proof of Expression (9) and (10)

Proof. Note that

[B.sub.w] = max(0, y - SW) (A.3)

From (1), (A.2), we readily get expected backorder at warehouse:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.4)

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.5)

From (A.3), we have

E([B.sup.2.sub.w]) = [[integral].sup.+[infinity].sub.SW][(y- SW).sup.2][f.sub.Y](y)dy (A.6)

Similarly, substituting (3) into (A6) to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.7)

Variance of BW is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.8)

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.9)

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Liu Jian

Huazhong University of Science & Technology, P.R. China

Yi Weiming

Huazhong University of Science & Technology, P.R. China

Table 1: Value of Safety Factor Optimization

Local [k.sup.+] [l.sup.+] [SW.sup.+]
optimization
 0.67 0.84 225

Global [k.sup.++] [l.sup.++] [SW.sup.++]
optimization
 1.22 0.84 252

Cost sharing a's feasible interval
 (0.50,0.61)

Local [SR.sup.+] [ICW.sup.+] [ICR.sup.+]
optimization
 72 62.9 52.3

Global [SR.sup.++] [ICW.sup.++] [ICR.sup.++]
optimization
 62 71.0 33.2

Cost sharing a t [RICW.sup.++]
 0.55 13.7 57.3

Local [TIC.sup.+]
optimization
 115.2

Global [TIC.sup.++]
optimization
 104.2

Cost sharing [RICR.sup.++]

 46.9

Notion:

(a.) authors assume that a is equal to 0.55, which depend on
negotiation between supplier and buyer in practice.

(b.) [k.sup.++] is numerical solution of the equation(26), which is
obtained through Mathematica program.

Table 2: Effect of Lead-time and Demand Uncertainties

[sigma] [[sigma].sub.L] [k.sup.++] [SW.sup.+] [SR.sup.+]

 1 2 1.26 208 62
 4 1.29 224 71
 6 1.30 240 81
 3 2 1.14 210 63
 4 1.22 225 72
 6 1.26 241 82

[sigma] [[sigma].sub.L] [SW.sup.++] [SR.sup.++] [TIC.sup.+]

 1 2 223 57 55.9
 4 254 61 109.8
 6 286 65 164.1
 3 2 223 58 65.8
 4 252 62 115.2
 6 284 66 167.8

[sigma] [[sigma].sub.L] [TIC.sup.++] [DELTA] TIC r (%)

 1 2 50.0 5.9 10.6
 4 97.4 12.4 11.4
 6 145.3 18.8 11.5
 3 2 61.2 4.6 6.9
 4 104.2 11.0 9.5
 6 150.1 17.7 10.5

Table 3: Effect of Warehouse's Cost parameters

[h.sub.w] [p.sub.w] [k.sup.+] [k.sup.++] [SW.sup.++]

 0.5 1 0.43 1.50 213
 3 1.07 1.64 245
 5 1.34 1.74 258
 1 1 0.00 1.00 192
 3 0.67 1.22 225
 5 0.97 1.36 240

[h.sub.w] [p.sub.w] [SR.sup.+] [SW.sup.++] [SR.sup.++]

 0.5 1 78 266 59
 3 64 273 58
 5 60 278 57
 1 1 93 242 65
 3 72 252 62
 5 65 259 60

[h.sub.w] [p.sub.w] [TIC.sup.+] [TIC.sup.++] [DELTA] TIC

 0.5 1 89.7 65.8 23.9
 3 76.7 68.2 8.5
 5 74.6 70.1 4.5
 1 1 122.0 97.6 24.4
 3 115.2 104.2 11.0
 5 115.0 108.7 6.3

[h.sub.w] [p.sub.w] r (%)

 0.5 1 26.6
 3 11.1
 5 6.0
 1 1 20.0
 3 9.5
 5 5.5

Table 4: Effect of Retailer's Cost Parameters

[h.sub.w] [p.sub.w] [k.sup.++] [l.sup.++] [SR.sup.+]

 2 3 1.08 0.25 60
 5 1.16 0.57 67
 7 1.21 0.77 70
 3 3 1.16 0.00 55
 5 1.26 0.32 62
 7 1.32 0.52 66

[h.sub.w] [p.sub.w] [SW.sup.++] [SR.sup.++] [TIC.sup.+]

 2 3 245 55 98.9
 5 249 59 107.5
 7 252 61 113.2
 3 3 249 51 107.7
 5 254 54 119.7
 7 257 56 127.7

[h.sub.w] [p.sub.w] [TIC.sup.++] [DELTA] TIC r (%)

 2 3 93.3 5.6 5.7
 5 99.2 8.3 7.7
 7 102.9 10.3 9.1
 3 3 99.3 8.4 7.8
 5 106.9 12.8 10.7
 7 111.8 15.9 12.5