Heuristics-based integrated decisionsfor logistics network systems

Journal (~/ Mam(/hcmrmg Systems Vol. 23/No. 1 2004

Heuristics-Based Integrated Decisions for Logistics Network Systems diafu

Tang, Dept. of Systems Engineering, Northeastern University, Shenyang, Liaoning, China

(E-mail: jftang @ mail.neu.edu.cn) and Dept. of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong (E-mail: mfjftang @polyu.edu.hk) Kai-Leung Yung and AndrewW.H. Ip, Dept. of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong. E-mail: [email protected]

Abstract In the global manufacturing environment, decisions involving production activities and distribution should be made in an integrative manner for substantial savings in the logistics costs. An integrated decision on production assignment, lot sizing, transportation, and order quantity for a multiple-suppliers/multiple-destinations logistics network in a global manufacturing system is discussed in this paper. The integrated decision process can be viewed as a two-layer decision. The first layer is the combined decisions of production assignment and lot sizing (APLS-M), while the second layer is the combined decisions of transportation and order quantity determination (TOQ-M). A two-layer decomposition method combining two heuristics is hence developed to solve the integrated decision model (IDM-M). An assignment heuristic and a transportation heuristic are designed to solve the decision layers APLS-M and TOQ-M, respectively. Simulations were conducted on a practical example in an electronics manufacturing enterprise together with different sizes of problems. The results indicate that the two-layer decomposition method with heuristics is effective and provides a practical way to solve medium and large-scale integrated decision problems.

Keywords: Production~Distribution Coordination, Heuristics, Two-Layer Decomposition, Supply Chain Management

Introduction With the development of network and information technology, global manufacturing is becoming the mainstream in the near future. A logistics network system as a component of the global manufacturing system is a production-distribution network where multiple products are produced at multiple suppliers/origins and then delivered through distribution facilities to satisfy requirements at multiple destinations. The suppliers may be parts plants, assembly plants, or warehouses, while the destinations may be assembly plants, distribution centers, or retailers. In the past, most investigation has been on the economic production lot size problem, transportation problem, and the order quantity problem sepa-

rately (Winston 1987). Within a global manufacturing enterprise, the logistics costs are major expenditures, and hence if the logistics decisions involving production activities and distribution are made in an integrative manner, there can be substantial savings in the logistics costs. Therefore, recent attention has been directed to integrated/coordinated production and distribution systems (Glover et al. 1979; Cohen and Lee 1988; Thomas and Griffin 1996; Tayur, Ganeshan, and Magazine 1999; Bhutta et al. 2003) in supply chain management (SCM). In particular, the integrated decisions for production and transportation (Blumenfeld, Bums, and Daganzo 1991; Hahm and Yano 1992; Chien 1993; Hall 1996; Fumero and Vercellis 1999), production and inventory (Williams 1981; Cohen and Lee 1988), and transportation and inventory (Speranza and Ukovich 1994; Bertazzi and Speranza 1999; Qu, Bookbinder, and Iyogun 1999) are important topics in SCM. For example, Fumero and Vercellis (1997) integrate decisions of allocating production volumes among the different manufacturing plants, assigning production quantities to alternative technologies within each plant, and determining production lot size for individual products on a particular type of machine within a production-distribution network. The problem is decomposed into five subproblems and subsequently solved by a Lagrangian relaxation. Fumero and Vercellis (1999) further apply a Lagrangian multiplier method to tackle the combined decisions in production planning, inventory allocation, and vehicle routing in a multiple-period horizon for a single plant logistic system. Sharafali and Co (2000) present several stochastic models of cooperation in making inventory-related decisions between the supplier and the buyer in a distribution system. Chen, Federgruen, and Zheng (2001) considered an integrated distribu-

Journal of Mamt,/?1cturing Systems Vol. 23/No. 1 2004

tion system with one supplier and multiple buyers and discussed mechanisms to coordinate decisions in order quantity, lot size, and price so as to improve supply chain efficiency. Jayaraman and Pirkul (2001) study an integrated logistics decision model that involves two essential decisions, that is, strategic and operations decisions. The strategic decision determines where to locate plants and warehouses, while the operational decision is influenced by the product mix at each plant, the shipments of raw material from vendors to manufacturing plants, and the distribution of finished products from the plants to the different customer zones through a set of warehouses. An efficient heuristic solution procedure combined with Lagrangian relaxation is developed. Tsiakis, Shah, and Pantelides (2001) considered an integrated decision model for a multiproduct, multi-echelon supply chain network. The number, location, and capacity of warehouses and distribution centers, transportation links to be established in the network, and flows and production rates of materials are entities requiring decisions to minimize the total annualized cost of the network. Kelle, Al-khateeb, and Miller (2003) consider two typical cases of production and distribution policies for a production distribution system and propose a joint optimal ordering and setup policy for different JIT scenarios. Minner (2003) presented a review of inventory models with multiple suppliers and discussed their contribution to supply chain management. Chiang and Robert (2004) proposed a method to address an integrated purchasing and routing problem in the gas supply chain by reconfiguring the supply network involving depot locations, tanker fleet sizing, and allocation of capacity at supply terminals. From formulation points of view, these logistics decision models can be classified to (1) single or multiple periods, (2) single or multiple retailers/customers, (3) single or multiple products, (4) identical or different demand rate in retailers, (5) deterministic or stochastic demands, and (6) consistent or inconsistent production and demand rate. The solution methods for the aforementioned integrated decision models are categorized into EOQbased types (Hahm and Yano 1992; Hall 1996; Qu, Bookbinder, and Iyogun 1999; Chen, Federgruen, and Zheng 2000; Sharafali and Co 2000; Kelle, A1khateeb, and Miller 2003); heuristics (Williams 1981; Benjamin 1989; Jayaraman and Pirku12001); decomposition methods (Sharp et al. 1970; Cohen and Lee

1988; Speranza and Ukovich 1994; Fumero and Vercellis 1997, 1999; Tsiakis, Shah, and Pantelides 2001); and evolutionary computation-based global optimization (Chiang and Robert 2004). Due to the fact that most real-life logistics networks involve multiple products, multiple suppliers, multiple warehouses, and multiple destinations, the integrated decision models are hence complex and difficult for which to obtain an optimal solution. Thus, among these solution methods, the decomposition methods and the heuristics become important and popular methods in solving integrated decisions for production, transportation, and inventory for logistics networks with multiple suppliers, multiple products, and multiple destinations. Benjamin (1989) discussed combined decisions for production, transportation, and order quantity in a production-distribution network to closely coordinate the functions of production, transportation, and inventory for a minimum total cost. The cases of single-supplier/single-destination and multiple suppliers/multiple destinations are analyzed. A decomposition-based solution procedure and heuristics were d e v e l o p e d to solve this n o n l i n e a r optimization problem. The formulation and solution procedures proposed are based on the assumption that the annual production at each supplier point is a prespecified constant. In real life, given the annual demands at individual destinations, the assignment of how much each individual supplier should produce within its capacity to meet these demands should be one of the decisions to be made and not a prespecified constant. Under this scenario, the problem becomes more complex than the one discussed by Benjamin (1989) because it has to determine not only the production lot size at the suppliers, amount of units annually shipped, and the order quantity per unit time from the suppliers to the destinations, but also the assignment of annual production quantity to individual suppliers according to the total demands and within their production capacity. In addition, Benjamin only considers the simple case of one product in an aggregate way. In a real production distribution network, each supplier often produces multiple products, such as parts, and serves multiple destinations in a year. Similarly, each destination, such as an assembly factory, demands multiple parts f r o m many suppliers. Hence, the integrated decisions of production, transportation,

Jourlud i~f"Mamlfitclzti"i~lg S)srem.c; -vbl. 23/No. 1 2004

and order quantity with multiproducts in a production distribution network system have to be considered in a holistic way. In summary, even though much attention has been paid to integrated production distribution networks, the models tend to focus on individually combined decisions in transportation scheduling and inventory control, production lot sizing and transportation scheduling, and production lot sizing and inventory policies. Due to the complexity and nonlinearity of the solution methods, the integrated decisions for production assignment, lot sizing, transportation, and order quantity determination in a coordinated logistic network are rarely solved. This paper focuses on integrated decisions in the assignment of annual production, lot sizing, transportation planning, and order quantity determination for multiple products in a production distribution network with multiple suppliers and multiple destinations. The paper's aim is to determine simultaneously the assignment of annual production quantity and lot size at the suppliers, and the annual shipment amounts and order quantity from the suppliers to the destinations to meet the demands with minimum total costs in the productiondistribution network.

Notation and Formulations In a real-life production distribution network, given the annual demands for specific products at individual destinations, the assignment of how much each individual supplier should produce within its capacity to meet the total demands should not be specified in advance, but should be a decision made together with the decisions on lot sizing, transportation, and order quantity. Assume that there is a set of 7n suppliers producing L types of products to meet market demands at n destinations in the logistics network system, where each individual supplier, destination, and product type is denoted by subscripts i, j, and k, respectively. Each supplier is given a production capacity that depends on the number of machines and production rate of each machine. For simplicity, it is assumed that multiple parts are produced on production lines at a given production rate at a supplier; that is to say, different parts are produced at the same production rate at a specified supplier. Each destination is given annual demands of products. The total production capacity of the suppliers can be greater than the total annual demands of the destinations. The unit

production cost of products at the suppliers differs with different locations. The integrated decision is in effect a two-layer decision. The upper layer is the combined decisions of assigning the production quantity for each supplier and the production lot size for the suppliers, while the lower layer is the combined decisions of transportation and order quantity between the suppliers and the destinations. For individual supplier i, destination j, and product k, the following notations and decision variables are applied.

Parameters Hik. = annual production inventory holding cost for product k at supplier point (S/unit) Pi = annum production rate at supplier point (unit), Pi = maxk{Qi/ui~} Qi = annual production capacity at supplier point (year) I)~. = ordering cost of product k at destination point j ($) Gjk = annual holding cost for product k at destination pointj (S/unit) C!j = unit transportation cost from supplier point i to destinationj ($) rik = unit p r o d u c t i o n cost of product k at supplier point i ($) u~ = production capacity needed to produce unit product k at supplier point i (years) Ki~ = setup cost for product k at supplier point i ($) Di~ = annual demand of product k at destination j (unit) D k = ~Di~., total annual demand of product k (unit) Decision Variables z;k = p r o d u c t i o n lot size of p r o d u c t k at supplier point i (unit) Sik = assigned annual production of product k at supplier point i (unit) Xij,. = order quantity of product k per time from supplier i to destination j (unit) Y,jk = amount of units of product k shipped a n n u a l l y f r o m supplier point i to destination j (unit) Integrated Decision Model with Multiple Products The cost components considered in this integrated decision problem include the production and inven-

JomTutl qf Mant(~wturingSystems Vol. 23/No. 1 2004 Inventory

Min

/b z,/e,

~ i ~ Prodactionclcle Zi~/S~h........y']'"

Time

W = FI(S,z,X)+ F2(Y)+ F~(X,Y)

(4a)

s.t. Z i ~ j k = Dix,Vj = 1, 2 ..... n;k = 1,2,...,L

(5a)

Z i ~ik = S/h, Vi = 1, 2 ..... m; k = 1, 2,..., L

(6a)

Z , Six = Z iD,~ = Dk' Vk = 1,2,...,L

(7a)

0 <_~kSikuix < Qi,Vi = 1,2,...,m

(8a)

0 < zix < S,k,Vi = 1,2,...m;k = 1,2 ..... L

(9a)

Figure 1 Average Cumulative Inventory Varies with Time in a Production Cycle for a Pair of Supplier and Product

tory costs at the suppliers, the transportation costs between the suppliers and the destinations, and the ordering costs and inventory costs at the destinations. These three parts of the costs are denoted by FI(S, z, X), F,( Y), and F3(X, Y), respectively (Benjamin 1989). When formulating the first part of the costs, the unit production cost, setup cost, and the cost for holding annual average inventory that resulted from the stock accumulation, ordering, and production are taken into the calculation. In a pair of specified supplier and product, due to the fact that the assigned annual production of a specific product is less than the annual production rate of the supplier (Six -< Pi), only a portion of the annual capacity is utilized for producing product k. During a production cycle, the production time for a lot size is zi~]P~and the idling time is z~flSix - z~flPi. It can be shown that, for a pair of supplier and product, the average inventory holding costs are (1 - Si~/ P~)z¢,H,x + - ~ ~i X,j, , where the former part is the average annual fiolding costs for stock accumulation and the latter part is the average holding cost for a customer's order, as illustrated in Figure 1. Hence, the production costs F~(S, z, X) at the suppliers are given as

FI(S.z,X)= ~,~,~ [r~kS~+[1-S,k/P~]H,~z,~+K~kS~Iz~ +(1/2)H,~ Xii,,} (la) Similarly, one can deduce the second and last parts of the combined costs as follows: (2a)

~(X,y)=~i~,[(1/2)Gjk~kXj~ + l j ~ , ~ / X i , ~ ]

(3a)

The integrated decisions of production and distribution with multiple products are expressed as the fol/owing model (IDM-M):

0 < X,jk _
Two-Layer Decomposition Solution Procedures with Heuristic The integrated decision model (IDM-M) is a fractional nonlinear programming model that is neither convex nor concave. It can be solved by traditional nonlinear programming techniques, such as GINO, gradient search methods, and quasi-Newton method, where only the local optimal solution may be found. The computation will also become more complex and difficult with increased number of suppliers, destinations, and product types. For example, for 10 suppliers, 10 destinations, and 50 products, the model has 11,000 decision variables and is difficult to be solved by traditional nonlinear programming techniques. On the other hand, this problem is essentially

Joztrnal o[' Mam{/~lclzls'ingSyslems Vol. 23/No. 1 2004

a combination of four decision subproblems--that is, the production assignment problem, lot sizing problem, transportation planning problem, and order quantity determination probleln--that can be viewed in two layers. A two-layer decomposition (TLD) based solution procedure is therefore more suitable for the integrated decision problems, particularly for large-scale problems. Using TLD, the IDM-M is solved by decomposing into combined decisions of production assignment and lot sizing, and combined decisions of transportation and order quantity. This is discussed in detail as follows.

Combined Decisions of Assigning Production and Lot Size The combined decisions of assigning production and lot size with multiple products (APLS-M) is given as

s.t. (7a), (8a), (9a) It determines the amount of each type of products to be produced and the lot size for each supplier to meet the total demands from the destinations at minimum total production costs. It can be decomposed into two related subproblems, that is, the assignment problem determining how to assign production capacity among product types for each supplier, and the lot size determination problem given a certain assigned production for each type of products. The objective function of the APLS-M is nonlinear and neither convex nor concave, hence the solution becomes complex. Thus, a two-step heuristic was applied to solve the APLS-M. The basic idea is that first the individual production lot size for each type of product and each supplier is determined in closed form by solving the local nonconstrained optimization with given annual production S~x.,and then combining it to determine the assigned production by solving the assignment problem. In a pair of product and supplier, the individual production lot size with closed form is given as

z,k = x/K~-ekek Sik/[(1 - Sik I P~)Hik ]

(12)

i = 1 , 2 . . . . . m; k= l , 2 , . . . , L and the constraint (9a) can be rewritten as

g,f),,, <_S~[1-S~k I PAH,,,Vi = 1,2..... m; k = 1,2..... L (13)

where 8i~ = 1 when Sik > 0, and 0 when Sik = 0. Let S~ be the value that Eq. (13) holds. In this case, production rate P~ is required to be greater than or equal to max {4KiJHik, k = 1, 2 ..... L}. Under this circumstance, (13) can be expressed as

Sj)ik _
(14)

Taking into account (14), the APLS-M model is equivalent to an assignment problem (AP-M) as follows: Mitt

Wj = ~ . ~ , ~ ,7eSi~+2~/KixS~e(1-[S~x/Pj])H,<

(15)

s.t. (7a), (8a), (14) From the AP-M, one can find that this assignment problem is also a nonlinear programming because the second term of the objective function is nonlinear. However, it has a similar structure as LR and hence an iterative LP-based heuristic is developed to solve it approximately. For a pair of supplier i and product k, the ratio of the former to the latter increases with the value of Six and it equals approximately to eik= (r,A-/2~-K-~ikHia.(;- Sik / Pi ))X[-~ik (16)

Let E~k = ( l + l / e i k ) r k then the AP-M is rewritten approximately as elk : ( r i k / 2 ~

Si, / Pi ) ) ~

(17)

s.t. (7a), (8a), (14) It can be solved by the simplex method (SM). Hence, combining the simplex method, an iterativebased heuristic is developed to solve the combined decision of assigning the production and lot sizing problem (APLS-M) with multiple products. This is described as follows:

Assignment Heuristics (AH-M) to Solve APLS-M Step 1 Set iteration account number t = 0. Find initial feasible solution S o by solving an upper-bound linear programming {min~,~,}~S~ + 4 ~ , s . t (7a),(aa),(14)}. Let its objective be W~° . Substitute S ikO into (12) 0 to determine z~x •

Journal of Manufiwturing Systems Vol. 23/No. 1 2004

Step 2 Determine feasible local optimal solutions for AP-M by: Step 2a Substituting S~a into (16) to determine the coefficients Elk, Step 2b Finding new feasible solution Sf~+~ by solvingAP-M (17) using the simplex method. Step 2c Substituting S~+j into (12) to deter'+~ , and calculate the corremine zi~. sponding objective function V¢~TM as in (15). Step 3 Check the termination condition. If S,~+l = Sf~ or t = Iter (Pre-specified largest iteration number), stop; otherwise, t = t + 1 and return to step 2. Step 4 Determine the optimal solution such that Wi = min{W(}.

Combined Decisions of Transportation and Order Quantity with Multiple Products After assigning annual production and production lot size for each suppliers and products, the next procedure is to determine the annual flow of each product and the quantity per order for pairs of suppliers and destinations. The combined transportation and order quantity problem (TOQ-M) is to determine the amount of units shipped annually and the order quantity per time between the supplier and the destination to meet the demand from the destinations at a minimum total cost of transportation, inventory, and ordering, within the annual assigned production and production lot size at the suppliers. This can be expressed as a nonlinear pro~amming model (TOQ-M): Min

W,=ZZjZ~[ -

~ik=42I.iaY,..ik(G.ik + H , ~ ) , V ( i , j , k )

(20)

Subsequently, the constraint (10a) is given similarly as

2Ijk/(Gja. + Hik)a(Y~ja.) _) = 1, when Y,.i~> 0, and 0 else. Taking into account (21), the TOQ-M model can be rewritten as

= Z ZZ;z,

+ CrYiik

(22)

s.t. (5a), (6a), (21) The property of the optimal solution for the TOQM model can be proved easily as follows. Property: For any triple of suppliers, destinations, and products, the sufficient and necessary condition Ct Y~ik

Cii

that F.,~ ---Hik + Gjk holds is Y,ik> 0, and the equation holds only when the left-hand side of the inequality in (21) exists. The above property indicates that for a triple of supplier, destination, and product with Y~ik> 0, transportation cost dominates the sum of inventory cost and ordering cost, and this dominance increases with the value of Bijk = C,:/(Hik + @~). For the triple of supplier, destination, and product, let w~jk = F~ik + C,..iY,:j~..It can be observed from the above p r o p e r t y that w~ _<(H;~ +Gjk +C~i)Y~ik , w h i c h leads us to deduce the upper bound of the IV,_ by solving an upper-bound transportation problem (TP) as follows:

(Hjk-+G,k)X~jk+ljk--+CuYgia. ] ( 1 8 )

i

"

X Ok

"

Min W;P= Z k £ , Z . i ( H i k

+Gik +C~i)Yiik

(23)

s.t. (5a), (6a), (10a) s.t. (5a), (6a), (21) TOQ-M is a fractional nonlinear programming model and, hence, effective heuristics are more acceptable than traditional nonlinear programming techniques for large-scale problems. Similar to the lot sizing in APLS-M, for any triple of supplier i, destination j, and product k, given Y,>, the order quantity X~jkand its objective function F,..i~ consisting of the first two terms of (18) are determined in closed form as follows:

Xii k = 42I.ia.Y~ik /(G.ik + H~k ), g(i, j, k)

(19)

The TP can be viewed as a classical balanced transportation problem with the addition of constraint (21). It can be solved by the transportation simplex method (TSM) or the simplex method, of which the optimal solution Yii~° is a feasible solution to the TOQ-M and gives an upper bound of the objective function. Given Xo~, the first parts of the objective W2 in (18) is a constant, and W2 is a linear function of Y,..j~.By neglecting the constant part of the objective function, the TOQ-M model is an LP model that is given as follows (TOQ-LP):

,Iota'hal

oJMam~fi~cmrilzg Systems Vol. 23/No, 1 2004

Min ~k~i2,A~ikYiik

(24)

using the two-layer decomposition method with heuristics can be derived and are described in detail as follows:

(25)

Overall Procedures

s.t. (5a), (6a), (21) where Aijk = 1jk / Xiik + C~j

Combining the closed form of the order quantity and solution of the TOQ-LP, a heuristic is developed to solve the TOQ-M model approximately. The main idea is as follows: Starting with the feasible solution Y j , one can deduce the corresponding order quantity X0k with closed form (19). The model TOQ-LP is then constructed by substituting the Xu~ to the model TOQ-M. By solving the TOQ-LP, an updated Yikiis generated. Repeating the above procedures, a nearoptimal solution to the TOQ-M is achieved and it converges to the optimal. The implementation is described in detail as follows.

Transportation Heuristics (TH-M) to Solve TOQ-M Problem Step 1 Set iteration number t = 0. Find initial feasible solution y0!ik by solving TP, substitute it into (19) to determine X~., and calculate the initial objective function W~° as (18).

Step 2 Determine feasible local optimal solutions to TOQ-M (18) by implementing Step 2a Substitute X'i# into (25) to determine the coefficients A~x.. Step 2b Find the new feasible solution Y'+~ by solving TOQ-LP with new coefficients A~k. Step 2c Substitute y,+l into (19) to determine iik X (i~ '÷1 and calculate the corresponding objective function W~+~ as in (18). Step 3 Check the termination condition. If Y.~[~ = g'i/~ or t = lter (Pre-specified largest iteration number), stop; otherwise, t = t + 1 and return to step 2. Step 4 Determine the optimal solution such that W2 = min {W2'} , '

Overall Procedures for Solving Combined Decision Model On the basis of the above analysis, the overall procedures to solve the integrated decision model

Step 1 D e t e r m i n e the lot size and annual production of products at the suppliers by solving the A P L S - M m o d e l with the assignment heuristics (AH-M). Let the optimal solution and objective function be

(sl;, w;). Step 2 S p e c i f y i n g SI~" as input of the annual production of the model TOQ-M. Step 3 Determining the annual ,amount of shipments and order quantity per time for products from the suppliers to the destinations by solving the c o m b i n e d transportation and order quantity p r o b l e m (TOQ-M) using the transportation heuristics (TH-M). Let the corresponding optimal solution and costs be

(<;,x;,w;) Step4 D e t e r m i n i n g the integrated objective function of integrated decision model (INN-M) as W" = W[ + W£~.

Simulation and Performance Analysis of the TLD with Heuristics In this section, the application of this model to an electronic appliance manufacturing enterprise in mainland China, and comparison of the heuristics and traditional NLP techniques (quasi-Newton method) for different sizes of problems are reported and analyzed. Example in an Electronic Appliance Enterprise An electronic appliance manufacturing enterprise produces more than 30 kinds and 150 series of electronic appliances, e.g., electric-thermo, electric iron, electric stove, electric fan, electric heater, and telephone. The headquarters of the electronic appliance manufacturing enterprise is located in a province within the southeastern region of China, while three manufacturing plants are located in the cities of Shenzhen, Xiamen, and Hanzhou, denoted by plants 1, 2, and 3, respectively. Even though each plant has its own special manufacturing technology, capacity, and brands of products, they can manufacture sev-

Journal of Mantt~cmring Systems Vol. 23/No. 1 2004

eral kinds of common products, e.g., the electric iron (EI), electric heater (EH), and electric fan (EF). The electronic appliances manufactured by the enterprise are distributed to customers all over China via more than 25 sales agents located in different regions of China. For simplicity, three types of products, the electric iron (EI), electric heater (EH), and electric fan (EF), with four sales agents, A, B, C, and D, all located in southern China are selected and cited in the illustrated example. The parameters of the model for the example are shown in Tables 1, 2, and 3. Table 1 presents the transportation cost between plants and sales agents. As indicated in Table 1, plant 1 is the nearest one to the four sales agents and hence the transportation costs from plant 1 in Shenzhen to the four sales agents are lower than those from plants 2 and 3 to the sales agents. Table 2 gives the data on demands, order costs, and holding costs of the three kinds of products at the sales agents. The information on unit costs, setup costs, holding costs, and unit capacity for products at each plant is given in Table 3. Annual capacity of these plants is 4200, 4500, and 6500 (hours), respectively, and their annual production rates are 16,000, 18,000, and 28,000 units, respectively. To avoid the unnecessary disturbance in

collecting business information, it is noted that the parameters related to products, e.g., unit cost, holding costs, setup costs, capacity, and so on, are historical data drawn from the enterprise. The integrated decisions of the abovementioned example using the TLD with heuristics is illustrated are Table 4. It can be seen from Table 4 that all of the demands for the electric iron (EI) are assigned to plant 3 with lot size 658, while the electric heater (EH) is assigned to plant 1 with lot size 530. The electric fan (EF) is assigned to plants 1, 2, and 3 with quantities of 5486, 12,857, and 7657 units, respectively, and lot sizes of 405, 940, and 455 units, respectively. The total logistic s costs spent is 2,111,890 RMB, of which 1,751,100 RMB is production-related expenditure and 360,790 RMB is distribution-related expenditure. The results have shown that if all the demands for a product are within the production capacity of a plant, the product can be assigned to just one of the plants. For example, the assigning of products EI and EH were to a single plant, which is reasonable and coincides with real-life operation. The results also have shown that the transportation flow of the electric iron only occurs from plant 3 to the four sales agents, with their individual annual flow being equal to the

Table 1 Transportation Costs, Annual Capacity, and Production Rate of the Example

Plants (i)

Transportation Costs (S/nni0 for Sales Agents (])

Annual Capacity

Annual Production Rate (units)

A

B

C

D

(hour)

1

5.85

3.55

4.30

2.25

4200

16000

2

8.70

3.80

5.58

3.71

4500

18000

3

6.55

0.88

4.6

2.65

6500

28000

Table 2 Annual Demands, Order Costs, and Unit Holding Costs for Products at the Sales Agents

Annual Demand (items) Product

Holding Costs ($)

Order Costs ($)

A

B

C

D

A

B

C

D

A

B

C

D

E1

5500

3000

4000

4500

21.0

21.0

21.0

21.0

250

250.0

250.0

250.0

EH

2500

2000

1500

3500

25.0

25.0

25.0

25.0

210.0

210.0

210.0

210.0

EF

6500

8000

6100

5400

20.0

20.0

20.0

20.0

285.0

285.0

285.0

285.0

Table 3 Unit Costs, Setup Costs, Holding Costs, and Unit Capacity for Products at the Plants

Plant

Unit Cost ($)

Setup Costs ($)

Holding Costs ($)

Unit Capacity (hours)

EI

EH

EF

E1

EH

EF

EI

EH

EF

EI

EH

EF

25.0

30.0

38.5

250

360

550

25

30

28

0.20

0.24

0. 35

2

23.5

35.5

37.0

250

360

550

25

30

28

0.20

0.24

0.35

3

23.0

32.0

41.0

250

360

550

25

30

28

0.20

0.24

0.35

1

doltrnal qf Mamflctctu;ing Systems

Vol. 23/No. 1 2004

quantities of demands at the specified agents. Transportation of the electric heater also exhibits similar characteristics. When compared with historical data, production managers consider the results acceptable and better than manual and arbitrary assignment. In particular, the results have pointed out the significance of assigning production among the plants. It reveals that business operations, including production and distribution among the plants in a distributed enterprise, should be considered in an integrative manner so as to reduce logistics costs and enhance the enterprises' competitiveness. Performance Analysis of TLD with Heuristics To prove the effectiveness of the two-layer decomposition (TLD) method and the two heuristics, the Quasi-Newton Method (QNM) and the TLD with QNM (called TLD/QNM hereafter in this paper) are applied to solve the integrated decision model. Using TLD/QNM, the integrated decision model is decomposed into combined decisions in two lay-

ers, that is, APLS-M and TOQ-M. Each of them is then solved by the QNM separately. Three parts of the computation experiments are conducted to evaluate the effectiveness of the TLD with heuristics as follows.

Comparison of TLD with Heuristics, TLD/QNM, and QNM on the Illustrated Example The comparison of the results of the above example for TLD with heuristics, TLD/QNM, and the QNM are illustrated in Table 4. In the table, the differences between the integrated decisions for the QNM and the TLD/QNM and that of the integrated decisions for the TLD with heuristics are marked with italic and bold fonts, respectively. The other results are the same for the three algorithms. It can be seen from Table 4 that the optimal decision for the above example is achieved through using the TLD with heuristics and TLD/QNM. A near-optimal solution with relative deviation of 0.919% is obtained from the best solutions after 10 runs by QNM with feasible initial solution. Table 4

Integrated Decisions of Example Using TLD with Heuristics, TLD/QNM, and QNM

Annual How and Order Quantity Between Production and Assembly Plants units) Link I-A

i giligfg#iiig; i:i!i iiiY!i!gi{;i=5 ; iNiii i;:ii;;£(! (:jigiiiii;i igiiiiiiiig:;ii:?:iiii(i

i{;:

Annual flow of product

Order quantity of product

EI

EH

EF

EI

EH

EF

0

2,500

5,486

0

138

255

1-B

0

2,000

0

0

124

0

1-C

0

1,500

0

0

107

0

1-D

0

3,500

0

0

164

0

0

Plant

EI

EH

EF

2-A

0

0

0

0

0

1

0

530

405

2-B

0

8,000

0

0

308

403 ~

361 ~

2-C

0

4,857

0

0

240

2

0

0

940 627 ~

3

658

0

494~

Total costs ($)

455 417 ~

Difference from

2-D

0

0

0

0

0

0

3-A

5,500

0

1,014

245

0

110

3-B

3,000

0

0

180

0

3-C

4,000

0

1,243

209

0

122

3-D

4,500

0

5,400

221

0

254

.

best one (%) TLD with QNM

2,111,890

0.000

TLD with heuristics

2,111,890

0.000

QNM

2,131,300

0.919

• /8t__._~

'~Difference of integrated decision of TLD/QNM from that of TLD with heuristics. b Difference of integrated decision of QNM from that of TLD with heuristics.

2_~'-' 2 3 '~

Journal of Manu.f~lcturing Systems

Vol. 23/No. 1 2004

Table 5 Comparison Results Between TLD with Heuristics and QNM for Different Size of Examples

Problem Size 5*5*5

Costs for IDM-M Model

CPU Time (sec)

TLD

QNM

Diff (%)

TLD

QNM

674,670

686,940

1.82

5

362

5"5"10

1,034,090

1,066,210

3.11

51

4,981

5"10"10

2,075,570

2,176,010

4.84

121

3,261

4,854,200

8.15

216

12,869

5"10"20

4,488,400

5"10"50

9,988,700

10"10"10

2,190,550

10"10"20

4,469,000

312

10"10"50

9,899,000

829

398 2,296,530

4.83

F r o m Table 4, sections of the production assignment, annual transportation flow, and order quantity of the t h r e e s o l u t i o n s for the T L D w i t h heuristics, TLD/QNM, and Q N M are identical and only the lot size of the solution for QNM is different from the ones for the TLD with heuristics and TLD/QNM. The QNM terminates after 79 iterations with a near-optimal solution on the condition that it starts with a feasible initial solution near the one obtained by TLD/QNM. The computation experiments also have shown that the Q N M easily terminates at a local optimal solution or an infeasible solution when it starts with a r a n d o m initial solution, particularly an infeasible one. Moreover, there were large differences between the best solutions obtained by QNM with different initial solutions. In terms of computation time, the TLD with heuristics can complete the computation after no more than five iterations in 5 seconds of CPU time, while the T L D / Q N M and the Q N M have taken about 10 times the C P U time of the T L D with heuristics. Hence, one can c o n c l u d e that the T L D m e t h o d is m o r e effective than T L D / Q N M and QNM in the case of the above example.

189

14,456

mands, ordering costs, and holding costs for products at the destinations are generated randomly in the range of [100, 1000], [150, 200], and [10, 30], respectively, while unit costs, setup costs, holding costs, and unit capacity of the products at the suppliers are randomly generated in the range of [20,50], [500, 1000], [10, 30], and [0.1, 0.5]. Due to the values related to the products being randomly generated, they may be inconsistent and could not reflect the actual situations. For example, a product with higher unit cost may have lower setup cost, holding costs, and unit capacity than another product with a lower unit cost. Hence, some adjustments should be made. In addition, the data on annual capacity of the suppliers should be checked so that the sum of these capacities of these suppliers is large enough to satisfy the demands from the respective destinations. When implementing the TLD with heuristics, the m a x i m u m number of iterations is set at 1500. The comparison between the TLD and the QNM with regard to the costs of the model and the CPU time with different sizes of examples are given in Table 5, where the solutions under QNM are the best of the five runs with initial feasible solutions. The CPU time is the time required to obtain the best solution using a Dell Optiplex GX 270 PC with Pentium-IV 2.6G Hz processor. It is noted that for examples 5 " 1 0 " 5 0 , 10* 10*20 and 10* 10*50, the QNM terminates with an infeasible solution and hence the best solution could not be obtained. From Table 5, one can see that the TLD is better than the QNM in these examples in terms of quality of the solution and CPU time. In terms of solution performance, the TLD saves 2% to 9% cost over than QNM, and this savings will further increase with the sizes of the problem, particularly with the number of

Comparison Between TLD with Heuristics and Q N M on Different Size Problems To illustrate the effectiveness of the TLD with heuristics, eight randomly generated examples with sizes 5*5*5, 5 " 5 " 1 0 , 5 " 1 0 " 1 0 , 5 " 1 0 " 2 0 , 5 " 1 0 " 5 0 , 10"10"10, 10"10"20, and 10"10"50 (suppliers * destinations * products) are cited in this section to make the comparison between the TLD with heuristics and the QNM. In these examples, the transportation costs between suppliers and destinations are randomly generated in the range [1, 10]. Product de-

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Sensitivity Analysis o f Heuristics with Parameters There are many parameters in the integrated decision model. Among them, the holding costs and setup costs at the suppliers and the holding costs and order costs at the destinations are important factors affecting the integrated decision. To analyze the sensitivity of the heuristics with each of these parameters, a set of coefficients 0.1, 0.5, 1, 5, and 10 for the basic data of setup costs/order costs and another set of coefficients 0.1, 1, 5, and 10 for the basic data of holding costs both at the suppliers and at the destinations were selected for simulation. The testing computations were conducted on the practical example as shown in Tables 1-3. These 20 cases broadly represent the variation of these parameters, and hence the simulation results can reflect the sensitivity and stability of the heuristics. The two heuristics and the QNM for the combinations of these parameters were tested and analyzed with the comparison results shown in Table 7. It can be observed from Table 7 that the optimal solutions for all of these 20 cases were obtained when the heuristics AH-M and TH-M were applied, while optimal solutions for 19 cases and 18 cases were obtained when the QNM was applied to APLS-M and TOQ-M models, respectively, with near-optimal solutions for other cases. An interesting fact from the computation is that, for the APLS-M model, each of the following groups: cases 3 and 6; cases 4, 7, and 11; cases 5, 8, and 16; cases 9, 13, and 17; cases 10 and 18; and cases 15 and 19 has the same solutions within the group. Furthermore, the product of the coefficient of the basic setup costs and the coefficient of the basic holding costs for these cases are identical. For example, cases

products. For CPU time, TLD requires less CPU time than QNM and this saving in computation becomes more vivid as the problem size increases. It can be observed that the CPU time increases slowly for TLD while quickly for QNM with the size of the problem, especially with the number of products. The computation experience also implies that the product size may have large impact on CPU time rather than the number of suppliers and destinations. The two heuristics AH-M and TH-M are the core and components of the TLD with heuristics for solving combined decision models in two layers, that is, APLS-M and TOQ-M of the joint decision model IDM-M, respectively. To compare the heuristics and the QNM for the two combined decision models in the two layers, that is, APLS-M and TOQ-M, the two heuristics and QNM are tested on the above examples, with their comparison results shown in Table 6. The computation experimentations have shown that the best solutions obtained by the heuristics AH-M and the TH-M for all the cited examples are better than those obtained by QNM, particularly with increased number of products. Computation-wise, both the AHM heuristics for the APLS-M and the TH-M heuristics for the TOQ-M have shown to reach the optimal solution quickly with no more than 500 iterations. On the other hand, the simulation has also shown that the optimal solution for the APLS-M and the TOQM when using QNM can only be obtained with good initial feasible solutions. The experimental computation indicates that the CPU time for the hemistic AH-M depends mainly on the iteration number and only slightly on the size of the problem, and hence it is suitable for application in large-scale problems.

Table 6 C o m p a r i s o n Results Between Individual Heuristics and QNM for Different Size of Examples

Problem Size

AH-M

Costs for APLS-M QNM Diff (%)

CPU Time (sec) AH-M QNM

Costs for TOQ-M TH-M

CPU Time (sec)

QNM

Diff (%)

TH-M

QNM

5*5*5

528,920

529,160

0.0452

2

35

145,750

148,280

1.735

5

58

5"5"10

883,780

884,070

0.0329

35

I18

150,310

154,750

2.95

42

251

5"10"10

1,714,900

1,715,100

0.0116

40

150

360,670

389,810

8.079

105

585

5"10"20

3,431,800

3,442,800

0.321

65

220

1,056,600

1,186,210

12.27

188

7,215

5"10"50

7,636,000

7,649,600

0.178

135

693

2,352,700

325

10"10"10

1,692,500

1,693,600

0.065

71

187

498,050

10"10"20

3,408,800

3,4t7,900

0.267

79

305

1,060,200

289

10"10"50

7,574,400

7,593,400

0.251

155

1,861

2,324,600

806

I1

529,830

6.381

165

6,289

Journal o[ Mantt[a('turing Systems

Vol. 23/No. 1 2004

Conclusions

3 and 6 are having the same solution of 1,700,300 RMB, while the product of their coefficient of the setup costs and their coefficient of the holding costs is equal to 0.1. Similarly, cases 5, 8, and 16 are having the same solution, while the product of their coefficient o f the setup costs and their coefficients of the holding cost is equal to 1. Further computations have led us to conclude that the optimal solution of the A P L S - M model could not change with the setup costs or the holding costs when the product of the setup costs and the holding costs remains constant. In fact, this conclusion coincides with the formulation of model AP-M (15). Hence, one can conclude that the two heuristics are effective to the two combined decision models A P L S - M and TOQ-M, respectively. In summary, the above simulations have shown that the TLD with heuristics is easily implemented and proved to be effective in the integrated decision problems for medium and large-scale decision applications.

One of the central problems of supply chain management is the coordination of production and distribution, especially in the integration of production, transportation, and inventory. This paper considers the integrated decisions of assigning production, lot sizing, transportation, and order quantity for multiple products in a production-distribution network system with multiple suppliers and multiple destinations. A two-layer decomposition method with heuristics is developed to solve this integrated decision model (IDM-M). The first layer is the combined decisions of assigning production and lot size (APLS-M) using an assignment heuristic. The second layer is the combined decisions in transportation and order quantity using a TH-M heuristic. The simulation results have shown that the TLD with heuristics is easily implemented and effective for the integrated decision problems. It provides a practical way to solve medium-scale and large-scale integrated decision problems but it could not guarantee the global optimum.

Table 7

Sensitivity Analysis of Heuristics and QNM for the Example

Case No.

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Parameters (at suppliers) Setup H o l d Cost* Costs* 0.1 0.1 0.5 0.1 1 0.1 5 0.1 10 0.1 0.1 1 0.5 1 1 1 5 1 10 1 0.1 5 0.5 5 1 5 5 5 10 5 0.1 10 0.5 10 1 10 5 10 10 10

AH-M & QNM for APLS-MModel AH-M

QNM

1,684,300 1,684,300 1,693,500 1,693,500 1,700,300 1,700,300 1,729,400 1,729,400 1,751,100 1,751,100 1,700,300 1,700,300 1,729,400 1,729,400 1,751,100 1,751,100 1,843,000 1,843,000 1,911,800 1,911,800 1,729,400 1,729,400 1,794,300 1,794,300 1,843,000 1,843,000 2,048,300 2,048,300 2,202,200 2,202,200 1,751,100

1,818,900

1,843,000 1,843,000 1,911,800 1,911,800 2,202,200 2,202,200 2,419,800 2,419,800

Parameters TH-M & QNM for TOQ-M Model (at destinations) Relative Order Holding TH-M QNM Relative Diff (%) # Diff (%)# Costs* Costs* 0.00 0.1 0.1 262,480 262,480 0.00 0.00 0.5 0.1 302,180 302,180 0.00 0.00 1 0.1 331,930 331,930 0.00 0.00 5 0.1 457,460 457 460 0.00 0.00 10 0.1 551,530 551 530 0.00 0.1 1 271,610 271 610 0.00 0.00 322 590 0.00 0.5 1 322,590 0.00 1 1 360,790 360 790 0.00 0.00 5 1 521,990 521 990 0.00 0.00 10 1 642,790 642 790 0.00 0.00 0.1 5 298,740 298 740 0.00 0.00 421 740 10.04 0.00 0.5 5 383,250 1 5 446,570 446,570 0.00 0.00 5 5 713,820 713,820 0.00 0.00 952,780 4.23 0.00 10 5 914,070 0.00 0.1 10 321,970 321,970 3.87 0.5 l0 435,190 435,190 0.00 0.00 1 10 520,040 520,040 0.00 0.00 5 10 878,090 878,090 0.00 0.00 10 10 1,146,400 1,t46,400 0.00 0.00

* Indicates number of multiples (coefficient)of basic data in Table 1. #Indicates differencesfrom best one in percentage.

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Journal qf Mam~in'ntr#lg Systems Vol. 23/No. 1 2004

Qu, w.w.: Bookbinder, J.H.; and Iyogun, R (1999l. "An integrated inventory-transportation system with modified periodic policy for multiple products." European Journal o/" Operafiomd Research (v115), pp245-269. Sharafali. M. and Co, H.C. (2000). "Some models for understanding the cooperation between the supplier and the buyer" hu'l JoutJlal o{'Production Resealvh (v38, n15), pp3425-3449. Sharp, J.E and Snyder, J.C., et al. (1970). "A decomposition algorithm for solving the multi-facility production-nansportation problem with nonlinear production costs." Econometrics (v38), pp490-506. Speranza, M.G. and Ukovich. W. (1994). "Minimizing transportation and inventory costs for several products on a single link." Operations Research (v42, n5), pp879-893. Tayur. S.; Ganeshan, R.; and Magazine, M. (1999). Quantitative Models for S~q)ply Chain Management. London: Kluwer Academic Publishers. Thomas, DJ. and Griffin, RM. (1996). "Coordinated supply chain management." European Journal qf Operational Resealvh (v94), pp 1-15. Tsiakis, E: Shah, N.: and Pantelides, C.C. (2001). "Design of multi-echelon supply chain networks under demand uncertainty." hMustrial and Engg. Chemistl3' Research (v40, nl6), pp3585-3604 Williams, J.E (t981). "Heuristics techniques for simultaneous scheduling of production and distribution in multi-echelon structures: theory and empirical comparisons." Mgmto Science (v27), pp336-352. Winston, W.L (1987). Operations Research, Applications andAIgorithms. Boston: PWS-Kent Publishing Co.

Acknowledgments The authors acknowledge that this project is cosupported financially by The Hong Kong Polytechnic University, the National Natural Science Foundation of China (70002009), the Key Program of Scientific Research of the Ministry of Education of China (104064), the Excellent Youth Teacher Program of MOE of China, and the Natural Science Foundation of Liaoning (20022019). References Benjamin, J. (1989). "An analysis of inventory and transportation costs in a constrained network." Transpoi'tation Science (v23. n3), pp177-183. Bertazzi, L. and Speranza, M.G. (1999). "'Models and algorithms for the minimization of inventory and transportation costs: A survey." New Trends in Distribution Logistics, Lecture Notes in Economics and Mathematical Systems 480, pp137-157. Bhutta, K.S.; Huq, E; Frazier, G.; and Mohanred, Z. (2003). "An integrated location, production, distribution and investment model for a multinational corporation." h~t'l Journal (~["Production Economics (v86, n3), pp201-216. Blumenfeld, D.E.; Burns, LD.: Diltz. J.D.; and Daganzo, C.E (1985). "Analyzing trade-offs between transportation, inventory and production costs on freight networks." Transportation Research, Part B (v 19), pp361-380. Blumenfeld, D.E.; Burns, LD." and Daganzo, C.E (1991). "Synchronizing production and transportation schedules." Transportation Research, Part B (v25, nl), pp23-37. Chen, E; Federgruen, A,: and Zheng, Y.S. (2001). "Coordination mechanisms for a distribution system with one supplier and multiple retailers." Mgmt. Science (v47, n5), pp693-708. Chiang, W.C. and Robert, A. (2004). "Integrating purchasing and routing in a propane gas supply chain." European Journal of"Operational Research (v154, n3), pp710-729. Chien, T.W. (1993). "Determining profit-maximizing production/shipping policies in a one-to-one direct shipping, stochastic demand environment." European Journal of Operational Research (v64), pp83-102. Cohen, M.A. and Lee, H.L (1988). "Strategic analysis of integrated production-distribution systems: models and methods." Operations Research (v36, n2), pp216-227. Fumero, E and Vercellis, C. (1997). "Integrating distribution, machine assignment and lot sizing via Lagrange Relaxation." btt'l Jou171al of Plvduction Economies (v49), pp45-54. Fumero, E and Vercellis, C. (1999). "Synchronized development of production, inventory and distribution schedule." Transportation Science (v33, n3), pp330-340. Glover, E; Jones, G.; and Kamey, D., et al. (1979). "'An integrated production, distribution and inventory planning system.'" hTtecfaces (v9), pp21-35. Hahm, J. and Yano, C.A. (1992). "The economic lot and delivery scheduling problem: the single item case." hlt'l Journal of Production Economics (v28), pp235-252. Hall, R.W. (1996). "On the integration of production and distribution: economic order and production quantity implications." Transportation Research, Pal7 B (v30, n5), pp387-403. Jayaraman, V. and Pirkul. H. (2001). "Planning and coordination of production and distribution facilities for multiple commodities." European JounTal (~ Operational Research (v133, n2), pp394-408. Kelle, E; Al-khateeb, E; and Miller. EA. (2003). "Partnership and negotiation support by joint optimal ordering/setup policies for JIT." hzt'l Journal of Ptvduction Economics (v81-82), pp431-441. Minner, S. (2003). "Multiple-supplier inventory models in supply chain management: a review." btt 'l Journal of Production Economics (v8182), pp265-279.

Authors' Biographies Jiafn Tang received the PhD degree in control theory and systems engineering from Northeastern University, Shenyang, China, in t999. He is currently a professor with the Dept. of Systems Engineering, School of Information Science & Engineering. Northeastern University, Shenyang, China, and now serves at the Dept. of Industrial & Systems Engineering at The Hong Kong Polytechnic University as a visiting resem'ch fellow. He has authored two books and has published 17 papers in international journals such as the International Journal of Production Research, Fuzzy Sets and Systems, Production Planning and ContJvl, International Jout71al of Advanced Mam(f~tcturing Technologies, Computelw & Operations Research, and Computers & Mathematics with Applications. His research interests include fuzzy m.odeling and optimization, supply chain planning and logistics management, and supplier-involved product development. K.L. Yung is currently a professor in the Dept, of Industrial and Systems Engineering at The Hong Kong Polytechnic University. He received the BSc degree in electronic engineering, MSc degree in automatic control, and PhD degree in microprocessor application fTom Brighton University (Brighton, UK), University of London, and Plymouth University (Plymouth, UK) in 1975, 1976, and 1985, respectively. He has working experience in companies such as B O C Advanced Welding Co. Ltd., British Ever Ready Group, and Cranfield Unit for Precision Engineering. He joined the Hong Kong Productive Council as a consultant in 1986 and then joined The Hong Kong Polytechnic University. His research interests include precision motion control and computer-integrated manufacturing systems and management. WoH. Ip is an associate professor in the Industrial and Systems Engineering Dept. of The Hong Kong Polytechnic UniversitS:Dr. Ip has more than 20 5,ears of experience in industry, education, mad consulting. He received his PhD from Loughborough University in the U.K. Dr. Ip also holds MBA, MSc, and LLB(Hons) degrees. He is a member of the Hong Kong Institution of Engineer, the Institution of Electrical Engineers, the Institution of Mechanical Engineers, and a fellow member of the Hong Kong Quality Management Association. Dr. Ip has published more than 100 intemationaljoumals and conference m'ticles.His research interests are operational research, logistics and supply chain mauagement, ERP. and MRR

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