PRODUCTION PLANNING & CONTROL, VOL. 14, NO. 5, JULY–AUGUST 2003, 425–436
Multi-site aggregate production planning with multiple objectives: a goal programming approach STEPHEN C. H. LEUNG, YUE WU and K. K. LAI
Keywords aggregate production planning, multiple criteria decision-making, optimization, multi-site planning Abstract. This paper addresses the problem of aggregate production planning (APP) for a multinational lingerie company in Hong Kong. The multi-site production planning problem considers the production loading plans among manufacturing factories subject to certain restrictions, such as production import/export quotas imposed by regulatory requirements of
different nations, the use of manufacturing factories/locations with regard to customers’ preferences, as well as production capacity, workforce level, storage space and resource conditions of the factories. In this paper, a multi-objective model is developed to solve the production planning problems, in which the profit is maximized but production penalties resulting from going over/under quotas and the change in workforce level are minimized. To enhance the practical implications of the proposed model, different managerial production loading
Authors: Stephen C. H. Leung, Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong, E-mail address:
[email protected] STEPHEN C. H. LEUNG is a Lecturer in the Department of Management Sciences at City University of Hong Kong. He received his PhD in Transportation Planning from City University of Hong Kong. Before joining the teaching profession, Dr Leung was a consultant in Wilbur Smith Associate Ltd. He had been involved in a variety of projects about transport modelling and traffic impact assessment. His current research interests are in logistics management, operations management and multiple-criteria decision-making.
YUE WU is a Senior Research Associate in the Department of Management Sciences at City University of Hong Kong. She received her PhD in Control Theory and Control Engineering from Northeastern University in China. Her research interests are in production planning and scheduling, computer-integrated manufacturing systems, intelligent computing and supply-chain management.
K. K. LAI is a Professor of Operations Research and Management Sciences at City University of Hong Kong, and he is also the Associate Dean of the Faculty of Business. Prior to his current post, he was an Operational Research Analyst at Cathay Pacific Airways and an Area Manager on Marketing Information Systems at Union Carbide Eastern. Professor Lai received his PhD at Michigan State University, USA. His main areas of research interests are in logistics and operations management, computer simulation and business decision modelling.
Production Planning & Control ISSN 0953–7287 print/ISSN 1366–5871 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0953728031000154264
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plans are evaluated according to changes in future policy and situation. Numerical results demonstrate the robustness and effectiveness of the developed model.
1. Introduction Because of rapid developments in the globalization of markets and international trade, managing operations in today’s competitive marketplace poses a significant challenge for companies. The concept of production management has evolved beyond the scope of a single manufacturing location. Increased competition, and the coordination and control of production activities of factories spread across countries have become more important than ever. Companies are increasingly devoting themselves to international expansion and the integration of functions such as production, marketing and R&D, as well as international collaboration and networking with other firms and institutions in order to gain competitive advantages (Ballou 1992). This study is particularly motivated by the problem faced by a multinational lingerie company manufacturing a number of product types, which has its headquarters in Hong Kong, product sales, R&D and customer service spread across North America and Europe, and manufacturing factories located in China, the Philippines, Thailand and other Southeast Asian countries. The management in Hong Kong headquarters collects orders through its American and European sales branch offices. The orders consist of the type of products, quantity, delivery date and location preference. Decision-makers then develop an initial aggregate production loading plan every three months based completely on their experience and judgement on yielding maximum revenue. In the planning process, they have to consider the manufacturing capacity, workforce level, inventory level, quota availability and other factors to fulfil forecasted demand. According to the production loading plans, the factories are assigned a list of products with quantities to be produced for each period of time. Obviously, such a decision-making process is costly and time-consuming. It is not surprising to see that existing production loading plans are significantly far different from optimal plans. Choosing the right production strategies at the right levels are highly complex decisions. Solving aggregate production planning problems has become a critical management task for the company. The aim of this paper is to formulate a model for aggregate production planning problems to maximize revenue as well as minimize production cost, labour cost, inventory cost, back-order cost, penalties for over/ under quota and other relevant costs. The characteristics
of the aggregate production planning problem are summarized as follows. . The headquarters of the multinational lingerie company is in Hong Kong and sales branches and manufacturing factories are located in different countries and regions. Guinet (2001) called several production units as sites that are located in various countries to supply irregular demands at lower costs. The costs are associated with processing cost, holding cost, set-up cost, delay cost, etc. The problem considered in this paper is a multi-site aggregate production planning problem. . Labour costs vary for different products and for different factories. The company’s products are mainly divided into two groups: cotton products which contain at least 90% cotton and silk products which are mainly made of silk. Naturally, labour costs for manufacturing silk products are high because the workers are required to be skilled and experienced. Nam and Logendran (1992) have stated that the value of worker is not homogeneous. The models assuming homogeneous workers’ value are not realistic. . Usually, the orders from North America are required to be processed in Chinese factories because of the required quality of materials from China such as silk. European customers also favour Chinese factories, but do not object to the use of the factories in the Philippines or Sri Lanka as substitutes. In many circumstances, the management needs to negotiate between factories and customers to ensure that the manufacturing location preference can be fulfilled. Manufacturing location preference plays an important role in the production planning process. Guinet (2001) claimed that, nowadays, faced with the open nature of international markets all over the world, competitive firms have to fulfil customer requirements which depend on country habits, and consider national legislation. . Main restrictions of the loading plan involve limits imposed by import quotas for the industry, the manufacturing country’s requirements, and the technical level of the factories. Import quotas, initially assigned by the importing country’s government, can be legally traded on the local market, from either other manufacturing or trading companies or agencies. Both the headquarters and some factories can buy quotas from these sources. If quotas are insufficient, the company has to postpone or reject orders, and this often happens in the period before a seasonal change. Product supply exceeding quotas would cause extra inventory costs or require the products to be sold at bargain prices on local
Multi-site aggregate production planning with multiple objectives markets. A similar situation has been considered in Tabucanon and Mukyangkoon (1985). By agreement among the factories in the industry and the government’s Board of Investment, the government set a specific production level that should be achieved in a year due to economic considerations. Then factories established production loading plans for their production goals. In addition, manufacturing location preferences also strongly conflicted with the quota availability of the nations involved. Traditionally, the objective of aggregate production planning is either to maximize profit or minimize cost and is formulated to a single-objective function in linear programming. Recently, many researchers and practitioners are increasingly aware of the presence of multiple objectives in real-life problems (Vincke 1992). Baykasoglu (2001) noted that the aggregate production planning may consider the minimization of cost, inventory levels, changes in workforce levels, production in overtime, subcontracting, changes in production rates, number of machine set-ups, plant/personnel idle time; and maximization of profits and customer service. Decision-makers always want to develop a model that can consider the real-life situation with multiple objectives. To achieve this, in this paper, a goal programming model is formulated to determine an optimal production loading plan. As opposed to linear programming, which directly optimizes objectives, goal programming is used to manage a set of conflict objectives by minimizing the deviations between the target values and the realized results (Rifai 1994). The original objectives are re-formulated as a set of constraints with target values and two auxiliary variables. Two auxiliary variables are called positive deviation d þ and negative deviation d , which represent the distance from this target value. The objective of goal programming is to minimize the deviations hierarchically so that the goals of primary importance receive firstpriority attention, those of second importance receive second-priority attention, and so forth. Then, the goals of first priority are minimized in the first phase. Using the obtained feasible solution result in the phrase, the goals of second priority are minimized, and so on. The explicit definition of goal programming was given by Charnes and Cooper (1961). This paper proposes a goal programming approach to multi-site aggregate production planning with multiple objectives such as the maximization of profit, minimization of the change of workforce level and maximization of utilization of import quota. By adjusting the hierarchy priority with respect to each objective, decision-makers and management will find the flexibility and robustness of the proposed model.
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This paper is organized as follows. After this introductory section, relevant researches on aggregate production planning are reviewed. A pre-emptive goal programming model is formulated for the aggregate production planning problems in section 3. In section 4, a set of data from Hong Kong is used to test the effectiveness and efficiency of the proposed model. In order to match future requirements, a variety of demand data and a change in production loading are also demonstrated so that decision-makers can handle unforeseen and complicated changes. Our conclusions are given in the final section.
2. Literature reviews The above-mentioned problem is well known as a multi-site aggregate production planning (APP) problem with multiple objectives. Nam and Logendran (1992) stated that APP is intended to translate forecasted sales demand and production capacity into future manufacturing plans for a family of products, in which the human and equipment resources of a company are fully utilized. The optimal plan consists of production, inventory and employment levels over a given finite planning horizon for all products so that the total forecasted sales demands are fulfilled with limited resources. Baykasoglu (2001) defined aggregate production planning as medium-term capacity planning over a 2–18-month planning horizon. As an aggregate level of planning, there is no need for APP to provide detailed material and capacity resource requirements for individual products and detailed schedules for facilities and personnel (Baykasoglu 2001). The most updated state-of-the-art review on models and methodologies for APP can be found in Nam and Logendran (1992), in which 140 journal articles and 14 books were categorized into optimal and near-optimal classifications with respect to models and methodologies. Although a wide variety of APP techniques have been developed since the early 1950s, there is still no widespread acceptance of these techniques in industry. One reason is that these techniques do not accurately reflect the APP process in the real world since they treat APP as a top-down constraint while managers often regard it as a bottom-up approach. Another reason is the underlying assumption that all products/product families are homogeneous and are capable of aggregation into some common measure. This complaint extends to the treatment of labour units as well. The assumption that they are equivalent contradicts real-life situations where some workers are more valuable than others and thus not equal where hiring and firing costs are concerned. A similar argument can be found in Mazzola et al. (1998).
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Masud and Hwang (1980) presented three criteria objective decision-making methods, which are goal programming (GP), step method (STEM) and sequential multiple objective problem-solving (SEMOPS), for an aggregate production planning problem with four objectives. The four objectives are maximization of contribution to profit, minimization of changes in workforce level, minimization of inventory investment and minimization of back orders. A set of data consisting of two products, a single production plant and eight periods of planning horizons was generated to compare the results. Wang and Fang (2001a) developed a linear programming method for solving Masud and Hwang’s model with fuzzy parameters such as fuzzy demand, fuzzy machine time, fuzzy machine capacity and fuzzy relevant costs. The proposed fuzzy linear programming was reformulated to a linear programming with crisp objective functions and crisp constraints using partial ordering of fuzzy numbers. The first six periods of planning horizon in Masud and Hwang’s data set was used to demonstrate the feasibility of applying the proposed method. Wang and Fang (2001b) also proposed a model for fuzzied objective coefficients and fuzzied decision variables in the APP. The solutions represented by fuzzy numbers provide more flexibility to the decision-maker with an uncertain environment. Baykasoglu (2001) extended Masud and Hwang’s model with additional constraints such as subcontractor selection and set-up decisions. The multiple-objective tabu search algorithm was designed to solve the preemptive goal programming model. An object-oriented program in Cþþ named as MOAPPS 1.0 (Multiple Objective Aggregate Production Planning Software) was developed and used to compare Masud and Hwang’s model and the extended model. Byrne and Bakir (1999) proposed a hybrid algorithm composed of mathematical programming and simulation models for the multi-period multi-product production planning problem, which gives a production plan with mathematical optimality and practical feasibility. Hung and Hu (1998) formulated a mixed integer programming model for the production planning problems with set-up decisions. Owing to the NP-hard of the problems, a heuristic algorithm was developed to simultaneously solve the objectives, which are maximization of revenue as well as minimization of inventory cost, back-order cost and set-up cost. In addition, a series of artificial intelligent approaches such as evolutionary algorithms (Hung et al. 1999), genetic algorithms (Wang and Fang 1997) and decision support systems (Ozdamar et al. 1998) are also widely used to solve production planning problems with additional constraints and limitations.
When it comes to real situations, it is not surprising to see that researchers and practitioners integrate production planning problems with other planning problems such as scheduling problems (Foote et al. 1988, Buxey 1993) and workforce planning problems (Mazzola et al. 1998). In this study, the proposed multi-objective model is an extended version of the APP problem developed by Masud and Hwang (1980) and is a modified version of the works by Baykasoglu (2001) with additional constraints according to the real situation in the production planning problem faced by the company in Hong Kong. For instance, apart from regular workers who work at normal time and can work overtime as considered in Masud and Hwang (1980), in order to increase production loads, a company will recruit a group of parttime workers who are not fully trained and do not have extensive experience. Therefore, the proposed model in this paper can be regarded as a generalization of Masud and Hwang’s model and Baykasoglu’s model.
3. Model formulation In this study, a multi-site aggregate production planning problem faced by a multinational lingerie company in Hong Kong is investigated. For cost effectiveness, the decision-makers have to determine the quantity of product i, i ¼ 1, 2, . . ., n, manufactured from factory j, j ¼ 1, 2, . . ., m, to fulfil future sales demand in each period of time t, t ¼ 1, 2, . . ., T. In addition, some customers may favour the products manufactured from Chinese factories, CF, where a higher quality of products is expected. In this section, the formulation of the model is described. The parameters and decision variables used throughout the paper are firstly defined. Then, the various goal constraints corresponding to various goals, and the lexicographical objective function are presented in section 3.2. At last, the system constraints are formulated in section 3.3.
3.1. Notations Indices: i : the index of the product j : the index of the plant t : the index of the planning period Sets: I J T CF
: : : :
the the the the
set set set set
of of of of
the the the the
products plants planning periods Chinese factories
Multi-site aggregate production planning with multiple objectives Parameters: ri : the unit revenue of product i ($/unit) C1ij : the unit production cost for product i manufactured from factory j by experienced workers at regular time ($/unit) C2ij : the unit production cost for product i manufactured from factory j by non-experienced workers at regular time ($/unit) C3ij : the unit production cost for product i manufactured from factory j by experienced workers at overtime ($/unit) C4j : the labour cost of experienced worker in factory j at regular time ($/man-period) C5j : the labour cost of experienced worker in factory j at overtime ($/man-hour) C6j : the labour cost of non-experienced worker in factory j ($/man-hour) C7j : the unit inventory cost to hold a product i in factory j at the end of each period ($/unit) C8j : the unit back-order cost for a product i in factory j at the end of each period ($/unit) C9j : the cost to hire one experienced worker at factory j ($/man) C10j : the cost to lay off one experienced worker at factory j ($/man) Iij0 : the initial inventory of product i at the start of planning horizon in factory j (units) Bij0 : the initial outstanding back-order of product i at the start of planning horizon in factory j (units) ui : the labour time for product i by experienced worker (man-hour/unit) vi : the labour time for product i by non-experienced worker (man-hour/unit) : the working hour of experienced worker in each period (man-hour/man-period) : the fraction of workforce available for overtime use in each period " : the fraction of workforce allowable for variation in each period i : the fraction of product i required to be manufactured from Chinese factories Mjt : the machine time capacity in factory j at period t (machine-hour) li : the machine time for product i operated by experienced worker (machine-hour/unit) i : the machine time for product i operated by non-experienced worker (machine-hour/unit) : the fraction of machine capacity available for overtime use in each period Sitmin : minimum known demand of product i in period t (units) Sitmax : maximum forecasted demand of product i in period t (units)
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WEjtmin : minimum worker-force level of experienced worker available in factory j in period t (man-period) WEjtmax : maximum worker-force level of experienced worker available in factory j in period t (man-period) BUFjt : maximum storage capacity in factory j in period t (units) STKit : maximum back order of product i in period t (units) bp : the aspiration level of profits to be achieved bw : the aspiration level of cost of hiring or layingoff workers : the aspiration level of quantity of import quota bi of product i Variables: : the quantity of product i sold in period t (units) Sit : the quantity of product i manufactured from xijt factory j by experienced worker at regular time in period t (units) : the quantity of product i manufactured from yijt factory j by non-experienced worker at regular time in period t (units) : the quantity of product i manufactured from zijt factory j by experienced worker at overtime in period t (units) : the number of experienced workers required in WEjt factory j in period t (man-period) : the number of experienced workers hired in Hjt factory j in period t (man-period) : the number of experienced workers laid-off in Ljt factory j in period t (man-period) : the overtime of experienced worker in factory j TEjt in period t (hour) : the labour time of non-experienced worker in TNjt factory j in period t (hour) : the inventory of product i in factory j at the end Iijt of period t (units) : the back order of product i in factory j at the Bijt end of period t (units) : the deviation variable of underachievement dp of the goal bp : the deviation variable of overachievement dpþ of the goal bp : the deviation variable of underachievement dw of the goal bw : the deviation variable of overachievement dpþ of the goal bw : the deviation variable of underachievement di of the goal bi diþ : the deviation variable of overachievement of the goal bi
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3.2. Goal constraints and objective functions The aim of this study is to find an optimal production loading plan with maximal profit by fulfilling market demand. The aggregate production loading plan consists of production quantities by experienced workers at regular time and at overtime, and by non-experienced workers for each period of time. With the plans, decision-makers can also determine inventory level, back-order level and workforce level. To achieve the optimal plan, three goals are considered in this study. Goal 1: Profit goal XX
ri Sit
XXX
i2I t2T
ð2Þ
j2J t2T
The first component in expression (2) is the cost of hiring extra workers at each period of time. The second component is the cost of laying-off redundant workers at each period of time. The two deviation variables, dwþand dw, represent the overachievement and underachievement of the aspiration level of hiring and layoff goal. This also gives dwþdw ¼ 0. Parameter bw denotes the desired cost management is willing to pay for hiring and laying-off workers. If the company wants to consider the costs for hiring or laying-off the non-experienced worker, equation (2) can be modified accordingly.
i2I j2J t2T
XX
XX C4j WEjt C5j TEjt
j2J t2T
j2J t2T
XX
XXX
C6j TNjt
j2J t2T
C1ij xijt þ C2ij yijt þ C3ij zijt
workers is formulated in (2): XX C9j Hjt þ C10j Ljt dwþ þ dw ¼ bw
XXX
C7ij Iijt
i2I j2J t2T
C8ij Bijt dpþ þ dp ¼ bp
ð1Þ
i2I j2J t2T
This first component in expression (1) is the total revenue based on the quantity of product sales. The second component is the production cost. The third component is the labour cost of experienced workers at regular time. The fourth component is the labour cost of experienced workers at overtime. The fifth component is the labour cost of non-experienced workers. The sixth and seventh components are the inventory cost and the back-order cost respectively. Parameter bp denotes the target profit that management in the company pursues to achieve. A positive deviational variable, dpþ, represents the overachievement of the aspiration level of profit goal, and a negative deviational variable, dp, represents the underachievement of the aspiration level of profit goal. This gives dpþdp ¼ 0. It is noted that no non-experienced will be recruited to work in overtime under current company policy in order to maintain the quality of products. Certainly, expression (1) can be modified accordingly if the company wants to consider the production by non-experienced workers at overtime. Goal 2: Hiring and layoff goal Subject to production loading and market demand, in each period of time, management has to determine how many additional experienced workers need to be recruited to handle extra production loading or redundant experienced workers to be laid-off to reduce overheads. The cost of hiring and laying-off experienced
Goal 3: Availability and utilization of import quota Some of the main restrictions in the loading plan involve limits imposed by import quotas for the industry, manufacturing country requirements and the technical level of the factories. Import quotas, initially assigned by the importing country’s government, can be legally traded on the local market, from either other manufacturing or trading companies or agencies. Both the headquarters and some factories can buy quotas from these sources. If quotas are insufficient, the company has to postpone or reject orders, and this often happens in the period before a seasonal change. Product supply exceeding quotas would cause extra inventory costs or would require the products to be sold at bargain prices on local markets. X Sit diþ þ di ¼ bi i 2 I ð3Þ t2T
The available import quota of product i is bi representing the aspiration level of utilization of import quota, and the corresponding deviation variables are diþ and di . The objective function of the model is to minimize the deviation variables corresponding to various goals. The company assigns the highest priority to its profit goal, to achieve or even exceed the target profit set by management. Therefore, the undesirable deviational variable in the first priority goal is underachievement of the goal bp, i.e., dp should be minimized. Then, the first priority function in the objective function is zp ¼ dp. In the second priority of strategic production plan, the company wishes for costs associated with hiring or laying-off of workers to be as low as possible. Therefore, the undesirable deviational variable in the second priority goal is overachievement of the goal bw i.e., dwþ should be minimized. Then, the second priority function in the objective function is zW ¼ dwþ. Finally, the deviational variables in the remaining goals may be clustered into K priority groups according to their importance for the availability
431
Multi-site aggregate production planning with multiple objectives or utilization of import quotas of different products. The variables within each priority group may be the combination of several overachievements and underachievements of goal bi , i.e., X þ di þ di for some i
should be minimized. Then, the third, fourth, . . ., Kth, (K þ 1)th and (K þ 2)th priority goals in the objective function are z1, . . ., zK, where X þ zj ¼ di þ di
and lower bound on the sales of each product in a period are provided by constraint (6). Constraint (7) ensures that the inventory cannot exceed storage space limitation. Constraint (8) ensures that the backorder of each product cannot exceed the specified amount in order to maintain a favorable customer service standard. Relationship among the number of workers: WEjt ¼ WEjt1 þ Hjt Ljt
j 2 J, t 2 T
ð9Þ
WEjtmin WEjt WEjtmax
j 2 J, t 2 T
ð10Þ
X
for some i
Correspondingly, the aspiration levels for these priority goals are the sum of bi in expression (3) and denoted by bz1 , . . . , bzK , where X bz1 ¼ bi
X
The objective functions formulated in the previous section are restricted by four sets of constraints. They are the inventory-level constraints, the relationship among the number of workers, the production capacity constraints and the non-negative constraints. The inventory-level constraints: X Iijt1 Bijt1 þ xijt þ yijt þ zijt Iijt þ Bijt j2J
i 2 I, t 2 T
ð5Þ
Sitmin Sit Sitmax i 2 I, t 2 T X Iijt BUFjt j 2 J, t 2 T
ð6Þ ð7Þ
j 2 J, t 2 T
ð12Þ
ui zijt ¼ TEjt
j 2 J, t 2 T
ð13Þ
TEjt WEjt ,
j 2 J, t 2 T
ð14Þ
i2I
Hjt þ Ljt "WEjt1
j 2 J, t 2 T
ð15Þ
Constraint (9) ensures that the available workforce in any period equals the workforce in the previous period plus the change of workforce level in the current period. The change of workforce level may be due to either hiring extra workers or laying-off redundant workers. It is noted that Hjt Ljt ¼ 0 because either net hiring or net laying-off of workers takes place in a period, but not both. Constraint (10) ensures the upper and lower bound of change in workforce level in a period are provided. Constraint (11) limits the regular time production to the available workers. Constraints (12) and (13) determine the labour time of non-experienced workers and overtime of experienced workers respectively. Constraint (14) limits the overtime hours of the available workers. Constraint (15) ensures that the change in workforce level cannot exceed the proportion of workers in the previous period. The production capacity constraints:
i2I
X
ð11Þ
vi yijt ¼ TNjt
X
The objective function of the multi-objective multi-site aggregate production planning problem is formulated in the following: Lexicographicallyminimize z ¼ zP , zW , z1 , . . . , zK
3.3. Constraints
j 2 J, t 2 T
i2I
for some i
¼ Sit
ui xijt WEjt
i2I
Bijt STKit
i 2 I, t 2 T
ð8Þ
j2J
Constraint (5) ensures that total sales of each product in a period plus the inventory (or back order) at the end of the period equal the total supply consisting of inventory (back order) from the previous period plus the regular and overtime production in the current period. It is assumed that there is no initial quantity of back order. Moreover, it is noted that IijtBijt ¼ 0 because either inventory or back order will be in solution. The upper
n X i¼1 n X i¼1
i xijt þ
n X
i yijt Mjt
j 2 J, t 2 T
ð16Þ
i¼1
i zijt Mjt
X
j 2 J, t 2 T
xijt þ yijt þ zijt ¼ i Si
i 2 I, t 2 T
ð17Þ ð18Þ
j2CF
Total production in each period of time by experienced workers and non-experienced workers and overtime production by experienced workers at regular
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S. C. H. Leung et al.
time are limited by the available production capacity by constraints (16) and (17) respectively. Constraint (18) ensures that an amount of products are manufactured from Chinese factories. Non-negative constraints: Sit , xijt , yijt , zijt , WEjt , TEjt , TNjt , Hjt , Ljt , Iijt , Bijt , dpþ , dp , dwþ , dw , diþ , di 0 i 2 I, j 2 J, t 2 T
ð19Þ
Constraint (19) ensures that all decision variables are non-negative.
4. Computational results In order to illustrate the flexibility of the proposed model for the multi-site aggregate production planning, we use the data provided by the multinational lingerie company in Hong Kong. The tactical/operational level of decision-making in the aggregate planning process was described. Based on the projection report from the company, a three-month planning horizon is determined. The company receives sales orders from its sales branches covering America and Europe. Each order may require one or more of six products, i ¼ 1, 2, . . ., 6. The products are manufactured from three main factories, j ¼ 1, 2, 3, located in China, the Philippines and Thailand. The production season of orders is the summer, from June to August, precisely 12 weeks, t ¼ 1, 2, . . ., 12. The production cost, inventory cost and back-order cost for
different products in each factory are shown in table 1. The labour cost, hiring cost and lay-off cost and the relevant data for the workforce with regard to factories are shown in table 2. For each weekly period, the required product quantities from the North American and European markets are shown in table 3 and the forecast for product quantities determined by the company are shown in table 4. The production capacities of three factories in the planning horizon are shown in table 5. The revenue of six products and import quotas obtained through different channels are also shown in table 6. Finally, the labour production time and machine time are shown in table 7. The other parameters used are ¼ 8, ¼ 0.3 and ¼ 0.4. The target value of profit, bp, is $48,000,000 and the target value of cost of hiring and laying-off of workers, bw, is $250,000. After consulting the company’s management, the deviation variables to be minimized in þ the first priority is zP ¼ dP ; the second priority zW ¼ dW . The remaining three priorities are z1 ¼ d6þ þ d6 ,z2 ¼
5 X
3 X diþ þ di and z3 ¼ diþ þ di
i¼4
i¼1
corresponding to the importance of the availability and utilization of import quotas of different products. Therefore, the aspiration level for z1 is bz1 ¼ b6 ¼ 25,010, z2 is bz2 ¼ b4 þ b5 ¼ 33,560 þ 15,780 ¼ 49,340 and for z3 is bz1 ¼ b1 þ b2 þ b3 ¼ 10,820 þ 39,840þ 11,470 ¼ 62,130. The ordered set of objective functions is listed in table 8.
Table 1. Unit production cost, unit holding cost, unit backorder cost (all in HK$, 1US$ ¼ 7.8HK$).
Factories, j 1
2
3
Product, i
Production cost of regular time, C1ij, $/unit
Production cost by non-experienced workers, C2ij, $/unit
Production cost of overtime, C3ij, $/unit
Inventory cost, C7ij, $/unit
Back-order cost, C8ij, $/unit
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
50 60 80 90 110 120 55 65 85 95 115 125 60 70 90 110 120 130
150 160 180 190 210 220 160 170 190 200 220 230 170 180 200 210 230 240
140 150 170 180 200 210 145 155 175 185 205 215 150 160 180 190 210 220
15 20 30 35 45 50 13 18 28 33 43 48 10 15 25 30 40 45
30 35 45 50 60 65 30 35 45 50 60 65 30 35 45 50 60 65
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Multi-site aggregate production planning with multiple objectives Table 2. Labor cost, hiring cost, layoff cost and workforce data. Labour cost (regular time), C5j, $/hour
Labour cost (regular time), C4j, $/man-period
Factories, j 1 2 3
250 225 200
Factories, j
Labour cost (overtime), C6j, $/hour
10 9 8
Layoff cost, C10j, $/man
100 90 80
120 110 100
8 7 6
Minimum workforce
Change rate, "
Initial workforce
300 300 300
0.40 0.45 0.50
300 300 300
1 2 3
Hire cost, C9j, $/man
Table 3. Minimum known demand Data. Product, i 1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
11
12
300 300 500 500 700 700
800 800 1,000 1,000 1,200 1,200
400 400 700 700 1,000 1,000
500 500 600 600 700 700
800 800 1,000 1,000 1,200 1,200
600 600 800 800 1,000 1,000
500 500 700 700 900 900
700 700 800 800 900 900
800 800 1,000 1,000 1,200 1,200
1,100 1,100 1,400 1,400 1,700 1,700
1,200 1,200 1,400 1,400 1,600 1,600
1,300 1,300 1,400 1,400 1,500 1,500
Table 4. Maximum forecasted demand data Product, i 1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
11
12
3,000 3,000 2,800 2,800 2,600 2,600
4,000 4,000 3,800 3,800 3,600 3,600
3,500 3,500 3,300 3,300 3,100 3,100
3,800 3,800 3,500 3,500 3,200 3,200
4,000 4,000 3,800 3,800 3,600 3,600
3,600 3,600 3,400 3,400 3,200 3,200
3,400 3,400 3,200 3,200 3,000 3,000
3,300 3,300 3,200 3,200 3,100 3,100
3,000 3,000 2,800 2,800 2,600 2,600
3,700 3,700 3,500 3,500 3,300 3,300
3,300 3,300 3,100 3,100 2,900 2,900
3,000 3,000 2,800 2,800 2,600 2,600
Table 5. Machine capacity data. Factories, j 1 2 3
1
2
3
4
5
6
7
8
9
10
11
12
5,000 4,000 3,000
4,000 3,500 3,000
4,500 3,500 2,500
5,500 5,000 4,500
4,000 3,000 2,000
4,500 3,500 2,500
5,000 4,000 3,000
4,000 3,500 3,000
5,000 4,000 3,000
5,500 4,500 3,500
4,000 3,000 2,000
4,000 3,500 3,000
In this paper, it is assumed that the weights are equal among one group for the quota’s goal priority. When it is assumed not equal, the decision-maker has to determine the relative importance of the availability and utilization of import quotas of different products. The analytical hierarchy process (AHP) developed by Saaty (1994) is one of the powerful tools to consider multiple criteria decision-making involving quantitative and qualitative factors. First, the decision-maker has to define the
group criteria for comparing three priorities, z1, z2 and z3. Then, pairwise comparison using a nine-point scale among the decision criteria and the evaluation of performance of priorities in each criterion are assigned by decision-makers. After calculation, a set of weight factors assigned to the three priorities can be determined. Using the data presented, the linear goal programming model was formulated as in section 3, and the optimal solution can be easily obtained using the simplex method.
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Many other packages such as LINDO (Linear Interactive and Discrete Optimization) can also solve the problem efficiently (Evans 1993). The results from the proposed model are described in the following. The highest-priority goal, zP, is to minimize the underachievement of profit. The negative deviation variable dp is 798,906. It means that the realized production loading does not fully meet the profit goal set. The solution yields the optimal profit the company can achieve, which is $47,201,095. The second-priority goal, zW , is to minimize the overachievement of the cost of hiring and laying-off of workers. The positive variable, dwþ , is 31,400. It represents that the cost of hiring and laying-off of workers is over-satisfied. The cost associated with hiring and layingoff of workers in the planning horizon is $284,400. Finally, the remaining priority goals, z1, z2 and z3, are to minimize the over- or under-utilization of import quotas. The objective values corresponding to the three groups of quota are z1 ¼ 3098, z2 ¼ 2540 and z3 ¼ 5235. It
results in the following production loads suggested by the model: 11,364 for product 1; 37,800 for product 2; 14,121 for product 3; 36,100 for product 4; 15,620 for product 5; and 28,108 for product 6. In order to investigate the implications of a different priority ranking of the two most important goals, the model that reverses the first two priorities groups is run (run 2). As can be seen in table 9, the results in run 2 show that this slightly reduces the cost of hiring and laying-off of workers by 0.1% ($31,400), while reducing the profits by approximately 0.3% ($120,747). It is noted that the maximal profit is $47,201,095 from the first two runs. It is expected that when the profit goal is the highest priority, then the profit is $47,201,095. It is reasonable to test when the profit is the highest priority, whereas the cost of hiring and laying-off of workers is the lowest priority. In run 3, the result is identical to that from run 1. It illustrates that the profit goal dominates the result whatever the remaining goals are. Compared with run 2, the cost associated with hiring and laying-off of workers and the availability and utilization of import quota are higher in those in run 1 and run 3. Although the profit is maximal, it is realized that the cost of hiring and laying-off of workers is also a crucial issue from the management’s point of view. In run 4, it is considered that when the cost of hiring and laying-off of workers is considered the highest priority, the profit is the lowest. The result shows that
Table 6. Revenue and import quota. Product, i
Sales revenue, ri
Import quota
300 350 450 500 600 650
10,820 39,840 11,470 33,560 15,780 25,010
1 2 3 4 5 6
Table 7. Machine time data.
Product, i
Labor production time, ai hour/unit, of experienced worker
Labour production time, bi hour/unit, of non-experienced worker
Machine time, li hour/unit, of experienced worker
Machine time, i hour/unit, of non-experienced worker
1 1 1.5 1.5 2 2
1.5 1.5 2 2 2.5 2.5
0.75 0.75 1 1 1.5 1.5
1.5 1.5 2 2 2.5 2.5
1 2 3 4 5 6
Table 8. Model description.
Run 1 2 3 4 5 6
Profit
Change of workforce level
First group of equally weighted quota
Second group of equally weighted quota
Third group of equally weighted quota
P1 P2 P1 P5 P4 P5
P2 P1 P5 P1 P5 P4
P3 P3 P2 P2 P1 P1
P4 P4 P3 P3 P2 P2
P5 P5 P4 P4 P3 P3
Pj is the pre-emptive priority factor which expresses the relative importance of various goals, Pj Pj þ 1.
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Multi-site aggregate production planning with multiple objectives Table 9. Result Run Profit Work Quota UQ1 OQ1 UQ2 OQ2 UQ3 OQ3 UQ4 OQ4 UQ5 OQ5 UQ6 OQ6
1
2
3
4
5
6
47,201,095 281,400 11,033 0 544 2,040 0 0 2,651 0 2,540 160 0 0 3,098
47,080,348 250,000 10,668 0 400 2,040 0 0 2,651 0 2,540 160 0 0 2,877
47,201,095 281,400 11,033 0 544 2,040 0 0 2,651 0 2,540 160 0 0 3,098
44,911,976 250,000 0 0 0 0 0 0 0 0 0 0 0 0 0
44,949,669 293,280 0 0 0 0 0 0 0 0 0 0 0 0 0
44,911,976 250,000 0 0 0 0 0 0 0 0 0 0 0 0 0
zW ¼ z1 ¼ z2 ¼ z3 ¼ 0. In this case, the profit is $44,911,976. It is supposed that when no overachievement of the cost of hiring and laying-off of workers goal and no deviation between import quotas are allowed, then the profit is $44,911,976. As can be seen in run 4, although z1, z2 and z3 are the second, third and fourth priority, the deviations are zero. It is of interest to test when these three goals are the first, second and third priority, which are the settings for run 5 and 6. In run 5, the profit goal is the fourth priority while the change in workforce level is the lowest, and in run 6 the setting is reversed. In these two runs, it is not surprising that z1 ¼ z2 ¼ z3 ¼ 0. Because of the higher priority of profit goal over the goal of cost of hiring and laying-off of workers, the profit is $44,949,669 and the cost of hiring and laying-off workers is $293,280. When the profit goal is the lowest, zW ¼ 0 and the profit is $44,911,976. It is identical to run 4. It is useful to foresee when the import quota is not available in the real world, the trade-off between profit and the cost of hiring and laying-off can be determined. For instance, in order to increase profit by 0.08%, $37,693, the company has to increase the cost associated with hiring and laying-off of workers by 0.17%, $43,280.
5. Conclusions In this paper, we discuss the aggregate production planning problems of a multinational lingerie company in a global supply chain setting, at a time when more and more companies are moving rapidly towards internationalization and re-engineering in order to establish a global network. A new operational strategy across nations and factories with model-based support for production planning problems is essential. It enables management to evaluate planning solutions with optimization. A multi-objective programming model for multi-site aggre-
gate production planning problem is presented, in which three major objectives with target values are optimized hierarchically. The three objectives are the maximization of profit, the minimization of costs associated with hiring and laying-off of workers, and the minimization of underutilization and over-utilization of import quota. The results of the runs illustrate the flexibility and robustness of the model so that management can estimate numerous scenarios regarding various strategic assumptions by changing priority rankings. Computerized production planning problems for a global supply chain will be a critical part of the company’s management system. It is expected that the proposed model will bring about a significant improvement in the company’s global supply chain management in the future.
Acknowledgments We thank the anonymous referees for their helpful comments. The work described in this paper was supported by the Strategic Research Grant from City University of Hong Kong (Project Number 7001516).
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