A goal programming model for joint decision ... - Semantic Scholar

Computers & Industrial Engineering 71 (2014) 1–9

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A goal programming model for joint decision making of inventory lot-size, supplier selection and carrier selection Devendra Choudhary ⇑, Ravi Shankar Department of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

a r t i c l e

i n f o

Article history: Received 15 November 2011 Received in revised form 3 February 2014 Accepted 6 February 2014 Available online 14 February 2014 Keywords: Goal programming Integer programming Lot-sizing Supplier selection Carrier selection Quantity discount

a b s t r a c t In this paper, we address a problem in which a storage space constrained buyer procures a single product in multiple periods from multiple suppliers. The production capacity constrained suppliers offer all-unit quantity discounts. The late deliveries and rejections are also incorporated in sourcing. In addition, we consider transportation cost explicitly in decision making which may vary because of freight quantity and distance of shipment between the buyer and a supplier. We propose a multi-objective integer linear programming model for joint decision making of inventory lot-sizing, supplier selection and carrier selection problem. In the multi-objective formulation, net rejected items, net costs and net late delivered items are considered as three objectives that have to be minimized simultaneously over the decision horizon. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve the multi-objective optimization problem using three variants of goal programming (GP) approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP. The solution of these models is compared at different service-level requirements using value path approach. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Firm can accomplish competitiveness by reducing total logistics costs through integration of its various internal activities of purchasing process. The purchasing function of a firm consists of three activities: lot-sizing decision, supplier selection decision, and carrier selection decision. While a lot-sizing decision aims to minimize inventory and shortage costs by optimizing timings and order sizes. The intent of supplier and carrier selection decisions is to minimize inbound logistics costs and to attain a high degree of quality and delivery performance. Due to the inherent interdependency among these three decisions, a firm cannot optimize them separately (Aissaoui, Haouari, & Hassini, 2007; Choudhary & Shankar, 2013). The value of scheduling orders over the multi-period horizon along with the supplier and carrier selections can be significantly higher than planning over a single period. In practice, suppliers offer price discounts for large order quantities. Per unit transportation cost also reduces with long shipment distances and/or large freight quantity (Russell & Krajewski, 1991; Swenseth & Godfrey,

⇑ Corresponding author. Tel.: +91 9818955052. E-mail addresses: [email protected] (D. Choudhary), [email protected] (R. Shankar). http://dx.doi.org/10.1016/j.cie.2014.02.003 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved.

2002; Mansini, Tocchella, & Savelsbergh, 2012). By considering multi-period horizon, a firm can aggregate orders to take advantage of economies of scale in procurement and transportation costs. In such a situation, however, inventory costs increase as excessive products need to be carried forward to future periods. The firm can sometime allow shortages with backordering to reduce inventory costs and increase order sizes, especially when a few customers are ready to wait. While purchasing functions need to consider cost minimization objective, yet in doing so one cannot compromise on quality and delivery related criteria. Nowadays, even quality and delivery related objectives are getting higher priority than cost criterion during purchasing decisions (Ho, Xu, & Dey, 2010). Suppliers’ performance on quality and delivery criteria has significant influence on the lot-sizing and total logistics costs (Choudhary & Shankar, 2011, 2013). In this study, we take into account above observations and then develop a multi-objective integer linear programming model for an integrated inventory lot-sizing, supplier selection and carrier selection problem. We investigate a problem in which a single product is procured from multiple suppliers in multiple periods considering suppliers’ capacity limitations, rejections and late deliveries. We also incorporate economies of scale concepts in purchasing and shipping costs. The model considers three important goals that

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D. Choudhary, R. Shankar / Computers & Industrial Engineering 71 (2014) 1–9

need to be minimized. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve this model using preemptive GP, non-preemptive GP and weighted max–min fuzzy GP approaches. Furthermore, the sensitivity of the GP methods with respect to each goal is assessed using value path approach. The paper is further organized as follows. Section 2 presents a brief literature review of the existing quantitative approaches related to supplier selection and lot-sizing problem. In Section 3, a multi-objective integer linear programming formulation is developed for inventory lot-sizing, supplier selection and carrier selection problem. A brief description of three GP methods: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP is also provided in this section. To demonstrate the effectiveness of the proposed approach, Section 4 presents an illustration. Finally, conclusions are drawn in Section 5. 2. Literature review In the supply chain literature, while lot-sizing is considered as a tactical decision, supplier selection is regarded as a strategic decision. The lot-sizing problem deals with determining order quantity and its timing by striking a tradeoff between ordering and storage costs. A comprehensive classification of the lot-sizing models can be found in Ben-Daya, Darwish, and Ertogral (2008) and Robinson, Narayanan, and Sahin (2009). The supplier selection problem has been extensively studied by the researchers from several perspectives. Ghodsypour and O’Brien (2001) have studied sourcing problem in a multi-criteria framework. Supplier selection and order allocation problem usually becomes complicated under quantity discount environment. Chaudhry, Forst, and Zydiak (1993) propose a mathematical formulation for supplier selection over a single period with quantity discounts. Tsai and Wang (2010) compare the influence upon the buying decisions considering two schemes of quantity discounts: all-unit discount, and incremental discount. Xu, Lu, and Glover (2000) develop a mathematical model for multi-item dynamic lotsize problem with joint business volume discount. Dahel (2003) proposes a multi-objective mixed integer programming model to determine order allocation of multiple products to multiple supplier considering volume discounts. Ravindran, Bilsel, Wadhwa, and Yang (2010) study supplier selection and order allocation considering incremental price breaks. Kokangul and Susuz (2009) apply an integrated AHP and non-linear integer programming approach considering quantity discounts to determine the best suppliers and optimal order quantities among them. Hassini (2008) studies a lotsizing and supplier selection problem when supplier capacity reservation and price discounts are both dependent on lead time. Burke, Carrillo, and Vakharia (2008) analyse the impact of linear discounts, incremental unit discounts, and all-unit discounts as well as capacity limitations on the optimal sourcing policy for a single period. Ebrahim, Razmi, and Haleh (2009) formulate a multi-objective mathematical model for a purchasing problem which considers different types of discount schemes such as all-unit discount, incremental discount, and total business volume discount. In the last decade, several researchers propose models that can simultaneously deal with lot-sizing and supplier selection decisions (Basnet & Leung, 2005; Demirtas & Ustun, 2009; Rezaei & Davoodi, 2008, 2011; Ustun & Demirtas, 2008a,b). A comprehensive discussion of these studies can be found in Choudhary and Shankar (2013). Liao and Rittscher (2007) propose a multi objective programming model for supplier selection, procurement lotsizing and carrier selection decisions. Jolai, Yazdian, Shahanaghi, and Khojasteh (2011) proposed a two-phase approach for supplier

selection and order allocation problem under fuzzy environment for multiple products from multiple suppliers in multiple periods. Razmi and Maghool (2010) propose a fuzzy bi-objective model for multiple items, multiple period, supplier selection and purchasing problem under capacity constraint and budget limitation. The literature review confirms that integrated lot-sizing and supplier selection problem has been studied sufficiently. But, in practice, it is observed that almost half of the total logistics cost of a product is due to transportation. Moreover, per unit transportation cost decreases for large quantity of cargo and/or long distance of shipment. Suppliers’ performance on delivery and rejection criteria has a significant influence on lot-sizing and total logistics costs. For example, in order to meet service-level requirement, a firm has to order larger quantity due to presence of defective and late delivery. Also, per unit purchasing and transportation costs increase when defective items in a procured lot go up. In other words, while higher rejection rate of the supplier eats up the savings obtained through economies of scale in purchasing and shipment costs, late deliveries result in increasing transportation costs and stock-outs. These realistic aspects of a purchasing process have not been adequately addressed so far in the literature. Our study aims to incorporate these aspects of a purchasing process by integrating lot-sizing, supplier and carrier selection decisions so as to minimize total logistics costs while attaining desired levels of quality and delivery performances.

3. Model development Consider Fig. 1 where a buyer procures a product from multiple suppliers. Buyer’s demand of the product in each period is deterministic and known in advance. Each supplier has limited production capacity and a different unit price of the product. In addition, each supplier offers all-unit quantity discounts to motivate the buyer for procuring large quantity. Products could be shipped by using different size carriers. A particular size carrier can ship any lot-size up to its full truck load (FTL) capacity. The transportation cost will be different for different carriers as well as for different suppliers because of carrier size and geographical distance of the buyer’s premises from sourcing locations. Over finite discrete time periods, shortages are allowed and backlog is permitted when available inventory plus procured lot-size for a period is less than buyer’s demand during that period. Alternatively, excessive products could be carried forward for use in subsequent periods, incurring storage cost. Both shortage and inventory are restricted by the service-level requirement and available storage space, respectively. The buyer needs to select one or more suppliers as well as carriers, and determine procurement timings and lot-sizes in these periods. The total procurement from the selected suppliers should satisfy the demand considering rejections and late deliveries, and allowing shortages with backlogging while minimizing net rejected items, net costs and net late delivered items. The following sub-sections provide model notations and a mathematical model. Assumptions considered in this study are the same as in Choudhary and Shankar (2013). 3.1. Model parameters and decision variables Indices i set m set t set j set

of of of of

suppliers, i = 1, . . . , I all-unit price break levels, m = 1, . . . , M discrete time periods, t = 1, . . . , T transportation carriers, j = 1, . . . , J

3


Suppliers

Carriers

Buyer

Customers

Fig. 1. General scheme of the problem under consideration.

Parameters dt buyer’s demand of the product in period t pimt cost of procuring one unit of product from supplier i at price break level m in period t bimt quantity at which all-unit price break m occur at supplier i in period t oit cost of ordering to supplier i in period t titj buyer’s transportation cost from supplier i in period t for carrier j qimt percentage of rejected items delivered by supplier i at price break level m in period t limt percentage of items late delivered by supplier i at price break level m in period t Cit capacity of supplier i in period t Xtj full truck load carrying capacity of carrier j in period t Vtj total numbers of carrier j of same FTL capacity are available in period t ht buyer’s unit inventory holding cost of the product in period t wt buyer’s storage capacity in period t ht buyer’s service-level in period t so (1 ht) is the proportion of end user demand that are not met and backordered for buyer in period t intermediate variable indicates inventory level in period Iþ t t I intermediate variable indicates the amount of the t shortage in period t Decision variables ximtj lot-size (number of units of product) that buyer procures from supplier i at the price break level m in period t and transports using carrier j yimtj binary variable used in separating price level m for a product in a transaction between buyer and supplier i in period t using carrier j uimtj binary variable indicating whether carrier j is selected or not for transporting procured lot-size from supplier i at the price level m in period t zit binary variable indicating whether supplier i gets order or not in period t

Min Z2 ¼

XXXX XX pimt ximtj þ oit zit m

i

t

j

t

i

XXXX X þ þ t itj uimtj þ ht I t m

i

Min Z3 ¼

t

ð2Þ

t

j

XXXX limt ximtj i

m

t

ð3Þ

j

The first objective function (1) minimizes the net rejected items. The objective function (2) consists of three parts: the purchasing costs, the transaction costs and the inventory holding costs. While the second term in Eq. (2) represents the total ordering costs, the third term refers to total transportation costs. Note that ordering cost depends on whether procurement takes place or not; therefore binary variable zit has been used in the expression of ordering cost. Similarly, transportation cost depends on which size carrier is selected and from whom the product is procured; therefore binary variable uimtj has been used in the expression of transportation cost. The third objective (3) refers to the minimization of net late delivered items over the decision horizon. The buyer wishes to optimize these objectives, subject to the following constraints: 1. The inventory level of the buyer: Prior researchers do not consider the impact of late deliveries and rejections on the inventory level at the end of each period. Under the multi-criteria framework, consideration of net rejection and net late delivered items as separate objectives is not sufficient because rejections and late deliveries significantly affect the inventory levels as well as total logistics cost. In this study, we consider the impact of suppliers’ performance on delivery and rejection criteria at the time of determining inventory levels. The exact inventory level (Iþ t ) or shortages (It ) of the buyer at þ time t is given by the stock (It1 ) or backlog (It1 ) at time t 1, plus the quantity procured (ximtj) and receipt of late delivered items (limtximtj) at time t, minus current period’s late deliveries (limtximtj), rejections (qimtximtj) and end-user demand (dt) at time t, that is

Iþt1 þ

XXX XXX XXX ximtj þ limk ximkj limt ximtj i

m

j

i

m

j

i

m

j

i

m

XXX qimt ximtj dt It1 ¼ Iþt It 8t; and k ¼ t 1

3.2. Mathematical formulation With above parameters and decision variables, a multi-objective optimization model is expressed as follows: Objective functions:

XXXX Min Z1 ¼ qimt ximtj i

m

t

j

ð1Þ

j

ð4Þ

2. Charging ordering cost constraints: According to these constraints, the buyer cannot procure without charging appropriate ordering cost.

ximtj

T X k¼t

!

dk zit

8i; 8m; 8t; 8j

ð5Þ

4


3. All-unit quantity scheme: Generally, the suppliers offer discounts to the buyer for purchasing greater than normal or usual quantity. All-unit discounts result in a decreased cost per unit, but storage costs increase. The Constraints (6)–(8) along with Constraints (4) help the buyer to achieve an optimal tradeoff between purchasing and storage costs.

biðm1Þt yimtj ximtj bimt yimtj

8i; 8m; 8t; 8j

xiMtj C it yiMtj 8i; 8t; 8j XX yimtj ¼ zit 8i; 8t m

ð6Þ ð7Þ ð8Þ

j

4. Carrier selection constraints: In practice, per unit transportation cost decreases in large quantity of cargo and/or long distance of the shipment. Per unit procurement cost also reduces when suppliers offer discounts for large order quantities. On the other hand, higher inventory levels of the buyer and greater rejection rate of the supplier may eat up the savings obtained through economies of scale in purchasing and shipment costs. This complex decision making process is modeled in Constraints (9)– (11). According to Constraints (9), a chosen carrier cannot ship a lot-size more than its FTL capacity. Constraints (10) ensure that only one carrier is used for shipping the lot-size if it is ordered in period t from the supplier i. Constraints (11) represent the restriction on availability of total numbers of carrier j of same FTL capacity in period t.

ximtj 6 Xtj uimtj 8i; 8m; 8t; 8j XX uimtj ¼ zit 8i; 8t

ð9Þ ð10Þ

j

8t; 8j

ð11Þ

m

It is interesting to note that binary variable zit activates three constraints simultaneously: optimal order lot-size Constraints (5), optimal price break level Constraints (8), and optimal carrier selection Constraints (10). Thus, this model integrates three purchasing decisions of a firm under the consideration of economies of scale in procurement and shipment costs. 5. Storage capacity constraints: These constraints guarantee that inventory level at the end of period t cannot be more than available storage space.

Iþt 6 wt

8t

ð12Þ

6. Service-level constraints: These constraints guarantee that the amount of stock-out cannot be more than buyer’s service-level requirement.

It 6 ð1 ht Þdt

8t

ð13Þ

7. Binary and non-negative constraints:

Iþt ; It

yimtj 2 f0; 1g 8i; 8m; 8t; 8j

ð16Þ

uimtj 2 f0; 1g 8i; 8m; 8t; 8j

ð17Þ

zit 2 f0; 1g 8i; 8t

ð18Þ

We solve the above multi-objective optimization model using three different variants of goal programming approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP.

Goal Programming (GP), introduced by Charnes, Cooper, and Ferguson (1955), is the most widely used approach in multi-criteria decision making and in multi-objective programming (Aouni & Kettani, 2001; Shankar & Vrat, 1999). In GP, each objective is assigned a target level for achievement and pre-specified priority of decision maker on achieving the target. The role of the GP is to minimize the unwanted deviations between the achievement of goals and their aspiration levels. Liao and Kao (2010) propose an integrated approach using Taguchi loss function, AHP and multichoice goal programming for solving the supplier selection problem. A brief description of three popular variants of GP methods is provided in following sub-sections. 3.3.1. Preemptive GP model In preemptive GP, problems are solved as a sequence of linear programs: one linear program for each priority level. At each stage of the procedure, a revision in the solution is permitted only if it causes no reduction in the achievement of higher priority goals. The preemptive GP formulation for the multi-period inventory lot-sizing with supplier selection and carrier selection problem is as follows: þ

P 0 8t

ð14Þ

þ

þ

Min Z ¼ p1 d1 þ p2 d2 þ p3 d3

ð19Þ

Subject to

XXXX þ qimt ximtj þ d1 d1 ¼ Net rejected items goal m

i

XX uimtj 6 V tj i

ð15Þ

3.3. Solution methodology

According to Constraints (6), a procured lot of the suppliers at a specific price break is in the discount interval offered. Constraints (7) represent the restriction on order lot-size due to the capacity of a supplier. It is assumed that the minimum capacity of the supplier is greater than the quantity at which price break level M is offered. Constraints (8) ensure that only one price break level is used for the lot-size if it is procured during period t from the supplier i.

m

ximtj P 0 and integer 8i; 8m; 8t; 8j

t

ð20Þ

j

XXXX XX XXXX X þ pimt ximtj þ oit zit þ t itj uimtj þ ht I t m

i

t

j

i

t

i

m

t

j

þ

þ d2 d2 ¼ Net cost goal

t

ð21Þ

XXXX þ limt ximtj þ d3 d3 ¼ Net late delivered items goal m

i

t

j

ð22Þ

dn ;

þ

dn P 0 8n 2 f1; . . . ; 3g dn

ð23Þ

þ dn

where and are the deviation variables representing how far the solution deviates from each goal in underachievement and overachievement side, respectively. Constraints (4)–(18) stated earlier will also be included in this model. 3.3.2. Non-preemptive GP model In non-preemptive GP, the buyer sets a goal value for each objective and preferences in achieving those goals are expressed as numerical weights. Here, the buyer sets following three goals: i. Limit the net rejected items with weight W1. ii. Limit the net cost with weight W2. iii. Limit the net late delivered items with weight W3.

5


The weights W1, W2 and W3 are assigned by the decision maker. The non-preemptive GP model can be formulated as follows: þ

þ

þ

Min Z ¼ W 1 d1 þ W 2 d2 þ W 3 d3

Period (t)

ð24Þ

Subject to the Constraints (4)–(18), (20)–(22) and (23).

Demand

3.3.3. Weighted max–min fuzzy GP model In a real case, decision makers do not have exact and complete information related to objective targets. In such cases, the theory of fuzzy sets is one of the best tools to handle uncertainty. Fuzzy GP uses the ideal values as targets. Zimmermann (1978) applies the fuzzy optimization technique to solve fuzzy multi-objective linear programming problem. Kumar, Vrat, and Shankar (2004) propose a mixed integer goal programming model for solving the vendor selection problem in a supply chain considering vague and imprecise values of multiple goals. In this paper weighted max–min fuzzy GP model, developed by Amid, Ghodsypour, and O’Brien (2011), has been used to enable the decision maker to achieve levels of fuzzy goals which are consistent with desirable relative weights. The advantage of weighted max–min fuzzy GP is that no target values have to be specified by the decision maker and it manages to find an optimal solution such that the ratio of the achievement levels is the same as the weights of the objectives. Based on the weighted max–min fuzzy GP model, the crisp single objective formulation for multi-period inventory lot-sizing with supplier selection and carrier selection problem is as follows:

Max k

Table 1 Demand data of a product in four periods.

1

2

3

4

650

520

500

650

Table 2 Problem data set for the illustrative example. Supplier

Quantity level

pimt

oit

limt

qimt

Cit

S1

Q < 150 150 6 Q < 300 300 6 Q

62 61 60

1000

0.14

0.10

500

S2

Q < 200 200 6 Q < 350 350 6 Q

72 71 70

1500

0.06

0.08

450

S3

Q < 250 250 6 Q < 400 400 6 Q

68 67 66

1400

0.12

0.06

620

The values in parenthesis in Table 3 provide per unit shipping cost from each supplier at the FTL capacity of each size carrier. One can observe that economies of scale in transportation cost can be realized for far away Suppliers 1 and 3 as compared to nearby Supplier 2.

ð25Þ 4.1. Results and discussion

s.t.

Wn k 6

zþn

zn ðxÞ zþn zn

8n 2 f1; . . . ; 3g

ð26Þ

where z n is a minimum value obtained through solving the multiobjective problem as a single objective. In other words, each objective is to be minimized separately. zþ n is the maximum value (worst solution) of negative objective (zn). Constraints (4)–(18) stated earlier will also be included in this model. Weight values Wn are the same as those used in the non-preemptive GP model. 4. An illustration The effectiveness of proposed multi-objective mixed integer linear programming model can be demonstrated through a case, which is presented below: The demand of a product in unit for a distributor over one month decision horizon, which comprises of four weeks /periods, is given in Table 1. Table 2 provides other data needed to model the multi-objective optimization problem where three suppliers offer all-unit price discounts. The distributor may procure the product once in a week from each supplier. Further, the distributor can hire at most two carriers in each period for shipping lot from each chosen supplier to its premises. The FTL capacity of each carrier and transportation cost for each supplier are different as given in Table 3. The distributor sells the product to the many local retailers throughout the week and store unsold items, if any, at the weekend. The available storage capacity is limited to 220 units. Storage cost per unit per period is 10. Shortages of the product are also backlogged. However, the distributor wishes to maintain at least 80% service-level in each period. Beginning and ending inventory as well as shortages are assumed to be zero for the decision horizon. The problem for the distributor here is to find which supplier to procure from, how much to procure from each selected supplier and in which periods, and which carrier to select for delivering procured lot from each chosen supplier over the decision horizon.

Three GP models discussed in Section 3.3 are used to solve the multi-objective inventory lot-size, supplier selection and carrier selection problem. Ideal solutions obtained by optimizing each objective separately in the proposed model as depicted by Eqs. (1)–(18) are given in Table 4 at three different service-level requirements. An ideal solution obtained by optimizing net rejections, Supplier 3 receives most of the orders because of the high quality rate. Supplier 2 also gets a few orders after the capacity of Supplier 3 is exhausted. Late delivered items optimization solution shows that Suppliers 2 and 3 receive all the orders due to the better performance on delivery criterion than Supplier 1. Cost optimization solution shows that Suppliers 1 and 2 receive all the orders at a higher price break level to take advantage of economies of scale in purchasing and shipping costs. Further, it is observed that only net cost increases with an increase in service-level requirement while net rejections and net late deliveries remains the same. Now, optimal solution obtained by each GP model is discussed below. 4.1.1. Preemptive GP solution In the preemptive GP model, net rejected items objective is given the highest priority, followed by net cost and net late delivered

Table 3 Transportation cost data. Supplier

Carrier1

Carrier 2

S1

3500 (10)

5000 (8.33)

S2

2000 (5.71)

3500 (5.83)

S3

7000 (20)

10,000 (16.66)

Capacity

350

600

6


Table 4 Ideal solution of the problem. Service level?

h = 0.80

h = 0.90

h = 0.99

Objectives? Ideal goal

Rejections 155.86

Cost 197787.6

Late items 196.32

Rejections 155.86

Cost 198073.4

Late items 196.32

Rejections 156.24

Cost 201579.4

Late items 196.32

Procurement Lot-size

x1141 = 1 x2111 = 33 x2131 = 114 x3312 = 600 x3322 = 600 x3332 = 600 x3342 = 600

x1312 = 476 x1322 = 500 x1332 = 500 x1342 = 485 x2311 = 350 x2341 = 350

x2312 = 450 x2322 = 450 x2332 = 450 x2342 = 450 x3111 = 163 x3141 = 7 x3221 = 314 x3231 = 252

x1141 = 1 x2111 = 147 x3312 = 600 x3322 = 600 x3332 = 600 x3342 = 600

x1312 = 500 x1322 = 499 x1332 = 492 x2211 = 314 x2331 = 350 x2342 = 427

x2312 = 450 x2322 = 450 x2332 = 450 x2342 = 450 x3111 = 244 x3121 = 237 x3131 = 248 x3141 = 7

x2111 = 177 x3312 = 600 x3322 = 599 x3332 = 596 x3342 = 573

x1312 = 498 x1322 = 500 x1332 = 500 x1342 = 373 x2121 = 71 x2311 = 350 x2341 = 350

x2312 = 450 x2322 = 450 x2332 = 450 x2342 = 450 x3211 = 313 x3221 = 330 x3131 = 86 x3141 = 7

Table 5 Preemptive GP solution of the problem. Service level? Objectives;

h = 0.80

h = 0.90

h = 0.99

Preemptive priorities

Objective targets

Obj. value achieved


Objective targets

Obj. value achieved


Objective targets

Obj. value Objectives achieved

Rejections Cost Late items

1 2 3

165 210,000 230

165 215,466 258.6

1 2 3

165 210,000 230

165 215,466 258.6

1 2 3

165 210,000 230

165 215,466 258.6


x2211 = 264, x2342 = 450, x3312 = 598, x3322 = 600, x3332 = 600

x2211 = 264, x2342 = 450, x3312 = 598, x3322 = 600, x3332 = 600

items, respectively. The highest priority is given to net rejected items objective by the distributor as the delivery of defective products in the purchased lot is a direct loss. Target values for net rejected items, net costs, and net late delivered items are 165, 210,000, and 230, respectively. Table 5 illustrates the solution using the preemptive GP model. Suppliers 2 and 3 are selected and target value of net rejected items objective is achieved, but at the expense of net cost and net late delivered goals. In order to minimize cost, most of the lot-sizes are procured at highest price break levels. It is interesting to observe that in the preemptive GP model, change in service-level requirement is not affecting optimal solution. 4.1.2. Non-preemptive GP solution In the non-preemptive GP model, weight W1, W2 and W3 obtained through expert opinions is applied. The same target values as in the preemptive GP model are used. The solution of non-preemptive GP model is shown in Table 6. We observe that net cost and net late delivery goals have been over-achieved. However, the net rejections goal is not achievable. Here, the optimal solution is changing with the change in service-level requirement. 4.1.3. Weighted max – min fuzzy GP solution In the weighted max – min fuzzy GP model, ideal values are used as targets or goal for three objectives. The linear membership

x2211 = 264, x2342 = 450, x3312 = 598, x3322 = 600, x3332 = 600

function is used for fuzzifying the objective functions. The data set of the values of the lower bounds and upper bounds of the objective functions is given in Table 7. Linear membership functions lzn ðxÞ are shown in Fig. 2. The optimal solution obtained using weighted max–min fuzzy GP model is shown in Table 8. Suppliers 2 and 3 are selected and Carrier 1 is used for delivering lot from Supplier 2 while Carrier 2 delivers from Supplier 3. Since, targets are set at the ideal values which are best values ever achievable. However, none of the targets are fully achieved. Here, net cost increases with an increase in service-level requirement while net rejections and net late deliveries remain the same.

4.2. Comparison of results The solution obtained through three GP models can be depicted graphically for decision maker to compare the three alternatives and their tradeoffs. The tradeoffs among GP methods can be demonstrated using a value path approach (VPA). Since the values obtained for objectives have different measurements, each objective value can be normalized by dividing with the minimum value for each associated objective. Therefore, the minimum value for each objective solution is one. The normalized values have dimensionless units; the smaller the resulting rating, the more preference it has in this model. Table 9 shows the objective values and VPA

Table 6 Non-preemptive GP solution of the problem. Service level? Objectives;

h = 0.80

h = 0.90

h = 0.99

Weights

Objective targets

Obj. value achieved

Weights

Objective targets

Obj. value achieved

Weights targets

Objective achieved

Obj. value


0.40 0.35 0.25

165 210,000 230

197.18 209725.4 212.06

0.40 0.35 0.25

165 210,000 230

199.96 210,000 221.46

0.40 0.35 0.25

165 210,000 230

198.30 209965.4 215.50


x1211 = 295, x2231 = 349, x2312 = 450, x2322 = 450, x2342 = 397, x3332 = 600

x1332 = 411, x2211 = 304, x2331 = 350, x2322 = 450, x2342 = 434, x3312 = 597

x1331 = 335, x2211 = 266, x2322 = 450, x2332 = 449, x2342 = 445, x3312 = 600

7

D. Choudhary, R. Shankar / Computers & Industrial Engineering 71 (2014) 1–9 Table 7 Data set of membership functions. Service level?

h = 0.80

h = 0.90

h = 0.99

Objectives;

l=1

l=0

l=1

l=0

l=1

l=0


155.86 197787.6 196.32

253.12 242988.2 360.6

155.86 198073.4 196.32

252.78 242431.4 359.88

156.24 201579.4 196.32

252.46 241,466 359.06

Fig. 2. Membership functions for rejections, costs and late delivered items.

Table 8 Weighted max–min fuzzy GP solution of the problem. Service level? Objectives;

h = 0.80

h = 0.90

h = 0.99

Weights

Objective targets

Obj. value achieved

Weights

Objective targets

Obj. value achieved

Weights

Objective targets

Obj. value achieved


0.40 0.35 0.25

155.86 197787.6 196.32

178 212804.8 223.5

0.40 0.35 0.25

155.86 198073.4 196.32

178 213,320 223.5

0.40 0.35 0.25

156.24 201579.4 196.32

177.76 213993.2 223.32


x2211 = 219, x2231 = 306, x2321 = 350, x2342 = 450, x3312 = 600, x3332 = 600

x2211 = 275, x2231 = 250, x2321 = 350, x2342 = 450, x3312 = 600, x3332 = 600

values for each GP model with their normalized values in the parenthesis at three different service-level requirements. The results are then plotted with net defective items, net cost, and net late delivered items on the horizontal axis and normalized objective values on the vertical axis as shown in Fig. 3. The output of the VPA can be used to determine tradeoffs among different compromise solutions. As per tradeoffs obtained from the VPA, it can be seen that preemptive GP solution does 19.50% better than the non-preemptive GP on the net rejections, but it does 2.74% worse on net cost and 21.95% worse on net late delivered items when each period’s service-level requirement is at least 80%. When the service-level requirement of each period is higher than 90%, the preemptive GP solution does 21.19% better than the non-preemptive GP on the net rejections, but it does 2.60% worse on net cost and 16.77% worse on net late delivered

x2211 = 326, x2231 = 242, x2321 = 350, x2342 = 404, x3312 = 600, x3332 = 600

items. VPA also allows assessing the sensitivity of optimal solutions with respect to each other. It can be seen from Fig. 3 that all GP methods perform close to one another in the net cost objective; thereby the distributor’s preference is very sensitive to the net cost criterion. GP methods provide a scattered performance pattern for the net rejected items objective and net late delivered items objective which suggests that the weight assigned by the decision maker is not very sensitive to these objectives. Fig. 4 illustrates the effect of service-level requirements and GP methods on multiple objectives. At all service-level requirement, both preemptive and weighted max–min fuzzy GP methods are sensitive to the preference order provided by the decision maker to the objectives. On the contrary, the nonpreemptive GP approach is less sensitive to the order in which weights are assigned to the objectives.

Table 9 Summary of GP solutions. Service level?

h = 0.80

GP methods;

Rejections

Cost

Delivery

h = 0.90 Rejections

Cost

Delivery

h = 0.99 Rejections

Cost

Delivery

Pre. GP

165 (1.0000)

215,466 (1.0274)

258.6 (1.2195)

165 (1.0000)

215,466 (1.0260)

258.6 (1.1677)

165 (1.0000)

215,466 (1.0262)

258.6 (1.2000)

N-pre. GP

197.18 (1.1950)

209725.4 (1.0000)

212.06 (1.0000)

199.96 (1.2119)

210,000 (1.0000)

221.46 (1.0000)

198.3 (1.2018)

209965.4 (1.0000)

215.5 (1.0000)

Fuzzy GP

178 (1.0788)

212804.8 (1.0147)

223.5 (1.0539)

178 (1.0788)

213,320 (1.0158)

223.5 (1.0092)

177.76 (1.0773)

213993.2 (1.0192)

223.32 (1.0363)

8

D. Choudhary, R. Shankar / Computers & Industrial Engineering 71 (2014) 1–9 Preemptive GP Non-preemptive GP Weighted max-min fuzzy GP

1.24 1.20

Preemptive GP Non-preemptive GP Weighted max-min fuzzy GP

1.24 1.20

1.24 1.16

1.16

1.16

1.12

1.12

1.12

1.08

1.08

1.08

1.04

1.04

1.04

1.00

1.00

1.00

0.96

0.96

0.96

Rejections

Cost

Late items

Preemptive GP Non-preemptive GP Weighted max-min fuzzy GP

1.20

Rejections

θ t = 0.80

Cost

Rejections

Late items

Cost

Late items

θ t = 0.99

θ t = 0.90 Fig. 3. Comparison of the GP methods using value path approach.

1.24

θ = 0.80

θ = 0.90

θ = 0.99

1.20

1.03

θ = 0.80

θ = 0.99

θ = 0.90

1.24

θ = 0.80

θ = 0.90

θ = 0.99

1.20

1.02

1.16

1.16

1.01

1.12

1.12

1.00

1.08

1.08

1.04

1.00

1.04

1.00

0.99

1.00

0.96

0.98 Preemptive GP

Non-preemptive GP

Weighted maxmin fuzzy GP

0.96

Preemptive GP

Rejections

Non-preemptive GP


Cost

Preemptive GP

Non-preemptive GP


Late items

Fig. 4. Comparison of the service-levels using value path approach.

5. Conclusions

Acknowledgements

Inventory lot-sizing with supplier selection and carrier selection decision process simultaneously determines the sourcing, carrier allocation, and lot-sizes over the decision horizon. Selecting the right suppliers and splitting lot-sizes to the selected suppliers become a major challenge for a decision maker when suppliers offer price discounts with a few defective items and certain delivery performance. Further, the problem becomes more realistic as well as interesting when transportation cost varies with carrier capacity and shipment distance between a downstream buyer and a sourcing firm. This paper presents a multi-objective integer linear programming (MOILP) approach for inventory lot-sizing with supplier selection and carrier selection problem. By formulating the inventory lot-sizing problem with sourcing and carrier selection as a MOILP, we have captured a realistic supply chain setting. The proposed model allows the decision maker to find which supplier to procure from, how much to procure from the selected suppliers and in which periods, and which carrier is to be chosen for delivering procured items over the decision horizon. The proposed formulation for problem of reasonable size can be solved using any commercial software in few seconds of computer time. The optimization package LINGO 12 took less than 20 s of computer time on IntelÒ CoreTM i3-380 Processor 2.53 GHz laptop to solve all problems illustrated in this paper. However, as the number of periods and price discount levels increase, the available commercial software may not be able to solve the proposed formulations in a reasonable time. Hence, future research work may explore heuristics and evolutionary algorithms such as genetic algorithm to solve larger size problems using proposed formulation.

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