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Single vs. multiple objective supplier selection in a make to order environment
ARTICLE IN PRESS Omega 38 (2010) 203–212
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Omega journal homepage: www.elsevier.com/locate/omega
Single vs. multiple objective supplier selection in a make to order environment Tadeusz Sawik AGH University of Science and Technology, Department of Operations Research and Information Technology, Al.Mickiewicza 30, 30-059 Krak´ ow, Poland
a r t i c l e in fo
abstract
Article history: Received 12 May 2009 Accepted 6 September 2009 Available online 12 September 2009
The problem of allocation of orders for custom parts among suppliers in make to order manufacturing is formulated as a single- or multi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order. The selection of suppliers is based on price and quality of purchased parts and reliability of on time delivery. The risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate can be unacceptable. Furthermore, the quantity or business volume discounts offered by the suppliers are considered. Numerical examples are presented and some computational results are reported. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Supplier selection Operational risk Discounts Multi-criteria decision making Mixed integer programming
1. Introduction In make-to-order environment customer-oriented manufacturers should be prepared to produce varieties of products to meet the different customer needs. Each product is typically composed of many common and non-common (custom) parts that can be sourced from different approved suppliers with different supply capacity. An important issue is how to best allocate the orders for parts among various part suppliers to fulfill all customer orders for products and to achieve a high customer service level at a low cost. The decision maker needs to decide from which supplier to purchase parts required to complete each customer order. The above decisions are based on price, quality (defect rate) and reliability (on time delivery) criteria that may conflict each other, e.g. the supplier offering the lowest price may not have the best quality or the supplier with the best quality may not deliver on time. Furthermore, to reduce the fixed ordering (transaction) costs the number of suppliers and the total number of orders should be minimized. On the other hand, the selection of more suppliers sometimes may divert the risk of unreliable supplies. In spite of the importance of supplier selection and order allocation problems, the decision making is not sufficiently addressed in the literature (for a recent review, see Aissaoui et al. [1]), in particular for make-to-order manufacturing environment, e.g. Murthy et al. [2], Sawik [3], Yue et al. [4]. Basically, the authors distinguish between single and multiple item models and supplier selection with single or multiple sourcing, where each
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supplier can fully meet all requirements (e.g. Akinc [5]) or none of the suppliers is able to satisfy the total requirements, respectively. The vast majority of the decision models are mathematical programming models either single objective, e.g. Kasilingam and Lee [6], Basnet and Leung [7], Jayaraman [8] or multiple objectives, e.g. Weber and Current [9], Xia and Wu [10], Demirtas and Ustun [11], Ustun and Demirtas [12], Pokharel [13]. The supplier selection is a complex decision making problem which includes both quantitative and qualitative factors and one of the disadvantages of the mathematical programming methods is their failure to account for qualitative factors that may affect suppliers performance. In order to consider both quantitative and qualitative factors some researchers propose hybrid approaches that combine different methods. For example, Sanayei et al. [14] propose an integration of multi-attribute utility theory and linear programming, first to rate and choose the best suppliers and then to find optimal allocation of order quantities among the selected suppliers to maximize total additive utility. The combined method allows both quantitative and qualitative factors under risk and uncertainty to be considered as well as to account for the probabilistic nature of supplier performance. Another integrated approach that combines analytic network process and multiobjective mixed integer programming is proposed in [11,12]. First, the potential suppliers are evaluated according to 14 criteria that are involved in the four clusters: benefits, opportunities, costs and risks, to calculate the priorities of each supplier. Then, the optimum quantities are allocated among selected suppliers to maximize total value of purchasing (using the calculated priorities) and to minimize the total cost and total defect rate. However, the disadvantages of the integrated methods usually may affect the performance of hybrid approaches. The other approaches that are also applied to solve the supplier selection
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problem are methods based on fuzzy sets, e.g. a fuzzy multiobjective integer programming Huo and Wei [15] and genetic algorithms (e.g. Liaoa and Rittscher [16], Che and Wang [17]). For example, in [16] a genetic algorithm with problem specific operator is developed to account for the inbound transportation and to combine supplier selection with carrier selection decisions. The fuzzy and genetic algorithms, however, are heuristics that do not guarantee optimality of a solution. The models developed for supplier selection and order allocation can be either single-period models (e.g. [6,8,9,11]) that do not consider inventory management or multi-period models (e.g. [3,7,12,16], Ghodsypour and O’Brien [18], Tempelmeier [19]) which consider the inventory management by lot-sizing and scheduling of orders. Since common parts can be efficiently managed by material requirement planning methods, this research is focused on custom parts that can be critical in make-to-order manufacturing. For custom-engineered products no inventory of custom parts can be kept on hand. Instead, the custom parts need to be requisitioned with each customer order and hence the custom parts inventory need not to be considered. This paper presents mixed integer programming models for single or multiple objective supplier selection in make-toorder manufacturing for a static supply portfolio in a nondiscount or discount environment, that is for the allocation of orders for parts among the suppliers without or with discount and with no timing decisions. In contrast to the dynamic portfolio, which is the allocation of orders among the suppliers combined with the allocation of orders among the planning periods. The major contribution of this paper is that it proposes a simple mixed integer programming approach for selection of supply portfolio under conditions of operational risk associated with uncertain quality and reliability of supplies. The integer programming models incorporate risk constraints where the risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate may exceed the maximum acceptable rates. The number of maximum delivery patterns and the corresponding maximum rates represent, respectively, the confidence level and the targeted rates above which a risk averse decision maker wants to limit the number of outcomes. The paper is organized as follows. In Section 2 description of the supplier selection problem in make-to-order manufacturing is provided. The mixed integer program for a single objective supplier selection in a non-discount environment is presented in Section 3. The model enhancements for the supplier selection with a business volume discount or quantity discount are presented in Section 4. The multiple objective approach is proposed in Section 5. Numerical examples and some computational results are provided in Section 6, and final conclusions are made in the last section.
2. Problem description In the supply chain under consideration various types of products are assembled by a single producer to satisfy customer orders, using custom parts purchased from multiple suppliers (for notation used, see Table 1). Each supplier can provide the producer with custom parts for all customer orders. However, the suppliers have different limited capacity and, in addition, differ in price and quality of offered parts and in reliability of on time delivery of parts. Let I ¼ f1; . . . ; mg be the set of m suppliers and J ¼ f1; . . . ; ng the set of n customer orders for the products, known ahead of time. Each order j A J is described by the quantity sj of required
Table 1 Notation. Indices i supplier, iA I ¼ f1; . . . ; mg j customer order, jA J ¼ f1; . . . ; ng t delivery pattern, t A T ¼ f1; . . . ; hg Input parameters ci capacity of supplier i oi cost of ordering parts from supplier i pij price of part for customer order j purchased from supplier i qit expected defect rate of supplier i for delivery date in pattern t rit expected late delivery rate of supplier i for delivery date in pattern t sj number of parts to be purchased for customer order j P D j A J sj total demand for parts q r v
the largest acceptable average defect rate of supplies the largest acceptable average late delivery rate of supplies the maximum allowed number of delivery patterns with the average defect rate or average late delivery rate of supplies greater than q or r, respectively
custom parts and requested delivery date, where the latter need not to be explicitly considered when selecting a supplier. Each supplier is assumed to have sufficient capacity to complete manufacturing and to deliver the ordered parts to the producer by the requested dates. All parts ordered from a supplier are shipped together with a single shipment at one of a series of fixed delivery dates (e.g. Hall et al. [20]). The parts are dispatched to the producer at the earliest fixed delivery date after the completion time of their manufacturing. Hence, for each supplier the delivery date and the corresponding reliability of supply depend on the completion time of manufacturing the ordered parts, which is unknown to the producer when the supplier selection decision is made. Likewise, the quality of supply may depend on the completion time. When the suppliers are selected, however, the risk of defective and unreliable supplies can be considered using past observations. Since different suppliers may complete manufacturing of ordered parts at different times, and then deliver the parts at different dates, a different risk can be associated with each combination of suppliers delivery dates. Let us call each combination of m fixed delivery dates, one delivery date for each supplier, a delivery pattern. Each delivery pattern must be feasible in respect to requested delivery dates. The total number of all feasible combinations of m fixed delivery dates consists of h delivery patterns and let T ¼ f1; . . . hg be the index set of all feasible delivery patterns. The probability that is assigned to the occurrence of each delivery pattern is identical and equals 1=h. Let ci be the capacity of supplier i A I, oi Fcost of ordering parts from supplier i A I, pij Fpurchasing price of part for customer order j A J from supplier iA I, and qit , rit Frespectively, the expected defect rate, the expected late delivery rate of supplier iA I for delivery date in pattern t A T. The rates qit and rit are based on past observations. We assume that the risk of defective or unreliable supplies from the selected suppliers can be measured by the number of delivery patterns for which the average defect rate or the late delivery rate of supplies are unacceptable. The decision maker needs to decide from which supplier to purchase custom parts required for each customer order to achieve a low unit cost and high quality and reliability of supplies.
3. Single objective supplier selection in a non-discount environment In this section a mixed integer program is proposed for a single-period supplier selection and order allocation problem in a
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non-discount environment, i.e. for determining a static portfolio of suppliers without discount. The static portfolio of suppliers is defined as ðx1 ; . . . ; xm Þ; where X xi ¼ 1 iAI
and 0 r xi r1 is the fraction of the total demand for parts ordered from supplier i (for definition of problem variables, see Table 2). When deciding on a static portfolio of suppliers it is assumed that the orders for all parts are simultaneously placed on selected suppliers (e.g. at time 0), and each supplier delivers all the ordered parts at the earliest possible delivery date. Therefore, the allocation of orders for parts among the suppliers is not combined with the allocation of orders among the planning periods. Nevertheless, given past observations, the static portfolio should be checked over time horizon against the risk of too low quality of purchased parts (too high defect rate) and too low reliability of supplies (too high late delivery rate), where both the quality and the reliability are randomly varying over time. Denote by q, r and v the maximal acceptable defect rate of portfolio, the maximal acceptable late delivery rate of portfolio and the maximum number of delivery patterns for which the average defect rate or the average late delivery rate of the portfolio can be above the threshold q or r, respectively. The parameters q, r and v are fixed by the decision maker to control the risk of defective and unreliable supplies. We assume that the decision maker is willing to accept only portfolios for which the number of delivery patterns with the average defect rate greater than the threshold q or with the average late delivery rate greater than the threshold r is not greater than v. The performance of the static portfolio can be measured by the following two criteria: X XX oi yi =D þ pij sj zij =D; ð1Þ f1 ðy; zÞ ¼ iAI
iAI jAJ
XX ðqit þ rit Þxi =h; f2 ðxÞ ¼
ð2Þ
iAI tAT
where f1 ðy; zÞ is the average ordering and purchasing cost per part, and f2 ðxÞ is the average defect and late delivery rate. The overall performance can be measured by the sum Fðy; zÞ of average cost per part of ordering, purchasing, defects and delays, in which the cost of a defective or delayed part is assumed to be identical with its price 0 1 ! X X X X X @ A oi yi þ pij sj zij D þ ðqit þrit Þ=h pij sj zij =D: Fðy; zÞ ¼ iAI
jAJ
iAI
tAT
jAJ
ð3Þ The second summation term in (3) can be interpreted as the purchasing cost of additional parts required to compensate for Table 2 Problem variables.
205
defective and delayed parts. Clearly, the assumption that cost of a defective or delayed part is identical with its price may be too relaxed in practice. The lack of all required parts may lead to unfulfilled customer orders and then the resulting shortage cost can be much higher than the purchasing cost of additional parts. On the other hand, parts not passing quality acceptance level may not be paid for, and parts delivered late may be paid for at a reduced price. The mixed integer program SP for the single objective supplier selection problem in a non-discount environment is formulated below. Model SP: Single objective portfolio of suppliers in a non-discount environment Minimize (3) subject to 1. Order assignment constraints:
for each customer order required parts are supplied by exactly one supplier,
for each supplier the total quantity of ordered parts cannot exceed its capacity. X zij ¼ 1; j A J;
ð4Þ
iAI
X sj zij r ci ;
i A I:
ð5Þ
jAJ
2. Portfolio selection constraints: the portfolio definition constraint (note that the order allocation variable xi is an auxiliary variable determined by zij ), for each delivery pattern t, the portfolio with average defect rate greater than q is not acceptable, for each delivery pattern t, the portfolio with average late delivery rate greater than r is not acceptable, the portfolio can be unacceptable for at most v delivery patterns, parts are ordered from supplier i, if at least one customer order is assigned to supplier i, X sj zij =D; i A I; ð6Þ xi ¼ jAJ
P vt Z P vt Z X
qit xi q ; 1q
iAI
i A I rit xi
r
1r
;
t A T;
ð7Þ
t A T;
ð8Þ
vt r v;
ð9Þ
tAT
c i y ; iA I; D i X zij ; iA I: yi r
xi r
ð10Þ ð11Þ
jAJ
3. Non-negativity and integrality conditions: vt ¼ 1, if for delivery pattern t the average defect rate or the average late delivery rate of supplies for the selected portfolio is, respectively greater than q or greater than r , otherwise vt ¼ 0 (portfolio selection variable) xi ¼ the fraction of total demand for parts ordered from supplier i (order allocation variable) yi ¼ 1, if an order for parts is placed on supplier i; otherwise yi ¼ 0 (supplier selection variable) zij ¼ 1, if parts required for customer order j are ordered from supplier i; otherwise zij ¼ 0 (customer order assignment variable)
vt A f0; 1g; xi Z 0;
t A T;
iA I;
ð12Þ ð13Þ
yi A f0; 1g;
i A I;
ð14Þ
zij A f0; 1g;
i A I; j A J:
ð15Þ
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Constraints (7) and (8) prevent the choice of portfolios whose average defect rate or whose average late delivery rate is above the fixed threshold q (7) or r (8), respectively. For each delivery P pattern t such that the average defect rate i A I qit xi is greater than the largest acceptable rate q (7) or the average late delivery P rate i A I rit xi is greater than the largest acceptable rate r (8), vt ¼ 1. Then all the delivery patterns with the average defect rate or with the average late delivery rate above the threshold are summed up in (9). If the result is greater than v, then the portfolio is infeasible. In view of risk constraints (7)–(9) and under the assumption of identical probability 1=h that each delivery pattern is realized, q and r can be considered as the targeted average defect and late delivery rates based on the a-percentile of these rates, where a ¼ ð1 v=hÞ represents the confidence level for the average rates across all delivery patterns. In other words, we allow 100ð1 aÞ% ¼ 100ðv=hÞ% of the delivery patterns to exceed q and/or r. If different probability pt is assigned to each delivery pattern t A T, then constraint (9) should be replaced with X pt vt r 1 a; ð90 Þ tAT
where the confidence level a A ð0; 1Þ is fixed by the decision maker to control the risk of defective and unreliable supplies, and the higher is the confidence level, the more risk averse is the selected supply portfolio. P P Notice that v ¼ 0 implies i A I qit xi r q and i A I rit xi rr 8t A T, i.e., the selected portfolio must be acceptable for each delivery pattern. In particular, for the stationary expected defect rates qit ¼ qi 8i A I; t A T and the stationary expected late delivery rates rit ¼ ri 8i A I; t A T, v can take on either value 0 or h, since the selected portfolio can be either acceptable or unacceptable over the entire set of all possible delivery patterns. Hence, in the stationary case variable v can be eliminated from the model.
4. Single objective supplier selection in a discount environment In make to order environment, in which custom parts are typically ordered in small lot sizes, supplier may sometimes offer discount that depends on total value of sales volume (business volume) or on total quantity of ordered parts, e.g. Xia and Wu [10], Tempelmeier [19], Chaudhry et al. [21], Schotanus et al. [22]. In the context of business volume discount the quantity or variety of purchased parts does not affect the offered price, while for the quantity discount the price does not depend on the total ‘‘euro’’ amount of sales volume. In this section the presence of price discounts offered by the suppliers is considered, first the supplier selection with business volume discount and then the supplier selection with quantity discount, respectively, based on the total value and the total quantity of the order (for notation used, see Table 3). Assume that each supplier i offers cumulative (all-units) price breaks having gi discount intervals ðbi0 ; bi1 ; . . . ; ðbi;gi 1 ; bi;gi according to total business volume (or total quantity), where bik is upper limit on the kth (k A Ki ¼ f1; . . . ; gi g) business volume (or quantity) discount interval ðbik1 ; bik and bi0 ¼ 0 8i. Let 0 o aik o1 be the discount rate (percentage of discount) associated with interval k of supplier i. If total business volume (total quantity) from supplier i falls on interval k, then the price of each part for customer order j is ð1 aik Þpij .
Table 3 Notation for a discount environment. k Ki aik bik
discount interval of supplier i, k A Ki ¼ f1; . . . ; gi g set of discount intervals of supplier i price discount rate associated with discount interval k of supplier i upper limit on discount interval k of supplier i, 0 ¼ bi0 r bi1 r r bigi
The following discount interval selection variable need to be added to enhance the supplier selection and order allocation problem SP for the discount environment: uik ¼ 1 if total business volume (or total quantity) from supplier i falls on the discount interval k; otherwise uik ¼ 0. Furthermore, the customer order assignment variable zij need to be modified as below. Zijk ¼ 1 if parts required for customer order j are ordered from supplier i and total business volume (or total quantity) of parts purchased from this supplier falls on the discount interval k; otherwise Zijk ¼ 0. The total business volume or total quantity of parts purchased P P P from supplier i is, respectively, jAJ k A Ki pij sj Zijk or jAJ P k A Ki sj Zijk . Notice that the total business volume or the total quantity purchased from each supplier falls on exactly one discount interval of this supplier. Now, the average purchasing and ordering cost per part can be expressed as follows: X XX X oi yi =Dþ ð1 aik Þpij sj Zijk =D: ð16Þ f1 ðy; ZÞ ¼ iAI
i A I j A J k A Ki
The overall performance of the static portfolio with discount can be measured by the sum Fðy; ZÞ of average cost per part of ordering, purchasing, defects and delays, in which the cost of a defective or delayed part is assumed to be identical with its regular price 0 1 XX X @oi yi þ ð1 aik Þpij sj Zijk AD Fðy; ZÞ ¼ iAI
j A J k A Ki
! XX X X ðqit þ rit Þ=h pij sj Zijk =D: þ iAI
tAT
ð17Þ
j A J k A Ki
The mixed integer program SPD for the single objective supplier selection problem with business volume discount or with quantity discount is formulated below. Model SPD: Single objective portfolio of suppliers with discount Minimize (17) subject to (7)–(10), (12)–(14) and 1. Order assignment constraints:
the parts required for each customer order are supplied by exactly one supplier in one discount interval,
for each supplier the total quantity of ordered parts cannot exceed its capacity. XX Zijk ¼ 1; j A J
ð18Þ
i A I k A Ki
XX
sj Zijk r ci ;
iAI
ð19Þ
j A J k A Ki
2. Portfolio selection constraints:
the portfolio definition constraint (note that the order allocation variable xi is an auxiliary variable determined by Zijk ),
ARTICLE IN PRESS T. Sawik / Omega 38 (2010) 203–212
an order for parts is placed on supplier i, if for at least one customer order the required parts are ordered from supplier i, XX sj Zijk =D; iA I; ð20Þ xi ¼
solution set of the bi-objective program MP and MPD can be found by the parameterization on l the mixed integer program MPl and MPDl, respectively.
j A J k A Ki
yi r
XX
207
Model MPl: Zijk ;
i A I:
ð21Þ
j A J k A Ki
minfd þ eðf1 ðy; zÞ þ f2 ðxÞÞg
ð30Þ
subject to (1), (2), (4)–(15) and
3. Discount constraints:
total business volume purchased from each selected supplier can be exactly in one discount interval X pij sj Zijk r bik uik ; i A I; k A Ki ðbi;k1 þ 1Þuik r
ð22Þ
lðf1 ðy; zÞ f 1 Þ r d;
ð31Þ
ð1 lÞðf2 ðxÞ f 2 Þ r d;
ð32Þ
d Z0;
ð33Þ
where 0 r l r1.
jAJ
for a business volume discount environment, or total quantity purchased from each selected supplier can be exactly in one discount interval X sj Zijk rbik uik ; i A I; k A Ki ð23Þ ðbi;k1 þ 1Þuik r jAJ
for a quantity discount environment,
parts are ordered from supplier i, if supplier i is selected and assigned at least one customer order, X uik ¼ yi ; i A I;
ð24Þ
k A Ki
Zijk ruik ;
iA I; j A J; kA Ki
ð25Þ
uik A f0; 1g;
i A I; k A Ki ;
ð26Þ
Zijk A f0; 1g;
i A I; j A J; k A Ki :
ð27Þ
5. Multiple objective supplier selection In the single objective approach for the supplier selection proposed in previous sections, two objective functions f1 (1) or (16) and f2 (2) were aggregated into a single cost function F (13) or (17), respectively for a non-discount or discount environment. In this section the two objective functions are considered independently, and a bi-objective mixed integer program MP or MPD is presented for the supplier selection in a non-discount or discount environment, respectively. Model MP: Multiple objective portfolio of suppliers in a nondiscount environment Minimize
In this section some computational examples are presented to illustrate possible applications of the proposed mixed integer programming approach for supplier selection in a non-discount or discount environment and to compare the single- and the multiple-objective approach. The following parameters have been used for the example problems:
ð28Þ
subject to (1), (2), (4)–(15).
Model MPD: Multiple objective portfolio of suppliers with discount Minimize f ðx; y; ZÞ ¼ ½f1 ðy; ZÞ; f2 ðxÞ
The programs MPl and MPDl are based on the augmented l-weighted Tchebycheff metric minfljf1 ðy; zÞ f 1 jþ eðf1 ðy; zÞ þ f2 ðxÞÞ; ð1 lÞjf2 ðxÞ f 2 jþ eðf1 ðy; zÞ þ f2 ðxÞÞg. Steuer [23] proved that there always exists e small enough that enable to reach all the nondominated solutions. However, for the mixed integer programs, there may be portions of the nondominated set (nearby weakly nondominated solution) that the above Tchebycheff program is unable to compute, even considering very small e. 6. Computational examples
4. Integrality conditions:
f ðx; y; zÞ ¼ ½f1 ðy; zÞ; f2 ðxÞ
Model MPDl: Minimize (30) subject to (2), (7)–(10), (12)–(14), (18)–(27), (31)–(33).
ð29Þ
subject to (2), (7)–(10), (12)–(14), (18)–(27).
5.1. Reference point based scalarizing program Let f ¼ ðf 1 ; f 2 Þ be a reference point in the criteria space such that f i o fi ; i ¼ 1; 2 for all feasible solutions satisfying (1), (2), (4)–(15) and (2), (7)–(10), (12)–(14), (18)–(27), respectively of MP and MPD. Denote by e a small positive value. The nondominated
h, the number of delivery patterns, was equal to 30; m, the number of suppliers, was equal to 20; n, the number of customer orders, was equal to 100; oi , the cost of ordering parts from supplier i, was equal to 5000 for each supplier i; pij , the unit price of parts required for each customer order j, purchased from each supplier i was uniformly distributed over [10,15] and reduced by the factor ð1 maxt A T ðqit þ rit ÞÞ to get a lower price for parts from the suppliers with higher defect and late delivery rates; qit and rit , the expected defect rate and the expected late delivery rate of each supplier i and delivery date in each pattern t, was uniformly distributed over (0,0.08) and (0,0.10), respectively; q and r, the largest acceptable average defect rate and late delivery rate, was equal to 0.05 and 0.06, respectively; v, the maximum allowed number of delivery patterns with the average defect or late delivery rates greater than, respectively q ¼ 0:05 or r ¼ 0:06, was equal to 0, 1 or 2; sj , the numbers of required parts for each customer order, were integers uniformly distributed over ½100; 5000; P ci , the capacity of each supplier i, was equal to d2 j A J sj =me, i.e., the total capacity of all suppliers was equal to the double total demand for parts; quantity discount: gi ¼ 3 discount intervals for each supplier i, with the upper limits bik ¼ dkci =gi e; iA I; k ¼ 1; . . . ; gi ; business volume discount: gi ¼ 3 discount intervals for each supplier i, with the upper limits bik ¼ dkp i ci =gi e; iA I; k ¼
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Table 4 Computational results for single objective approach. v
Var.
Bin.
Cons.
Non-discount environment (model SP) 0 2040 2020 241 1 2070 2050 242 2 2070 2050 242 Business volume discount (model SPD) 0 6120 6100 6381 1 6130 6110 6382 2 6130 6110 6382 Quantity discount (model SPD) 0 6120 6100 6381 1 6130 6110 6382 2 6130 6110 6382
F ðf1 ; f2 Þa, no. of suppliersb
Table 6 Computational results for multiple objective approach: discount environment. CPUc
9.108 (8.366, 0.090), 13 9.027 (8.321, 0.087), 13 9.004 (8.283, 0.089), 12
17 7 15
8.335 (7.592, 0.090), 12 8.239 (7.528, 0.087), 12 8.207 (7.494, 0.087), 11
2191 1265 292
8.392 (7.580, 0.090), 12 8.226 (7.515, 0.087), 13 8.204 (7.495, 0.087), 12
234 380 140
a F ðf1 ; f2 ÞFthe average cost (the average ordering and purchasing cost per part, the average defect and late delivery rate), respectively. b The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11.
Table 5 Computational results for multiple objective approach: non-discount environment (model MPl). Var.
Bin.
Cons.
f1 ; f2 a, no. of suppliersb
CPUc
2041 2071 2071
2020 2050 2050
242 243 243
8.401, 0.089, 13 8.386, 0.086, 11 8.424, 0.085, 12
459 262 48
2041 2071 2071
2020 2050 2050
243 244 244
8.390, 0.090, 13 8.328, 0.087, 13 8.321, 0.087, 13
746 338 33
2041 2071 2071 l ¼ 0:95 0 2041 1 2071 2 2071 l¼1 0 2041 1 2071 2 2071
2020 2050 2050
243 244 244
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
239 72 73
2020 2050 2050
244 244 244
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
134 460 22
2020 2050 2050
242 243 243
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
232 53 102
v
l¼0 0 1 2
l ¼ 0:05 0 1 2
l ¼ 0:5 0 1 2
a f1 ; f2 —the average ordering and purchasing cost per part, the average defect and late delivery rate, respectively ðf 1 ¼ 8; f 2 ¼ 0:08Þ. b The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11.
P 1; . . . ; gi , where p i ¼ j A J pij =n is the average unit price of parts from supplier i; aik , discount coefficients, were equal to 0:05ðk 1Þ; i A I; k ¼ 1; . . . ; gi , i.e., the maximum percentage discount is 10%.
The computational results for the three risk levels represented by the maximum number of delivery patterns v ¼ 1; 2; 3 for which the average defect rate or late delivery rate of supplies can be unacceptable are summarized in Tables 4 and 5, 6, respectively for single- and multiple-objective approach. The size of the integer programs SP, MPl and SPD, MPDl, respectively, for the supplier selection in a non-discount and discount (business volume or quantity) environment are represented by the total number of variables, Var., number of binary variables, Bin., and number of constraints, Cons. The last two columns of the tables present the solution values and CPU time in seconds required to prove optimality of the solution. The subsets of nondominated
v
Var.
Bin.
Cons.
f1 ; f2 a, no. of suppliersb
Business volume discount (model MPDl) l¼0 0 6101 6080 6382 7.594, 0.090, 12 1 6131 6110 6383 7.571, 0.086, 11 2 6131 6110 6383 7.674, 0.085, 11 l ¼ 0:05 0 6101 6080 6383 7.593, 0.091, 12 1 6131 6110 6384 7.528, 0.087, 12 2 6131 6110 6383 7.481, 0.091, 11 l A f0:5; 0:95; 1g 0 6101 6080 6382 7.592, 0.091, 12 1 6131 6110 6383 7.505, 0.091, 11 2 6131 6110 6383 7.481, 0.091, 11 Quantity discount (model MPDl) l¼0 0 6121 6100 6382 7.636, 0.089, 13 1 6151 6130 6383 7.568, 0.086, 11 2 6151 6130 6383 7.604, 0.085, 12 l ¼ 0:05 0 6121 6100 6383 7.580, 0.091, 12 1 6131 6110 6384 7.528, 0.087, 12 2 6131 6110 6383 7.495, 0.087, 12 l A f0:5; 0:95; 1g 0 6121 6100 6382 7.580, 0.090, 12 1 6151 6130 6383 7.496, 0.092,11 2 6151 6130 6383 7.480, 0.091, 12
CPUc
1403 747 247 1148 823 246 2495–3587 2245–3005 250–500
723 57 107 179 189 74 294–625 226–549 142–312
a
ðf 1 ¼ 7:4; f 2 ¼ 0:08Þ. The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11. b
solutions of the bi-objective programs MP and MPD were computed by parameterization the corresponding Tchebycheff l A f0; 0:05; 0:5; 0:95; 1g. The computational programs on experiments were performed using AMPL programming language and the CPLEX v.11 solver (with the default settings) on a laptop with Pentium IV processor running at 1.8 GHz and with 1 GB RAM. The solver was capable of finding proven optimal solutions within the limit of 3600 CPU seconds for all examples. The results indicate that while the single objective approach yields a single proven optimal solution in a very short CPU time, the multiple objective approach requires the much longer computation time to produce a very small subset of nondominated solutions. For example, Table 5 shows that in a non-discount environment only three nondominated solutions ðf1 ; f2 Þ were found for each risk level v: (8.401, 0.089), (8.390, 0.090), (8.363, 0.091) for v ¼ 0, (8.386, 0.086), (8.328, 0.087), (8.304, 0.092) for v ¼ 1, and (8.424, 0.085), (8.321, 0.087), (8.284, 0.089) for v ¼ 2. In particular, only one solution for each v was found for l A f0:5; 0:95; 1g. In addition, each single objective solution weakly dominates one of the nondominated multiple objective solutions: (8.366, 0.090) weakly dominates (8.390, 0.090) for v ¼ 0, (8.321, 0.087) weakly dominates (8.328, 0.087) for v ¼ 1 and (8.283, 0.089) weakly dominates (8.284, 0.089). for v ¼ 3. Similarly, only two or three nondominated solutions were obtained for each risk level in a business volume discount or quantity discount environment (see, Table 6). In the computational experiments e ¼ 0:001. For the mixed integer programs, however, there may be portions of the nondominated set that the Tchebycheff program is unable to compute, even considering very small e [23]. For the single objective approach, the optimal allocation of demand for parts among the suppliers in a non-discount or discount environment is shown in Fig. 1. The best leveled allocation of demand among the selected suppliers (the allocation with a minimum difference between the highest and the lowest fraction of allocated demand) has been achieved for the
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Fig. 1. Optimal portfolios for single objective approach.
quantity discount, with xi ranging from 0.068 to 0.098. The most uneven demand allocation was found for the business volume discount at the lowest risk level (v ¼ 0), with xi ranging from 0.028 to 0.098.
For the multiple objective approach and a non-discount environment, the allocation of demand for parts among the suppliers for the three nondominated solutions and the three risk levels is shown Fig. 2. The number of selected suppliers varies
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Fig. 2. Optimal portfolios for multiple objective approach: non-discount environment.
between 11 and 13, and the allocated fraction of total demand for parts varies most (from xi ¼ 0:028 to xi ¼ 0:098) for the lowest risk level (v ¼ 0).
For the multiple objective approach and the quantity discount environment, the allocation of demand for parts among the suppliers for the two or three nondominated solutions obtained
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211
Fig. 3. Optimal portfolios for multiple objective approach: quantity discount.
for each risk level is shown Fig. 3. The number of selected suppliers varies between 11 and 13, and the allocated fraction of total demand for parts varies most (from xi ¼ 0:024 to xi ¼ 0:10) for the lowest risk level (v ¼ 0).
The computational results demonstrate that the greater is the allowed number of delivery patterns v with the average defect and late delivery rates above the thresholds (i.e., the higher is the risk of defective or delayed supplies), the lower is the average cost. The
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largest average cost is achieved for the lowest risk level (v ¼ 0), i.e. where the average defect and late delivery rates never exceed the acceptable rates. Simultaneously, for the lowest risk level the allocation of demand among the selected suppliers is the most uneven.
known ahead of time. The last assumption can be relaxed, and the approach can also be used in a dynamic case where orders arrive irregularly over time. In this case, the supplier selection decisions can be made periodically over a rolling planning horizon, upon arrivals of a number of orders in a specific time interval.
7. Conclusion Acknowledgments The problem of optimal allocation of orders for parts among a set of approved suppliers in make-to-order manufacturing has been modeled as a mixed integer program, in which two different objective functions were either aggregated into a single cost function to be minimized or considered as independent functions of a dual objective optimization problem. In particular, the risk level of defective or unreliable supplies that the decision maker is disposed to accept is controlled by the maximum number v of delivery patterns for which the average defect rate or late delivery rate can be unacceptable. Under the assumption of identical probability associated with each of the h delivery patterns, the decision parameters q and r can be considered as the targeted average defect and late delivery rates based on the ð1 v=hÞpercentile of these rates. In other words, we allow 100ðv=hÞ% of the delivery patterns to exceed q and/or r, and the smaller is v, the more risk averse is the selected supply portfolio. In a single objective approach the corresponding costs of defective or delayed parts are directly introduced in a minimized cost function, while in a multiple objective approach the average defect rate and the average late delivery rate are aggregated and considered as a separate objective function to be minimized. A single objective approach is capable of finding a single optimal portfolio of suppliers that minimizes the total average cost of ordering, purchasing, defects and delays of parts. The proposed aggregation of the different types of cost into a single cost function should account for a specific business environment and the decision maker preferences. In practice, cost of a defective or delayed part may sometimes exceed its price. In particular, when the resulting lack of all required parts leads to a much higher cost of unfulfilled customer orders. On the other hand, a subset of nondominated portfolios can be found, applying the multiple objective Tchebycheff approach to minimize the average ordering and purchasing cost of parts and the average defect and late delivery rate of supplies. The computational results indicate that while the single objective approach yields a single proven optimal solution in a very short CPU time, the multiple objective approach requires the much longer computation time to produce a very small subset of nondominated solutions. In addition, a single objective solution weakly dominates some of the nondominated multiple objective solutions, which further justifies the usefulness of the proposed single objective approach. In the models proposed various simplifying assumptions have been introduced. For example, it has been assumed that each supplier is capable of manufacturing all required part types. In a more general setting, each supplier may only be prepared to manufacture a subset of part types and provide with the parts the corresponding subset of customer orders. Furthermore, the quantity discount offered by the supplier has been assumed to be based on the total quantity of all ordered parts. In practice, independent quantity discounts may be offered for the individual part types. The limited computational experiments have indicated that the proposed approach requires a relatively small CPU time to find the optimal solution in a static case, where all customer orders are
The author is grateful to two anonymous reviewers for reading the manuscript carefully and providing constructive comments which helped to improve this paper. This work has been partially supported by research grant of MNiSzW and by AGH.
References [1] Aissaoui N, Haouari M, Hassini E. Supplier selection and order lot sizing modeling: a review. Computers and Operations Research 2007;34:3516–40. [2] Murthy NN, Soni S, Ghosh S. A framework for facilitating sourcing and allocation decisions for make-to-order items. Decision Sciences 2004;35: 237–259. [3] Sawik T. A cyclic versus flexible approach to materials ordering in make-toorder assembly. Mathematical and Computer Modelling 2005;42:279–90. [4] Yue J, Xia Y, Tran T. Selecting sourcing partners for a make-to-order supply chain. Omega. The International Journal of Management Science, available online 5 August 2009, in press. [5] Akinc U. Selecting a set of vendors in a manufacturing environment. Journal of Operations Management 1993;11:107–22. [6] Kasilingam RG, Lee CP. Selection of vendors—a mixed-integer programming approach. Computers and Industrial Engineering 1996;31:347–50. [7] Basnet C, Leung JMY. Inventory lot-sizing with supplier selection. Computers and Operations Research 2005;32:1–14. [8] Jayaraman V, Srivastava R, Benton WC. Supplier selection and order quantity allocation: a comprehensive model. Journal of Supply Chain Management 2006;35(2):50–8. [9] Weber CA, Current JR. A multiobjective approach to vendor selection. European Journal of Operational Research 1993;68:173–84. [10] Xia W, Wu Z. Supplier selection with multiple criteria in volume discount environments. Omega. The International Journal of Management Science 2007;35:494–504. [11] Demirtas EA, Ustun O. An integrated multiobjective decision making process for supplier selection and order allocation. Omega. The International Journal of Management Science 2008;36:76–90. [12] Ustun O, Demirtas EA. An integrated multi-objective decision making process for multi-period lot sizing with supplier selection. Omega. The International Journal of Management Science 2008;36:509–21. [13] Pokharel S. A two-objective model for decision making in a supply chain. International Journal of Production Economics 2008;111:378–88. [14] Sanayei A, Mousavi SF, Abdi MR, Mohaghar A. An integrated group decisionmaking process for supplier selection and order allocation using multiattribute utility theory and linear programming. Journal of the Franklin Institute 2008;345(7):731–47. [15] Huo H, Wei Z. Suppliers selection and order allocation in the environment of supply chain based on fuzzy multi-objective integer programming model. In: The IEEE international conference on service operations and logistics, and informatics, vol. 2, Beijing, China, October 12–15, 2008. p. 2299–304. [16] Liaoa Z, Rittscher J. Integration of supplier selection, procurement lot sizing and carrier selection under dynamic demand conditions. International Journal of Production Economics 2007;107:502–10. [17] Che ZH, Wang HS. Supplier selection and supply quantity allocation of common and non-common parts with multiple criteria under multiple products. Computers and Industrial Engineering 2008;55:110–33. [18] Ghodsypour SH, O’Brien C. The total cost of logistics in supplier selection under conditions of multiple sourcing, multiple criteria and capacity constraints. International Journal of Production Economics 2001;73:15–27. [19] Tempelmeier H. A simple heuristic for dynamic order sizing and supplier selection with time-varying data. Production and Operations Management 2002;11:499–515. [20] Hall NG, Lesaoana M, Potts CN. Scheduling with fixed delivery dates. Operations Research 2001;49:134–44. [21] Chaudhry SS, Forst FG, Zydiak JL. Vendor selection with price breaks. European Journal of Operational Research 1993;70:52–66. [22] Schotanus F, Telgen J, de Boer L. Unraveling quantity discounts. Omega. The International Journal of Management Science 2009;37:510–21. [23] Steuer RE. Multiple criteria optimization: theory, computation and application. New York: Wiley; 1986.
Contents lists available at ScienceDirect
Omega journal homepage: www.elsevier.com/locate/omega
Single vs. multiple objective supplier selection in a make to order environment Tadeusz Sawik AGH University of Science and Technology, Department of Operations Research and Information Technology, Al.Mickiewicza 30, 30-059 Krak´ ow, Poland
a r t i c l e in fo
abstract
Article history: Received 12 May 2009 Accepted 6 September 2009 Available online 12 September 2009
The problem of allocation of orders for custom parts among suppliers in make to order manufacturing is formulated as a single- or multi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order. The selection of suppliers is based on price and quality of purchased parts and reliability of on time delivery. The risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate can be unacceptable. Furthermore, the quantity or business volume discounts offered by the suppliers are considered. Numerical examples are presented and some computational results are reported. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Supplier selection Operational risk Discounts Multi-criteria decision making Mixed integer programming
1. Introduction In make-to-order environment customer-oriented manufacturers should be prepared to produce varieties of products to meet the different customer needs. Each product is typically composed of many common and non-common (custom) parts that can be sourced from different approved suppliers with different supply capacity. An important issue is how to best allocate the orders for parts among various part suppliers to fulfill all customer orders for products and to achieve a high customer service level at a low cost. The decision maker needs to decide from which supplier to purchase parts required to complete each customer order. The above decisions are based on price, quality (defect rate) and reliability (on time delivery) criteria that may conflict each other, e.g. the supplier offering the lowest price may not have the best quality or the supplier with the best quality may not deliver on time. Furthermore, to reduce the fixed ordering (transaction) costs the number of suppliers and the total number of orders should be minimized. On the other hand, the selection of more suppliers sometimes may divert the risk of unreliable supplies. In spite of the importance of supplier selection and order allocation problems, the decision making is not sufficiently addressed in the literature (for a recent review, see Aissaoui et al. [1]), in particular for make-to-order manufacturing environment, e.g. Murthy et al. [2], Sawik [3], Yue et al. [4]. Basically, the authors distinguish between single and multiple item models and supplier selection with single or multiple sourcing, where each
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supplier can fully meet all requirements (e.g. Akinc [5]) or none of the suppliers is able to satisfy the total requirements, respectively. The vast majority of the decision models are mathematical programming models either single objective, e.g. Kasilingam and Lee [6], Basnet and Leung [7], Jayaraman [8] or multiple objectives, e.g. Weber and Current [9], Xia and Wu [10], Demirtas and Ustun [11], Ustun and Demirtas [12], Pokharel [13]. The supplier selection is a complex decision making problem which includes both quantitative and qualitative factors and one of the disadvantages of the mathematical programming methods is their failure to account for qualitative factors that may affect suppliers performance. In order to consider both quantitative and qualitative factors some researchers propose hybrid approaches that combine different methods. For example, Sanayei et al. [14] propose an integration of multi-attribute utility theory and linear programming, first to rate and choose the best suppliers and then to find optimal allocation of order quantities among the selected suppliers to maximize total additive utility. The combined method allows both quantitative and qualitative factors under risk and uncertainty to be considered as well as to account for the probabilistic nature of supplier performance. Another integrated approach that combines analytic network process and multiobjective mixed integer programming is proposed in [11,12]. First, the potential suppliers are evaluated according to 14 criteria that are involved in the four clusters: benefits, opportunities, costs and risks, to calculate the priorities of each supplier. Then, the optimum quantities are allocated among selected suppliers to maximize total value of purchasing (using the calculated priorities) and to minimize the total cost and total defect rate. However, the disadvantages of the integrated methods usually may affect the performance of hybrid approaches. The other approaches that are also applied to solve the supplier selection
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problem are methods based on fuzzy sets, e.g. a fuzzy multiobjective integer programming Huo and Wei [15] and genetic algorithms (e.g. Liaoa and Rittscher [16], Che and Wang [17]). For example, in [16] a genetic algorithm with problem specific operator is developed to account for the inbound transportation and to combine supplier selection with carrier selection decisions. The fuzzy and genetic algorithms, however, are heuristics that do not guarantee optimality of a solution. The models developed for supplier selection and order allocation can be either single-period models (e.g. [6,8,9,11]) that do not consider inventory management or multi-period models (e.g. [3,7,12,16], Ghodsypour and O’Brien [18], Tempelmeier [19]) which consider the inventory management by lot-sizing and scheduling of orders. Since common parts can be efficiently managed by material requirement planning methods, this research is focused on custom parts that can be critical in make-to-order manufacturing. For custom-engineered products no inventory of custom parts can be kept on hand. Instead, the custom parts need to be requisitioned with each customer order and hence the custom parts inventory need not to be considered. This paper presents mixed integer programming models for single or multiple objective supplier selection in make-toorder manufacturing for a static supply portfolio in a nondiscount or discount environment, that is for the allocation of orders for parts among the suppliers without or with discount and with no timing decisions. In contrast to the dynamic portfolio, which is the allocation of orders among the suppliers combined with the allocation of orders among the planning periods. The major contribution of this paper is that it proposes a simple mixed integer programming approach for selection of supply portfolio under conditions of operational risk associated with uncertain quality and reliability of supplies. The integer programming models incorporate risk constraints where the risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate may exceed the maximum acceptable rates. The number of maximum delivery patterns and the corresponding maximum rates represent, respectively, the confidence level and the targeted rates above which a risk averse decision maker wants to limit the number of outcomes. The paper is organized as follows. In Section 2 description of the supplier selection problem in make-to-order manufacturing is provided. The mixed integer program for a single objective supplier selection in a non-discount environment is presented in Section 3. The model enhancements for the supplier selection with a business volume discount or quantity discount are presented in Section 4. The multiple objective approach is proposed in Section 5. Numerical examples and some computational results are provided in Section 6, and final conclusions are made in the last section.
2. Problem description In the supply chain under consideration various types of products are assembled by a single producer to satisfy customer orders, using custom parts purchased from multiple suppliers (for notation used, see Table 1). Each supplier can provide the producer with custom parts for all customer orders. However, the suppliers have different limited capacity and, in addition, differ in price and quality of offered parts and in reliability of on time delivery of parts. Let I ¼ f1; . . . ; mg be the set of m suppliers and J ¼ f1; . . . ; ng the set of n customer orders for the products, known ahead of time. Each order j A J is described by the quantity sj of required
Table 1 Notation. Indices i supplier, iA I ¼ f1; . . . ; mg j customer order, jA J ¼ f1; . . . ; ng t delivery pattern, t A T ¼ f1; . . . ; hg Input parameters ci capacity of supplier i oi cost of ordering parts from supplier i pij price of part for customer order j purchased from supplier i qit expected defect rate of supplier i for delivery date in pattern t rit expected late delivery rate of supplier i for delivery date in pattern t sj number of parts to be purchased for customer order j P D j A J sj total demand for parts q r v
the largest acceptable average defect rate of supplies the largest acceptable average late delivery rate of supplies the maximum allowed number of delivery patterns with the average defect rate or average late delivery rate of supplies greater than q or r, respectively
custom parts and requested delivery date, where the latter need not to be explicitly considered when selecting a supplier. Each supplier is assumed to have sufficient capacity to complete manufacturing and to deliver the ordered parts to the producer by the requested dates. All parts ordered from a supplier are shipped together with a single shipment at one of a series of fixed delivery dates (e.g. Hall et al. [20]). The parts are dispatched to the producer at the earliest fixed delivery date after the completion time of their manufacturing. Hence, for each supplier the delivery date and the corresponding reliability of supply depend on the completion time of manufacturing the ordered parts, which is unknown to the producer when the supplier selection decision is made. Likewise, the quality of supply may depend on the completion time. When the suppliers are selected, however, the risk of defective and unreliable supplies can be considered using past observations. Since different suppliers may complete manufacturing of ordered parts at different times, and then deliver the parts at different dates, a different risk can be associated with each combination of suppliers delivery dates. Let us call each combination of m fixed delivery dates, one delivery date for each supplier, a delivery pattern. Each delivery pattern must be feasible in respect to requested delivery dates. The total number of all feasible combinations of m fixed delivery dates consists of h delivery patterns and let T ¼ f1; . . . hg be the index set of all feasible delivery patterns. The probability that is assigned to the occurrence of each delivery pattern is identical and equals 1=h. Let ci be the capacity of supplier i A I, oi Fcost of ordering parts from supplier i A I, pij Fpurchasing price of part for customer order j A J from supplier iA I, and qit , rit Frespectively, the expected defect rate, the expected late delivery rate of supplier iA I for delivery date in pattern t A T. The rates qit and rit are based on past observations. We assume that the risk of defective or unreliable supplies from the selected suppliers can be measured by the number of delivery patterns for which the average defect rate or the late delivery rate of supplies are unacceptable. The decision maker needs to decide from which supplier to purchase custom parts required for each customer order to achieve a low unit cost and high quality and reliability of supplies.
3. Single objective supplier selection in a non-discount environment In this section a mixed integer program is proposed for a single-period supplier selection and order allocation problem in a
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non-discount environment, i.e. for determining a static portfolio of suppliers without discount. The static portfolio of suppliers is defined as ðx1 ; . . . ; xm Þ; where X xi ¼ 1 iAI
and 0 r xi r1 is the fraction of the total demand for parts ordered from supplier i (for definition of problem variables, see Table 2). When deciding on a static portfolio of suppliers it is assumed that the orders for all parts are simultaneously placed on selected suppliers (e.g. at time 0), and each supplier delivers all the ordered parts at the earliest possible delivery date. Therefore, the allocation of orders for parts among the suppliers is not combined with the allocation of orders among the planning periods. Nevertheless, given past observations, the static portfolio should be checked over time horizon against the risk of too low quality of purchased parts (too high defect rate) and too low reliability of supplies (too high late delivery rate), where both the quality and the reliability are randomly varying over time. Denote by q, r and v the maximal acceptable defect rate of portfolio, the maximal acceptable late delivery rate of portfolio and the maximum number of delivery patterns for which the average defect rate or the average late delivery rate of the portfolio can be above the threshold q or r, respectively. The parameters q, r and v are fixed by the decision maker to control the risk of defective and unreliable supplies. We assume that the decision maker is willing to accept only portfolios for which the number of delivery patterns with the average defect rate greater than the threshold q or with the average late delivery rate greater than the threshold r is not greater than v. The performance of the static portfolio can be measured by the following two criteria: X XX oi yi =D þ pij sj zij =D; ð1Þ f1 ðy; zÞ ¼ iAI
iAI jAJ
XX ðqit þ rit Þxi =h; f2 ðxÞ ¼
ð2Þ
iAI tAT
where f1 ðy; zÞ is the average ordering and purchasing cost per part, and f2 ðxÞ is the average defect and late delivery rate. The overall performance can be measured by the sum Fðy; zÞ of average cost per part of ordering, purchasing, defects and delays, in which the cost of a defective or delayed part is assumed to be identical with its price 0 1 ! X X X X X @ A oi yi þ pij sj zij D þ ðqit þrit Þ=h pij sj zij =D: Fðy; zÞ ¼ iAI
jAJ
iAI
tAT
jAJ
ð3Þ The second summation term in (3) can be interpreted as the purchasing cost of additional parts required to compensate for Table 2 Problem variables.
205
defective and delayed parts. Clearly, the assumption that cost of a defective or delayed part is identical with its price may be too relaxed in practice. The lack of all required parts may lead to unfulfilled customer orders and then the resulting shortage cost can be much higher than the purchasing cost of additional parts. On the other hand, parts not passing quality acceptance level may not be paid for, and parts delivered late may be paid for at a reduced price. The mixed integer program SP for the single objective supplier selection problem in a non-discount environment is formulated below. Model SP: Single objective portfolio of suppliers in a non-discount environment Minimize (3) subject to 1. Order assignment constraints:
for each customer order required parts are supplied by exactly one supplier,
for each supplier the total quantity of ordered parts cannot exceed its capacity. X zij ¼ 1; j A J;
ð4Þ
iAI
X sj zij r ci ;
i A I:
ð5Þ
jAJ
2. Portfolio selection constraints: the portfolio definition constraint (note that the order allocation variable xi is an auxiliary variable determined by zij ), for each delivery pattern t, the portfolio with average defect rate greater than q is not acceptable, for each delivery pattern t, the portfolio with average late delivery rate greater than r is not acceptable, the portfolio can be unacceptable for at most v delivery patterns, parts are ordered from supplier i, if at least one customer order is assigned to supplier i, X sj zij =D; i A I; ð6Þ xi ¼ jAJ
P vt Z P vt Z X
qit xi q ; 1q
iAI
i A I rit xi
r
1r
;
t A T;
ð7Þ
t A T;
ð8Þ
vt r v;
ð9Þ
tAT
c i y ; iA I; D i X zij ; iA I: yi r
xi r
ð10Þ ð11Þ
jAJ
3. Non-negativity and integrality conditions: vt ¼ 1, if for delivery pattern t the average defect rate or the average late delivery rate of supplies for the selected portfolio is, respectively greater than q or greater than r , otherwise vt ¼ 0 (portfolio selection variable) xi ¼ the fraction of total demand for parts ordered from supplier i (order allocation variable) yi ¼ 1, if an order for parts is placed on supplier i; otherwise yi ¼ 0 (supplier selection variable) zij ¼ 1, if parts required for customer order j are ordered from supplier i; otherwise zij ¼ 0 (customer order assignment variable)
vt A f0; 1g; xi Z 0;
t A T;
iA I;
ð12Þ ð13Þ
yi A f0; 1g;
i A I;
ð14Þ
zij A f0; 1g;
i A I; j A J:
ð15Þ
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Constraints (7) and (8) prevent the choice of portfolios whose average defect rate or whose average late delivery rate is above the fixed threshold q (7) or r (8), respectively. For each delivery P pattern t such that the average defect rate i A I qit xi is greater than the largest acceptable rate q (7) or the average late delivery P rate i A I rit xi is greater than the largest acceptable rate r (8), vt ¼ 1. Then all the delivery patterns with the average defect rate or with the average late delivery rate above the threshold are summed up in (9). If the result is greater than v, then the portfolio is infeasible. In view of risk constraints (7)–(9) and under the assumption of identical probability 1=h that each delivery pattern is realized, q and r can be considered as the targeted average defect and late delivery rates based on the a-percentile of these rates, where a ¼ ð1 v=hÞ represents the confidence level for the average rates across all delivery patterns. In other words, we allow 100ð1 aÞ% ¼ 100ðv=hÞ% of the delivery patterns to exceed q and/or r. If different probability pt is assigned to each delivery pattern t A T, then constraint (9) should be replaced with X pt vt r 1 a; ð90 Þ tAT
where the confidence level a A ð0; 1Þ is fixed by the decision maker to control the risk of defective and unreliable supplies, and the higher is the confidence level, the more risk averse is the selected supply portfolio. P P Notice that v ¼ 0 implies i A I qit xi r q and i A I rit xi rr 8t A T, i.e., the selected portfolio must be acceptable for each delivery pattern. In particular, for the stationary expected defect rates qit ¼ qi 8i A I; t A T and the stationary expected late delivery rates rit ¼ ri 8i A I; t A T, v can take on either value 0 or h, since the selected portfolio can be either acceptable or unacceptable over the entire set of all possible delivery patterns. Hence, in the stationary case variable v can be eliminated from the model.
4. Single objective supplier selection in a discount environment In make to order environment, in which custom parts are typically ordered in small lot sizes, supplier may sometimes offer discount that depends on total value of sales volume (business volume) or on total quantity of ordered parts, e.g. Xia and Wu [10], Tempelmeier [19], Chaudhry et al. [21], Schotanus et al. [22]. In the context of business volume discount the quantity or variety of purchased parts does not affect the offered price, while for the quantity discount the price does not depend on the total ‘‘euro’’ amount of sales volume. In this section the presence of price discounts offered by the suppliers is considered, first the supplier selection with business volume discount and then the supplier selection with quantity discount, respectively, based on the total value and the total quantity of the order (for notation used, see Table 3). Assume that each supplier i offers cumulative (all-units) price breaks having gi discount intervals ðbi0 ; bi1 ; . . . ; ðbi;gi 1 ; bi;gi according to total business volume (or total quantity), where bik is upper limit on the kth (k A Ki ¼ f1; . . . ; gi g) business volume (or quantity) discount interval ðbik1 ; bik and bi0 ¼ 0 8i. Let 0 o aik o1 be the discount rate (percentage of discount) associated with interval k of supplier i. If total business volume (total quantity) from supplier i falls on interval k, then the price of each part for customer order j is ð1 aik Þpij .
Table 3 Notation for a discount environment. k Ki aik bik
discount interval of supplier i, k A Ki ¼ f1; . . . ; gi g set of discount intervals of supplier i price discount rate associated with discount interval k of supplier i upper limit on discount interval k of supplier i, 0 ¼ bi0 r bi1 r r bigi
The following discount interval selection variable need to be added to enhance the supplier selection and order allocation problem SP for the discount environment: uik ¼ 1 if total business volume (or total quantity) from supplier i falls on the discount interval k; otherwise uik ¼ 0. Furthermore, the customer order assignment variable zij need to be modified as below. Zijk ¼ 1 if parts required for customer order j are ordered from supplier i and total business volume (or total quantity) of parts purchased from this supplier falls on the discount interval k; otherwise Zijk ¼ 0. The total business volume or total quantity of parts purchased P P P from supplier i is, respectively, jAJ k A Ki pij sj Zijk or jAJ P k A Ki sj Zijk . Notice that the total business volume or the total quantity purchased from each supplier falls on exactly one discount interval of this supplier. Now, the average purchasing and ordering cost per part can be expressed as follows: X XX X oi yi =Dþ ð1 aik Þpij sj Zijk =D: ð16Þ f1 ðy; ZÞ ¼ iAI
i A I j A J k A Ki
The overall performance of the static portfolio with discount can be measured by the sum Fðy; ZÞ of average cost per part of ordering, purchasing, defects and delays, in which the cost of a defective or delayed part is assumed to be identical with its regular price 0 1 XX X @oi yi þ ð1 aik Þpij sj Zijk AD Fðy; ZÞ ¼ iAI
j A J k A Ki
! XX X X ðqit þ rit Þ=h pij sj Zijk =D: þ iAI
tAT
ð17Þ
j A J k A Ki
The mixed integer program SPD for the single objective supplier selection problem with business volume discount or with quantity discount is formulated below. Model SPD: Single objective portfolio of suppliers with discount Minimize (17) subject to (7)–(10), (12)–(14) and 1. Order assignment constraints:
the parts required for each customer order are supplied by exactly one supplier in one discount interval,
for each supplier the total quantity of ordered parts cannot exceed its capacity. XX Zijk ¼ 1; j A J
ð18Þ
i A I k A Ki
XX
sj Zijk r ci ;
iAI
ð19Þ
j A J k A Ki
2. Portfolio selection constraints:
the portfolio definition constraint (note that the order allocation variable xi is an auxiliary variable determined by Zijk ),
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an order for parts is placed on supplier i, if for at least one customer order the required parts are ordered from supplier i, XX sj Zijk =D; iA I; ð20Þ xi ¼
solution set of the bi-objective program MP and MPD can be found by the parameterization on l the mixed integer program MPl and MPDl, respectively.
j A J k A Ki
yi r
XX
207
Model MPl: Zijk ;
i A I:
ð21Þ
j A J k A Ki
minfd þ eðf1 ðy; zÞ þ f2 ðxÞÞg
ð30Þ
subject to (1), (2), (4)–(15) and
3. Discount constraints:
total business volume purchased from each selected supplier can be exactly in one discount interval X pij sj Zijk r bik uik ; i A I; k A Ki ðbi;k1 þ 1Þuik r
ð22Þ
lðf1 ðy; zÞ f 1 Þ r d;
ð31Þ
ð1 lÞðf2 ðxÞ f 2 Þ r d;
ð32Þ
d Z0;
ð33Þ
where 0 r l r1.
jAJ
for a business volume discount environment, or total quantity purchased from each selected supplier can be exactly in one discount interval X sj Zijk rbik uik ; i A I; k A Ki ð23Þ ðbi;k1 þ 1Þuik r jAJ
for a quantity discount environment,
parts are ordered from supplier i, if supplier i is selected and assigned at least one customer order, X uik ¼ yi ; i A I;
ð24Þ
k A Ki
Zijk ruik ;
iA I; j A J; kA Ki
ð25Þ
uik A f0; 1g;
i A I; k A Ki ;
ð26Þ
Zijk A f0; 1g;
i A I; j A J; k A Ki :
ð27Þ
5. Multiple objective supplier selection In the single objective approach for the supplier selection proposed in previous sections, two objective functions f1 (1) or (16) and f2 (2) were aggregated into a single cost function F (13) or (17), respectively for a non-discount or discount environment. In this section the two objective functions are considered independently, and a bi-objective mixed integer program MP or MPD is presented for the supplier selection in a non-discount or discount environment, respectively. Model MP: Multiple objective portfolio of suppliers in a nondiscount environment Minimize
In this section some computational examples are presented to illustrate possible applications of the proposed mixed integer programming approach for supplier selection in a non-discount or discount environment and to compare the single- and the multiple-objective approach. The following parameters have been used for the example problems:
ð28Þ
subject to (1), (2), (4)–(15).
Model MPD: Multiple objective portfolio of suppliers with discount Minimize f ðx; y; ZÞ ¼ ½f1 ðy; ZÞ; f2 ðxÞ
The programs MPl and MPDl are based on the augmented l-weighted Tchebycheff metric minfljf1 ðy; zÞ f 1 jþ eðf1 ðy; zÞ þ f2 ðxÞÞ; ð1 lÞjf2 ðxÞ f 2 jþ eðf1 ðy; zÞ þ f2 ðxÞÞg. Steuer [23] proved that there always exists e small enough that enable to reach all the nondominated solutions. However, for the mixed integer programs, there may be portions of the nondominated set (nearby weakly nondominated solution) that the above Tchebycheff program is unable to compute, even considering very small e. 6. Computational examples
4. Integrality conditions:
f ðx; y; zÞ ¼ ½f1 ðy; zÞ; f2 ðxÞ
Model MPDl: Minimize (30) subject to (2), (7)–(10), (12)–(14), (18)–(27), (31)–(33).
ð29Þ
subject to (2), (7)–(10), (12)–(14), (18)–(27).
5.1. Reference point based scalarizing program Let f ¼ ðf 1 ; f 2 Þ be a reference point in the criteria space such that f i o fi ; i ¼ 1; 2 for all feasible solutions satisfying (1), (2), (4)–(15) and (2), (7)–(10), (12)–(14), (18)–(27), respectively of MP and MPD. Denote by e a small positive value. The nondominated
h, the number of delivery patterns, was equal to 30; m, the number of suppliers, was equal to 20; n, the number of customer orders, was equal to 100; oi , the cost of ordering parts from supplier i, was equal to 5000 for each supplier i; pij , the unit price of parts required for each customer order j, purchased from each supplier i was uniformly distributed over [10,15] and reduced by the factor ð1 maxt A T ðqit þ rit ÞÞ to get a lower price for parts from the suppliers with higher defect and late delivery rates; qit and rit , the expected defect rate and the expected late delivery rate of each supplier i and delivery date in each pattern t, was uniformly distributed over (0,0.08) and (0,0.10), respectively; q and r, the largest acceptable average defect rate and late delivery rate, was equal to 0.05 and 0.06, respectively; v, the maximum allowed number of delivery patterns with the average defect or late delivery rates greater than, respectively q ¼ 0:05 or r ¼ 0:06, was equal to 0, 1 or 2; sj , the numbers of required parts for each customer order, were integers uniformly distributed over ½100; 5000; P ci , the capacity of each supplier i, was equal to d2 j A J sj =me, i.e., the total capacity of all suppliers was equal to the double total demand for parts; quantity discount: gi ¼ 3 discount intervals for each supplier i, with the upper limits bik ¼ dkci =gi e; iA I; k ¼ 1; . . . ; gi ; business volume discount: gi ¼ 3 discount intervals for each supplier i, with the upper limits bik ¼ dkp i ci =gi e; iA I; k ¼
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Table 4 Computational results for single objective approach. v
Var.
Bin.
Cons.
Non-discount environment (model SP) 0 2040 2020 241 1 2070 2050 242 2 2070 2050 242 Business volume discount (model SPD) 0 6120 6100 6381 1 6130 6110 6382 2 6130 6110 6382 Quantity discount (model SPD) 0 6120 6100 6381 1 6130 6110 6382 2 6130 6110 6382
F ðf1 ; f2 Þa, no. of suppliersb
Table 6 Computational results for multiple objective approach: discount environment. CPUc
9.108 (8.366, 0.090), 13 9.027 (8.321, 0.087), 13 9.004 (8.283, 0.089), 12
17 7 15
8.335 (7.592, 0.090), 12 8.239 (7.528, 0.087), 12 8.207 (7.494, 0.087), 11
2191 1265 292
8.392 (7.580, 0.090), 12 8.226 (7.515, 0.087), 13 8.204 (7.495, 0.087), 12
234 380 140
a F ðf1 ; f2 ÞFthe average cost (the average ordering and purchasing cost per part, the average defect and late delivery rate), respectively. b The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11.
Table 5 Computational results for multiple objective approach: non-discount environment (model MPl). Var.
Bin.
Cons.
f1 ; f2 a, no. of suppliersb
CPUc
2041 2071 2071
2020 2050 2050
242 243 243
8.401, 0.089, 13 8.386, 0.086, 11 8.424, 0.085, 12
459 262 48
2041 2071 2071
2020 2050 2050
243 244 244
8.390, 0.090, 13 8.328, 0.087, 13 8.321, 0.087, 13
746 338 33
2041 2071 2071 l ¼ 0:95 0 2041 1 2071 2 2071 l¼1 0 2041 1 2071 2 2071
2020 2050 2050
243 244 244
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
239 72 73
2020 2050 2050
244 244 244
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
134 460 22
2020 2050 2050
242 243 243
8.363, 0.091, 13 8.304, 0.092, 11 8.284, 0.089, 12
232 53 102
v
l¼0 0 1 2
l ¼ 0:05 0 1 2
l ¼ 0:5 0 1 2
a f1 ; f2 —the average ordering and purchasing cost per part, the average defect and late delivery rate, respectively ðf 1 ¼ 8; f 2 ¼ 0:08Þ. b The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11.
P 1; . . . ; gi , where p i ¼ j A J pij =n is the average unit price of parts from supplier i; aik , discount coefficients, were equal to 0:05ðk 1Þ; i A I; k ¼ 1; . . . ; gi , i.e., the maximum percentage discount is 10%.
The computational results for the three risk levels represented by the maximum number of delivery patterns v ¼ 1; 2; 3 for which the average defect rate or late delivery rate of supplies can be unacceptable are summarized in Tables 4 and 5, 6, respectively for single- and multiple-objective approach. The size of the integer programs SP, MPl and SPD, MPDl, respectively, for the supplier selection in a non-discount and discount (business volume or quantity) environment are represented by the total number of variables, Var., number of binary variables, Bin., and number of constraints, Cons. The last two columns of the tables present the solution values and CPU time in seconds required to prove optimality of the solution. The subsets of nondominated
v
Var.
Bin.
Cons.
f1 ; f2 a, no. of suppliersb
Business volume discount (model MPDl) l¼0 0 6101 6080 6382 7.594, 0.090, 12 1 6131 6110 6383 7.571, 0.086, 11 2 6131 6110 6383 7.674, 0.085, 11 l ¼ 0:05 0 6101 6080 6383 7.593, 0.091, 12 1 6131 6110 6384 7.528, 0.087, 12 2 6131 6110 6383 7.481, 0.091, 11 l A f0:5; 0:95; 1g 0 6101 6080 6382 7.592, 0.091, 12 1 6131 6110 6383 7.505, 0.091, 11 2 6131 6110 6383 7.481, 0.091, 11 Quantity discount (model MPDl) l¼0 0 6121 6100 6382 7.636, 0.089, 13 1 6151 6130 6383 7.568, 0.086, 11 2 6151 6130 6383 7.604, 0.085, 12 l ¼ 0:05 0 6121 6100 6383 7.580, 0.091, 12 1 6131 6110 6384 7.528, 0.087, 12 2 6131 6110 6383 7.495, 0.087, 12 l A f0:5; 0:95; 1g 0 6121 6100 6382 7.580, 0.090, 12 1 6151 6130 6383 7.496, 0.092,11 2 6151 6130 6383 7.480, 0.091, 12
CPUc
1403 747 247 1148 823 246 2495–3587 2245–3005 250–500
723 57 107 179 189 74 294–625 226–549 142–312
a
ðf 1 ¼ 7:4; f 2 ¼ 0:08Þ. The number of selected suppliers. c CPU seconds for proving optimality on a PC Pentium IV, 1.8 GHz, RAM 1 GB/ CPLEX 11. b
solutions of the bi-objective programs MP and MPD were computed by parameterization the corresponding Tchebycheff l A f0; 0:05; 0:5; 0:95; 1g. The computational programs on experiments were performed using AMPL programming language and the CPLEX v.11 solver (with the default settings) on a laptop with Pentium IV processor running at 1.8 GHz and with 1 GB RAM. The solver was capable of finding proven optimal solutions within the limit of 3600 CPU seconds for all examples. The results indicate that while the single objective approach yields a single proven optimal solution in a very short CPU time, the multiple objective approach requires the much longer computation time to produce a very small subset of nondominated solutions. For example, Table 5 shows that in a non-discount environment only three nondominated solutions ðf1 ; f2 Þ were found for each risk level v: (8.401, 0.089), (8.390, 0.090), (8.363, 0.091) for v ¼ 0, (8.386, 0.086), (8.328, 0.087), (8.304, 0.092) for v ¼ 1, and (8.424, 0.085), (8.321, 0.087), (8.284, 0.089) for v ¼ 2. In particular, only one solution for each v was found for l A f0:5; 0:95; 1g. In addition, each single objective solution weakly dominates one of the nondominated multiple objective solutions: (8.366, 0.090) weakly dominates (8.390, 0.090) for v ¼ 0, (8.321, 0.087) weakly dominates (8.328, 0.087) for v ¼ 1 and (8.283, 0.089) weakly dominates (8.284, 0.089). for v ¼ 3. Similarly, only two or three nondominated solutions were obtained for each risk level in a business volume discount or quantity discount environment (see, Table 6). In the computational experiments e ¼ 0:001. For the mixed integer programs, however, there may be portions of the nondominated set that the Tchebycheff program is unable to compute, even considering very small e [23]. For the single objective approach, the optimal allocation of demand for parts among the suppliers in a non-discount or discount environment is shown in Fig. 1. The best leveled allocation of demand among the selected suppliers (the allocation with a minimum difference between the highest and the lowest fraction of allocated demand) has been achieved for the
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209
Fig. 1. Optimal portfolios for single objective approach.
quantity discount, with xi ranging from 0.068 to 0.098. The most uneven demand allocation was found for the business volume discount at the lowest risk level (v ¼ 0), with xi ranging from 0.028 to 0.098.
For the multiple objective approach and a non-discount environment, the allocation of demand for parts among the suppliers for the three nondominated solutions and the three risk levels is shown Fig. 2. The number of selected suppliers varies
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Fig. 2. Optimal portfolios for multiple objective approach: non-discount environment.
between 11 and 13, and the allocated fraction of total demand for parts varies most (from xi ¼ 0:028 to xi ¼ 0:098) for the lowest risk level (v ¼ 0).
For the multiple objective approach and the quantity discount environment, the allocation of demand for parts among the suppliers for the two or three nondominated solutions obtained
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211
Fig. 3. Optimal portfolios for multiple objective approach: quantity discount.
for each risk level is shown Fig. 3. The number of selected suppliers varies between 11 and 13, and the allocated fraction of total demand for parts varies most (from xi ¼ 0:024 to xi ¼ 0:10) for the lowest risk level (v ¼ 0).
The computational results demonstrate that the greater is the allowed number of delivery patterns v with the average defect and late delivery rates above the thresholds (i.e., the higher is the risk of defective or delayed supplies), the lower is the average cost. The
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largest average cost is achieved for the lowest risk level (v ¼ 0), i.e. where the average defect and late delivery rates never exceed the acceptable rates. Simultaneously, for the lowest risk level the allocation of demand among the selected suppliers is the most uneven.
known ahead of time. The last assumption can be relaxed, and the approach can also be used in a dynamic case where orders arrive irregularly over time. In this case, the supplier selection decisions can be made periodically over a rolling planning horizon, upon arrivals of a number of orders in a specific time interval.
7. Conclusion Acknowledgments The problem of optimal allocation of orders for parts among a set of approved suppliers in make-to-order manufacturing has been modeled as a mixed integer program, in which two different objective functions were either aggregated into a single cost function to be minimized or considered as independent functions of a dual objective optimization problem. In particular, the risk level of defective or unreliable supplies that the decision maker is disposed to accept is controlled by the maximum number v of delivery patterns for which the average defect rate or late delivery rate can be unacceptable. Under the assumption of identical probability associated with each of the h delivery patterns, the decision parameters q and r can be considered as the targeted average defect and late delivery rates based on the ð1 v=hÞpercentile of these rates. In other words, we allow 100ðv=hÞ% of the delivery patterns to exceed q and/or r, and the smaller is v, the more risk averse is the selected supply portfolio. In a single objective approach the corresponding costs of defective or delayed parts are directly introduced in a minimized cost function, while in a multiple objective approach the average defect rate and the average late delivery rate are aggregated and considered as a separate objective function to be minimized. A single objective approach is capable of finding a single optimal portfolio of suppliers that minimizes the total average cost of ordering, purchasing, defects and delays of parts. The proposed aggregation of the different types of cost into a single cost function should account for a specific business environment and the decision maker preferences. In practice, cost of a defective or delayed part may sometimes exceed its price. In particular, when the resulting lack of all required parts leads to a much higher cost of unfulfilled customer orders. On the other hand, a subset of nondominated portfolios can be found, applying the multiple objective Tchebycheff approach to minimize the average ordering and purchasing cost of parts and the average defect and late delivery rate of supplies. The computational results indicate that while the single objective approach yields a single proven optimal solution in a very short CPU time, the multiple objective approach requires the much longer computation time to produce a very small subset of nondominated solutions. In addition, a single objective solution weakly dominates some of the nondominated multiple objective solutions, which further justifies the usefulness of the proposed single objective approach. In the models proposed various simplifying assumptions have been introduced. For example, it has been assumed that each supplier is capable of manufacturing all required part types. In a more general setting, each supplier may only be prepared to manufacture a subset of part types and provide with the parts the corresponding subset of customer orders. Furthermore, the quantity discount offered by the supplier has been assumed to be based on the total quantity of all ordered parts. In practice, independent quantity discounts may be offered for the individual part types. The limited computational experiments have indicated that the proposed approach requires a relatively small CPU time to find the optimal solution in a static case, where all customer orders are
The author is grateful to two anonymous reviewers for reading the manuscript carefully and providing constructive comments which helped to improve this paper. This work has been partially supported by research grant of MNiSzW and by AGH.
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