A hierarchical planning approach for a production ...

International Journal of Production Research

ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20

A hierarchical planning approach for a productiondistribution system Linet Ozdamar & Tulin Yazgac To cite this article: Linet Ozdamar & Tulin Yazgac (1999) A hierarchical planning approach for a production-distribution system, International Journal of Production Research, 37:16, 3759-3772, DOI: 10.1080/002075499190031 To link to this article: http://dx.doi.org/10.1080/002075499190031

Published online: 14 Nov 2010.

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int. j. prod. res., 1999, vol. 37, no. 16, 3759± 3772

A hierarchical planning approach for a production-distribution system

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È ZDAMAR{* and TU È LIN YAZGACË{ LINET O A production-distribution model involving production and transportation decisions in a central factory and its warehouses is developed. The model is based on the operating system of a multi-national company producing detergents in a central factory from which products are distributed to geographically distant warehouses. The overall system costs are optimized considering factory and warehouse inventory costs and transportation costs. Constraints include production capacity, inventory balance and ¯ eet size integrity. Here, a hierarchical approach is adopted in order to make use of medium range aggregate information, as well as to satisfy weekly ¯ uctuating demand with an optimal ¯ eet size. Thus, a model which involves an aggregation of products, demand, capacity, and time periods is solved. In the next planning phase, the aggregate decisions are disaggregated into re® ned decisions in terms of time periods, product families, inventory and distribution quantities related to warehouses. Consistency between the aggregate and disaggregation models is obtained by imposing additional constraints on the disaggregation model. Infeasibilities in the disaggregated solution are resolved through an iterative constraint relaxation scheme which is activated in response to infeasible solutions pertaining to di erent causes. Here, we investigate the robustness of the hierarchical model in terms of infeasibilities occurring due to the highly ¯ uctuating nature of demand in the re® ned time periods and also due to the aggregation process itself.

1.

Introduction

A production system consists of a chain of subsystems starting from the supplier subsystem up to the distribution subsystem. Here, we consider the manufacturing subsystem linked to the distribution system. Distribution is the dispersal of goods from producers to customers. This dispersal can be constructed either in organizational terms, as the successive transfer of ownership along a distribution channel composed of producers, wholesalers and retailers, or in terms of the physical movement of goods from factories through warehouses to shops. Physical distribution is the collective term for the series of interrelated functions involved in the physical transfer of goods from producer to consumer. The principal task of physical distribution is to ensure that products are available at the right place, at the right time and at the quantities which satisfy customer demand. The distribution subsystem can be conceptualized as bridging the gap between production and consumption. Considering these facts, we can say that it is not possible to separate these two subsystems and in order to minimize total costs, planning for manufacturing should also involve distribution decisions. In the literature several attempts have been carried out to treat production and distribution systems in an integrated manner. The Thomas and Gri n (1996) extenRevision received January 1999. { Istanbul KuÈltuÈr University, Department of Computer Engineering, Istanbul, Turkey. * To whom correspondence should be addressed. e-mail: [email protected] International Journal of Production Research ISSN 0020± 7543 print/ISSN 1366± 588X online # 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/prs.htm http://www.taylorandfrancis.com/JNLS/prs.htm


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L . OÈzdamar and T. Y azgacË

sive review on coordinated supply chain management mentions strategic issues related to production-distribution systems as well as mathematical models developed in the literature. Here, we discuss a few models which are closely related to the problem considered. Pyke and Cohen (1992) develop a model of an integrated production and distribution system composed of a single station model of a factory, a stockpile of ® nished goods (FG), and a single retailer. A reorder point approach is adopted to manage the system. Chandra (1993) developed an integrated model to determine replenishment policies at a warehouse during a ® nite planning horizon of discrete time periods where demand at each customer location for every period is deterministic. In the solution approach, the problem is decomposed into the Warehouse Ordering Problem (WOP) and the Distribution Planning Problem (DP). Fleischman (1993) designed a distribution system with transport economies of scale. Factories, central warehouses, trans-shipment points and customers are the types of node in the distribution network. The topology of the network as well as locations and capacities of warehouses are the decisions in the system. Cohen and Lee (1988) developed a modelling framework to predict the performance of a ® rm with respect to several criteria such as product costs, service level and system responsiveness. Chandra and Fisher (1994) made a computational study to investigate the positive e ects of coordinating production and distribution scheduling rather than decomposing them. Edghill et al. (1988) used a simulation model to analyse the dynamics and sensitivity of a production-distribution system. Zuo et al. (1991) model a large scale agricultural production-distribution system including decisions on allocation of products to factories and their transportation to customers. Burns et al. (1985) propose shipment strategies for minimizing transportation and inventory costs. Here, we develop a planning strategy for a production-distribution system and show how it works by using the data of a company manufacturing liquid and powder detergents in a central factory in Izmir (Turkey) which distributes ® nished goods to four distant warehouses all over Turkey. Although the number of vehicles is attempted to be optimized besides inventory costs, we do not consider vehicle routing issues since the warehouses are far apart. The integrated production-distribution problem is hard to solve due to the number of integer variables representing vehicles and to the binary variables indicating major setup. Thus, a monolithic model which provides a detailed optimal production-distribution plan for a given set of re® ned time periods is very di cult to obtain. Consequently, here, we adopt a pure top down hierarchical production planning (HPP) approach in order to provide a viable plan for the whole system. Instead of decomposing the planning problem into several components, we solve the monolithic problem at di erent levels of aggregation. HPP partitions the larger decision-making domain into hierarchical levels in agreement with the organizational structure of companies (Schneeweiû 1995). HPP was ® rst proposed by Hax and Meal (1975) and Gabbay (1975). Researchers continue to carry out theoretical work on HPP (Erschler et al. 1986, Pienkosz and È zdamar et al. 1996, Kira et al. 1997), because this approach is Toczylowski 1993, O still valid in many industrial sectors such as the tile industry (Liberatore and Miller È zdamar and Birbil 1997), steel manufacturing (Bowers and Jarvis 1992), 1985, O È zdamar et al. 1997), shoe production (Caravilla and metal can manufacturing (O È zdamar et al. de Sousa 1995), motor industry (Tsubone and Sugawara 1987, O 1998), milk powder manufacturing (Rutten 1993) and paint and colourant manufacturing (Venkataraman and Smith 1996).


Hierarchical distribution system

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In the hierarchical approach proposed here, an aggregate planning model which re¯ ects the system’s physical constraints and provides a rough plan over the planning horizon is solved ® rst. In the aggregate model, product families are aggregated into product types, and capacity consumption by setups is not considered. Rather, an approximate percentage of capacity is allocated to setups. Next, the plan is re® ned into shorter time periods and product families are discriminated in the disaggregation model which is solved for the ® rst period of the planning horizon. In this model, setup variables are also included to indicate major setups on production lines. There might be inconsistencies or infeasibilities in the disaggregated solution due to the following causes: when re® ning the aggregate time period into shorter time periods there might exist high demand ¯ uctuations between re® ned periods, and capacity becomes insu cient to result in backorders; the disaggregated solutions obtained on a rolling horizon basis might lead to an accumulation of discrepancies due to the aggregation of certain parameters such as demand. We aim to show that using the HPP approach and the relaxation scheme proposed here, the planning system can cope with these inconsistencies. We further test the robustness of the system for various levels of demand ¯ uctuations during re® ned time periods. 2.

Description of the case

A multi-national company which produces liquid and powder detergents is located in Izmir. With its wide distribution area and big market share all over Turkey, the plant has to manufacture (in some periods) near to full capacity in order to satisfy the customer demand due to the intensive use of detergents in hot seasons. The plant in Izmir supplies products for ® ve main warehouses and the chain stores. The eastern warehouse, which is located in Adana, meets the demand of the customers in Adana and the other small warehouses at the eastern cities of Turkey (Antakya, Mersin, Antep and Malatya). The Middle Anatolian warehouse which is located in Ankara supplies the demands of central Anatolia and eastern Black Sea region (Samsun, Trabzon, Konya, Erzurum, Zonguldak, Kayseri, Sivas and Tokat). Istanbul which is the biggest city in Turkey has two warehouses, on the Europe and Anatolian sides. The Istanbul Europe warehouse supplies products for Trakya region (Edirne, Tekirdag and the other small cities). The Istanbul Anatolian warehouse meets the demand of the west Marmara region (Adapazari, Bursa, Eskisehir, and Izmit). The plant in Izmir has also a large warehouse providing products for the Aegean region (Izmir, Balikesir, Aydin, Denizli, Mugla, Usak, Manisa, Afyon, and Antalya). Figure 1 demonstrates the topology of the distribution system. The manufacturing plant consists of two separate lines for the powder and liquid detergents. Although there are numerous product groups, for planning purposes one can aggregate them into three major powder (A, B, C) and three major liquid (D, E, F) product families, each with di erent ingredients, and manufactured on separate lines. End items within product families di erentiate according to package size. Since the packaging department does not constitute a resource bottleneck, we concentrate only on the process of mixing departments which are active 24 hours a day. The total mixing capacity for powder product families A, B, and C is approximately 20 000 tons per year and 25 000 tons per year for product families D, E, and F. Average setup time is 3 hours and 8 hours for the powder and liquid product families respectively. However, there are no signi® cant di erences among the processing times of families in the same group, because the mixing procedure is similar. The latter


3762


Figure 1. Topology of the distribution system.

property is important for a sound aggregation of product families into powder and liquid product types. The products are transported to the four main warehouses by long vehicles. The vehicle capacity is about 2250 boxes which is equivalent to 37.5 tons. Transportation costs per vehicle depend on the distance of the warehouses to Izmir and the cost of transportation from the plant to the warehouse in Izmir is zero. Inventory holding costs di er among warehouses due to di erent standards of life in various geographic areas. Among warehouses, Istanbul constitutes the most expensive area in terms of rents, personnel, safety keeping and other functions of warehousing. In calculating inventory holding costs these factors are included, as well as the cost of capital tied to ® nished goods. Although backorders are not allowed in the aggregate model, this restriction is removed in the disaggregation phase if no feasible solution can be identi® ed. We assume that backorder costs are 10% higher than inventory holding costs, because they also represent the loss of delayed capital income plus intangible costs. Customer orders are conveyed to the plant by an on-line computer system, so the manufacturer can develop a production plan for at least one aggregate period. The plan for the whole planning horizon is based on rough sales forecasts obtained from the warehouses. The transportation lead time is at most 72 hours, so, given that the smallest planning period is a week, the transportation lead time may be omitted from the model. 3.

A hierarchical planning approach

We present a hierarchical planning approach in order to deal with the development of the production-distribution plan for a planning horizon of a year. In this approach, we try to avoid backorders as much as possible, because of the ® rm’s sales policy. However, we include a backorder capability in both aggregate and disaggregation models when capacity is not su cient to meet demand peaks. Thus, to obtain a re® ned solution in a certain time period, we carry out an iterative procedure re-solving both aggregate and disaggregation models until a feasible solution is identi® ed.

3763



3.1. The aggregate model The aggregate model is solved for two product types, the ® rst of which includes families A, B, and C, and the second including families D, E and F. An aggregate time period is de® ned as two months in order to reduce the number of integer variables for the vehicles. Setup variables are not included to avoid binary variables, but they are represented implicitly in the model by reserving a percentage of aggregate capacity for each product type to enable setup considerations in the disaggregation model. The aggregate model is solved on a rolling horizon basis where in each time period the current inventory levels are updated by the detailed inventory information provided by the disaggregated solution in the previous period and this information is aggregated into demand information of the corresponding product type. If positive inventory exists for a certain family, the corresponding demands of families are deducted. Otherwise, the demands are augmented by the backordered amounts. In this manner, possible infeasibilities arising from incorrect information are eliminated. In the following, we provide the mathematical formulation for the aggregate model. The objective (expression 0) is to minimize transportation costs of vehicles sent to warehouses from the factory and warehouse inventory costs. If backorders are permitted, the objective also includes costs of backordered amounts in warehouses. In expression (0), k is the warehouse index k ˆ 1; . . . ; 5), j indicates product type j ˆ 1; 2 for powder and liquid) and t is the period index t ˆ 1; . . . ; 6, an aggregate time bucket consists of two months). Nkt is the number of vehicles sent from factory to warehouse k in period t, Ijkt and Bjkt are the inventory and backorder quantity held of product j in warehouse k in period t, respectively. The coe cients of these variables are corresponding unit costs. min z ˆ

XX k

t

TCk Nkt ‡

XXX j

k

t

Hjkt Ijkt ‡

XXX j

k

t

Cjkt Bjkt

…0†

Expression (1) restricts the consumption of capacity by the production quantity Xjt to its available limit CAPj in each period. pj is the unit process time of product j in hours. pj Xjt

CAPj ;

8j ; t

…1†

8j; t:

…2†

In expression (2), the production quantity of each product type j is equated to the sum of transported quantities T jkt over all warehouses k. This expression implies that no inventories are held in the factory and ® nished goods are transported to their destinations within the period that they are produced. Xjt ˆ

X k

T jkt ;

In expression (3), the inventory balance equations are given for each warehouse, where Djkt is the demanded quantity of product j in warehouse k in period t. Again, when backorders are not permitted, the corresponding variables are omitted. Ijk;t¡1 ¡ Ijkt ¡ Bjk;t¡1 ‡ Bjkt ‡ T jkt ˆ Djkt ;

8j; k; t:

…3†

In expression (4), an integer number of vehicles are required to deliver the total quantity to be transported to warehouse k in period t. Here, V is the capacity of one vehicle. In expression (5), ending conditions are imposed on the backorders of last period.


3764

X j

T jkt

Nkt V ;

Bjk6 ˆ 0;

8k; t:

…4†

8j ; k

…5†

Expressions (6) and (7) indicate nonnegativity and integrality. Xjt; Ijkt ; Bjkt ; Nkt ; T jkt

0:

…6†


Nkt integer:

…7†

3.2. The disaggregation model In the disaggregation model the ® rst aggregate time period in the planning horizon is re® ned into 8 weeks. The six product families and setup variables are explicitly considered in the constraints. Here, j ˆ 1; . . . ; 6; k ˆ 1; . . . ; 5, and t ˆ 1; . . . ; 8. The production, transportation, backorder, inventory variables and the number of trucks are indicated in small CAPS to represent families and the weekly time bucket. The cost coe cients of inventories and backorders are also recalculated according to the re® ned time bucket. In expression (8) the objective function is given. min z 0 ˆ

XX k

t

TCk nkt ‡

XXX j

k

t

hjkt invjkt ‡

XXX j

k

t

cjkt bjkt

…8†

Expression (9) indicates the restriction of capacity capG for each group of families, G, G ˆ 1; 2; powder and liquid). But here, capG indicates the actual weekly capacity for production line G, i.e., capacity is not reserved for setup times. sj indicates setup time of family j and wjt is the binary setup indicator. Expression (10) links the production quantity of family j in period t with the corresponding binary variable (M is a big number). X j2G

… pj xjt ‡ sj wjt†

capG;

8G; t:

xjt

Mwjt

8j ; t

…9† …10†

Expression (11) achieves the equality between production, xjt, and transportation quantities, trjkt , whereas (12) implies inventory balance of each family j in warehouse k in period t: djkt is the weekly demand of family j in warehouse k in period t. The backorder variables are optional in (12). In (13) transported quantity is linked to the number of vehicles. xjt ˆ

X

trjkt ;

k

invjk;t¡1 ¡ invjkt ¡ bjk;t¡1 ‡ bjkt ‡ trjkt ˆ djkt X j

trjkt

nkt V

8j; t:

…11†

8j ; k ; t

…12†

8k; t

…13†

Expression (14) achieves the consistency of the sum of production quantities over all families in group G with the optimal aggregate production quantity XAG in the ® rst aggregate (2 months) time period of the planning horizon. Expression (15) provides a similar type of consistency between the sum of transported family quantities to warehouse k and the aggregate transported quantity TAGk obtained from the aggre-


3765

gate model in the ® rst aggregate period. Expression (16) achieves the consistency between the aggregate number of trucks NAk sent to warehouse k in the ® rst aggregate period and the sum of trucks sent to warehouse k over 8 weeks. As (16) involves an integer number of trucks for each week, the consistency equation turns into an inequality to obtain a feasible solution over 8 weeks. XX t

XX


t

xjt

XAG;

8G:

…14†

trjkt

TAGk

8k; G

…15†

NAk ;

8k:

…16†

j2G

j2G

X t

nkt

If backorders are permitted in the disaggregation model, then (17) is used to restrict the maximum level of backorders in any period by a percentage BP of demand in that period. Expression (18) imposes ending conditions on backorders. bjkt

BPdjkt ;

bjk8 ˆ 0;

8 j; k; t: 8j; k:

Expressions (19) and (20) indicate non-negativity and integrality. xjt; invjkt ; bjkt ; nkt ; trjkt nkt integer, wjt binary:

0:

…17† …18† …19† …20†

3.3. An iterative hierarchical approach 3.3.1. Analysis of the aggregate solution In obtaining an aggregate solution incorporating a 6-period planning horizon, we ® rst omit backorder variables from expressions (0) and (3). If a feasible solution is identi® ed, the next step is to read X11 and X21, and equate them to XA1 and XA2. Similarly, Nk1 is equated to NAk , T 1k1 is equated to TA1k and T 2k1 is equated to TA2k . Then, we proceed to ® nd a disaggregate solution. If a feasible aggregate solution cannot be obtained, then backorder variables are included in expressions (0) and (3), and a solution is obtained. Analysing the utilization rates of the capacity constraints, the planner becomes aware of future resource bottlenecks. Thus, an aggregate solution with backorders does not necessarily mean that there exist backorders in the ® rst planning period. 3.3.2. Analysis of the disaggregation The disaggregation model is solved with the aggregate consistency parameters provided by the aggregate solution. If the aggregate solution does not contain backorders, then the e orts of solving the disaggregate solution are concentrated on eliminating backorders as much as possible. Initially, in (17), the percentage BP is set to zero. Thus, backorder variables become ine ective in (8) and (12). If a feasible solution cannot be obtained in this manner, then BP is increased to a value of 8 (because, 8 weeks of backorder can be accumulated within one aggregate period and backorders carried from the previous aggregate period should be compensated for in the current period) and the disaggregation model is re-solved. If the solution is now


3766


feasible, the latter implies that there are weekly backorders due to ¯ uctuations in demand, but the consistency between the aggregate and disaggregate solutions is preserved in terms of production and transported quantities. Otherwise, (18) is omitted and the solution is allowed to have backorders at the end of the aggregate period. If no feasible solution is found with the latter interventions, then it can be argued that (17) should be omitted. However, this course of action leads to undesirable backorder levels. Therefore, instead of omitting (17), we convert expression (15) into an inequality of type . The reason is that too low inventory levels or backorders of some product families are carried from the previous aggregate period due to the myopic single period disaggregation model (no future demand information is provided). Hence, the transportation variables which act as production variables in the inventory balance constraints of warehouses cannot meet the augmented demands in the current aggregate planning period, and (15) has to be relaxed. If a feasible solution is not yet available, then (14) is now converted into an inequality of type . This implies that either capacity is not su cient to produce the imposed aggregate quantities or the implicit e ect of relaxing (15) is transmitted by (11) to (14). Inequality (16) is not relaxed in order to force the disaggregate formulation to result in logical production quantities. The reason is that transportation costs are incomparably higher than backorder costs and if these constraints are released the optimization package chooses to backorder almost all demand. However, the solution is analysed and if there exist redundant vehicles the cost of the solution is reduced accordingly. On the other hand, if the aggregate solution contains backorders in the ® rst period, then BP is set to eight and (18) is omitted from the disaggregation model and backorders are no longer restricted. If no feasible disaggregate solution is found, then the relaxation scheme described above is applied. In the disaggregate solution, there might be positive and negative weekly inventories due to high ¯ uctuations among weekly demands. Therefore, the levels of inventory over the 8-period planning horizon are optimized, but the cost of the disaggregate solution is inevitably higher than the corresponding period’s aggregate solution cost. The above set of rules to ® nd a feasible disaggregate solution are summarized below. 3.3.3.

Rules for solving the disaggregation model

(1) If the aggregate solution contains backorders for the ® rst aggregate period, then set BP to eight and go to Step 3. Else, go to Step 2. (2) Set BP to zero in the disaggregation model and solve it. If the solution is feasible, stop. Else, set BP to eight and re-solve it. If the solution is feasible, then stop. Else, go to Step 3. (3) Omit (18) in the disaggregation model and re-solve it. If the solution is feasible, stop. Else, convert (15) into an inequality of type and re-solve it. If the solution is feasible, stop. Else, convert (14) into an inequality of type and re-solve it. Stop, the solution obtained is feasible. 4.

Implementatio n of the hierarchical planning approach

The hierarchical planning approach described here is implemented on the data provided by the detergent company. Aggregate demand data is obtained from the



3767

past year’s sales data. However, weekly data are generated randomly by taking the weekly demand average (ˆ aggregate demand of 2 months/8 weeks) and adding or subtracting to each week’s demand, a random amount, which is held at three di erent levels equal to 0.25 weekly demand average, 0.5 weekly demand average and 1.0 weekly demand average, respectively. We impose the latter weekly demand ¯ uctuations in order to show the e ects of weekly ¯ uctuations on the robustness of the proposed approach. Tables 1, 2 and 3 demonstrate the inventory and backorder levels at the product family/warehouse detail, the aggregate inventory and the number of trucks sent to each warehouse, and the single period aggregate and disaggregate costs obtained from applying both models on a rolling horizon basis. However, here, new demand information is not added to the end of the planning horizon in order to discriminate the inconsistencies arising from the top down hierarchical approach rather the ones due to the incorporation of new information. In exploring the disaggregate solution in each period, 150 000 iterations are permitted in the execution of the GAMS optimization software (it takes 400± 600 CPU s. on a Pentium 233 MMX machine). Unfortunately, even a million iterations are insu cient to verify the optimal solution as the number of integer and binary variables is not small. Therefore, we take the best integer solution identi® ed within 150 000 iterations. In the tabulated results, we also indicate which of the rules are imposed in the solution of the disaggregated formulation. In the corresponding notation, the ® rst rule is to set BP= 8; the second rule adds on the omitting of the ending conditions on backorders; the third one adds on the relaxation of transportation consistency; and the fourth one adds on the relaxation of the production quantity consistency. We remark that the fourth rule is only applied once in all tables, and that is period 5 in table 1. In this period, the di erences between the aggregate and disaggregate production quantities are 155 tons and 302 tons for the two product types, respectively. For this period only, we also tested the case where (17) is omitted rather than relaxing (15). As expected, the cost of the disaggregated solution turned out to be 9924 697 which is higher than that of the proposed approach, 9 138 593 and ending backorders are at an unacceptable level. The third rule is applied once in all tables. The most common way to resolve inconsistencies in the distribution of product type production quantities among families is to relax the ending conditions on backorders. It is also indicated in the aggregate results wherever backorders are permitted. It is possible that the aggregate solution does not result in backorders in the current planning period, but projects them into future periods. In this case, we start the solution of the disaggregate formulation by assuming that no backorders exist in the aggregate solution. We remark that in table 1, the ® rst aggregate solution’s annual total cost is equal to 47 180 100 ( 10 000 TL) whereas the total aggregate cost on a rolling horizon basis is 48 028 440 TL. The total disaggregate solution cost which is re® ned in terms of capacity, time periods and product groups is equal to 49 111 384 TL. which is 4.09% higher than the aggregate solution obtained in the ® rst period. In our opinion, the relative performance of the disaggregated solutions is satisfactory in spite of the ¯ uctuations of demand during the re® ned periods which lead to additional inventory and backorders. In all tables, the percentages of total inventory and backorder quantities to total demand are provided both for the aggregate solution and the disaggregate one over the whole planning horizon. Although there is some discre-


3768

Periods


Aggregate solution

Inv. of prod. type 1 (tons) Inv. of prod. type 2 (tons) No. of trucks (Ist/Eur) No. of trucks (Ist/Ana) No. of trucks (Ankara) No. of trucks (Adana) No. of trucks (Izmir) (trans. cost = 0) Total number of trucks Cost of single period agg. sol. (* back. permitted) % of total inv./back. to total demand (type 1) % of total inv./back. to total demand (type 2) Total no. of trucks Total aggregat e cost

1

2

3

4

5

6

0 113 22 22 22 10 12

716 303 24 26 30 10 62

620.5 707.5 34 40 40 18 60

0 212.5 40 50 56 24 12

88 5 150 800

152 6 411 700

192 9 433 300*

182 11 410 670*

188 9 164 770*

428

100.5

7 28

711 784.5

0

7284

36 42 42 16 52

0 0 26 32 28 12 40 138 6 457 200

4.3/0.0 6.2/1.3 940 48 028 440

Disaggregat e solution

Family A Inventory Inventory Inventory Inventory Inventory

(Ist/Eur) (tons) (Ist/Ana) (tons) (Ankara) (tons) (Adana) (tons) (Izmir) (tons)

Family B Inventory Inventory Inventory Inventory Inventory


Family C Inventory (Ist/Eur) (tons) Inventory (Ist/Ana) (tons) Inventory (Ankara) (tons) Inventory (Adana) (tons) Inventory (Izmir) (tons) % of total inv./back. to total demand (type 1) Family D Inventory Inventory Inventory Inventory Inventory


Family E Inventory Inventory Inventory Inventory Inventory


Family F Inventory (Ist/Eur) (tons) Inventory (Ist/Ana) (tons) Inventory (Ankara) (tons) Inventory (Adana) (tons) Inventory (Izmir) (tons) % of total inv./back. to total demand (type 2) Number of Trucks in Disagg. Sol. Cost of Single Period Disagg. Sol. I:BP= 8; II: BP= 8+ (18); III: (II)+ (15); IV: (III)+ (14) Total no. of trucks Total disaggregate cost

7 14

93

7.1/1.9

718 719 713 711 718.833

11

38 29 6.7/5.8 112 5 601 181

7 14

770 768 740 714 783.667

35

35 232 160 6 682 675

716 758

344

225

104 87 53 25 794 198 9 471 516 II

711 75.5

714

7448

7176

10

716.5 78 760

748 725

720 715 736.5 78 735 723 734 717

198 11 522 682 II

192 9 138 593 IV

144 6 694 737

1004 49 111 384

Table 1. Results of aggregate and disaggregate solutions on a rolling horizon basis (demand variability = average 0.25). Costs are in 10 000 TL.

3769

Hierarchical distribution system Periods


Aggregat e solution


1

2

3

4

0 113 22 22 22 10 12

716 303 24 26 30 10 62

686 652 34 40 40 12 64

7 20 7303

756

88 5 150 800

152 6 411 700

190 8 927 500*

192 11 696 580*

192 9 407 780*

154

377

73 75

40 48 54 28 22

5

6

36

0 0 26 30 26 12 34

36 42 44 18 52

130 6 208 500

9.0/0.1 5.0/1.7 942 47 802 860










Family F Inventory (Ist/Eur) (tons) Inventory (Ist/Ana) (tons) Inventory Ankara Inventory (Adana) (tons) Inventory (Izmir) (tons) % of total inv./back. to total demand (type 2) Number of Trucks in Disagg. Sol. Cost of Single Period Disagg. Sol. I:BP= 8; II: BP= 8+ (18); III: (II)+ (15); IV: (III)+ (14) Total no. of trucks Total disaggregate cost

220

8.6/0.6

35 22 232

14 26 10.3/7.7 164 6 749 981

722 721 75

712

18 14 713 777 237

34

114 5 146 931

3

19

715

342

49

4

107

72

7111 7173

59 11 103 45 219

7165 7176 7189 7113

726 77 724

120

137 287 140 113 753

733 74 779 79 7211

200 8 867 479 III

202 12 433 402 II

19 15 730 238 72 72 795

740

202 9 952 425 II

134 6 306 693

1016 49 456 911

Table 2. Results of aggregate and disaggregate solutions on a rolling horizon basis (demand variability = average 0.50). Costs are in 10 000TL.


3770

Periods


Aggregate solution


1

2

3

0 113 22 22 22 10 12

716 303 24 26 30 10 62

711 744 34 40 40 18 62

88 5 150 800

152 6 411 700

4

5 0

0

¡161

¡66

194 9 497 400

192 11 088 220*

190 9 148 195*

713.5

108.667

7 24 7 91 75.667 73 7 20

40 48 56 22 26

36 44 42 16 52

6 0 0 26 30 26 12 28 132 6 208 500

8.2/0.0 4.4/0.3 938 47 504 815










Family F Inventory (Ist/Eur) (tons) Inventory (Ist/Ana) (tons) Inventory (Ankara) (tons) Inventory (Adana) (tons) Inventory (Izmir) (tons) % of total inv./back. to total demand (type 2) Number of trucks in disagg. sol. Cost of single period disagg. sol. I: BP= 8; II: BP= 8+ (18); III: (II)+ (15); IV: (III)+ (14) Total no. of trucks Total disaggregate cost

425

73 719

5.667 84

220.333

720

2 91

7.7/1.6

49

207

197

14

21

43

35 232

3

7 5.5

722

719

7177

716.5 758 7166.5

733

23 292.657

713

33 22 4.8/1.1 118 5 260 987

160 6 573 993

328.333

7 18

48.5 281.5

204 9 583 785 I

202 11 628 527 II

194 9 616 182 II

132 6 276 525

1010 48 939 999

Table 3. Results of aggregate and disaggregate solutions on a rolling horizon basis (demand variability = average 1.0). Costs are in 10 000 TL.


3771

pancy between aggregate and disaggregate percentages for the ® rst product type, this di erence is higher for the second product type. As for the number of trucks sent in the disaggregate solution, 5 more trucks are used on the average in each aggregate period due to integrality constraints imposed in each of 8 weeks. We further remark that the least discrepancy between the aggregate and disaggregate solutions is encountered in table 3 where weekly demand ¯ uctuations are highest. Thus, we observe that demand ¯ uctuations do not have signi® cant e ects on the inconsistency between the aggregate and disaggregate solutions and the proposed hierarchical approach is quite robust against such external factors.


5.

Conclusion

We propose a hierarchical production-distribution planning approach for a factory which transports goods to geographically distant warehouses. In this system, warehousing and transportation costs are parts of the planning problem. The hierarchical planning approach attempts to solve the problem optimally by aggregating the time periods and product families while omitting detailed capacity consumption by setup. The aggregate model’s optimal solution is disaggregated for a single period on a rolling horizon basis with the aim of reducing problem size. This approach is tested on data obtained from a detergent manufacturer which distributes ® nished goods among ® ve warehouses. The simulated implementation of the approach demonstrates that the sub-optimality of the hierarchical approach is at most 3± 4% when compared with the aggregate solution’s. References BOWERS, M. R. and JARVIS, J. P., 1992, A hierarchical production planning and scheduling model. Decision Sciences, 23, 144± 159. BURNS, L. D., HALL, R. W., BLUMENFELD, D. E. and DAGANZO, C. F., 1985, Distribution strategies that minimize transportation and inventory costs. Operations Research, 33, 469± 490. CARAVILLA, M. A. and DE SOUSA, J. P., 1995, Hierarchical production planning in a make-toorder company: A case study. European Journal of Operational Research, 86, 43± 56. CHANDRA, P., 1993, A dynamic distribution model with warehouse customer replenishment requirements. Journal of Operational Research Society, 44 , 681± 692. CHANDRA, P. and FISHER, M. L., 1994, Coordination of production planning and distribution planning. European Journal of Operational Research, 72, 503± 517. COHEN, M. A. and LEE, H. L., 1988, Strategic analysis of integrated production-distribution systems: models and methods. Operations Research, 36 , 216± 229. EDGHILL, J., OLSMATS, C. and TOWILL, D., 1988, Industrial case-study on the dynamics and sensitivity of a close-coupled production-distribution system. International Journal Production Research, 26, 1681± 1693. ERSCHLER, J., FONTAN, G. and MERCE, C., 1986, Consistency of the disaggregation process in hierarchical planning. Operations Research, 34 , 464± 469. FLEISCHMAN, B., 1993, Designing distribution systems with transport economies of scale. European Journal of Operational Research, 31± 42. HAX, A. C. and MEAL, H. C., 1975, Hierarchical integration of production planning and scheduling. In M.Geisler (ed.) TIMS Studies in Management Science, vol. 1, L ogistics, (New York: Elsevier). GABBAY, H., 1975, A hierarchical approach to production planning. Technical report no. 120, Operations Research Center, MIT, USA. KIRA, D., KUSY, M. and RAKITA, I., 1997, A stochastic linear programming approach to hierarchical production planning. Journal of the Operational Research Society, 48, 207± 211.


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