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A goal programming model for production planning of perishable products with postponement
Computers & Industrial Engineering 53 (2007) 531–541 www.elsevier.com/locate/dsw
A goal programming model for production planning of perishable products with postponement Stephen C.H. Leung *, Wan-lung Ng Department of Management Sciences, City University of Hong Kong, Hong Kong Received 3 May 2006; received in revised form 28 May 2007; accepted 28 May 2007 Available online 2 June 2007
Abstract One of the important characteristics of perishable products that a decision-maker has to take into account seriously is that the price will drop significantly after a day, or a season. Hence, over-production and storage of such products is not recommended. In this paper, the production process for perishable products is proposed to be divided into two phases by applying the concept of postponement. Consequently, three production activities – direct production, master production and final assembly – will be considered. A preemptive goal programming model to solve aggregate production planning for perishable products is developed, in which three objectives are optimized hierarchically. A set of Hong Kong data has been used to test the effectiveness and the efficiency of the proposed model. Results demonstrate that the decision-makers can find the flexibility and robustness of the proposed model by adjusting the goal priorities with respect to the importance of each objective and the aspiration level with respect to desired target values. 2007 Elsevier Ltd. All rights reserved. Keywords: Production management; Goal programming; Perishable products; Postponement
1. Introduction Postponement of production refers to a common intermediate product being manufactured in the first phase and, according to the differentiating options (such as colors, sizes and types), production line activities such as dyeing, compounding, final assembling, packaging and so on being postponed to a second phase – i.e. until customer orders are received (Aviv & Federgruen, 2001a; Aviv & Federgruen, 2001b; Van Hoek, 2001). Lee and Billington (1994) redesigned a European DeskJet Printer line of Hewlett Packard using the postponement strategy. Pagh and Cooper (1998) stated that the advantage of postponement is reduction or full elimination of risk and uncertainty in manufacturing and logistics operations. Garg and Tang (1997) investigated two points of postponement in the manufacturing stage – early postponement and late postponement – and investigated the importance of demand variabilities and correlations between, and relative magnitudes, of lead times in determining the appropriate points of differentiation. Aviv and Federgruen (2001b) studied the benefit *
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0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.05.010
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of postponement with unknown demand distributions. Recently, Van Hoek (2001) reviewed the literature on postponement and identified postponement opportunities in operations. In this paper, a postponement strategy is employed to solve production planning for perishable products in a situation with limited resources in production and dramatic growth in demand. Demand for perishable products is time-sensitive because the demand dramatically increases as the day approaches the end of lifecycle, such as the Christmas Day. On the other hand, a shortage of perishable products while the product is saleable may result in significant loss of revenue because perishable products cannot be profitable after a certain day. For instance, in manufacturing industries, people want to buy a Christmas gift in or before December only. Controlling the inventory of perishable products is crucial. However, there is little research that addresses aggregate production planning for perishable products. 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 presence of multiple objectives in real-life problems (Vincke, 1992). Baykasoglu (2001) noted that the aggregate production planning may consider minimization of costs, inventory levels, changes in work force levels, overtime wages in production, subcontracting, changes in production rates, number of machine set-ups, plant/personnel idle time and maximization of profits and superior customer service. Decision-makers always want to develop a model that can consider real-life situations with multiple objectives. To achieve this, in this paper, a preemptive goal programming model is formulated to determine optimal production loading plans. As opposed to linear programming, which directly optimizes objectives, preemptive goal programming is used to manage a set of conflicting objectives by minimizing 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+ and negative deviation d, which represent the distance from this target value. The objective of preemptive goal programming is to minimize the deviations hierarchically so that the goals of primary importance receive first-priority attention, those of secondary importance receive second-priority attention, and so on and so forth. Then, the goals of first-priority are minimized in the first phase. Using the obtained feasible solution result in the phase, the goals of second priority are minimized, and so on. With fast computational growth, both linear and non-linear GP can be solved using well-developed software such as Linear Interactive and Discrete Optimization (LINDO) or meta-heuristics such as simulated annealling, genetic algorithms, tabu search and so on (Jones, Mirrazavi, & Tamiz, 2002). An explicit definition of goal programming was given by Charnes and Cooper (1961). The purpose of this study is to develop a preemptive goal programming model to optimize the production planning problem for perishable products under an uncertain environment, from which we can determine (1) how many finished products should be produced from raw materials directly (direct production), (2) how many semi-finished products should be produced from raw materials (master production), and (3) how many finished products should be produced from semi-finished products (final assembly) so that resources can be better utilized to meet any dramatic growth in demand and total costs (consisting of setup costs, production costs, labor costs, inventory costs, hiring costs and lay-off costs) can be minimized. The organization of this paper is as follows. After this introduction, characteristics and issues of current operations in the toy company under investigation are reviewed. Then a preemptive goal programming model is formulated to solve the production planning problem for perishable products using a postponement strategy in Section 3, and a set of data from the toy company is used to test the effectiveness and efficiency of the proposed model in Section 4. Our conclusions are given in the final section. 2. Characteristics and issues of current operations This study is particularly motivated by the problems faced by a company manufacturing plush toys, whose headquarters is in Hong Kong and the production plant is in China. The finished products are exported to the US and Europe. The company mainly produces two plush toy products – animal-type plush toy products with light songs (duck, bear, hop and dog) and Christmas-theme plush toy products with Christmas songs (snowman and Christmas tree). From the sales report, it is known that animal-type plush toy products can be sold
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throughout the year, whilst Christmas-theme plush toy products can only be sold in November and December as Christmas gifts. Furthermore, demand for the Christmas-theme plush toy products during these 2 months comprises over 50% of total sales volume for the entire year. In addition, demand for Christmas-theme plush toy products increases dramatically as Christmas approaches. Based on the reports and from past experience, the current annual production planning strategy is to produce animal-type products from January to October and Christmas-theme products in the remaining 2 months. The company’s main problem is how to design the production loading plan for Christmas-theme plush toy products in November and December, which is constrained by limited resources, including machine time, workforce level and inventory space. In order to fulfill the substantial increase in demand in December, currently more Christmas-theme plush toy products are produced and stored in November in order to compensate for the limited resources in December. However, the management has discovered that this approach leads to a huge inventory cost and takes up space. The characteristics of the production planning problem with perishable products (Christmas-theme plush toy products) in the company are summarized as follows: • The demand for Christmas-theme plush toy products is time-sensitive. The demand dramatically increases as 25th December approaches. Aviv and Federgruen (2001a) realized that the continual increase of sales volumes will result in a capacity bottleneck. The production plant is not able to produce enough products to fulfill market demand. Currently, the company produces more Christmas-theme plush toy products in November and stores them for future demand. Moreover, it may not be practical for the company to subcontract production to other manufacturers due to constraints of quality control, special technologies, plant specifications and client contracts. In addition, adding more machines leads to higher capital and maintenance costs and the machines may be under-utilized during non-peak seasons. • The Christmas-theme plush toy product is perishable, which is like fashion and high-technology products that have a short life-cycle. A backorder strategy is not appropriate here since this will not only lower customer satisfaction, damaging the company’s goodwill and customer loyalty, but also does not deal with the fact that demand falls after the special event is over. No customer is willing to send Christmas gifts at any time other than Christmas. Aviv and Federgruen (2001b) noted that, subject to dynamic and competitive market forces, demand distribution for short life-cycle products might not be accurately determined. • The finished product is comprised of two components – a musical box mounted on a printed circuit board (PCB) and an outside body with plush fabric finish. The final assembly is a spliced fabric body with a musical box. From here on, we refer to the musical box as the semi-finished product and the outer body with musical box as the finished product. Clearly, the space used to store finished products is usually larger than space needed to store semi-finished products. Therefore, the inventory cost for finished products is usually quite large.
3. Model formulation In this study, the aggregate production planning problem for perishable products faced by a plush toy company in Hong Kong is investigated. For cost effectiveness, the decision-makers have to determine the quantity of product i, i = 1, 2, . . . , n, manufactured over each period of time t, t = 1, 2, . . . , T, to fulfill market demands. The production loading plan consists of: (1) the quantity of finished products to be produced from raw materials directly (direct production), (2) the quantity of semi-finished products to be produced from raw materials (master production), and (3) the quantity of finished products to be produced from semi-finished products (final assembly) in each period of time. 3.1. Notations 3.1.1. Parameters Dit the demand for product i in period t It the storage space limitation in period t vFi the space occupied by one unit of finished product i
534
vSi Wt d adi am i afi bdi bm i bfi kW t kM t Mt CK di CK m i CK fi CWt CHt CLt CP di CP m i CP fi COdi COm i COfi CC Fi CC Si CBi bOC bIC bHC
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the space occupied by one unit of semi-finished product i the maximum number of workers available in period t the regular working hours of labor in each period the man hours required to produce one unit of finished product i from direct production the man hours required to produce one unit of semi-finished product i from master production the man hours required to produce one unit of finished product i from final assembly the machining time required to produce one unit of finished product i from direct production the machining time required to produce one unit of semi-finished product i from masterproduction the machining time required to produce one unit of finished product i from final assembly the fraction of regular workforce available for over-time in period t the fraction of regular machine capacity available for over-time use in period t the maximum regular time machine capacity in period t the setup cost of producing finished product i from direct production the setup cost of producing semi-finished product i from master production the setup cost of producing finished product i from final assembly the labor cost in period t the cost of hiring one worker in period t the cost of lay-off of one worker in period t the regular-time per unit cost of producing one unit of finished product i from direct production the regular-time per unit cost of producing one unit of semi-finished product i from master production the regular-time per unit cost of producing one unit of finished product i from final assembly the overtime per unit cost of producing one unit of finished product i from direct production the overtime per unit cost of producing one unit of semi-finished product i from master production the overtime per unit cost of producing one unit of finished product i from final assembly the carrying cost for one unit of finished product i the carrying cost for one unit of semi-finished product i the backorder cost for one unit of product i aspiration level of ‘operating cost goal’ to be achieved aspiration level of ‘inventory cost goal’ to be achieved aspiration level of ‘hiring and layoff cost goal’ to be achieved
3.1.2. Decision variables P dit the number of finished products i produced from direct production during regular time in period t Pm the number of semi-finished products i produced from master production during regular time in perit iod t P fit the number of finished products i produced from final assembly during regular time in period t Odit the number of finished products i produced from direct production during overtime in period t Om the number of semi-finished products i produced from master production during overtime in period t it Ofit the number of finished products i produced from final assembly during overtime in period t K dit theindicator for producing finished product i from direct production in period t 1; if P dit > 0 ¼ 0; otherwise Km theindicator for producing semi-finished product i from master production in period t it 1; if P m it > 0 ¼ 0; otherwise K fit theindicator for producing finished product i from final assembly in period t f ¼ 1; if P it > 0 0; otherwise Ht the number of workers hired in period t Lt the number of workers laid-off in period t
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Wt I Fit I Sit Bit
the the the the
535
number of workers in period t inventory level of finished product i in period t inventory level of semi-finished product i in period t backorder of product i in period t
3.1.3. Auxiliary variables d deviation of underachievement of bOC OC dþ deviation of overachievement of bOC OC d deviation of underachievement of bIC IC dþ deviation of overachievement of bIC IC d deviation of underachievement of bHC HC þ d HC deviation of overachievement of bHC
3.2. Goal constraints and objective functions The aim of this study is to find an optimal production loading plan that minimizes operating costs, inventory costs and hiring and layoff costs hierarchically. With postponement strategy, the aggregate production loading plan consists of production quantities of direct production, master production and final assembly during regular time and at overtime. Along with these plans, decision-makers can also determine inventory levels, backorder levels and workforce levels. To achieve the optimal plan, three goals are considered in this study. Goal 1: Operating cost goal T X n X t¼1
T X n X d d CK di K dit þ CK mi K mit þ CK fi K fit þ CP i P it þ CP mi P mit þ CP fi P fit
i¼1
þ
t¼1
T X n X t¼1
COdi Odit þ COmi Omit þ COfi Ofit þ
i¼1
i¼1 T X
CW t W t d þ OC þ d OC ¼ bOC
ð1Þ
t¼1
The first term in expression (1) is the setup cost. The second and third terms are the regular-time and over-time production costs, which comprise costs associated with direct production, master production and final assembly. The labor cost associated with regular-time workers is formulated in the fourth term of expression (1). Goal 2: Inventory cost goal T X n X t¼1
i¼1
T X n X CC Fi I Fit þ CC Si I Sit þ CBi Bit d þ IC þ d IC ¼ bIC t¼1
ð2Þ
i¼1
The inventory cost consists of carrying costs and backorder costs. The first component in expression (2) is the sum of carrying cost of finished products and semi-finished products associated with the storage of units of products in warehouses for a given period of time. The second component is the backorder cost associated with under-fulfillment of demand. Goal 3: Hiring and layoff goal T X
ðCH t H t þ CLt Lt Þ d þ HC þ d HC ¼ bHC
ð3Þ
t¼1
Subject to production loading and market demand, in each period of time, the management has to determine how many additional workers would be needed to be recruited to handle extra production loading or how many workers would be needed to be laid off to reduce overheads. The cost of hiring and laying-off workers is expressed in (3).
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The achievement function of the multi-objective production planning of perishable products problem is formulated in the following: þ þ Lexicographically minimize fd þ OC ; d IC ; d HC g 3.3. Constraints The objective functions formulated in the previous section are restricted by four sets of constraints. They are – inventory level constraints, the relationship among the number of workers, the production capacity constraints and the non-negative constraints. 3.3.1. The inventory level constraints I Fit Bit ¼ I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit Dit I Sit ¼ n X
I Sit1
þ
P mit
þ
Omit
ðvFi I Fit þ vSi I Sit Þ 6 I t
P fit
Ofit
i ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; n; t ¼ 1; 2; . . . ; T
t ¼ 1; 2; . . . ; T
t ¼ 1; 2; . . . ; T
ð4Þ ð5Þ ð6Þ
i¼1
Constraint (4) determines either the quantity of finished products stored in the warehouse or the shortfall in meeting market demand. If the total quantity of finished products produced at the company’s plants and transferred to semi-finished products during period t plus previous stock at period t 1 minus previous backorder at period t 1 (i.e. I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit ) is greater than market demand Dit, then the stock at period t will be equal to I Fit ¼ I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit Dit and, under minimization, the deviation Bit = 0; whereas if I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit is less than market demand, then I Fit ¼ 0 and Bit ¼ Dit I Fit1 P dit Odit P fit Ofit , indicating that market demand is not satisfied. It is noted that backorder is considered in the model, so Bit is carried over as extra demand to the next period. Constraint (5) determines the quantity of semi-finished products stored in the warehouse. The total quantity of semi-finished products produced at the company’s plants during period t plus previous stock at period t 1 (i.e. I Sit1 þ P mit þ Omit ) must be equal to the semi-finished products stored in the warehouse for period t plus the quantity of semi-finished products used to perform final assembly. The physical storage space during period t is limited by constraint (6). 3.3.2. Relationship among the number of workers W t ¼ W t1 þ H t L Wt 6Wt
t ¼ 1; 2; . . . ; T
ð7Þ
t ¼ 1; 2; . . . ; T
ð8Þ
Constraint (7) ensures that the available workforce in any period equals the workforce from the previous period plus or minus any change in workforce level during the current period. The change in workforce level may be due to either hiring extra workers or laying-off redundant workers. It is noted that Ht Æ Lt = 0 because either the net hiring or the net laying-off of workers takes place over a period, but not both. Constraint (8) ensures that the upper-bounds of change in workforce level over a period are provided. 3.3.3. Production capacity constraints n X
adi P dit þ ami P mit þ afi P fit 6 dW t
t ¼ 1; 2; . . . ; T
ð9Þ
i¼1 n X
adi Odit þ ami Omit þ afi Ofit 6 kWt dW t
t ¼ 1; 2; . . . ; T
ð10Þ
i¼1 n X i¼1
bdi P dit þ bmi P mit þ bfi P fit 6 M t
t ¼ 1; 2; . . . ; T
ð11Þ
S.C.H. Leung, W. Ng / Computers & Industrial Engineering 53 (2007) 531–541 n X d d bi Oit þ bmi Omit þ bfi Ofit 6 kM t Mt
537
t ¼ 1; 2; . . . ; T
ð12Þ
i¼1
P dit þ Odit 6 PK dit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð13Þ
PK mit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð14Þ
P fit þ Ofit 6 PK fit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð15Þ
P mit
þ
Omit
6
Constraints (9) and (10) limit the labor working hours during regular time and overtime, respectively. Similarly, constraints (11) and (12) limit the machining time during regular time and overtime, respectively. Constraints (13)–(15) ensure that setup costs will be incurred only when the corresponding production activities start. 3.3.4. Non-negative constraints W t ; H t ; Lt ; P dit ; P mit ; P fit ; Odit ; Omit ; Ofit I Fit ; I Sit ; Bit P 0 K dit ; K mit ; K fit
¼ f0; 1g
i ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð16Þ
t ¼ 1; 2; . . . ; T
ð17Þ
where P is a large positive number. Constraint (16) ensures that all decision variables are non-negative. Boolean constraints (17) are used for the setup indications of the production activities. 4. Computational results In this section, a data set provided by the plush toy company in Hong Kong is used to illustrate the flexibility of the proposed preemptive goal programming model for the aggregate production planning problem with postponement strategy for perishable products. The tactical/operational level of decision-making in the production planning process is described below. The company receives sales orders from its sales branches covering America and Europe. Based on the company’s projections report, a 2-month planning horizon is determined (November and December) for Christmas products. Each order may require two types of products, i = 1, 2 covering 8 weeks, t = 1, 2, . . . , 8. The setup cost and production cost, as well as labor and machine requirements of different production activities, are given in Table 1. Table 2 shows the limitations of workforce, machine capacity and inventory spaces in each period. Table 3 lists regular time labor cost and the hiring and laying-off cost associated with changes in the workforce level. The unit space occupation for finished products and semi-finished products is provided in Table 4. The carrying cost and backorder costs are shown in Tables 5 and 6. It is noted that, owing to the characteristics of the products, the backorder cost is time-sensitive and dramatically increases as the time approaches the event kick-off period (i.e. the ending period). For each weekly period, demand for products required by the markets is shown in Table 7.
Table 1 Operating and costs data (in HK$, 1US$ = 7.8HK$)
Setup cost ($) Production cost (regular time) Production cost (overtime) Labor time (h) Machining time (h)
Product
Direct finished product production
Semi-finished product production
Transfer production
1 2 1 2 1 2 1 2 1 2
2000 2500 50 60 50 60 0.5 0.6 0.5 0.6
1000 1000 30 30 30 30 0.35 0.35 0.4 0.4
1500 2000 30 40 30 40 0.15 0.25 0.1 0.2
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Table 2 Inventory, production and workforce capacity 1000 m3 1000 16,000 0.3 0.4
Inventory space limitation, I it Maximum workforce level, W t Maximum machine capacity, Mt Fraction of workforce available for over-time, kW t Fraction of machine capacity available for over-time, kM t
Table 3 Labor costs and hiring and laying-off costs Period
Labor cost per worker per period, CWt($) Hiring cost per worker, cHt($) Laying-off cost per worker, cLt ð$Þ
1
2
3
4
5
6
7
8
80 80 120
80 80 120
80 100 120
80 100 120
80 100 120
80 150 120
80 150 120
80 150 120
Table 4 Space occupation Product
Finished product inventory space occupied (m3)
Semi-finished product inventory space occupied (m3)
1 2
1.0 1.0
0.3 0.3
Table 5 Inventory costs (in $HK) Product, i
Finished product
Semi-finished product
1 2
50 50
5 5
Table 6 Backorder cost Product, i
Period, t
1 2
1
2
3
4
5
6
7
8
100 120
110 132
121 145
133 160
146 176
161 193
177 213
195 234
Table 7 Market demand data Product, i
1 2
Period, t 1
2
3
4
5
6
7
8
3000 4800
6600 5100
3750 5700
4350 6300
5100 7200
6600 8550
9450 10,650
17,850 15,150
In preemptive goal programming formulation, the aspiration level for each goal has to be defined. After consulting the company’s management, the target value of ‘Operating cost goal’ bOC is $6,500,000; the target value of ‘Inventory cost goal’ bIC is $500,000; and the target value of ‘Hiring and layoff cost goal’ bHC is $60,000 (Table 8). Moreover, the deviation variables to be minimized are prioritized as follows: first d þ OC , þ second d þ and third d (Run 1). IC HC
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Table 8 Target value with priority Run
Operating cost, $
Inventory cost, $
Hiring and layoff cost, $
1 2 3 4 5
6,500,000 6,500,000 7,000,000 7,000,000 7,000,000
500,000 500,000 500,000 500,000 600,000
60,000 60,000 60,000 60,000 60,000
(P1) (P2) (P1) (P2) (P2)
(P2) (P1) (P2) (P1) (P1)
(P3) (P3) (P3) (P3) (P3)
Note: Numbers in parenthesis indicate the preemptive priority. Pj is the preemptive priority factor which expresses the relative importance of various goals, Pj Pj+1.
Using the data presented, the preemptive goal programming model was formulated as in Section 3; the optimal solution can be easily obtained using the simplex method. Many other packages such as Linear Interactive and Discrete Optimization (LINDO) can also solve the problem efficiently (Evans, 1993). Results from the proposed model are shown in Tables 9–11. First, Table 9 reports that operating costs, inventory costs and hiring and layoff costs are $6,500,000, $2,688,190 and $103,037, respectively. In this run, the highest-priority goal is to minimize the overachievement of operating cost. The positive deviation variable d þ OC is 0. The goal for operating cost has been achieved. The second-priority goal is to minimize the overachievement of inventory costs. The positive deviation variable d þ IC is $2,188,190. This shows that inventory cost is extremely overachieved. The inventory cost, which is the sum of carrying cost ($86,559) and backorder cost ($2,601,631) is $2,688,190. Finally, the remaining priority goal is to minimize the over-achievement of hiring and layoff cost. The positive deviation variable d þ HC is $43,037. The hiring costs and layoff costs are $81,287 and $21,750, respectively. In Table 10, products 1 and 2 produced in this run are 43,358 units and 63,450 units, respectively. On the other hand, the under-fulfillment of products 1 and 2 are 13,342 units and 0, respectively. It is of interest to determine the average cost per product which is the total cost consisting of operating cost, inventory cost and hiring and layoff cost divided by total units of product produced. The average cost in this run is $87. Under the current result, it is observed that the backorder cost is extremely high, resulting in high inventory cost (over $2,188,190 more than the target value). The main reason is that, since the minimization of operating cost is ranked as the first priority, producing more products is not encouraging. Hence, the market demand is under-satisfied. From Table 10, it can seen that under-fulfillment of demand for products 1 and 2 are 13,342 Table 9 Results Run
Operating cost,a $
Inventory cost,
1 2 3 4 5
6,500,000 7,373,513 7,000,000 7,373,513 7,333,075
2,688,190 500,000 1,407,411 500,000 600,000
a
a
Hiring and layoff costa
$
103,037 86,066 86,856 86,066 86,066
Numbers have been rounded off to the nearest integer.
Table 10 Product produced and under-fulfillment Product produced, units
1 2 3 4 5
Product under-fulfillment, units
1
2
1
2
43,358 56,236 52,052 56,236 55,549
63,450 63,379 62,333 63,379 63,450
13,342 464 4648 464 1151
0 71 1117 71 0
Average cost, $
87.0 66.5 81.2 71.0 67.4
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Table 11 Production loading plan Product, i
Direct production
Regular time Overtime
Master production
Regular time Overtime
Final assembly
Regular time Overtime
Workforce level Hiring Laying-off
Period, t 1
2
3
4
5
6
7
8
1 2 1 2
978 4800 4800 2022
6600 2654 2653 0
1306 5700 5700 2444
4350 6300 6300 0
5100 7200 7200 0
6600 7833 7833 1908
9450 5458 5458 4000
0 11,493 11,493 0
1 2 1 2
0 0 0 0
0 0 0 0
0 0 0 0
1689 0 5611 0
3229 0 6857 0
0 0 0 3586
0 0 0 0
0 0 0 0
1 2 1 2
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1386 3586 16,000 0
421
612
509
818
1000
1000
1000
1000
0 79
190 0
0 102
309 0
182 0
0 0
0 0
0 0
units and 0; and hence a higher backorder cost is incurred in this run. It is obvious to state that this result is not acceptable. Therefore, it is reasonable to test when the ‘Inventory cost goal’ is the first-priority and the ‘Operating cost goal’ is the second-priority (Run 2). As can be seen in Table 9, the results in Run 2 show that the ‘Inventory cost goal’ is achieved and the inventory cost is decreased by 81.4% ($2,188,190), while increasing the operating cost by 13.4% ($873,513). At the same time, the hiring and layoff cost is also dropped to $86,066 (16.5% reduction). From Table 10, products 1 and 2 produced are 56,236 units and 63,379 units, respectively, and the under-fulfillment are 464 units and 71 units, respectively. Although more products are produced, the average cost for each product in Run 2 is $66, which is significantly less than that in Run 1. The production loading plan of Run 2 is shown in Table 11. It can be seen that the majority of products are produced using regular-time labour. In order to meet the growth in demand during the last two periods, the management is recommended to produce semi-finished products during periods 4–6 and to perform final assembly in period 8. The majority of resources consumed in period 8 are used to perform final assembly of product 2. It is shown that, using the postponement strategy, more products can be produced, particularly in period 8. Lastly, the workforce level does not attain the upper-bound limit in each period. The corresponding number of workers hired and laid off can also be found in Table 11. As can be seen, Runs 1 and 2 have the same set of target values but different priorities. It is of interest to test different sets of target values keeping the same priority structure for goals. In Runs 3 and 4, the target value of ‘Operating cost goal’ bOC is $7,000,000 and the other two target values are kept unchanged. Moreover, the priority structure of goals for Runs 3 and 4 are the same as in Runs 1 and 2, respectively. From Table 9, the results of Run 3 show that operating costs, inventory costs and hiring and layoff costs are $7,000,000, $1,407,411 and $86,856, respectively. Moreover, the ‘Operating cost goal’ is achieved. Compared with the results in Run 1, both inventory cost and hiring and layoff cost are reduced significantly, which are 47.6% for inventory cost and 15.7% for hiring and layoff cost. It is because more products can be produced as target value of ‘Operating cost goal’ is increased. It results in reduction of backorder cost and hence inventory cost is also reduced. Therefore, although the operating cost is increased by $500,000, the inventory cost is reduced by $1,280,779. The total products 1 and 2 produced are 52,052 and 62,333 units. It is noted that more quantities of product 1 are produced. The average cost for each product is $81.2, which is a little smaller than that in Run 1. Although the inventory cost is reduced significantly, it is not a preferable solution. Hence, with regard to the results in Runs 1 and 2, it is reasonable to swap the ranking of priority between ‘Operating cost goal’ and ‘Inventory cost goal’. It is noted that the target values set are the same as in Run 3. The results show that all
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are the same as Run 2. It means that if ‘Inventory cost goal’ is minimized, a non-inferior solution is obtained (Romero, 1991). As mentioned before, target value plays an important role in finding the optimal solution in the preemptive goal programming formulation. It is reasonable to change target value of ‘Inventory cost goal’ as $600,000 (20% increased). The results show that ‘Inventory cost goal’ is achieved, but operating cost is reduced only marginally, by 0.5%. It is claimed that it is not economically attractive to increase the target value of ‘Inventory cost goal’ to $600,000. 5. Conclusions In this paper, a preemptive goal programming model is proposed to deal with the aggregate production planning problem in a toy company in Hong Kong. Due to demand characteristics, the product studied is regarded as a perishable product which cannot be sold or the demand for which drops significantly after Christmas. Three major objectives with target values are optimized hierarchically. The three objectives are the minimization of operating cost consisting of setup cost, production cost and labour cost, the minimization of inventory cost consisting of carrying cost and backorder cost, and the minimization of hiring and layoff cost to reduce the change of workforce level. The model can effectively find an optimal production loading plan consisting of three production activities which are direct production, master production and final assembly. It is also able better to control inventory levels with surplus storage and shortage. A real case in an existing Hong Kong-based plush toy manufacturing company is studied in this paper. Some useful findings are observed. With postponement strategy, it is recommended to produce semi-finished products in the earlier planning horizon, to be used to perform final assembly in the later period of planning horizon in order to meet the dramatically increases in demand. The results of the runs illustrate the flexibility and robustness of the model so that the management can examine numerous scenarios regarding various strategic assumptions by changing priority rankings. This paper illustrates the effectiveness and the efficiency of the proposed model. However, there is still much room for improvement and investigation for the proposed model with regard to its application to real-world situations. Real data from other manufacturing companies can be used to validate the model and to analyze its sensitivity to changes in production management strategies. It should be noted that the computation and analysis of the model under different scenarios would lead to different outcomes. References Aviv, Y., & Federgruen, A. (2001a). Capacitated multi-item inventory systems with random and seasonally fluctuating demand: implications for postponement strategies. Management Science, 47, 512–531. Aviv, Y., & Federgruen, A. (2001b). Design for postponement: a comprehensive characterization of its benefits under unknown demand distribution. Operations Research, 49, 578–598. Baykasoglu, A. (2001). MOAPPS 1.0: aggregate production planning using the multiple-objective tabu search. International Journal of Production Research, 39, 3685–3702. Charnes, A., & Cooper, W. W. (1961). Management models and industrial applications of linear programming. New York: Wiley. Evans, J. R. (1993). Applied production and operations management. St. Paul: West Group. Garg, A., & Tang, C. S. (1997). On postponement strategies for product families with multiple points of differentiation. IIE Transactions, 29, 641–650. Jones, D. F., Mirrazavi, S. K., & Tamiz, M. (2002). Multi-objective meta-heuristics: An overview of the current state-of-the-art. European Journal of Operational Research, 137, 1–9. Lee, H. L., & Billington, C. (1994). Design products and processes for postponement. In S. Dasu & C. Eastman (Eds.), Management of design: Engineering and management perspectives (pp. 105–122). Boston: Kluwer. Pagh, J. D., & Cooper, M. C. (1998). Supply chain postponement and speculation strategies: How to choose the right strategy. Journal of Business Logistics, 19, 13–33. Rifai, A. K. (1994). A note on the structure of the goal programming model: Assessment and evaluation. International Journal of Operations and Production Management, 16, 40–49. Romero, C. (1991). Handbook of critical issues in goal programming. Oxford: Pergamon Press. Van Hoek, R. I. (2001). The rediscovery of postponement a literature review and directions for research. Journal of Operations Research, 19, 161–184. Vincke, P. (1992). Multicriteria decision-aid. Chichester: Wiley.
A goal programming model for production planning of perishable products with postponement Stephen C.H. Leung *, Wan-lung Ng Department of Management Sciences, City University of Hong Kong, Hong Kong Received 3 May 2006; received in revised form 28 May 2007; accepted 28 May 2007 Available online 2 June 2007
Abstract One of the important characteristics of perishable products that a decision-maker has to take into account seriously is that the price will drop significantly after a day, or a season. Hence, over-production and storage of such products is not recommended. In this paper, the production process for perishable products is proposed to be divided into two phases by applying the concept of postponement. Consequently, three production activities – direct production, master production and final assembly – will be considered. A preemptive goal programming model to solve aggregate production planning for perishable products is developed, in which three objectives are optimized hierarchically. A set of Hong Kong data has been used to test the effectiveness and the efficiency of the proposed model. Results demonstrate that the decision-makers can find the flexibility and robustness of the proposed model by adjusting the goal priorities with respect to the importance of each objective and the aspiration level with respect to desired target values. 2007 Elsevier Ltd. All rights reserved. Keywords: Production management; Goal programming; Perishable products; Postponement
1. Introduction Postponement of production refers to a common intermediate product being manufactured in the first phase and, according to the differentiating options (such as colors, sizes and types), production line activities such as dyeing, compounding, final assembling, packaging and so on being postponed to a second phase – i.e. until customer orders are received (Aviv & Federgruen, 2001a; Aviv & Federgruen, 2001b; Van Hoek, 2001). Lee and Billington (1994) redesigned a European DeskJet Printer line of Hewlett Packard using the postponement strategy. Pagh and Cooper (1998) stated that the advantage of postponement is reduction or full elimination of risk and uncertainty in manufacturing and logistics operations. Garg and Tang (1997) investigated two points of postponement in the manufacturing stage – early postponement and late postponement – and investigated the importance of demand variabilities and correlations between, and relative magnitudes, of lead times in determining the appropriate points of differentiation. Aviv and Federgruen (2001b) studied the benefit *
Corresponding author. Tel.: +852 2788 8650; fax: +852 2788 8560. E-mail address: [email protected] (S.C.H. Leung).
0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.05.010
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of postponement with unknown demand distributions. Recently, Van Hoek (2001) reviewed the literature on postponement and identified postponement opportunities in operations. In this paper, a postponement strategy is employed to solve production planning for perishable products in a situation with limited resources in production and dramatic growth in demand. Demand for perishable products is time-sensitive because the demand dramatically increases as the day approaches the end of lifecycle, such as the Christmas Day. On the other hand, a shortage of perishable products while the product is saleable may result in significant loss of revenue because perishable products cannot be profitable after a certain day. For instance, in manufacturing industries, people want to buy a Christmas gift in or before December only. Controlling the inventory of perishable products is crucial. However, there is little research that addresses aggregate production planning for perishable products. 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 presence of multiple objectives in real-life problems (Vincke, 1992). Baykasoglu (2001) noted that the aggregate production planning may consider minimization of costs, inventory levels, changes in work force levels, overtime wages in production, subcontracting, changes in production rates, number of machine set-ups, plant/personnel idle time and maximization of profits and superior customer service. Decision-makers always want to develop a model that can consider real-life situations with multiple objectives. To achieve this, in this paper, a preemptive goal programming model is formulated to determine optimal production loading plans. As opposed to linear programming, which directly optimizes objectives, preemptive goal programming is used to manage a set of conflicting objectives by minimizing 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+ and negative deviation d, which represent the distance from this target value. The objective of preemptive goal programming is to minimize the deviations hierarchically so that the goals of primary importance receive first-priority attention, those of secondary importance receive second-priority attention, and so on and so forth. Then, the goals of first-priority are minimized in the first phase. Using the obtained feasible solution result in the phase, the goals of second priority are minimized, and so on. With fast computational growth, both linear and non-linear GP can be solved using well-developed software such as Linear Interactive and Discrete Optimization (LINDO) or meta-heuristics such as simulated annealling, genetic algorithms, tabu search and so on (Jones, Mirrazavi, & Tamiz, 2002). An explicit definition of goal programming was given by Charnes and Cooper (1961). The purpose of this study is to develop a preemptive goal programming model to optimize the production planning problem for perishable products under an uncertain environment, from which we can determine (1) how many finished products should be produced from raw materials directly (direct production), (2) how many semi-finished products should be produced from raw materials (master production), and (3) how many finished products should be produced from semi-finished products (final assembly) so that resources can be better utilized to meet any dramatic growth in demand and total costs (consisting of setup costs, production costs, labor costs, inventory costs, hiring costs and lay-off costs) can be minimized. The organization of this paper is as follows. After this introduction, characteristics and issues of current operations in the toy company under investigation are reviewed. Then a preemptive goal programming model is formulated to solve the production planning problem for perishable products using a postponement strategy in Section 3, and a set of data from the toy company is used to test the effectiveness and efficiency of the proposed model in Section 4. Our conclusions are given in the final section. 2. Characteristics and issues of current operations This study is particularly motivated by the problems faced by a company manufacturing plush toys, whose headquarters is in Hong Kong and the production plant is in China. The finished products are exported to the US and Europe. The company mainly produces two plush toy products – animal-type plush toy products with light songs (duck, bear, hop and dog) and Christmas-theme plush toy products with Christmas songs (snowman and Christmas tree). From the sales report, it is known that animal-type plush toy products can be sold
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throughout the year, whilst Christmas-theme plush toy products can only be sold in November and December as Christmas gifts. Furthermore, demand for the Christmas-theme plush toy products during these 2 months comprises over 50% of total sales volume for the entire year. In addition, demand for Christmas-theme plush toy products increases dramatically as Christmas approaches. Based on the reports and from past experience, the current annual production planning strategy is to produce animal-type products from January to October and Christmas-theme products in the remaining 2 months. The company’s main problem is how to design the production loading plan for Christmas-theme plush toy products in November and December, which is constrained by limited resources, including machine time, workforce level and inventory space. In order to fulfill the substantial increase in demand in December, currently more Christmas-theme plush toy products are produced and stored in November in order to compensate for the limited resources in December. However, the management has discovered that this approach leads to a huge inventory cost and takes up space. The characteristics of the production planning problem with perishable products (Christmas-theme plush toy products) in the company are summarized as follows: • The demand for Christmas-theme plush toy products is time-sensitive. The demand dramatically increases as 25th December approaches. Aviv and Federgruen (2001a) realized that the continual increase of sales volumes will result in a capacity bottleneck. The production plant is not able to produce enough products to fulfill market demand. Currently, the company produces more Christmas-theme plush toy products in November and stores them for future demand. Moreover, it may not be practical for the company to subcontract production to other manufacturers due to constraints of quality control, special technologies, plant specifications and client contracts. In addition, adding more machines leads to higher capital and maintenance costs and the machines may be under-utilized during non-peak seasons. • The Christmas-theme plush toy product is perishable, which is like fashion and high-technology products that have a short life-cycle. A backorder strategy is not appropriate here since this will not only lower customer satisfaction, damaging the company’s goodwill and customer loyalty, but also does not deal with the fact that demand falls after the special event is over. No customer is willing to send Christmas gifts at any time other than Christmas. Aviv and Federgruen (2001b) noted that, subject to dynamic and competitive market forces, demand distribution for short life-cycle products might not be accurately determined. • The finished product is comprised of two components – a musical box mounted on a printed circuit board (PCB) and an outside body with plush fabric finish. The final assembly is a spliced fabric body with a musical box. From here on, we refer to the musical box as the semi-finished product and the outer body with musical box as the finished product. Clearly, the space used to store finished products is usually larger than space needed to store semi-finished products. Therefore, the inventory cost for finished products is usually quite large.
3. Model formulation In this study, the aggregate production planning problem for perishable products faced by a plush toy company in Hong Kong is investigated. For cost effectiveness, the decision-makers have to determine the quantity of product i, i = 1, 2, . . . , n, manufactured over each period of time t, t = 1, 2, . . . , T, to fulfill market demands. The production loading plan consists of: (1) the quantity of finished products to be produced from raw materials directly (direct production), (2) the quantity of semi-finished products to be produced from raw materials (master production), and (3) the quantity of finished products to be produced from semi-finished products (final assembly) in each period of time. 3.1. Notations 3.1.1. Parameters Dit the demand for product i in period t It the storage space limitation in period t vFi the space occupied by one unit of finished product i
534
vSi Wt d adi am i afi bdi bm i bfi kW t kM t Mt CK di CK m i CK fi CWt CHt CLt CP di CP m i CP fi COdi COm i COfi CC Fi CC Si CBi bOC bIC bHC
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the space occupied by one unit of semi-finished product i the maximum number of workers available in period t the regular working hours of labor in each period the man hours required to produce one unit of finished product i from direct production the man hours required to produce one unit of semi-finished product i from master production the man hours required to produce one unit of finished product i from final assembly the machining time required to produce one unit of finished product i from direct production the machining time required to produce one unit of semi-finished product i from masterproduction the machining time required to produce one unit of finished product i from final assembly the fraction of regular workforce available for over-time in period t the fraction of regular machine capacity available for over-time use in period t the maximum regular time machine capacity in period t the setup cost of producing finished product i from direct production the setup cost of producing semi-finished product i from master production the setup cost of producing finished product i from final assembly the labor cost in period t the cost of hiring one worker in period t the cost of lay-off of one worker in period t the regular-time per unit cost of producing one unit of finished product i from direct production the regular-time per unit cost of producing one unit of semi-finished product i from master production the regular-time per unit cost of producing one unit of finished product i from final assembly the overtime per unit cost of producing one unit of finished product i from direct production the overtime per unit cost of producing one unit of semi-finished product i from master production the overtime per unit cost of producing one unit of finished product i from final assembly the carrying cost for one unit of finished product i the carrying cost for one unit of semi-finished product i the backorder cost for one unit of product i aspiration level of ‘operating cost goal’ to be achieved aspiration level of ‘inventory cost goal’ to be achieved aspiration level of ‘hiring and layoff cost goal’ to be achieved
3.1.2. Decision variables P dit the number of finished products i produced from direct production during regular time in period t Pm the number of semi-finished products i produced from master production during regular time in perit iod t P fit the number of finished products i produced from final assembly during regular time in period t Odit the number of finished products i produced from direct production during overtime in period t Om the number of semi-finished products i produced from master production during overtime in period t it Ofit the number of finished products i produced from final assembly during overtime in period t K dit theindicator for producing finished product i from direct production in period t 1; if P dit > 0 ¼ 0; otherwise Km theindicator for producing semi-finished product i from master production in period t it 1; if P m it > 0 ¼ 0; otherwise K fit theindicator for producing finished product i from final assembly in period t f ¼ 1; if P it > 0 0; otherwise Ht the number of workers hired in period t Lt the number of workers laid-off in period t
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Wt I Fit I Sit Bit
the the the the
535
number of workers in period t inventory level of finished product i in period t inventory level of semi-finished product i in period t backorder of product i in period t
3.1.3. Auxiliary variables d deviation of underachievement of bOC OC dþ deviation of overachievement of bOC OC d deviation of underachievement of bIC IC dþ deviation of overachievement of bIC IC d deviation of underachievement of bHC HC þ d HC deviation of overachievement of bHC
3.2. Goal constraints and objective functions The aim of this study is to find an optimal production loading plan that minimizes operating costs, inventory costs and hiring and layoff costs hierarchically. With postponement strategy, the aggregate production loading plan consists of production quantities of direct production, master production and final assembly during regular time and at overtime. Along with these plans, decision-makers can also determine inventory levels, backorder levels and workforce levels. To achieve the optimal plan, three goals are considered in this study. Goal 1: Operating cost goal T X n X t¼1
T X n X d d CK di K dit þ CK mi K mit þ CK fi K fit þ CP i P it þ CP mi P mit þ CP fi P fit
i¼1
þ
t¼1
T X n X t¼1
COdi Odit þ COmi Omit þ COfi Ofit þ
i¼1
i¼1 T X
CW t W t d þ OC þ d OC ¼ bOC
ð1Þ
t¼1
The first term in expression (1) is the setup cost. The second and third terms are the regular-time and over-time production costs, which comprise costs associated with direct production, master production and final assembly. The labor cost associated with regular-time workers is formulated in the fourth term of expression (1). Goal 2: Inventory cost goal T X n X t¼1
i¼1
T X n X CC Fi I Fit þ CC Si I Sit þ CBi Bit d þ IC þ d IC ¼ bIC t¼1
ð2Þ
i¼1
The inventory cost consists of carrying costs and backorder costs. The first component in expression (2) is the sum of carrying cost of finished products and semi-finished products associated with the storage of units of products in warehouses for a given period of time. The second component is the backorder cost associated with under-fulfillment of demand. Goal 3: Hiring and layoff goal T X
ðCH t H t þ CLt Lt Þ d þ HC þ d HC ¼ bHC
ð3Þ
t¼1
Subject to production loading and market demand, in each period of time, the management has to determine how many additional workers would be needed to be recruited to handle extra production loading or how many workers would be needed to be laid off to reduce overheads. The cost of hiring and laying-off workers is expressed in (3).
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The achievement function of the multi-objective production planning of perishable products problem is formulated in the following: þ þ Lexicographically minimize fd þ OC ; d IC ; d HC g 3.3. Constraints The objective functions formulated in the previous section are restricted by four sets of constraints. They are – inventory level constraints, the relationship among the number of workers, the production capacity constraints and the non-negative constraints. 3.3.1. The inventory level constraints I Fit Bit ¼ I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit Dit I Sit ¼ n X
I Sit1
þ
P mit
þ
Omit
ðvFi I Fit þ vSi I Sit Þ 6 I t
P fit
Ofit
i ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; n; t ¼ 1; 2; . . . ; T
t ¼ 1; 2; . . . ; T
t ¼ 1; 2; . . . ; T
ð4Þ ð5Þ ð6Þ
i¼1
Constraint (4) determines either the quantity of finished products stored in the warehouse or the shortfall in meeting market demand. If the total quantity of finished products produced at the company’s plants and transferred to semi-finished products during period t plus previous stock at period t 1 minus previous backorder at period t 1 (i.e. I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit ) is greater than market demand Dit, then the stock at period t will be equal to I Fit ¼ I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit Dit and, under minimization, the deviation Bit = 0; whereas if I Fit1 Bit1 þ P dit þ Odit þ P fit þ Ofit is less than market demand, then I Fit ¼ 0 and Bit ¼ Dit I Fit1 P dit Odit P fit Ofit , indicating that market demand is not satisfied. It is noted that backorder is considered in the model, so Bit is carried over as extra demand to the next period. Constraint (5) determines the quantity of semi-finished products stored in the warehouse. The total quantity of semi-finished products produced at the company’s plants during period t plus previous stock at period t 1 (i.e. I Sit1 þ P mit þ Omit ) must be equal to the semi-finished products stored in the warehouse for period t plus the quantity of semi-finished products used to perform final assembly. The physical storage space during period t is limited by constraint (6). 3.3.2. Relationship among the number of workers W t ¼ W t1 þ H t L Wt 6Wt
t ¼ 1; 2; . . . ; T
ð7Þ
t ¼ 1; 2; . . . ; T
ð8Þ
Constraint (7) ensures that the available workforce in any period equals the workforce from the previous period plus or minus any change in workforce level during the current period. The change in workforce level may be due to either hiring extra workers or laying-off redundant workers. It is noted that Ht Æ Lt = 0 because either the net hiring or the net laying-off of workers takes place over a period, but not both. Constraint (8) ensures that the upper-bounds of change in workforce level over a period are provided. 3.3.3. Production capacity constraints n X
adi P dit þ ami P mit þ afi P fit 6 dW t
t ¼ 1; 2; . . . ; T
ð9Þ
i¼1 n X
adi Odit þ ami Omit þ afi Ofit 6 kWt dW t
t ¼ 1; 2; . . . ; T
ð10Þ
i¼1 n X i¼1
bdi P dit þ bmi P mit þ bfi P fit 6 M t
t ¼ 1; 2; . . . ; T
ð11Þ
S.C.H. Leung, W. Ng / Computers & Industrial Engineering 53 (2007) 531–541 n X d d bi Oit þ bmi Omit þ bfi Ofit 6 kM t Mt
537
t ¼ 1; 2; . . . ; T
ð12Þ
i¼1
P dit þ Odit 6 PK dit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð13Þ
PK mit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð14Þ
P fit þ Ofit 6 PK fit
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð15Þ
P mit
þ
Omit
6
Constraints (9) and (10) limit the labor working hours during regular time and overtime, respectively. Similarly, constraints (11) and (12) limit the machining time during regular time and overtime, respectively. Constraints (13)–(15) ensure that setup costs will be incurred only when the corresponding production activities start. 3.3.4. Non-negative constraints W t ; H t ; Lt ; P dit ; P mit ; P fit ; Odit ; Omit ; Ofit I Fit ; I Sit ; Bit P 0 K dit ; K mit ; K fit
¼ f0; 1g
i ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; n;
t ¼ 1; 2; . . . ; T
ð16Þ
t ¼ 1; 2; . . . ; T
ð17Þ
where P is a large positive number. Constraint (16) ensures that all decision variables are non-negative. Boolean constraints (17) are used for the setup indications of the production activities. 4. Computational results In this section, a data set provided by the plush toy company in Hong Kong is used to illustrate the flexibility of the proposed preemptive goal programming model for the aggregate production planning problem with postponement strategy for perishable products. The tactical/operational level of decision-making in the production planning process is described below. The company receives sales orders from its sales branches covering America and Europe. Based on the company’s projections report, a 2-month planning horizon is determined (November and December) for Christmas products. Each order may require two types of products, i = 1, 2 covering 8 weeks, t = 1, 2, . . . , 8. The setup cost and production cost, as well as labor and machine requirements of different production activities, are given in Table 1. Table 2 shows the limitations of workforce, machine capacity and inventory spaces in each period. Table 3 lists regular time labor cost and the hiring and laying-off cost associated with changes in the workforce level. The unit space occupation for finished products and semi-finished products is provided in Table 4. The carrying cost and backorder costs are shown in Tables 5 and 6. It is noted that, owing to the characteristics of the products, the backorder cost is time-sensitive and dramatically increases as the time approaches the event kick-off period (i.e. the ending period). For each weekly period, demand for products required by the markets is shown in Table 7.
Table 1 Operating and costs data (in HK$, 1US$ = 7.8HK$)
Setup cost ($) Production cost (regular time) Production cost (overtime) Labor time (h) Machining time (h)
Product
Direct finished product production
Semi-finished product production
Transfer production
1 2 1 2 1 2 1 2 1 2
2000 2500 50 60 50 60 0.5 0.6 0.5 0.6
1000 1000 30 30 30 30 0.35 0.35 0.4 0.4
1500 2000 30 40 30 40 0.15 0.25 0.1 0.2
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Table 2 Inventory, production and workforce capacity 1000 m3 1000 16,000 0.3 0.4
Inventory space limitation, I it Maximum workforce level, W t Maximum machine capacity, Mt Fraction of workforce available for over-time, kW t Fraction of machine capacity available for over-time, kM t
Table 3 Labor costs and hiring and laying-off costs Period
Labor cost per worker per period, CWt($) Hiring cost per worker, cHt($) Laying-off cost per worker, cLt ð$Þ
1
2
3
4
5
6
7
8
80 80 120
80 80 120
80 100 120
80 100 120
80 100 120
80 150 120
80 150 120
80 150 120
Table 4 Space occupation Product
Finished product inventory space occupied (m3)
Semi-finished product inventory space occupied (m3)
1 2
1.0 1.0
0.3 0.3
Table 5 Inventory costs (in $HK) Product, i
Finished product
Semi-finished product
1 2
50 50
5 5
Table 6 Backorder cost Product, i
Period, t
1 2
1
2
3
4
5
6
7
8
100 120
110 132
121 145
133 160
146 176
161 193
177 213
195 234
Table 7 Market demand data Product, i
1 2
Period, t 1
2
3
4
5
6
7
8
3000 4800
6600 5100
3750 5700
4350 6300
5100 7200
6600 8550
9450 10,650
17,850 15,150
In preemptive goal programming formulation, the aspiration level for each goal has to be defined. After consulting the company’s management, the target value of ‘Operating cost goal’ bOC is $6,500,000; the target value of ‘Inventory cost goal’ bIC is $500,000; and the target value of ‘Hiring and layoff cost goal’ bHC is $60,000 (Table 8). Moreover, the deviation variables to be minimized are prioritized as follows: first d þ OC , þ second d þ and third d (Run 1). IC HC
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Table 8 Target value with priority Run
Operating cost, $
Inventory cost, $
Hiring and layoff cost, $
1 2 3 4 5
6,500,000 6,500,000 7,000,000 7,000,000 7,000,000
500,000 500,000 500,000 500,000 600,000
60,000 60,000 60,000 60,000 60,000
(P1) (P2) (P1) (P2) (P2)
(P2) (P1) (P2) (P1) (P1)
(P3) (P3) (P3) (P3) (P3)
Note: Numbers in parenthesis indicate the preemptive priority. Pj is the preemptive priority factor which expresses the relative importance of various goals, Pj Pj+1.
Using the data presented, the preemptive goal programming model was formulated as in Section 3; the optimal solution can be easily obtained using the simplex method. Many other packages such as Linear Interactive and Discrete Optimization (LINDO) can also solve the problem efficiently (Evans, 1993). Results from the proposed model are shown in Tables 9–11. First, Table 9 reports that operating costs, inventory costs and hiring and layoff costs are $6,500,000, $2,688,190 and $103,037, respectively. In this run, the highest-priority goal is to minimize the overachievement of operating cost. The positive deviation variable d þ OC is 0. The goal for operating cost has been achieved. The second-priority goal is to minimize the overachievement of inventory costs. The positive deviation variable d þ IC is $2,188,190. This shows that inventory cost is extremely overachieved. The inventory cost, which is the sum of carrying cost ($86,559) and backorder cost ($2,601,631) is $2,688,190. Finally, the remaining priority goal is to minimize the over-achievement of hiring and layoff cost. The positive deviation variable d þ HC is $43,037. The hiring costs and layoff costs are $81,287 and $21,750, respectively. In Table 10, products 1 and 2 produced in this run are 43,358 units and 63,450 units, respectively. On the other hand, the under-fulfillment of products 1 and 2 are 13,342 units and 0, respectively. It is of interest to determine the average cost per product which is the total cost consisting of operating cost, inventory cost and hiring and layoff cost divided by total units of product produced. The average cost in this run is $87. Under the current result, it is observed that the backorder cost is extremely high, resulting in high inventory cost (over $2,188,190 more than the target value). The main reason is that, since the minimization of operating cost is ranked as the first priority, producing more products is not encouraging. Hence, the market demand is under-satisfied. From Table 10, it can seen that under-fulfillment of demand for products 1 and 2 are 13,342 Table 9 Results Run
Operating cost,a $
Inventory cost,
1 2 3 4 5
6,500,000 7,373,513 7,000,000 7,373,513 7,333,075
2,688,190 500,000 1,407,411 500,000 600,000
a
a
Hiring and layoff costa
$
103,037 86,066 86,856 86,066 86,066
Numbers have been rounded off to the nearest integer.
Table 10 Product produced and under-fulfillment Product produced, units
1 2 3 4 5
Product under-fulfillment, units
1
2
1
2
43,358 56,236 52,052 56,236 55,549
63,450 63,379 62,333 63,379 63,450
13,342 464 4648 464 1151
0 71 1117 71 0
Average cost, $
87.0 66.5 81.2 71.0 67.4
540
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Table 11 Production loading plan Product, i
Direct production
Regular time Overtime
Master production
Regular time Overtime
Final assembly
Regular time Overtime
Workforce level Hiring Laying-off
Period, t 1
2
3
4
5
6
7
8
1 2 1 2
978 4800 4800 2022
6600 2654 2653 0
1306 5700 5700 2444
4350 6300 6300 0
5100 7200 7200 0
6600 7833 7833 1908
9450 5458 5458 4000
0 11,493 11,493 0
1 2 1 2
0 0 0 0
0 0 0 0
0 0 0 0
1689 0 5611 0
3229 0 6857 0
0 0 0 3586
0 0 0 0
0 0 0 0
1 2 1 2
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1386 3586 16,000 0
421
612
509
818
1000
1000
1000
1000
0 79
190 0
0 102
309 0
182 0
0 0
0 0
0 0
units and 0; and hence a higher backorder cost is incurred in this run. It is obvious to state that this result is not acceptable. Therefore, it is reasonable to test when the ‘Inventory cost goal’ is the first-priority and the ‘Operating cost goal’ is the second-priority (Run 2). As can be seen in Table 9, the results in Run 2 show that the ‘Inventory cost goal’ is achieved and the inventory cost is decreased by 81.4% ($2,188,190), while increasing the operating cost by 13.4% ($873,513). At the same time, the hiring and layoff cost is also dropped to $86,066 (16.5% reduction). From Table 10, products 1 and 2 produced are 56,236 units and 63,379 units, respectively, and the under-fulfillment are 464 units and 71 units, respectively. Although more products are produced, the average cost for each product in Run 2 is $66, which is significantly less than that in Run 1. The production loading plan of Run 2 is shown in Table 11. It can be seen that the majority of products are produced using regular-time labour. In order to meet the growth in demand during the last two periods, the management is recommended to produce semi-finished products during periods 4–6 and to perform final assembly in period 8. The majority of resources consumed in period 8 are used to perform final assembly of product 2. It is shown that, using the postponement strategy, more products can be produced, particularly in period 8. Lastly, the workforce level does not attain the upper-bound limit in each period. The corresponding number of workers hired and laid off can also be found in Table 11. As can be seen, Runs 1 and 2 have the same set of target values but different priorities. It is of interest to test different sets of target values keeping the same priority structure for goals. In Runs 3 and 4, the target value of ‘Operating cost goal’ bOC is $7,000,000 and the other two target values are kept unchanged. Moreover, the priority structure of goals for Runs 3 and 4 are the same as in Runs 1 and 2, respectively. From Table 9, the results of Run 3 show that operating costs, inventory costs and hiring and layoff costs are $7,000,000, $1,407,411 and $86,856, respectively. Moreover, the ‘Operating cost goal’ is achieved. Compared with the results in Run 1, both inventory cost and hiring and layoff cost are reduced significantly, which are 47.6% for inventory cost and 15.7% for hiring and layoff cost. It is because more products can be produced as target value of ‘Operating cost goal’ is increased. It results in reduction of backorder cost and hence inventory cost is also reduced. Therefore, although the operating cost is increased by $500,000, the inventory cost is reduced by $1,280,779. The total products 1 and 2 produced are 52,052 and 62,333 units. It is noted that more quantities of product 1 are produced. The average cost for each product is $81.2, which is a little smaller than that in Run 1. Although the inventory cost is reduced significantly, it is not a preferable solution. Hence, with regard to the results in Runs 1 and 2, it is reasonable to swap the ranking of priority between ‘Operating cost goal’ and ‘Inventory cost goal’. It is noted that the target values set are the same as in Run 3. The results show that all
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are the same as Run 2. It means that if ‘Inventory cost goal’ is minimized, a non-inferior solution is obtained (Romero, 1991). As mentioned before, target value plays an important role in finding the optimal solution in the preemptive goal programming formulation. It is reasonable to change target value of ‘Inventory cost goal’ as $600,000 (20% increased). The results show that ‘Inventory cost goal’ is achieved, but operating cost is reduced only marginally, by 0.5%. It is claimed that it is not economically attractive to increase the target value of ‘Inventory cost goal’ to $600,000. 5. Conclusions In this paper, a preemptive goal programming model is proposed to deal with the aggregate production planning problem in a toy company in Hong Kong. Due to demand characteristics, the product studied is regarded as a perishable product which cannot be sold or the demand for which drops significantly after Christmas. Three major objectives with target values are optimized hierarchically. The three objectives are the minimization of operating cost consisting of setup cost, production cost and labour cost, the minimization of inventory cost consisting of carrying cost and backorder cost, and the minimization of hiring and layoff cost to reduce the change of workforce level. The model can effectively find an optimal production loading plan consisting of three production activities which are direct production, master production and final assembly. It is also able better to control inventory levels with surplus storage and shortage. A real case in an existing Hong Kong-based plush toy manufacturing company is studied in this paper. Some useful findings are observed. With postponement strategy, it is recommended to produce semi-finished products in the earlier planning horizon, to be used to perform final assembly in the later period of planning horizon in order to meet the dramatically increases in demand. The results of the runs illustrate the flexibility and robustness of the model so that the management can examine numerous scenarios regarding various strategic assumptions by changing priority rankings. This paper illustrates the effectiveness and the efficiency of the proposed model. However, there is still much room for improvement and investigation for the proposed model with regard to its application to real-world situations. Real data from other manufacturing companies can be used to validate the model and to analyze its sensitivity to changes in production management strategies. It should be noted that the computation and analysis of the model under different scenarios would lead to different outcomes. References Aviv, Y., & Federgruen, A. (2001a). Capacitated multi-item inventory systems with random and seasonally fluctuating demand: implications for postponement strategies. Management Science, 47, 512–531. Aviv, Y., & Federgruen, A. (2001b). Design for postponement: a comprehensive characterization of its benefits under unknown demand distribution. Operations Research, 49, 578–598. Baykasoglu, A. (2001). MOAPPS 1.0: aggregate production planning using the multiple-objective tabu search. International Journal of Production Research, 39, 3685–3702. Charnes, A., & Cooper, W. W. (1961). Management models and industrial applications of linear programming. New York: Wiley. Evans, J. R. (1993). Applied production and operations management. St. Paul: West Group. Garg, A., & Tang, C. S. (1997). On postponement strategies for product families with multiple points of differentiation. IIE Transactions, 29, 641–650. Jones, D. F., Mirrazavi, S. K., & Tamiz, M. (2002). Multi-objective meta-heuristics: An overview of the current state-of-the-art. European Journal of Operational Research, 137, 1–9. Lee, H. L., & Billington, C. (1994). Design products and processes for postponement. In S. Dasu & C. Eastman (Eds.), Management of design: Engineering and management perspectives (pp. 105–122). Boston: Kluwer. Pagh, J. D., & Cooper, M. C. (1998). Supply chain postponement and speculation strategies: How to choose the right strategy. Journal of Business Logistics, 19, 13–33. Rifai, A. K. (1994). A note on the structure of the goal programming model: Assessment and evaluation. International Journal of Operations and Production Management, 16, 40–49. Romero, C. (1991). Handbook of critical issues in goal programming. Oxford: Pergamon Press. Van Hoek, R. I. (2001). The rediscovery of postponement a literature review and directions for research. Journal of Operations Research, 19, 161–184. Vincke, P. (1992). Multicriteria decision-aid. Chichester: Wiley.