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Simultaneous determination of multiproduct batch and full truckload shipment schedules
ARTICLE IN PRESS Int. J. Production Economics 118 (2009) 111–117
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Simultaneous determination of multiproduct batch and full truckload shipment schedules Avijit Banerjee Department of Decision Sciences, Drexel University, Philadelphia, PA 19063, USA
a r t i c l e in fo
abstract
Available online 20 August 2008
A fundamental premise of the well-known economic lot-scheduling problem (ELSP) is that the finished products are consumed at continuous rates, i.e. their respective cycle inventories are depleted on the basis of unit transactions. In today’s supply chains, however, employing complex distribution networks, finished goods inventories from manufacturing plants are usually shipped in bulk to succeeding stages along the distribution process. Moreover, existing transport economies often tend to favor full truckload (TL), rather that partial or less than truckload (LTL) shipments, for economical movement of such goods. The scenario examined here, however, involves a set of products, for which individual TL shipments are uneconomical. As a remedy, we construct a model for taking advantage of TL rates by combining LTL quantities of the items into a full load. We adopt the common cycle approach for the ELSP, in conjunction with a common replenishment cycle, as a coordination mechanism that is simple to analyze and implement. This is integrated with a periodic full truckload shipping schedule. Such effective coordination of production and shipment schedules is likely to result in a more streamlined supply chain. The concepts developed are illustrated through a simple numerical example. & 2008 Elsevier B.V. All rights reserved.
Keywords: Economic lot scheduling Full truckload shipments Supply chain coordination Inventory model
1. Introduction The classical economic lot-scheduling problem (ELSP) involves the production of multiple products in a single facility or machine, which can process only one item at a time. The determination of each product’s lot size and a feasible production schedule at minimum total relevant cost are the fundamental issues of the ELSP. The ELSP has received considerable research attention over the years. Past attempts to solve this problem optimally have utilized various mathematical programming techniques, such as linear programming (Maxwell, 1964), dynamic programming (Bomberger, 1966; Elmaghrabi, 1978) and mixed integer programming (Delporte and Thomas, 1977). Efforts along these lines, however, have experienced Tel.: +1 215 895 1449; fax: +1 215 895 2907.
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increasing computational inefficiency as the problem size gets larger and have concluded that the optimization of the ELSP becomes either inefficient or impossible for even relatively small problems. Insuring the feasibility of a schedule in the ELSP appears to be a major factor of complexity, since it is NP-hard (Luenberger, 1973). The NP class of problems is solvable in nondeterministic polynomial time. There are no efficient algorithms known for this class of problems. This class of problems can be (a) solved by polynomialtime algorithms if NP is identical to P (polynomial class), or (b) proved to be permanently intractable. Therefore, most efforts in solving the ELSP during the last three decades have been dedicated towards developing heuristics for obtaining near-optimal solutions, rather than optimization (e.g. Madigan, 1968; Stankard and Gupta, 1969; Doll and Whybark, 1973; Goyal, 1973; Haessler and Hogue, 1976; Saipe, 1977; Elmaghrabi, 1978; Haessler,
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1979; Park and Yun, 1984; Panayiotopoulos, 1983; Geng and Vickson, 1988; Davis, 1990) and by constructing feasibility conditions (e.g. Vemuganti, 1978; Boctor, 1982; Hsu, 1983; Davis, 1990). Needless to say that within the framework of supply chain management, the ELSP has an important role to play in terms of coordinating the activities of the various members of such systems. In recent years, a great deal of research has focused on issues of coordinating supply and demand within this context (see, e.g. Goyal and Gupta (1989) and Thomas and Griffin (1996) for surveys). Much of the existing research on ELSP, however, assumes that the inventory depletion rate for each of the multiple items is uniform, which would be the case if customer demands are satisfied directly from the manufacturing facility. In most of today’s global firms, supply chains consist of complex distribution networks, involving production plants, vehicle terminals, warehouses, distribution centers, retail outlets, etc. In such cases, the notion of uniform product demands at a production facility is an incorrect representation of the real world. In reality, inventory depletions at a manufacturing plant occur in discrete, sizeable lots, as a result of bulk shipments, that take advantage of transportation economies of scale. Thus, for achieving streamlined supply chain structures, it is necessary to link or coordinate the production schedule with the outbound shipment schedule. Unfortunately, as mentioned above, there is a dearth of research that addresses the ELSP in terms of such linkages. This paper is an attempt to fill this research gap and reexamine the ELSP for developing a procedure for integrating the production schedule of multiple items into a shipment plan. In other words, the primary focus here lies in coordinating the manufacturing process with the transportation function. In shipping goods from manufacturing facilities to subsequent stages of the supply chain, truck shipments are perhaps the most commonly used means. Such transportation can be either full truckload (TL) or less than truckload (LTL) shipments. Although truck shipping rate structures are often complex, generally speaking, TL shipments are substantially less expensive, on a cost per unit basis, than LTL shipments (see, e.g. Chopra and Meindl, 2004). Under certain circumstances, though, the demand rate of a product may not be sufficiently large to warrant periodic individual TL shipments, since such large delivery lots may result in excessive inventory holding costs, negating the advantage of low transportation costs. Nevertheless, if there is a group of such relatively low-demand products that are shipped from a source to various demand locations, it may be possible to combine several partial truckloads of individual products to constitute a full truckload, thus obtaining the relative advantage of TL rates, in conjunction with lower inventory levels at the destination locations. In this study, we examine a scenario such as this. In our proposed procedure for coordinating the production schedule of multiple products (in an ELSP environment) with their shipments to several demand locations, the common production cycle approach (see,
e.g. Maxwell, 1964) is combined with the notion of a delivery (or replenishment) cycle that is common to all the individual items. For simplicity of implementation, we restrict the production cycle to an integer multiple of the delivery cycle. Under many circumstances, as illustrated later through a numerical example and subsequent sensitivity analysis, coordinated production and shipment decisions, such as the one suggested in this paper, may result in a more coherent and efficient supply chain. 2. Assumptions and notation 2.1. Assumptions The following assumptions, describing the manufacturing-distribution scenario, are made in this paper: 1. The operating environment is deterministic. 2. Multiple products are produced in a capacitated batch production environment, with different production rates for the various products. We assume, without loss of generality, zero setup times for all the products. If, in reality, the setup times are nonzero, they can be easily incorporated into our model, outlined later. 3. Only a single product may be produced at any given time. 4. Stockouts or shortages are not permitted. 5. We adopt the common cycle solution to the ELSP, where each product is produced exactly once in every production cycle. Although this is a restrictive assumption, the rationale for its adoption is threefold. First, this approach ensures a feasible solution if sufficient capacity to produce all the products exists. Secondly, from a practitioner’s viewpoint, the common cycle method is relatively easy to understand and implement, especially when stable production and transportation schedules need to be coordinated on a routine basis. Finally, the common cycle solution can indeed be optimal or near-optimal under certain conditions (see, Jones and Inman (1989) for details). 6. Each of the products is transported via truck and sold at one or more given demand location(s). For any of these products, however, its market demand rate is not sufficiently high to warrant full TL shipments. On the other hand, LTL shipments are relatively more expensive and undesirable. 7. Thus, TL shipments are made, where each truckload (capacitated by total weight and/or volume) contains a mix of all the products, which are delivered at the appropriate location(s). 8. For each TL shipment, a fixed cost is incurred, regardless of the mix of products carried. In contrast, LTL shipment rate structures often entail a unit variable shipping cost, based on the weight and/or the volume of shipment. 9. During each production cycle, an integer number, K, of these TL shipments are made at equal intervals of time.
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10. At each of the various demand locations, stocks are replenished via a periodic review, order-up-to level inventory control system. For coordination purposes, all the items at all the demand locations share a common fixed review period. 11. The product lot sizes, Qi, are treated as continuous variables for all i.
2.2. Notation The following notational scheme applies throughout this paper: i Di Pi Ai hi Qi K T KQi KT C wi Ii TRC
an index used to denote a specific product, i ¼ 1, 2, y, n the demand rate for product i (units/time unit) the production rate for product i (units/time unit) manufacturing setup cost per production batch for product i ($/batch) inventory holding (carrying) cost for product i ($/unit/time unit) amount of product i contained in each TL shipment (units) a positive integer, representing the number of shipments per production cycle; the shipment interval in time units (common to all products and locations) the production lot size (in units) for product i production cycle length in time units the TL capacity, i.e. maximum total load (or volume) allowable per truckload weight (or volume) of each unit of product i average inventory level (units) of product i total relevant cost ($) per time unit
3. Model development For illustrative and model development purposes, the inventory-time plots pertaining to a scenario involving three products (n ¼ 3) and three full TL shipments per production cycle (K ¼ 3), of length KT time units, are depicted in Fig. 1. From this figure it is clear that for a common delivery cycle time of T (i.e. T ¼ QiDi, 8i), each truckload contains Qi units of product i, i ¼ 1, 2, 3. The total weight (or volume) in a load, w1Q1+w2Q2+w3Q3, is obviously limited to the truck capacity, C. Also, the products should be sequenced to minimize the total inventory holding cost. The average inventory values for the three items are computed as follows:
I1 ¼
1 KQ 1 KQ 2 Q 3 Q1 KQ 1 þ þ KQ 1 þ ðK 1Þ Q1 2 P1 P2 P3 D1 Q1 Q1 þðK 2Þ Q1 þ þ Q 1 =½KQ 1 =D1 D1 D1
113
Q1 Q Q3 Q ¼ KD1 þ 2 þ D1 þ ðK 1Þ 1 2P 1 P2 P3 2 1 KQ 2 Q3 Q2 KQ 2 I2 ¼ þ KQ 2 þ ðK 1Þ Q2 2 P2 P3 D2 Q2 Q2 Q2 þ þ Q 2 =½KQ 2 =D2 þðK 2Þ D2 D2 Q2 Q3 Q ¼ KD2 þ D2 þ ðK 1Þ 2 2P 2 P3 2 Q 3 D3 I3 ¼ ð2 KÞ þ K 1 2 P3 (for derivation see, e.g. Joglekar, 1988). In general, for the n-products case: 2 3 n1 n1 n1 X X Q j5 Q n X Qi 4 þ Ii ¼ K Di þ D 2P i j¼iþ1 Pj P n i¼1 i i¼1 þ ðK 1Þ
n1 X
Q i =2;
for
i ¼ 1; 2; . . . ; n 1
i¼1
and
Q Dn In ¼ n ð2 KÞ þ K 1 2 Pn
3.1. Objective function The objective of our optimization model is to minimize the total relevant cost per time unit, which is a function of Q1, Q2, y, Qn (denoted by Q ) and K. Thus, using the results obtained above, the objective function is 2 3 n n1 n1 X X X Q j5 Di Ai Qi 4 Minimize TRCðQ ; KÞ ¼ þK Di hi þ KQ i 2Pi j¼iþ1 P j i¼1 i¼1 n1 n1 X Qn X Q i hi Di hi þ ðK 1Þ P n i¼1 2 i¼1 Q n hn Dn ð2 KÞ þ K 1 þ 2 Pn
þ
(1)
The first term on the RHS of (1) represents the total setup cost per time unit for the n products. The remaining terms, capturing the inventory holding cost per time unit for items 1, 2, y, n1 and n, respectively, are obtained by adding the average inventory levels, I¯i , for i ¼ 1, 2, y, n1, and I¯n , derived above, weighted by the respective holding cost parameters, hi and hn. 3.2. Constraints The relevant model constraints are outlined below: 1. The delivery cycle time is common to all products: Q1 Q2 Q ¼ ¼ ¼ n ¼ T D1 D2 Dn or Q i ¼ TDi ; i ¼ 1; 2; . . . ; n
(2)
2. The production capacity of the system is limited, i.e. n X Di i¼1
Pi
p1
(3)
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Inventory Product 1
KQ
Time
Q =T D
KQ P
KQ = KT D
Inventory Product 2
KQ
KQ P
Q =T D
Time
Q =T D
Time
KQ = KT D
Inventory Product 3
KQ
Q P
KQ P
KQ = KT D
Fig. 1. Inventory-time plots (n ¼ 3, K ¼ 3).
3. Schedule feasibility, i.e. total production time needed cannot exceed the manufacturing cycle time: K
n1 X Qi i¼1
or
Pi
n1 X i¼1
þ
Qn pKT Pn
Qi Q þ n pT Pi KP n
enables us to considerably simplify the non-linear mixed integer programming problem expressed by (1)–(5). From (2) Q i ¼ TDi ; 8i
or
Qj ¼
Q1 D; D1 j
j ¼ 2; 3; . . . ; n
(4)
Note that this constraint can be easily modified for nonzero setup times by adding these to its left-hand side. 4. Truck capacity is limited by weight or volume, i.e. n X wi Q i ¼ C (5)
Substituting the Qi values for all i in (5), we obtain Q1 Q1 D2 þ þ wn Dn ¼ C D1 D 9 1 n P > > = wi Di or Q 1 ¼ CD1 =
w1 Q 1 þ w2
i¼1
i¼1
and Q j ¼ TDj ;
> ; j ¼ 2; 3; . . . ; n: >
(6)
3.3. Model simplification The coordination mechanism of an equal inventory replenishment (or delivery) cycle time for all products
Recalling that Di for i ¼ 1,2, y, n are known parameters, T and all Qi values can be determined a priori and recursively using (6). Now, define the parameters a, b
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and g as follows: 9 n DA P i i > > ¼a > > > i¼1 Q i > > " # > > n1 n1 n1 = P Qj P P Q i hi Q n hn Qi Dn þ 1 ¼b Di hi þ þ 2Pi j¼iþ1 P j 2 Pn > i¼1 i¼1 2 > > > > n1 n1 > P P Qn Q i hi Dn 1 > > > þ Q n hn and Di h i 2 ¼g ; P n i¼1 2 P n i¼1 (7) Substituting the values Qi, i ¼ 1, 2, y, n, obtained from (6), into (7) and eliminating constraints (2) and (5), the original model expressed by (1)–(5) can be considerably simplified to Minimize TRCðKÞ ¼ a=K þ bK þ g
i¼1
Pi
n1 X Qi i¼1
Pi
4. Numerical example and sensitivity analysis For illustrative purposes, consider the following data pertaining to a three-product example: Product Di Pi Ai hi wi (i) (units/year) (units/year) ($/setup) ($/unit/year) (lbs./unit) 1 2 3
8000 12,000 15,000
30,000 50,000 40,000
500 1000 800
2 3 3
20 50 40
Truck capacity: C ¼ 50,000 lb. The steps of the solution procedure outlined above yield the following results:
p1; i ¼ 1; 2; . . . ; n
þ
5. Find K* from conditions (13), i.e. K n ðK n 1Þpða=bÞ pK n ðK n þ 1Þ. Pn1 6. Check constraint (10): i¼1 ðQ i =Pi Þ þ ðQ n =KPn ÞpT. P 7. If violated, reset K* to Q n =ðPn ½ n1 i¼1 Q i =P i TÞ.
(8)
Subject to n X Di
115
Qn pT KPn
K ¼ 1; 2; . . . ::; etc.
(9)
(10)
(11)
Note that (8) is strictly convex in the integer K. Therefore, if TRC(K) is minimized at K ¼ K*, then TRCðK n ÞpTRCðK n 1Þ and TRCðK n ÞpTRCðK n þ 1Þ
(12)
Substituting (12) into (8) we obtain the following optimality conditions:
a
K n ðK n 1Þp pK n ðK n þ 1Þ
b
(13)
3.4. Solution procedure Utilizing the optimality conditions (13), the simplified model, (8)–(11), can now be easily solved by a simple procedure, which is outlined below: 1. For incurring minimal inventory holding cost, sequence the order of production by rank ordering and numbering the n products [1], [2], y, [n] based on h½1 D½1 ph½2 D½2 p ph½n D½n . P 2. Check constraint (9) n1 i¼1 ðDi =P i Þp1. If this constraint is violated, increase some Pi values, if possible, or reduce the item set, such that (9) is satisfied. 3. Compute T, Q1 and all other Qi values using (6). 4. Calculate the parameters a and b using the definitions in (7).
1. The three items are already arranged in the sequence h1D1oh2D2oh3D3. Pn 2. i¼1 ðDi =P i Þ ¼ 0:882; i.e. sufficient production capacity is available. 3. From (6), Q1 ¼ 294, T ¼ 0.03675 year, Q2 ¼ 441 and Q3 ¼ 551 units. 4. Using (7), a ¼ 62560 and b ¼ 1851.3. 5. The ratio a/b is calculated to be 33.8. 6. Thus, K* ¼ 6 satisfies conditions (13). Based on these results, the TL delivery cycle time is 0.03675 year and, based on K*, the production cycle time is 0.2205 year. Also each TL shipment consists of, respectively, 294 units (5880 lb), 441 units (22,050 lb) and 551 units (22,040 lb) of products 1, 2 and 3, resulting in a total load of 49,970 lb. Finally, the parameter g is computed to be 445.8, yielding a total relevant cost of about $21,089 per year using (8). In order to compare this coordinated approach to the results yielded by the traditional ELSP common cycle solution, without any attempt to link the production plan to TL shipments, suppose the following shipping rate structure applies: the average cost of a TL shipment is $500 and an LTL shipment cost is based on a variable cost of $0.015/lb shipped. Thus, for the suggested coordinated policy, employing only TL shipments, an additional $13,600 per year are incurred in shipping cost, bringing the total relevant cost to about $34,689. On the other hand, if the common cycle policy is not linked to shipments, the average number of production cycles per year would be 5.93 (derived from the classical common cycle model). As a result, the lot sizes for products 1, 2 and 3 would be 1349 units (26,980 lb), 2024 units (101,200 lb) and 2529 units (101,160 lb), respectively. The resulting average annual setup and holding cost at the plant would be about $16,151. The cost of transporting these batches to the retail stage via a mix of TL and LTL shipments adds approximately another
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$14,470/year. Thus, the total cost, so far, for the uncoordinated policy is about $30,621 a year, which is approximately $4068 less per year than our suggested approach, in spite of an increase of $870 in shipping charges. However, suppose that at the retail level, the inventory holding costs for products 1, 2 and 3 are, respectively, $3, $4 and $4 per unit per year. Now, the average annual retail inventory holding costs for the coordinated and the uncoordinated policies are, respectively, $2426 and $11,129. Therefore, from the standpoint of the entire supply chain (production and retail facilities) the coordinated policy saves about $4645 (in excess of 11%) a year. It is clear from this illustration that from a supply chain perspective, the penalties of not having coordinated production and shipping schedules arise out of additional transportation and retail inventory holding costs. 4.1. Sensitivity analysis In order to explore the effectiveness of the coordinated model developed here, as opposed to an uncoordinated approach, we solved 23 additional versions of the numerical example discussed above under a variety of operating conditions. The results of this sensitivity analysis are summarized in Table 1. Needless to say that, for purposes of equitable comparison, both of these approaches are based on the notion of a common production cycle. Keeping the finished goods inventory holding cost parameters for the three items at the plant level fixed at 2, 3 and 3, respectively, the manufacturing setup costs sets are varied from relatively low (250, 500 and 400) to high (1000, 2000 and 1600) values. Due to the value-added concept, it
is likely that holding costs at the retail level are likely to be higher than those at the production stage. Consequently, two sets of these parameters, i.e. (2.1, 3.1, 3.1), about 3–5% higher, and (3, 4, 4), about 30–50% higher than those at the plant level, are chosen. Three levels of cost for a TL shipment (300, 500 and 1000) are considered. The corresponding variable per lb. LTL shipping rates are kept proportionately fixed at 0.009, 0.015 and 0.03, respectively. All transportation costs are arbitrarily allocated to the retail stage. Table 1 indicates that under the various parametric conditions examined, the coordinated policy tends to yield higher total setup and holding cost at the plant or manufacturing stage. The total shipping and holding cost at the retail stage, however, are significantly higher under the uncoordinated approach, yielding annual net savings ranging from over 6% to almost 16% as a result of coordinating production with transportation, in particular taking advantage of the economies of full truckload shipping. Not unexpectedly, such savings tend to be magnified under higher retail stage holding costs. While lower production setup costs generally tend to yield greater savings for our coordinated policy, the combination of higher production setup, transportation and retail stage carrying costs also tend to make this approach more advantageous. Interestingly, under the latter combination of parameters, the performance gap between coordinated and uncoordinated policies tend to become smaller, albeit still substantial, as the differential between the holding cost parameters at the two stages of the supply chain narrows. We examined a final scenario involving a modified version of the original numerical example provided
Table 1 Sensitivity analysis results Setup cost parameters
Coordinated common cycle shipment policy Transportation cost parameters Fixed
250, 500, 400 (K ¼ 4)
500, 1000, 800 (K ¼ 6)
1000, 2000, 1600 (K ¼ 8)
Uncoordinated shipment policy
Percent savings (%)
Retail-stage holding cost parameters
Total plant cost
Total retail stage cost
Total supply chain cost
Total plant cost
Total retail stage cost
Total supply chain cost
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
14,780 14,780 14,780 14,780 14,780 14,780
10,007 10,586 15,447 16,026 29,047 29,626
24,787 25,366 30,227 30,806 43,827 44,406
10,634 11,420 10,634 11,420 10,634 11,420
16,659 17,593 23,314 24,076 39,950 40,283
27,293 29,013 33,948 35,496 50,584 51,703
9.18 12.57 10.96 13.21 13.36 14.11
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
21,089 21,089 21,089 21,089 21,089 21,089
10,007 10,586 15,447 16,026 29,047 29,626
31,096 31,675 36,536 37,115 50,136 50,715
15,038 16,151 15,038 16,151 15,038 16,151
18,489 19,811 24,520 25,599 39,599 40,068
33,527 35,962 39,558 41,750 54,637 56,219
7.25 11.92 7.64 11.10 8.24 9.79
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
30,005 30,005 30,005 30,005 30,005 30,005
10,007 10,586 15,447 16,026 29,047 29,626
40,012 40,591 45,452 46,031 59,052 59,631
21,267 22,841 21,267 22,841 21,267 22,841
21,895 25,281 27,590 31,642 41,827 47,545
43,162 48,122 48,857 54,483 63,094 70,386
7.30 15.65 6.97 15.51 6.41 15.28
Variable
300
0.009
500
0.015
1000
0.03
300
0.009
500
0.015
1000
0.03
300
0.009
500
0.015
1000
0.03
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earlier. The holding cost parameters at the manufacturing and retail stages are set to be the same, i.e. 2, 3 and 3. Additionally, the fixed cost of a TL shipment and the variable unit cost for LTL transportation, respectively, are set at 2000 and 0.04. Thus, for the given truck capacity of 50,000, there is no apparent economic advantage for TL shipments, since the per unit transportation charge is the same for both shipping modes. Even under these conditions, the coordinated policy yielded close to a 2% savings in total cost, in comparison with the uncoordinated policy. This observation, in addition to the results discussed above, appears to highlight the inherent economic advantage (in addition to other practical merits) of supply chain coordination. 5. Conclusions We have developed a mathematical model to align the production schedule of multiple products in a manufacturing facility with a periodic full truckload shipping plan. Since it is uneconomical to ship any individual product on a TL basis, each truckload is composed of a mix of LTL quantities of the individual items. As a result of deploying a common cycle production plan, as well as a common delivery cycle, inventory control, production planning and shipping decisions are streamlined and the supply chain as a whole becomes more coherent from an operational perspective. Exploitation of the properties of the coordination mechanism allows us to simplify the model structure considerably, yielding a solution without the use of a computer or specialized software. The economic desirability of coordination is demonstrated through numerical analysis. Our approach, however, is limited in two respects. First of these is the adoption of the common or rotational cycle technique for scheduling the production of multiple products. Although this method does not guarantee optimality, it yields optimal or near-optimal solutions to the ELSP under many real-world-operating conditions (see Jones and Inman (1989) for a detailed discussion on this topic). Thus, our model is likely to find applicability under such circumstances. In addition, this approach has considerable practical appeal. From a managerial standpoint, the simplicity of this method renders it readily understandable and easy to implement. The notion of producing each product regularly once every cycle leads to identical and routine production and shipping decisions, without variation. Such stability is often a highly desirable characteristic for practitioners. Furthermore, if adequate capacity exists for manufacturing all the products towards fulfilling their demands, the common cycle procedure guarantees a feasible solution. This feasibility issue, as mentioned before, is a major complicating factor in solving the ELSP optimally (see Hsu (1983) or Davis (1990), for instance). A second limitation of this study is the assumption of a deterministic environment. In addition to exploring
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procedures other than rotational cycles, future research should attempt to develop coordinated decisions under stochastic conditions. Nevertheless, we hope that this study has been able to identify some of the major benefits of coordinating production and shipping in supply chains and will prove to be of some value in terms of future research efforts along these lines. Finally, it is hoped that our proposed model will be helpful for practitioners in deriving and implementing innovative real-world supply chain solutions. References Boctor, F.F., 1982. The two-product, single-machine, static demand, infinite horizon lot scheduling. Management Science 28, 798–807. Bomberger, E., 1966. A dynamic programming approach to a lot size scheduling problem. Management Science 12, 778–784. Chopra, S., Meindl, P., 2004. Supply Chain Management: Strategy, Planning and Operation, Second ed. Pearson, Prentice-Hall, Upper Saddle River, NJ. Davis, S.G., 1990. Scheduling economic lot size production runs. Management Science 36, 985–998. Delporte, C.M., Thomas, L.J., 1977. Lot sizing and sequence for N products on one facility. Management Science 23, 1070–1079. Doll, C., Whybark, D., 1973. An iterative procedure for the single-machine multi-product lot scheduling problem. Management Science 20, 50–55. Elmaghrabi, S.E., 1978. The economic lot-scheduling problem (ELSP) review and extensions. Management Science 24, 587–598. Jones, P.C., Inman, R.R., 1989. When is the economic lot scheduling problem easy? IIE Transactions 21, 11–20. Geng, P.C., Vickson, R.G., 1988. Two heuristics for the economic lot scheduling problem: An experimental study. Naval Research Logistics 35, 605–617. Goyal, S.K., 1973. Scheduling a multi-product single machine system. Operational Research Quarterly 24, 261–269. Goyal, S.K., Gupta, Y., 1989. Integrated inventory models: The buyer– vendor coordination. European Journal of Operational Research 41, 261–269. Haessler, R.W., 1979. An improved extended basic period procedure for solving the economic lot scheduling problem. AIIE Transactions 11, 336–340. Haessler, R.W., Hogue, S.L., 1976. A note on the single-machine multi-product lot scheduling problem. Management Science 22, 909–912. Hsu, W.L., 1983. On the general feasibility test of scheduling lot sizes for several products on one machine. Management Science 29, 93–105. Joglekar, P.N., 1988. Comments on ‘‘a quantity discount pricing model to increase vendor profits. Management Science 34, 1391–1398. Luenberger, D.G., 1973. Introduction to Linear and Nonlinear Programming. Addison-Wesley Publishing Co., Reading, MA. Madigan, J.G., 1968. Scheduling a multi-product single machine system for an infinite planning period. Management Science 14, 713–719. Maxwell, W.L., 1964. The scheduling of economic lot sizes. Naval Research Logistics Quarterly 11, 89–124. Panayiotopoulos, J.C., 1983. The generalized multi-item lot size scheduling. European Journal of Operational Research 14, 59–65. Park, K., Yun, D.K., 1984. A stepwise partial enumeration algorithm for the economic lot scheduling problem. IIE Transaction 16, 363–370. Saipe, A., 1977. Production runs for multiple products: the two-product heuristic. Management Science 12, 1321–1327. Stankard, M.F., Gupta, S.K., 1969. A note on Bomberger’s approach to lot size scheduling: Heuristic proposed. Management Science 15, 449–452. Thomas, D.J., Griffin, P.M., 1996. Coordinated supply chain management. European Journal of Operational Research 94, 1–15. Vemuganti, R.R., 1978. On the feasibility of scheduling lot sizes for two products on one machine. Management Science 24, 1668–1673.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Simultaneous determination of multiproduct batch and full truckload shipment schedules Avijit Banerjee Department of Decision Sciences, Drexel University, Philadelphia, PA 19063, USA
a r t i c l e in fo
abstract
Available online 20 August 2008
A fundamental premise of the well-known economic lot-scheduling problem (ELSP) is that the finished products are consumed at continuous rates, i.e. their respective cycle inventories are depleted on the basis of unit transactions. In today’s supply chains, however, employing complex distribution networks, finished goods inventories from manufacturing plants are usually shipped in bulk to succeeding stages along the distribution process. Moreover, existing transport economies often tend to favor full truckload (TL), rather that partial or less than truckload (LTL) shipments, for economical movement of such goods. The scenario examined here, however, involves a set of products, for which individual TL shipments are uneconomical. As a remedy, we construct a model for taking advantage of TL rates by combining LTL quantities of the items into a full load. We adopt the common cycle approach for the ELSP, in conjunction with a common replenishment cycle, as a coordination mechanism that is simple to analyze and implement. This is integrated with a periodic full truckload shipping schedule. Such effective coordination of production and shipment schedules is likely to result in a more streamlined supply chain. The concepts developed are illustrated through a simple numerical example. & 2008 Elsevier B.V. All rights reserved.
Keywords: Economic lot scheduling Full truckload shipments Supply chain coordination Inventory model
1. Introduction The classical economic lot-scheduling problem (ELSP) involves the production of multiple products in a single facility or machine, which can process only one item at a time. The determination of each product’s lot size and a feasible production schedule at minimum total relevant cost are the fundamental issues of the ELSP. The ELSP has received considerable research attention over the years. Past attempts to solve this problem optimally have utilized various mathematical programming techniques, such as linear programming (Maxwell, 1964), dynamic programming (Bomberger, 1966; Elmaghrabi, 1978) and mixed integer programming (Delporte and Thomas, 1977). Efforts along these lines, however, have experienced Tel.: +1 215 895 1449; fax: +1 215 895 2907.
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increasing computational inefficiency as the problem size gets larger and have concluded that the optimization of the ELSP becomes either inefficient or impossible for even relatively small problems. Insuring the feasibility of a schedule in the ELSP appears to be a major factor of complexity, since it is NP-hard (Luenberger, 1973). The NP class of problems is solvable in nondeterministic polynomial time. There are no efficient algorithms known for this class of problems. This class of problems can be (a) solved by polynomialtime algorithms if NP is identical to P (polynomial class), or (b) proved to be permanently intractable. Therefore, most efforts in solving the ELSP during the last three decades have been dedicated towards developing heuristics for obtaining near-optimal solutions, rather than optimization (e.g. Madigan, 1968; Stankard and Gupta, 1969; Doll and Whybark, 1973; Goyal, 1973; Haessler and Hogue, 1976; Saipe, 1977; Elmaghrabi, 1978; Haessler,
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1979; Park and Yun, 1984; Panayiotopoulos, 1983; Geng and Vickson, 1988; Davis, 1990) and by constructing feasibility conditions (e.g. Vemuganti, 1978; Boctor, 1982; Hsu, 1983; Davis, 1990). Needless to say that within the framework of supply chain management, the ELSP has an important role to play in terms of coordinating the activities of the various members of such systems. In recent years, a great deal of research has focused on issues of coordinating supply and demand within this context (see, e.g. Goyal and Gupta (1989) and Thomas and Griffin (1996) for surveys). Much of the existing research on ELSP, however, assumes that the inventory depletion rate for each of the multiple items is uniform, which would be the case if customer demands are satisfied directly from the manufacturing facility. In most of today’s global firms, supply chains consist of complex distribution networks, involving production plants, vehicle terminals, warehouses, distribution centers, retail outlets, etc. In such cases, the notion of uniform product demands at a production facility is an incorrect representation of the real world. In reality, inventory depletions at a manufacturing plant occur in discrete, sizeable lots, as a result of bulk shipments, that take advantage of transportation economies of scale. Thus, for achieving streamlined supply chain structures, it is necessary to link or coordinate the production schedule with the outbound shipment schedule. Unfortunately, as mentioned above, there is a dearth of research that addresses the ELSP in terms of such linkages. This paper is an attempt to fill this research gap and reexamine the ELSP for developing a procedure for integrating the production schedule of multiple items into a shipment plan. In other words, the primary focus here lies in coordinating the manufacturing process with the transportation function. In shipping goods from manufacturing facilities to subsequent stages of the supply chain, truck shipments are perhaps the most commonly used means. Such transportation can be either full truckload (TL) or less than truckload (LTL) shipments. Although truck shipping rate structures are often complex, generally speaking, TL shipments are substantially less expensive, on a cost per unit basis, than LTL shipments (see, e.g. Chopra and Meindl, 2004). Under certain circumstances, though, the demand rate of a product may not be sufficiently large to warrant periodic individual TL shipments, since such large delivery lots may result in excessive inventory holding costs, negating the advantage of low transportation costs. Nevertheless, if there is a group of such relatively low-demand products that are shipped from a source to various demand locations, it may be possible to combine several partial truckloads of individual products to constitute a full truckload, thus obtaining the relative advantage of TL rates, in conjunction with lower inventory levels at the destination locations. In this study, we examine a scenario such as this. In our proposed procedure for coordinating the production schedule of multiple products (in an ELSP environment) with their shipments to several demand locations, the common production cycle approach (see,
e.g. Maxwell, 1964) is combined with the notion of a delivery (or replenishment) cycle that is common to all the individual items. For simplicity of implementation, we restrict the production cycle to an integer multiple of the delivery cycle. Under many circumstances, as illustrated later through a numerical example and subsequent sensitivity analysis, coordinated production and shipment decisions, such as the one suggested in this paper, may result in a more coherent and efficient supply chain. 2. Assumptions and notation 2.1. Assumptions The following assumptions, describing the manufacturing-distribution scenario, are made in this paper: 1. The operating environment is deterministic. 2. Multiple products are produced in a capacitated batch production environment, with different production rates for the various products. We assume, without loss of generality, zero setup times for all the products. If, in reality, the setup times are nonzero, they can be easily incorporated into our model, outlined later. 3. Only a single product may be produced at any given time. 4. Stockouts or shortages are not permitted. 5. We adopt the common cycle solution to the ELSP, where each product is produced exactly once in every production cycle. Although this is a restrictive assumption, the rationale for its adoption is threefold. First, this approach ensures a feasible solution if sufficient capacity to produce all the products exists. Secondly, from a practitioner’s viewpoint, the common cycle method is relatively easy to understand and implement, especially when stable production and transportation schedules need to be coordinated on a routine basis. Finally, the common cycle solution can indeed be optimal or near-optimal under certain conditions (see, Jones and Inman (1989) for details). 6. Each of the products is transported via truck and sold at one or more given demand location(s). For any of these products, however, its market demand rate is not sufficiently high to warrant full TL shipments. On the other hand, LTL shipments are relatively more expensive and undesirable. 7. Thus, TL shipments are made, where each truckload (capacitated by total weight and/or volume) contains a mix of all the products, which are delivered at the appropriate location(s). 8. For each TL shipment, a fixed cost is incurred, regardless of the mix of products carried. In contrast, LTL shipment rate structures often entail a unit variable shipping cost, based on the weight and/or the volume of shipment. 9. During each production cycle, an integer number, K, of these TL shipments are made at equal intervals of time.
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10. At each of the various demand locations, stocks are replenished via a periodic review, order-up-to level inventory control system. For coordination purposes, all the items at all the demand locations share a common fixed review period. 11. The product lot sizes, Qi, are treated as continuous variables for all i.
2.2. Notation The following notational scheme applies throughout this paper: i Di Pi Ai hi Qi K T KQi KT C wi Ii TRC
an index used to denote a specific product, i ¼ 1, 2, y, n the demand rate for product i (units/time unit) the production rate for product i (units/time unit) manufacturing setup cost per production batch for product i ($/batch) inventory holding (carrying) cost for product i ($/unit/time unit) amount of product i contained in each TL shipment (units) a positive integer, representing the number of shipments per production cycle; the shipment interval in time units (common to all products and locations) the production lot size (in units) for product i production cycle length in time units the TL capacity, i.e. maximum total load (or volume) allowable per truckload weight (or volume) of each unit of product i average inventory level (units) of product i total relevant cost ($) per time unit
3. Model development For illustrative and model development purposes, the inventory-time plots pertaining to a scenario involving three products (n ¼ 3) and three full TL shipments per production cycle (K ¼ 3), of length KT time units, are depicted in Fig. 1. From this figure it is clear that for a common delivery cycle time of T (i.e. T ¼ QiDi, 8i), each truckload contains Qi units of product i, i ¼ 1, 2, 3. The total weight (or volume) in a load, w1Q1+w2Q2+w3Q3, is obviously limited to the truck capacity, C. Also, the products should be sequenced to minimize the total inventory holding cost. The average inventory values for the three items are computed as follows:
I1 ¼
1 KQ 1 KQ 2 Q 3 Q1 KQ 1 þ þ KQ 1 þ ðK 1Þ Q1 2 P1 P2 P3 D1 Q1 Q1 þðK 2Þ Q1 þ þ Q 1 =½KQ 1 =D1 D1 D1
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Q1 Q Q3 Q ¼ KD1 þ 2 þ D1 þ ðK 1Þ 1 2P 1 P2 P3 2 1 KQ 2 Q3 Q2 KQ 2 I2 ¼ þ KQ 2 þ ðK 1Þ Q2 2 P2 P3 D2 Q2 Q2 Q2 þ þ Q 2 =½KQ 2 =D2 þðK 2Þ D2 D2 Q2 Q3 Q ¼ KD2 þ D2 þ ðK 1Þ 2 2P 2 P3 2 Q 3 D3 I3 ¼ ð2 KÞ þ K 1 2 P3 (for derivation see, e.g. Joglekar, 1988). In general, for the n-products case: 2 3 n1 n1 n1 X X Q j5 Q n X Qi 4 þ Ii ¼ K Di þ D 2P i j¼iþ1 Pj P n i¼1 i i¼1 þ ðK 1Þ
n1 X
Q i =2;
for
i ¼ 1; 2; . . . ; n 1
i¼1
and
Q Dn In ¼ n ð2 KÞ þ K 1 2 Pn
3.1. Objective function The objective of our optimization model is to minimize the total relevant cost per time unit, which is a function of Q1, Q2, y, Qn (denoted by Q ) and K. Thus, using the results obtained above, the objective function is 2 3 n n1 n1 X X X Q j5 Di Ai Qi 4 Minimize TRCðQ ; KÞ ¼ þK Di hi þ KQ i 2Pi j¼iþ1 P j i¼1 i¼1 n1 n1 X Qn X Q i hi Di hi þ ðK 1Þ P n i¼1 2 i¼1 Q n hn Dn ð2 KÞ þ K 1 þ 2 Pn
þ
(1)
The first term on the RHS of (1) represents the total setup cost per time unit for the n products. The remaining terms, capturing the inventory holding cost per time unit for items 1, 2, y, n1 and n, respectively, are obtained by adding the average inventory levels, I¯i , for i ¼ 1, 2, y, n1, and I¯n , derived above, weighted by the respective holding cost parameters, hi and hn. 3.2. Constraints The relevant model constraints are outlined below: 1. The delivery cycle time is common to all products: Q1 Q2 Q ¼ ¼ ¼ n ¼ T D1 D2 Dn or Q i ¼ TDi ; i ¼ 1; 2; . . . ; n
(2)
2. The production capacity of the system is limited, i.e. n X Di i¼1
Pi
p1
(3)
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Inventory Product 1
KQ
Time
Q =T D
KQ P
KQ = KT D
Inventory Product 2
KQ
KQ P
Q =T D
Time
Q =T D
Time
KQ = KT D
Inventory Product 3
KQ
Q P
KQ P
KQ = KT D
Fig. 1. Inventory-time plots (n ¼ 3, K ¼ 3).
3. Schedule feasibility, i.e. total production time needed cannot exceed the manufacturing cycle time: K
n1 X Qi i¼1
or
Pi
n1 X i¼1
þ
Qn pKT Pn
Qi Q þ n pT Pi KP n
enables us to considerably simplify the non-linear mixed integer programming problem expressed by (1)–(5). From (2) Q i ¼ TDi ; 8i
or
Qj ¼
Q1 D; D1 j
j ¼ 2; 3; . . . ; n
(4)
Note that this constraint can be easily modified for nonzero setup times by adding these to its left-hand side. 4. Truck capacity is limited by weight or volume, i.e. n X wi Q i ¼ C (5)
Substituting the Qi values for all i in (5), we obtain Q1 Q1 D2 þ þ wn Dn ¼ C D1 D 9 1 n P > > = wi Di or Q 1 ¼ CD1 =
w1 Q 1 þ w2
i¼1
i¼1
and Q j ¼ TDj ;
> ; j ¼ 2; 3; . . . ; n: >
(6)
3.3. Model simplification The coordination mechanism of an equal inventory replenishment (or delivery) cycle time for all products
Recalling that Di for i ¼ 1,2, y, n are known parameters, T and all Qi values can be determined a priori and recursively using (6). Now, define the parameters a, b
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and g as follows: 9 n DA P i i > > ¼a > > > i¼1 Q i > > " # > > n1 n1 n1 = P Qj P P Q i hi Q n hn Qi Dn þ 1 ¼b Di hi þ þ 2Pi j¼iþ1 P j 2 Pn > i¼1 i¼1 2 > > > > n1 n1 > P P Qn Q i hi Dn 1 > > > þ Q n hn and Di h i 2 ¼g ; P n i¼1 2 P n i¼1 (7) Substituting the values Qi, i ¼ 1, 2, y, n, obtained from (6), into (7) and eliminating constraints (2) and (5), the original model expressed by (1)–(5) can be considerably simplified to Minimize TRCðKÞ ¼ a=K þ bK þ g
i¼1
Pi
n1 X Qi i¼1
Pi
4. Numerical example and sensitivity analysis For illustrative purposes, consider the following data pertaining to a three-product example: Product Di Pi Ai hi wi (i) (units/year) (units/year) ($/setup) ($/unit/year) (lbs./unit) 1 2 3
8000 12,000 15,000
30,000 50,000 40,000
500 1000 800
2 3 3
20 50 40
Truck capacity: C ¼ 50,000 lb. The steps of the solution procedure outlined above yield the following results:
p1; i ¼ 1; 2; . . . ; n
þ
5. Find K* from conditions (13), i.e. K n ðK n 1Þpða=bÞ pK n ðK n þ 1Þ. Pn1 6. Check constraint (10): i¼1 ðQ i =Pi Þ þ ðQ n =KPn ÞpT. P 7. If violated, reset K* to Q n =ðPn ½ n1 i¼1 Q i =P i TÞ.
(8)
Subject to n X Di
115
Qn pT KPn
K ¼ 1; 2; . . . ::; etc.
(9)
(10)
(11)
Note that (8) is strictly convex in the integer K. Therefore, if TRC(K) is minimized at K ¼ K*, then TRCðK n ÞpTRCðK n 1Þ and TRCðK n ÞpTRCðK n þ 1Þ
(12)
Substituting (12) into (8) we obtain the following optimality conditions:
a
K n ðK n 1Þp pK n ðK n þ 1Þ
b
(13)
3.4. Solution procedure Utilizing the optimality conditions (13), the simplified model, (8)–(11), can now be easily solved by a simple procedure, which is outlined below: 1. For incurring minimal inventory holding cost, sequence the order of production by rank ordering and numbering the n products [1], [2], y, [n] based on h½1 D½1 ph½2 D½2 p ph½n D½n . P 2. Check constraint (9) n1 i¼1 ðDi =P i Þp1. If this constraint is violated, increase some Pi values, if possible, or reduce the item set, such that (9) is satisfied. 3. Compute T, Q1 and all other Qi values using (6). 4. Calculate the parameters a and b using the definitions in (7).
1. The three items are already arranged in the sequence h1D1oh2D2oh3D3. Pn 2. i¼1 ðDi =P i Þ ¼ 0:882; i.e. sufficient production capacity is available. 3. From (6), Q1 ¼ 294, T ¼ 0.03675 year, Q2 ¼ 441 and Q3 ¼ 551 units. 4. Using (7), a ¼ 62560 and b ¼ 1851.3. 5. The ratio a/b is calculated to be 33.8. 6. Thus, K* ¼ 6 satisfies conditions (13). Based on these results, the TL delivery cycle time is 0.03675 year and, based on K*, the production cycle time is 0.2205 year. Also each TL shipment consists of, respectively, 294 units (5880 lb), 441 units (22,050 lb) and 551 units (22,040 lb) of products 1, 2 and 3, resulting in a total load of 49,970 lb. Finally, the parameter g is computed to be 445.8, yielding a total relevant cost of about $21,089 per year using (8). In order to compare this coordinated approach to the results yielded by the traditional ELSP common cycle solution, without any attempt to link the production plan to TL shipments, suppose the following shipping rate structure applies: the average cost of a TL shipment is $500 and an LTL shipment cost is based on a variable cost of $0.015/lb shipped. Thus, for the suggested coordinated policy, employing only TL shipments, an additional $13,600 per year are incurred in shipping cost, bringing the total relevant cost to about $34,689. On the other hand, if the common cycle policy is not linked to shipments, the average number of production cycles per year would be 5.93 (derived from the classical common cycle model). As a result, the lot sizes for products 1, 2 and 3 would be 1349 units (26,980 lb), 2024 units (101,200 lb) and 2529 units (101,160 lb), respectively. The resulting average annual setup and holding cost at the plant would be about $16,151. The cost of transporting these batches to the retail stage via a mix of TL and LTL shipments adds approximately another
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$14,470/year. Thus, the total cost, so far, for the uncoordinated policy is about $30,621 a year, which is approximately $4068 less per year than our suggested approach, in spite of an increase of $870 in shipping charges. However, suppose that at the retail level, the inventory holding costs for products 1, 2 and 3 are, respectively, $3, $4 and $4 per unit per year. Now, the average annual retail inventory holding costs for the coordinated and the uncoordinated policies are, respectively, $2426 and $11,129. Therefore, from the standpoint of the entire supply chain (production and retail facilities) the coordinated policy saves about $4645 (in excess of 11%) a year. It is clear from this illustration that from a supply chain perspective, the penalties of not having coordinated production and shipping schedules arise out of additional transportation and retail inventory holding costs. 4.1. Sensitivity analysis In order to explore the effectiveness of the coordinated model developed here, as opposed to an uncoordinated approach, we solved 23 additional versions of the numerical example discussed above under a variety of operating conditions. The results of this sensitivity analysis are summarized in Table 1. Needless to say that, for purposes of equitable comparison, both of these approaches are based on the notion of a common production cycle. Keeping the finished goods inventory holding cost parameters for the three items at the plant level fixed at 2, 3 and 3, respectively, the manufacturing setup costs sets are varied from relatively low (250, 500 and 400) to high (1000, 2000 and 1600) values. Due to the value-added concept, it
is likely that holding costs at the retail level are likely to be higher than those at the production stage. Consequently, two sets of these parameters, i.e. (2.1, 3.1, 3.1), about 3–5% higher, and (3, 4, 4), about 30–50% higher than those at the plant level, are chosen. Three levels of cost for a TL shipment (300, 500 and 1000) are considered. The corresponding variable per lb. LTL shipping rates are kept proportionately fixed at 0.009, 0.015 and 0.03, respectively. All transportation costs are arbitrarily allocated to the retail stage. Table 1 indicates that under the various parametric conditions examined, the coordinated policy tends to yield higher total setup and holding cost at the plant or manufacturing stage. The total shipping and holding cost at the retail stage, however, are significantly higher under the uncoordinated approach, yielding annual net savings ranging from over 6% to almost 16% as a result of coordinating production with transportation, in particular taking advantage of the economies of full truckload shipping. Not unexpectedly, such savings tend to be magnified under higher retail stage holding costs. While lower production setup costs generally tend to yield greater savings for our coordinated policy, the combination of higher production setup, transportation and retail stage carrying costs also tend to make this approach more advantageous. Interestingly, under the latter combination of parameters, the performance gap between coordinated and uncoordinated policies tend to become smaller, albeit still substantial, as the differential between the holding cost parameters at the two stages of the supply chain narrows. We examined a final scenario involving a modified version of the original numerical example provided
Table 1 Sensitivity analysis results Setup cost parameters
Coordinated common cycle shipment policy Transportation cost parameters Fixed
250, 500, 400 (K ¼ 4)
500, 1000, 800 (K ¼ 6)
1000, 2000, 1600 (K ¼ 8)
Uncoordinated shipment policy
Percent savings (%)
Retail-stage holding cost parameters
Total plant cost
Total retail stage cost
Total supply chain cost
Total plant cost
Total retail stage cost
Total supply chain cost
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
14,780 14,780 14,780 14,780 14,780 14,780
10,007 10,586 15,447 16,026 29,047 29,626
24,787 25,366 30,227 30,806 43,827 44,406
10,634 11,420 10,634 11,420 10,634 11,420
16,659 17,593 23,314 24,076 39,950 40,283
27,293 29,013 33,948 35,496 50,584 51,703
9.18 12.57 10.96 13.21 13.36 14.11
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
21,089 21,089 21,089 21,089 21,089 21,089
10,007 10,586 15,447 16,026 29,047 29,626
31,096 31,675 36,536 37,115 50,136 50,715
15,038 16,151 15,038 16,151 15,038 16,151
18,489 19,811 24,520 25,599 39,599 40,068
33,527 35,962 39,558 41,750 54,637 56,219
7.25 11.92 7.64 11.10 8.24 9.79
2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4 2.1, 3.1, 3.1 3, 4, 4
30,005 30,005 30,005 30,005 30,005 30,005
10,007 10,586 15,447 16,026 29,047 29,626
40,012 40,591 45,452 46,031 59,052 59,631
21,267 22,841 21,267 22,841 21,267 22,841
21,895 25,281 27,590 31,642 41,827 47,545
43,162 48,122 48,857 54,483 63,094 70,386
7.30 15.65 6.97 15.51 6.41 15.28
Variable
300
0.009
500
0.015
1000
0.03
300
0.009
500
0.015
1000
0.03
300
0.009
500
0.015
1000
0.03
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earlier. The holding cost parameters at the manufacturing and retail stages are set to be the same, i.e. 2, 3 and 3. Additionally, the fixed cost of a TL shipment and the variable unit cost for LTL transportation, respectively, are set at 2000 and 0.04. Thus, for the given truck capacity of 50,000, there is no apparent economic advantage for TL shipments, since the per unit transportation charge is the same for both shipping modes. Even under these conditions, the coordinated policy yielded close to a 2% savings in total cost, in comparison with the uncoordinated policy. This observation, in addition to the results discussed above, appears to highlight the inherent economic advantage (in addition to other practical merits) of supply chain coordination. 5. Conclusions We have developed a mathematical model to align the production schedule of multiple products in a manufacturing facility with a periodic full truckload shipping plan. Since it is uneconomical to ship any individual product on a TL basis, each truckload is composed of a mix of LTL quantities of the individual items. As a result of deploying a common cycle production plan, as well as a common delivery cycle, inventory control, production planning and shipping decisions are streamlined and the supply chain as a whole becomes more coherent from an operational perspective. Exploitation of the properties of the coordination mechanism allows us to simplify the model structure considerably, yielding a solution without the use of a computer or specialized software. The economic desirability of coordination is demonstrated through numerical analysis. Our approach, however, is limited in two respects. First of these is the adoption of the common or rotational cycle technique for scheduling the production of multiple products. Although this method does not guarantee optimality, it yields optimal or near-optimal solutions to the ELSP under many real-world-operating conditions (see Jones and Inman (1989) for a detailed discussion on this topic). Thus, our model is likely to find applicability under such circumstances. In addition, this approach has considerable practical appeal. From a managerial standpoint, the simplicity of this method renders it readily understandable and easy to implement. The notion of producing each product regularly once every cycle leads to identical and routine production and shipping decisions, without variation. Such stability is often a highly desirable characteristic for practitioners. Furthermore, if adequate capacity exists for manufacturing all the products towards fulfilling their demands, the common cycle procedure guarantees a feasible solution. This feasibility issue, as mentioned before, is a major complicating factor in solving the ELSP optimally (see Hsu (1983) or Davis (1990), for instance). A second limitation of this study is the assumption of a deterministic environment. In addition to exploring
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procedures other than rotational cycles, future research should attempt to develop coordinated decisions under stochastic conditions. Nevertheless, we hope that this study has been able to identify some of the major benefits of coordinating production and shipping in supply chains and will prove to be of some value in terms of future research efforts along these lines. Finally, it is hoped that our proposed model will be helpful for practitioners in deriving and implementing innovative real-world supply chain solutions. References Boctor, F.F., 1982. The two-product, single-machine, static demand, infinite horizon lot scheduling. Management Science 28, 798–807. Bomberger, E., 1966. A dynamic programming approach to a lot size scheduling problem. Management Science 12, 778–784. Chopra, S., Meindl, P., 2004. Supply Chain Management: Strategy, Planning and Operation, Second ed. Pearson, Prentice-Hall, Upper Saddle River, NJ. Davis, S.G., 1990. Scheduling economic lot size production runs. Management Science 36, 985–998. Delporte, C.M., Thomas, L.J., 1977. Lot sizing and sequence for N products on one facility. Management Science 23, 1070–1079. Doll, C., Whybark, D., 1973. An iterative procedure for the single-machine multi-product lot scheduling problem. Management Science 20, 50–55. Elmaghrabi, S.E., 1978. The economic lot-scheduling problem (ELSP) review and extensions. Management Science 24, 587–598. Jones, P.C., Inman, R.R., 1989. When is the economic lot scheduling problem easy? IIE Transactions 21, 11–20. Geng, P.C., Vickson, R.G., 1988. Two heuristics for the economic lot scheduling problem: An experimental study. Naval Research Logistics 35, 605–617. Goyal, S.K., 1973. Scheduling a multi-product single machine system. Operational Research Quarterly 24, 261–269. Goyal, S.K., Gupta, Y., 1989. Integrated inventory models: The buyer– vendor coordination. European Journal of Operational Research 41, 261–269. Haessler, R.W., 1979. An improved extended basic period procedure for solving the economic lot scheduling problem. AIIE Transactions 11, 336–340. Haessler, R.W., Hogue, S.L., 1976. A note on the single-machine multi-product lot scheduling problem. Management Science 22, 909–912. Hsu, W.L., 1983. On the general feasibility test of scheduling lot sizes for several products on one machine. Management Science 29, 93–105. Joglekar, P.N., 1988. Comments on ‘‘a quantity discount pricing model to increase vendor profits. Management Science 34, 1391–1398. Luenberger, D.G., 1973. Introduction to Linear and Nonlinear Programming. Addison-Wesley Publishing Co., Reading, MA. Madigan, J.G., 1968. Scheduling a multi-product single machine system for an infinite planning period. Management Science 14, 713–719. Maxwell, W.L., 1964. The scheduling of economic lot sizes. Naval Research Logistics Quarterly 11, 89–124. Panayiotopoulos, J.C., 1983. The generalized multi-item lot size scheduling. European Journal of Operational Research 14, 59–65. Park, K., Yun, D.K., 1984. A stepwise partial enumeration algorithm for the economic lot scheduling problem. IIE Transaction 16, 363–370. Saipe, A., 1977. Production runs for multiple products: the two-product heuristic. Management Science 12, 1321–1327. Stankard, M.F., Gupta, S.K., 1969. A note on Bomberger’s approach to lot size scheduling: Heuristic proposed. Management Science 15, 449–452. Thomas, D.J., Griffin, P.M., 1996. Coordinated supply chain management. European Journal of Operational Research 94, 1–15. Vemuganti, R.R., 1978. On the feasibility of scheduling lot sizes for two products on one machine. Management Science 24, 1668–1673.