以不同的績效評估標準來決定通路分配系統之最佳存貨與價格決策

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行政院國家科學委員會專題研究計畫成果報告

計畫類別：個別型計畫

計畫編號： NSC91-2213-E-011-118-

執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日執行單位：國立臺灣科技大學資訊管理系

計畫主持人：陳正綱

報告類型：精簡報告

處理方式：本計畫可公開查詢

中華民國 92 年 11 月 24 日

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行政院國家科學委員會補助專題研究計畫 ■ 成果報告

□期中進度報告

（計畫名稱）

以不同的績效評估標準來決定通路分配系統之最佳存貨與價格決策 Optimal Inventory and Pricing Policies for a Channel Distribution

System under Various Performance Criteria

計畫類別：■ 個別型計畫 □ 整合型計畫計畫編號：NSC 91－2213－E－011－118

執行期間： 91 年 8 月 1 日至 92 年 7 月 31 日

計畫主持人：陳正綱共同主持人：

計畫參與人員：林克維、張寶文、王玳琪

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執行單位：國立台灣科技大學資訊管理學系

中華民國九十二年十一月二十四日

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行政院國家科學委員會專題研究計畫成果報告

以不同的績效評估標準來決定通路分配系統之最佳存貨與價格決策 Optimal Inventory and Pricing Policies for a Channel Distribution

System under Various Performance Criteria

計畫編號：NSC 91－2213－E－011－118 執行期限：91 年 8 月 1 日至 92 年 7 月 31 日

主持人：陳正綱執行機構及單位名稱：國立台灣科技大學資訊管理學系計畫參與人員：林克維、張寶文、王玳琪

1. Introduction 中文摘要

In this paper, we propose an inventory model under return on inventory investment (ROII) maximization for an intermediate firm to determine: (1) the quality level, (2) the selling quantity, and (3) the purchasing price of a product. The selling price and the supply rate of the product as well as the fixed selling setup cost are assumed to be power functions of one or more of these three decision variables. Under these assumptions, the geometric programming techniques are effectively utilized to derive the global optimal closed-form solution for the proposed inventory model. In addition, interesting sensitivity analyses on primal and dual geometric programming problems for the inventory model are presented.

本研究計劃的目的，是考慮在不同的績效指標（如：成本、利潤、投資報酬或剩餘所得）下，為通路分配系統中不同的成員(如：製造商與零售商)，透過數學模式的建立，分別來推演並決定其最佳存貨與價格決策。從推演的過程與所得的結果中，我們除了希望能提出一些可供管理者遵循的法則外，也希望能對一般管理經濟學上的認知，做一印證或比較。

就通路分配系統中的存貨與價格管理決策來看，本研究計劃的重要性在於，拓展原本對該領域的認識。不僅以生產與存貨系統來思考，更考慮在通路分配系統中，通路成員以不同的績效評估標準，來決定存貨與價格策略，以期建立更符合現實環境的模式。

In Chen and Min (1991), an intermediate firm is defined to be “an economic agent that purchases products from numerous independent producers and sells those purchased products to other firms that process or utilize the products (or to the public) at a given market price”. For the definition of an intermediate firm in this paper, we will follow the one shown in Chen and Min (1991) except that the market (selling) price of the product is assumed to be a function of the selling quantity and the quality level rather than a given parameter.

關鍵詞：存貨理論、價格決策、績效指標、

市場通路。

Abstract

The objective of this proposal is to devise and develop the optimal inventory and pricing policies in a channel distribution system under various performance measurement criteria (e.g., cost, profit, rate of return, or residual income).

From the development of our models and the results of this project, several decision-making rules, managerial insights, and economic implications are expected to be obtained.

In practice, the intermediate firms described in this paper can be easily found in numerous industries. For example, the personal computer assembly firms purchase electronic components from independent producers. In addition, Chen and Min (1991) presents two examples of Keywords: Channel Distribution, Inventory,

Pricing, Game Theory, Performance Measurement.

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intermediate firms in the agricultural industry as well as the garment and apparel industry.

Traditionally, cost minimization (or profit maximization) is widely utilized as the performance measure of inventory models (see e.g., Silver et al. (1998), Smith (1958), or Ladany and Sternlieb (1974)). In Schroeder and Krishnan (1976), an inventory model is proposed under an alternative performance measure of return on investment (ROI) maximization. Rosenberg (1991) considers the logarithmic concave demand functions and examines the inventory models under profit maximization vs. return on inventory investment maximization. More recently, Toshitsugu et al. (1999) constructs and analyzes inventory and investment in setup operations policies under return on investment maximization. We note that the last three ROII-related papers deal with the inventory models from the perspectives of a retailer or a producer. In contrast to the perspectives of a retailer or a producer, the proposed inventory model in this paper is constructed and analyzed from the perspectives of an intermediate firm.

In this paper, by return on inventory investment, we mean the ratio of profit to the average inventory investment. In the expression of the profit, the only positive term is the revenue that results from selling those purchased products to the public (or to other firms). The negative terms (i.e., relevant costs) in the profit expression contains: (1) a fixed selling setup cost to sell those purchased products, (2) the variable purchasing costs of those purchased products, and (3) the inventory holding costs in carrying those purchased products. The average inventory investment is equal to the average inventory level multiplied by unit-purchasing price that an intermediate firm pays to independent producers.

We note that the quality level of the product is a critical decision variable in the proposed model. However, in the literature of production/operations management, there have been several ways to define the term of

“quality”. Porteus (1986) defines “quality”

as the probability that the production process is becoming out of control and discuss the issue of quality improvement. In Cheng

(1991), the term of “quality” means the percentage of a production lot that is acceptable. Min and Oren (1995) as well as Soesilo and Min (1996) define “quality” as one type of product characteristics which can be expressed in terms of a numerical measurement such as the speed in MHz of CPU. The meaning of “quality” utilized in this paper is the same as the ones shown in Min and Oren (1995) as well as Soesilo and Min (1996). We note that, from the perspectives of an intermediate firm, this type of quality will affect the cost and price structures of the product.

For this type of quality, we assume that the supply rate, the selling price as well as the fixed selling setup cost are affected by the quality level of the product. In this paper, the relations of the quality, price and cost of the product are expressed by power functions. In such a case, the geometric programming techniques can be directly applied to solve the proposed problem.

The geometric programming is an efficient and effective method to solve nonlinear programming problems with the terms in power functional form in the objective function and constraints (Duffin et al. (1967)). In the literature of

production/operations management, applications of geometric programming

techniques were well documented. Cheng (1989a) proposed an economic order quantity model with demand-dependent unit cost and derive the optimal solution by employing geometric programming techniques. In Cheng (1989b), the geometric programming techniques were applied to solve an economic production quantity model with flexibility and reliability considerations.

Lee (1993) utilized geometric programming techniques to determine the selling price and order quantity for a retailer. More recently, Hariri and Abou-El-Ata (1997) as well as Abou-El-Ata and Kotb (1997) applied geometric programming techniques to investigate multi-item economic order quantity models under different cost assumptions and environment restrictions.

In this paper, due to the nature of the proposed model in this paper, the global optimal closed-form solution can be obtained

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by utilizing geometric programming techniques

The rest of this paper is organized as follows. First, we mathematically characterize the model environments for the intermediate firms. Next, we formulate the problem in the primal and dual forms of geometric programming. Then, the global optimal closed-form solution for the problem is derived by way of geometric programming techniques. Finally, sensitivity analyses are presented and concluding remarks are made.

2. Model Environments of Intermediate Firms

The unit-purchasing price that an intermediate firm pays to independent producers of the product is denoted by r.

The quality level of the product is given by q.

The supply rate of the product, S, from the independent producers to the intermediate firm, is assumed to be a function of the unit price r and the quality level q of the product.

Specifically,

S = S q r( , ) =bq r⁻^{φ θ}, where b is the scaling parameter for S, b > 0;

φ is the supply rate elasticity with respect to the quality level, i.e.,

/ , /

q S

q S ∂

φ = ∂ φ > 0,

and θ is the supply rate elasticity with respect to the purchasing price of the product, i.e.,

/ , / r S

r S ∂

θ = ∂ θ > 0. We note that the supply rate of the product is a decreasing power function of the quality level and an increasing power function of the purchasing price. Hence, if the quality level of the product increases, the corresponding supply rate per unit time decreases. On the other hand, if the purchasing price of the product increases, the corresponding supply rate per unit time increases.

The purchased products are stored in the warehouse at a cost of h per unit per unit time. Since the performance criterion in this paper is return on inventory investment, the inventory holding cost per unit per unit time h does not include any opportunity cost (Rosenberg (1991)). Once an amount of Q

units accumulates, all Q units are sold to another firm that processes or utilizes the products (or sell them to the public) at a selling price of P per unit. Moreover, we assume that the selling price P is a function of the selling quantity Q and the quality level q. Specifically,

P= P Q q( , )=aQ q⁻^{α β}, where a is the scaling parameter for P, a > 0;

α is the price elasticity with respect to the selling quantity, i.e.,

/ , /

Q P

Q P ∂

α = ∂ α > 0,

and β is the price elasticity with respect to the quality level of the product, i.e.,

/ , / q P

q P ∂

β = ∂ β > 0 .

We note that the price per unit is a decreasing power function of the selling quantity and an increasing power function of the quality level. Hence, if the selling quantity of the product increases, the corresponding market price per unit decreases. On the other hand, if the quality level of the product increases, the corresponding market price per unit increases.

The fixed setup cost incurred by the intermediate firm in selling the accumulated products is represented by K and is assumed to be a function of the quality level of the product (Soesilo and Min (1996)).

Specifically,

K =K q( )=cq^η,

where c is the scaling parameter for K, c > 0;

η is the fixed selling cost elasticity with respect to the quality level of the product, i.e.,

/ , /

q K

q K ∂

η= ∂ η> 0 .

We note that the fixed selling setup cost is an increasing power function of the quality level of the product. Hence, if the quality level of the product increases, the corresponding fixed selling cost increases.

3. Model Formulation and Derivation of the Optimal Solution

Under the model environments

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described in section 2, the objective of the intermediate firm is to maximize the return on inventory investment (ROII), where the three decision variables are the selling quantity Q, the quality level q, and the purchasing price r. Before introducing the return on inventory investment maximization formulation, we first formulate the profit per unit time for the intermediate firm as follows.

Profit = revenue - (fixed selling setup cost + purchasing cost + inventory holding cost).

The revenue, fixed selling setup cost, purchasing cost and inventory holding cost per unit time are given by PS, KS

Q , rS and 2

hQ , respectively. Therefore, the mathematical formulation of the profit per unit time for the intermediate firm is

2 )

( hQ

Q rS KS

PS− + +

π = (1)

hQ

r q bcQ r

bq r q abQ

5 . 0

1 1

−

= ⁻^α ^β⁻^φ ^θ ⁻^φ ^θ⁺ ⁻ ^η⁻^φ ^θ

(1) We can easily find out that there is no

negative term in the objective function and constraints in formulation (4). Hence, formulation (4) is a posynomial problem.

There are five terms (i.e.,

, ,

, , and

) and four variables (i.e., z, Q, q, and r) in formulation (4). Therefore, the degree of difficulty is equal to zero (= 5 – 4 – 1).

From the perspectives of geometric programming, a posynomial problem with zero degree of difficulty is guaranteed to have a global optimal solution. In order to obtain the optimal solution for (4), we first construct the corresponding dual formulation as follows.

θ β φ

α+ − −

−

−1 1 1 1

5 .

0 za b Q q r

β η α− −

− cQ q

a ¹ ¹ 0.5a⁻¹b z⁻¹

r q Q a⁻¹ ^α ⁻^β

θ β φ− − +¹q r

α

−¹hQ

where ,

and .

S =S q r( , ) =bq r⁻^{φ θ} cqη

=

β αq aQ P= ⁻ K

The return on inventory investment is equal to the profit per unit time divided by the average inventory investment, which is equal to

2

rQ. Therefore, the formulation of the objective function is

2 )

, (

, ROII rQ

MAX

r q Q

= π

1 2

1 1

2 2

−

=

hr r

q bcQ

r q abQ

θ φ η

θ φ β α

(2)

1

2 1

−

−q r

bQ ^φ ^θ

We note that, from the perspectives of geometric programming, the above objective function (2) is the primal formulation, which is an unconstrained signomial problem with zero degree of difficulty. Since the geometric programming fails to guarantee the global optimal solution for signomial problems, we take the following two steps to convert formulation (2) into a posynomial

problem (Duffin et al. (1967), Lee (1993), and Soesilo and Min (1996)). First, we transform problem (2) into the following equivalent formulation.

z MAXQ,q,r,z

subject to :

(3) z hr r

q bcQ r

q bQ

r q abQ

≥

−

1 1 2

1

1 1

2 2

2

θ φ η θ

φ θ φ β α

where z, Q, q, and r are decision variables.

However, (3) is still a signomial problem. Second, by inverting the objective function z and performing some manipulations on the constraint in (2), we can have the following equivalent formulation.

min z⁻¹ subject to :

β η α

β α θ

β φ α

−

− +

−

− + +

q cQ a

r q Q a r q Q b za

1 1

5 1

. 0

(4) 1

5 .

0 ¹ ¹ ¹ ≤

+ a⁻b⁻ hQ^α⁺ q^φ⁻^βr⁻^θ

MAX d(ω) =

4 3

2

1 0

5 ) . (0 ) ( ) (

5 ) . (0 1 ) (

4 1 1

3 1

2 1

1 1 1

0

ω ω

ω

ω ω

ω λ ω

λ ω

λ

ω λ ω

h b a c

a a

b a

−

subject to : ω₀ = 1

−ω₀+ω₁ =0

0 ) 1 ( ) 1 ( )

1

(α + ω₁ +αω₂ + α − ω₃ + α + ω₄ =

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0 ) ( ) ( )

(φ −β ω₁ −βω₂ + η−β ω₃ + φ −β ω₄ = 0

) 1

( −θ ω₁ +ω₂ −θω₄ = λ ω ω= ₁+ ₂ +ω₃+ω₄

where ω = )(ω₀,ω₁,ω₂,ω₃,ω₄ and λ and ω are positive variables.

The first constraint represents the normality condition, and the second, third, fourth and fifth constraints represent the orthogonality conditions of a geometric programming dual problem. Since there are six variables in a set of six linear equations in the dual formulation (4), the corresponding solution can be easily obtained. Namely, six variables in a set of six linear equations in the dual formulation (4), the corresponding solution can be easily obtained. Namely, ω₀ = 1

ω₁ = 1

φ η αη αθη βθ

αφ β

αφ β φ η ω αη

−

− + +

−

− +

= + 2

2

αφ β

β ω αφ

−

− + +

= +

3 2 (6) m= 2β +αφ +βθ −αθη−αη−η−φ

αφ β

β βθ αφ η φ ω αθη

−

− + +

−

− +

= +

4 2

αφ β

η λ φ

−

− + +

= + 2

We note that ω₀,ω₁,ω₂,ω₃??and??ω₄must be all positive so as to guarantee the feasibility of the dual variables. From the dual feasible solution (ω,λ), the following weights can be defined:

δ ω λ

j

= j , j = 1, 2, 3 and 4. (7) _*

where ∑δ ^and

.

j j=

=

1 4

1 ω_j λ

j=

∑ ⁼

1 4

According to Beightler and Phillips (1976), these weights represent the proportions of the profit (δ₁), the variable purchasing cost (δ₂), the fixed selling cost (δ₃), and the inventory holding cost (δ₄ ) to the total revenue.

Hence, the following expressions for these weights can be obtained.

θ β φ

δ₁ =0.5za⁻¹b⁻¹Qα⁺¹q ⁻ r¹⁻ r

q Q

a ^α ^β

δ₂ = ⁻¹ ⁻ δ₃ = a cQ⁻¹ ^α⁻¹q^η⁻^β

θ β φ

δ₄ =0.5a⁻¹b⁻¹hQα⁺¹q ⁻ r⁻

Furthermore, by solving (8), we can obtain the corresponding primal solution , , and

Q* q^* r in closed-form as follows. *

h m

c b Q a

1

4 3

2

* 2











= ₋ ₋ ₋ ₋

−

− +

−

η β βθ β φ ηθ βθ

β η φ βθ β η β θη η φ η β

δ δ

δ ⁽⁹⁾

( )

h m

c b q a

1

1 4 1 3 1 2

1 1 1 2 1

* 2











= ₋ ₋₋ ₋ ₋

− + +

−

α αθ α α

θ

α αθ α α θ α

δ δ

δ (10)

h m

c b r a

1

4 3

2 2

* 2











= ₊ ₋ ₋ ₋ ₋ ₋ ₋

− +

−

β αη β αφ φ η αη αφ β

αη β β αφ β αη φ η β αη

δ δ

δ ⁽¹¹⁾

where

The optimal dual objective function value can be obtained by substituting the optimal dual variables , j

= 1, 2, 3, 4, and (as shown in (6)) into the objective function of the dual formulation (5). In addition, the corresponding optimal weights can be computed from (7).

*

*)

( d

d ω =

*

ωj

λ*

On the other hand, the optimal return on inventory investment can be obtained by substituting optimal primal decision variables , , and

ROII*

*

Q* q r (shown ^* from (9)-(11)) into the objective function of the primal formulation (2). Also, there exists another way to obtain the optimal return on inventory investment, . According to the duality theorem of the geometric programming,

ROII

*

* 1

= z

*

*) ( d ω

*

=d ,

where is the optimal objective function value of (2). Since is the optimal return on inventory investment, is simply equal to

z*

z

*

1

d (i.e.,

*

* 1

ROII =z = d ).

ROII

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We illustrate some of the features of the proposed model by the following numerical example.

Example 1: The values of parameters are given as follows α =0.5, β=2.1, φ=2.9,

θ =2.5, η =2.7, a=100, b=300, c=1300, and h=3.

By substituting the values of the relevant parameters into (5), the optimal dual variables can be obtained as follows: = 1,

= 1, = 2.26087, = 6.17391, = 0.30435, and = 9.73913. In addition, from (6), the corresponding optimal weights are = 0.10268, = 0.23214, = 0.63393, and = 0.03125.

*

ω0

δ

*

ω1 ω^*₂ ω₃^*

* 2

*

ω4

* 3

λ*

*

δ4

*

δ1 δ

From (8)-(10), the corresponding primal optimal variables can be obtained as follows:

= 4706.61680, = 7.48317, and

Q* q^* r = ^*

23.17290. Hence, the corresponding selling price per unit P = 99.82172, the supply rate ^* per unit time = 2263.20947, and the setup cost

S*

K = 297833.96875. In addition, we *

note that the cycle time (i.e., _*

*

S Q π*

) is given by 2.07962 and the profit level is given by 23196.87109. Finally, the optimal return on inventory investment is calculated to be 0.425373 by substituting ,

, and

ROII*

Q*

q* r into the objective function of the ^* primal formulation (1).

4. Conclusion

In this paper, we proposed an inventory model under return on inventory investment maximization to determine the quality level, selling quantity, and purchasing price for an intermediate firm. We formulated the inventory problem and converted it into a posynomial geometric programming problem.

By applying the geometric programming techniques, the global optimal closed-form solution was obtained. Also, the sensitivity analyses were presented.

One possible extension of the proposed inventory model in this paper is to consider the marketing effects such as advertising on the optimal decisions of the intermediate firm.

Another possible extension is to incorporate more sophisticated features into the model such as supply shortages, multiple products, and capacity constraints.

References

Abou-El-Ata, M. O., and Kotb, K. A. M., 1997, Multi-item EOQ inventory model with varying holding cost under two restrictions: a geometric programming approach, Production Planning and Control, 8, 608-611.

Beightler, C. S., and Phillips, D. T., 1976, Applied Geometric Programming (New York : Wiley).

Chen, C. K., and Min, K. J., 1991, Optimal selling quantity and purchasing price for intermediary firms, International Journal of Operations and Production Management, 11, 64-68.

Cheng, T. C. E., 1989a, An economic production quantity model with demand-dependent unit cost, European Journal of Operational Research, 40, 252-256.

Cheng, T. C. E., 1989b, An economic production quantity model with flexibility and reliability considerations, European Journal of Operational Research, 39, 174-179.

Cheng, T. C. E., 1991, An economic order quantity model with demand-dependent unit production cost and imperfect production processes, IIE Transactions, 23, 23-28.

Duffin, R. J., Peterson, E. L., and Zener, C., 1967, Geometric Programming-Theory and Application (New York: Wiley).

Hariri, A. M. A., and Abou-El-Ata, M. O., 1997, Multi-item production lot-size inventory model with varying order cost under a restriction: a geometric programming approach, Production Planning and Control, 8, 179-182.

Ladany, S. and Sternlieb, A., 1974, The interaction of economic ordering quantities and marketing policies, AIIE Transactions, 6, 35-40.

Lee, W. J., 1993, Determining order quantity and selling price by geometric programming: optimal solution, bounds, and sensitivity, Decision Sciences, 24,

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76-87. 研究計劃成果部分：

Min, K. J., and Oren, S. S., 1995, Economic determination of spec. levels and allocation priorities of semiconductor products, IIE Transactions, 27, 321-331.

●提出了一些可供管理決策者遵循的法則。

●已將計劃研究成果與一般管理經濟學上的認知，做一驗證或比較。

●已將研究成果融入在管理科學、作業研究與生產管理等科目的教材內容中。

Porteus, E. L., 1986, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research,

34, 137-144. 工作人員方面：

Rosenberg, D., 1991, Optimal price-inventory decisions: profit vs.

ROII, IIE Transactions, 23, 17-22.

●培養工作人員對作業研究、工業工程與管理科學等領域的學術研究興趣。

Schroeder, R. G., and Krishnan, R., 1976, Return on investment as a criterion for inventory models, Decision Sciences, 7, 697-704.

●培養工作人員對學術研究的嚴謹態度。

●訓練工作人員建立模式與解答問題的技巧。

Smith, W. M., 1958, An investigation of some quantitative relationships between breakeven point analysis and economic lot size theory, AIIE Transactions, 9, 52-57.

●訓練工作人員運用電腦資訊科技來協助解答問題。

3)學術上具體之貢獻

期刊發表：有２篇論文在準備中，也將於近期被投稿至相關的國際期刊。

Soesilo, D., and Min, K. J., 1996, An inventory model with variable levels of quality attributes via geometric programming, International Journal of Systems Science, 27, 379-386.

從(1)完成之工作項目，(2)完成之具體成果，及(3)學術上具體之貢獻等三方面來說。前兩項在計劃進行至此時，已經有具體的成果。至於第３項，我們則期待本研究計劃的論文可以通過國際學術期刊嚴格的審查標準，進而發表在國際學術期刊上。最後，與本計劃相關且具有研究潛力的主題尚有很多，也期望國科會能再次於財務上支持相關的研究計劃。

Silver, E. A., Pyke, D. F., and Peterson, R., 1998, Inventory Management and Production Planning and Scheduling (New York: John Wiley & Sons).

Toshitsugu, O., Min, K. J., and Chen, C. K., 1999, Inventory and investment in setup operations under return on investment maximization, Computers and Operations Research, 26, 883-899.

計劃結果自評

就本研究計劃已完成之工作項目與具體成果分述如下：

1)完成之工作項目

●最新文獻資料的收集。

●基本模式的建立。

●基本模式的擴展。

●推導模式的解答過程。

●數值資料的驗證。

●電腦程式撰寫與電腦模擬驗證。

●撰寫報告。

2)完成之具體成果

(10)

9