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International Journal of Industrial Engineering Computations 6 (2015) 391–404 Contents lists available at GrowingScience International Journal of Industrial Engineering Computations homepage: www.GrowingScience.com/ijiec Dealing with customers enquiries simultaneously under contingent situation Sujan Piya* Department of Mechanical and Industrial Engineering, College of Engineering, Sultan Qaboos University, AL-Khod 123, Muscat, Sultanate of Oman CHRONICLE Article history: Received September 14 2014 Received in Revised Format January 10 2015 Accepted February 10 2015 Available online February 11 2015 Keywords: Quotation Contingent order Negotiation Make-to-Order ABSTRACT This paper proposes a method to quote the due date and the price of incoming orders to multiple customers simultaneously when the contingent orders exist. The proposed method utilizes probabilistic information on contingent orders and incorporates some negotiation theories to generate quotations. Rather than improving the acceptance probability of quotation for single customer, the method improves the overall acceptance probability of quotations being submitted to the multiple customers. This helps increase the total expected contribution of company and acceptance probability of entire new orders rather than increasing these measures only for a single customer. Numerical analysis is conducted to demonstrate the working mechanism of proposed method and its effectiveness in contrast to sequential method of quotation. © 2015 Growing Science Ltd. All rights reserved 1. Introduction Companies that operate in Make-to-Order (MTO) environment must respond to enquiries by preparing quotations that are attractive to their customers and feasible for them to fill. Dealing properly with customers enquiries has a huge impact on orders being confirmed by the customer. Lots of researchers have contributed to the area of quotation. However, from the relevance of industrial domain, due date and price quotation are the most researched areas. With an objective of minimizing expected aggregate cost per job, Seidman and Smith (1981) have proposed a method to identify due date for incoming order. On the other hand, Bertrand (1983) and Baker (1984) have considered the objective of minimizing the average weighted due date quoted to the customers. Ragatz and Mabert (1984), Shantikumar and Sumita (1988), Wein (1991) and Chand and Chhajed (1992) have advocated that due date based on shop-floor congestion information achieves better delivery performance. Most of these researchers have dealt with due date and/or pricing decisions under a single-product and single-stage production system. Enns (1995) and Van and Bertrand (2001) have investigated the setting of cost optimal internal due dates for determining the priorities on the shop floor, and in determining the external due dates that can be quoted to the customer. Lawrence (1995) has developed a model for due date estimation using empirical distribution of forecasted error in production settings with machine breakdowns. Elhafse (2000) has proposed exact and heuristic algorithms for determining the lead-time and price to be quoted to a single order by dividing total orders into regular and rushed orders. Proposing two different models, Keskinocak * Corresponding author. E-mail: sujan@squ.edu.om (S. Piya) © 2015 Growing Science Ltd. All rights reserved. doi: 10.5267/j.ijiec.2015.2.003 392 et al. (2001) has shown that different strategies for quotation are needed for different categories of customers. Charnsirisakskul et al. (2006) has presented an optimization model for due date and price sensitive customers under deterministic demand function. Liu et al. (2007) has considered pricing and lead-time decisions from a supplier-retailer perspective where demand is deterministic and sensitive to price and lead-time decisions. Similarly, Kaminsky and Kaya (2008) have developed due date quotation model in the supply chain environment where the model is affected by the performance of supplier. Based on the result of empirical analysis, Zorzini et al. (2008) has proposed a method of quoting due date that took into account the known average lead time. Slotnick (2011) has presented stochastic dynamic programming model for lead time quotation for a production system in which bottleneck process requires a minimum batch size. Chaharsooghi et al. (2011) has extended the model of Charnsirisakskul et al. (2006) to include multi class customers with stochastic demand. Recently, Xianfei et al. (2013) has studied production scheduling problem with an objective of maximizing profit when customers are sensitive to quoted delivery time. When an enquiry about new order arrives, the company may already have several orders either in the form of confirmed orders or contingent orders. The time between customer enquiry and order acceptance (OA) decision can be extremely long and exhaustive enquiry may, unfortunately, not be converted into confirmed order. As a matter of fact, the completion time and production costs required to fill new order that arrive, when contingent orders exist, cannot be anticipated with any degree of precision. This complicates matters for company that have to prepare quote for new order. Easton and Moodie (1999) is the first paper to explicitly coin the effect of contingent orders on quotation. The paper dealt with this problem in a single resource and single job production environment. The same problem was later expanded into a multiple job environment with different routines for each job by Cakravastia and Takahashi (2003). To prepare quotation for different class of customer under contingency environment EDD and FCFS method of scheduling was utilized in Watanapa and Techanitisawad (2005a). Further, Watanapa and Techanitisawad (2005b) have proposed GA based quotation method where limited number of tardy jobs is allowed. In the same line of research, Corti et al. (2006) has developed analytical tool to compare capacity requirement of order pool, including contingent orders, with respect to actual capacity of shop floor. All these papers have defined contingent orders as potential orders that were awaiting customer confirmation on whether to accept or reject quotation submitted by company. These papers utilized S-shaped logit model to calculate the probability of quotation being accepted by new customer. The model was constructed without utilizing any probabilistic information on contingent orders. Also, as contingent orders are a source of uncertainty in capacity, these papers fail to propose appropriate strategy to hedge against this uncertainty in case contingent order cannot be converted into confirmed order. Quotation represents initial phase in OA decisions (Sujan et al., 2009). OA decisions, in MTO systems, are often the consequence of negotiations between the customer and the company over contested issues (Moodie and Bobrowski, 1999). Sujan et al. (2009) has proposed a method that can simultaneously quote the due date and the price by implementing the concept of negotiation margin. Further, defining contingent order as a customer engaged in negotiation with a company to reach an agreement, Piya et al. (2009b) has proposed a probabilistic method with a negotiation structure to counter the uncertainty created by contingent orders and prepare quotes for a new order. All these mentioned papers with contingent orders have considered generating quotation only for single customer at a time. However, once new customer makes an inquiry, there will be some time lapses before the company prepares quotation and submit it. Within the lapsed time, other new customers may arrive in the systems for an inquiry. In such situation, the company has to prepare quotation for the customer under the influence of uncertainties created by both the contingent orders and the other new orders. To cope with such circumstances, this paper proposes a method that can prepare quotations simultaneously for multiple new customers. The method aims to increase the probability of quotations being accepted by their customers. To the best of author’s knowledge, this is the first paper to consider the research on quotations for multiple customers simultaneously under contingency effect. The method will first tackle the effect of contingent orders and then generate quotations for multiple customers simultaneously. S. Piya / International Journal of Industrial Engineering Computations 6 (2015) 393 The rest of the paper is structured as follows. Section 2 describes the problem that will be addressed in this research. Section 3 explains the proposed method in detail. Section 4 discusses and presents the results of the numerical analysis. Finally, concluding remarks and future research directions are highlighted in section 5. 2. Problem Description This paper reflects the method of quotation as a function of negotiation between the company and the customers of contingent orders. In MTO systems, once customer has arrived with some technical specifications, the company is asked to give quote, which basically will be on the due date and the price. After submitting quotation, the customer and the company may engage in negotiations to reach an agreement, if the quoted value is not acceptable to the customer. Negotiations begin with a customer counter-offering another due date and/or price against the quotation submitted by the company. Such orders represent contingent orders in this paper. The negotiations may continue for several rounds during which the company proposes new offer on due date and/or price and the customer opposes it by counteroffering another due date and/or price within their limit levels. In an attempt to reach an agreement, they move in the opposite direction which will reduce the distance between them on the contested issues. While undergoing negotiation with contingent orders, if a new order arrives, it is difficult for the company to quote the due date and the price for new order. This is because the company cannot be sure at this stage whether agreement can be reached on orders involving ongoing negotiations. Also, while preparing quote for new order, more orders may arrive in the system within small time window for quotations. This will further complicate situation for the company on assigning the due dates and the prices for all the new orders. As it is believed that customer prefers early delivery of an order at a cheaper price, in such situation, the company must try to minimize the average due dates and prices to be quoted to the customers by taking into account the possible outcome of negotiations with the contingent orders. But no quotation should be so ludicrous that they will send the company bankrupt or tarnish its reputation merely for the sake of appeasing customers. This paper assumes above problem in a flow shop environment consisting of various work centers. An order consists of batch of one product type that will enter from the initial work center and exit through the last work center. Each order involves operations at all the work centers. The paper aims at providing company with a capability to manage contingent orders while generating quotes for multiple customers simultaneously, without compromising on profits they expect to achieve. For the purpose, following assumptions are considered. i) There would be at least one offer from the company and one counter-offer from all the contingent orders when the new orders arrive. ii) There are many factors that affect the length of negotiation (Bac, 2001). Before negotiation begins, the negotiating parties may basically fix a deadline for negotiation or the number of negotiation rounds. Based on this, it is assumed that the maximum rounds of negotiation (R) with contingent orders would be defined before starting negotiation. iii) All the orders accumulated within period {tz, (t+1)z} will be processed at period (t+1). 3. Proposed Method The proposed method can be explained by two steps; steps 1 and 2. These steps will be utilized to manage the contingent orders and prepare quotes for the new orders respectively. Step 1: Managing contingent orders To offset the effect of contingent orders on quotation, Piya et al. (2009b) proposed a method to classify these orders into different sets, namely negotiation set and rejection set. Proposed method is based on the statistical data on acceptance probability of orders that arrived in the past. The major shortcoming of this method is the subjective nature of standard deviation. If its value is very high or low, the method will 394 either classify all the contingent orders into negotiation set or rejection set. To overcome above shortcoming, this paper proposes new method that helps select best contingent orders in terms of expected contribution and acceptance probability. As shown in equation (1), the objective here is to find the combination and processing sequence of confirmed and contingent orders such that their expected contribution and acceptance probability lies at the minimum distance from the maximum expected contribution and acceptance probability. 1 (1) s 2 s 2 { min D = ( Z max − Z ) + ( Amax − A ) } 2 To define the maximum expected contribution Zmax, at first, expected contribution Zs for each combination and processing sequence of confirmed and contingent orders is calculated by Eq. (2). The expected contribution consists of selling price, production cost and acceptance probability. Nomenclature Index o: Order (o = 1, 2,……, O); c, k, n, l c: Confirmed order k: Contingent order, n: New order l: Latest accepted order t: Time/ period (t = te, ts, tz, to) te: End of the period t ts: Start of period t tz: Any time within period t j: Issues ∈ due date (d), price (p) r: Rounds of negotiation (r= 1,….., R) i : Operation (i = 1,……., I) f : Unit in the order ( f = 1, 2,….., F ) m: Machine (m=1,…., M) g: Processing sequence of new orders (g=1, 2, ….., G) s: Processing sequence of confirmed and contingent orders (s=1, 2,….,S) Variable D: Distance Zs: Expected contribution by sequence s As: Acceptance probability of sequence s Zmax: Maximum expected contribution limjo: Limit level on issue j of order o Qjo: Quoted value on issue j of order o w cj : Prodo: Total cost of producing order o Expected weight of customer on issue j prodo: Total production time of order o vj : 0 when Yjor= limjo; 1 otherwise (Aor)s: Acceptance probability for r negotiation round of order o with sequence s (Xjor)s: Offer on issue j for r negotiation round of order o with sequence s Yjor: Counter-offer on issue j for r negotiation round of order o hos : Inventory holding cost of order with sequence s ( hos ∈ hoIP ; In process, hoFGI ; Finished goods) ERoif / CToif: Earliest release date/ Completion time for the operation i and unit f of order o 395 S. Piya / International Journal of Industrial Engineering Computations 6 (2015) Parameter qo: Total units in the order o Amax: Maximum acceptance probability RMo: Raw material cost for order o SUoi: Set up cost for i operation of order o capm: Capacity of machine m poi: Processing time for i operation of order o N: Total new orders Poi: Cost of processing operation i of order o α: Smoothing constant wj: Weight assigned to issue j by the company Wl(t):Workload at time t λo: Profit margin coefficient of order o gfirst: Fist order in a sequence g NMjo: Negotiation margin on issue j of order o slast: Last order in a sequence s ajo: agreed value on issue j of order o imax: Operation with maximum processing time Cdjo: Cumulative difference on issue j of order o po: Price of order o δor: Aspiration level of order o at r negotiation round Zs =   I   I   s s  s qo ( X por ) −  ∑ SU oi + qo RM o + qo ∑ poi Poi  + ho  ( Aor )  o∈c , k  i =1      i =1  ∑ (2) i) Selling price (Xpor)s: The selling price of confirmed order, for any sequence s, will be fixed and equal to the agreed price (po). However, the selling price for the contingent order will be equal to the price that can be offered to the customer for the next round of negotiation. Basically, it depends on the strategy of negotiation used by the manufacturer. As proposed by Piya et al. (2009b), selling price will be calculated by considering a linear trade-off relationship between the due date and the price at a fixed aspiration level. Aspiration level here represents a level of benefit sought by the company for a particular round of negotiation (Cakravastia & Nakamura, 2002). pe S lo 2 Price Qpo 1.0 Quotation * (Xjor)s2 Sl . e1 op (Xjor)s1 . Limit level δ oR limdo δ o2 limpo δ o1 ∇ο δ o0 ∇o Qdo Due date Fig. 1. Different new offer on price for different due date at the same aspiration level Let the point (Qdo, Qpo) in Fig. 1 be the point of quotation for contingent order o and let the aspiration level at this point be the maximum aspiration level, δo0, of company. Normalizing this point and the point of limit level (limdo, limpo) within (1,0) and reducing aspiration level with the progress of negotiation by a fixed step size to approach nearer to the counter-offer of customer, aspiration level at any round can be 396 calculate by Eq. (3). (3) (δ − δ ) δ or = δ o( r −1) − o0 oR R As the completion time of contingent order will be different for the different combination and processing sequence s of confirmed and contingent orders, the price Xpor in equation (2) for contingent order depends on Xdor. Suppose, by sequence s1, if the due date of contingent order is (Xdor)s1 in Fig. 1, then (Xpor)s1 will be the price that will be offered if the aspiration level for round r is δo2. Based on this concept, the selling price for contingent order is given by the following equation. ( X por ) s = w {( X dor ) s − lim do }(Q po − lim po )  1  δ or (Q po − lim po ) − d  + lim po wp  (Qdo − lim do )    , ∀o ∈ k (4) From above equation, it can be noted that the price (Xpor)s also depends on the weight assigned by the company (wj) on the given issue. The weight affects the slope of aspiration level (Piya et al., 2010). The details on the method for calculating the limit level (limjo) and quotation (Qjo) on the two issues will be discussed in step 2 of proposed method. ii) Production cost: Production cost consists of set up cost, raw material cost, cost of processing the order and inventory holding cost. Except inventory holding cost, all the other costs are fixed. I hos = ∑ ∑ (ERo(i +1) f F i =1 f =1 ) − CToif hoIP + ∑ {( X dor ) s − CToIf }hoFGI F f =1 (5) As shown in Eq. (5), both in process (IP) and finished good (FGI) inventory holding cost makes up the total cost of carrying inventory. For contingent orders, FGI cost will not be affected by the processing sequence as it is assumed that, during the negotiation process with contingent order, the order will be delivered to the customer as soon as its processing is completed. On the other hand, for confirmed orders both IP cost and FGI cost depends on the order processing sequence s. The earliest release date and completion time in Eq. (5) can be calculated by using iterative method that considers precedence relationship between the operations of order and the shop floor information. iii) Acceptance probability (Aor)s: To calculate the probability of customer accepting a quotation submitted by the company, Easton and Moodie (1999) and Cakravastia and Takahashi (2003) have used an S-shaped logit model. This model was constructed without utilizing any information on contingent order. In the negotiation process, it is possible to obtain information on negotiating issues from the customer of contingent order. Use of such information will increase the authenticity of calculated acceptance probability. Proposed method utilizes information such as counter-offer on due date and price received from the customer, along with the due date and price that can be offered by the company, to calculate the acceptance probability of contingent orders. Therefore, acceptance probability here indicates the probability that the new offer of company will be accepted by the customer of contingent order. −1   r −1      1−     ( X ) s (lim − Y )  R    jor jo    jor   , ∀o ∈ k ( Aor ) s = exp ∑  wcj v j   lim jo (Q jo ) j∈d , p            (6) If the counter-offer, Yjor, on any issue is more than or equal to the limit level, limjo, the binary variable vj in Eq. (6) will be 0, thus resulting in an acceptance probability equal to 1. This is because the limit level S. Piya / International Journal of Industrial Engineering Computations 6 (2015) 397 in this research indicates the level at which the company is willing to accept the order. From the equation note that the acceptance probability will be different for different rounds of negotiation r. It depends on the latest counter-offer of customer, the offer that will be submitted next by the company, the expected weight of customer on different issues and the remaining rounds of negotiation. It should be noted that the acceptance probability for confirmed order will be equal to 1. For the method to calculate expected weight of customer in Eq. (6), refer to Piya et al. (2010). After calculating expected contribution for all the possible combination and sequence of confirmed and contingent orders, the expected contribution that has the highest value will be selected as the maximum expected contribution Zmax in Eq. (1). (7) Z max = max(Z s ) Next, the acceptance probability of sequence s in Eq. (1) is calculated by taking the product of acceptance probability of all the orders that is included in the given sequence. O As = ∏ ( Aor ) s , ∀o ∈ c, k (8) o =1 The maximum acceptance probability (Amax) in equation (1) can be directly assigned the value of 1.0. This is because acceptance probability for the combination of only confirmed orders with any sequence s will be 1.0. As discussed before, the expected contribution and acceptance probability will be calculated for each combination and processing sequence of confirmed and contingent orders. The number of combinations will be equal to 2k and for each combination the possible sequence will be equal to (k+c)!. It means that even for small problem instance, the possible combination and sequence will be very large. To avoid generation of such large combinations, two important constraints are employed. lim do ≤ CToIF ≤ d o , ∀o ∈ c s X dor ≥ max(lim do , CToIF ), (9) ∀o ∈ k (10) Constraint in Eq. (9) indicates that the completion time of confirmed order should be greater than or equal to its limit level on due date and less than or equal to agreed due date. Similarly, constraint in Eq. (10) indicates that the new offer on due date should be more than or equal to the maximum value between its limit level on due date or completion time. Other constraints include the precedence constraint, capacity constraint and constraint on total weight as indicated by Eq. (11), Eq. (12) and Eq. (13), respectively. ERo (i +1) f ≥ CToif , I ∀i, ∀f , ∀o ∈ c, k (11) M ∑ ∑ ( poi ){t z ,(t +1) z } ≤ ∑ (capm )(t s ,te ) (12) ∑ wj =1 (13) o∈c , k i =1 m =1 j∈d , p Step 2: Generating quotes Step 1 helps identify the set of confirmed and contingent orders, with their processing sequence, which gives the maximum expected contribution and acceptance probability. The result of step1 is then used in step 2. From here onwards, sequence s means the sequence obtained for confirmed and contingent orders in step 1 and k represents only those contingent orders that are included in the sequence s. The objective of step 2 is to minimize the average of due dates and prices to be quoted to the multiple new customers. As shown in Eq. (14), the average of quoted values on two issues i.e., the due dates and the prices are 398 integrated by vector normalization method (Van & Nijkamp, 1977). min V = Avg (Q jo ) g ∑ j∈d , p 1 22 , ∀o ∈ n (14) { ( ) } G  ∑ Avg Q jo  g =1 g  O Avg (Q jo ) g = ∑ (Q jo ) g o =1 N (15) ∀o ∈ n , Sequence g in Eqs. (14) means the sequence of new orders. The starting time of sequence g will be equal to the completion time of sequence s. It means that the model considers all the possible sequence g of new orders, at fixed sequence s of confirmed and contingent orders, to find the minimum value of V. As shown in Eq. (16), the quoted value on issue j consists of limit level and negotiation margin on the given issue. It will be different for the different processing sequence g of new orders. (16) (Q jn ) g = (lim jn ) g + ( NM jn ) g i) Limit Level (limjn): Limit level represents the level below which company will not negotiate with customers on either issues i.e., due date and price. (a) Due date: The limit level on due date, limdn, is the date below which it is not possible to complete the processing of an order. It is calculated by subtracting the completion time of new order at given sequence g with the product of probability of rejection and the total production time of contingent orders. (lim dn ) g = (CTniF ) g − [1 − A s ] ∑ prod o (17) o∈k Completion time of new order in Eq. (17) depends on its position in the sequence g. As shown in Eq. (18), if the position of new order lies in the beginning of sequence g, the completion time will be the summation of its total production time and the completion time of first operation of order o that lies at the end of sequence s. Otherwise, it will be the summation of its total production time and the completion time of first operation of other new order whose position lies before order n in the sequence g. g (18) (CT ) first = (CT ) slast + prod , ∀o ∈ c, k niF oif n (19) (CTniF ) g nth = (CTnif ) g nth −1 + prod n The total production time in the Eq. (17), Eq. (18), and Eq. (19) can be calculated based on the operation of order with the maximum processing time. As shown in Eq. (20), the first and second portions at the right hand side of equation are used to calculate the processing time without including service delays. However, the third and fourth portions add on the respective service delays before and after operation with maximum processing time. Service delays incurs if machine on which operation of an order is to be performed is occupied by the operation of any other order. I imax −1 imax +1 i =1 i =1 i =1 prodo = (qo − 1) max poi + ∑ poi + i ∑ (ERo(i +1)1 − ERoi1 − poi ) + ∑ (CToiF − CTo(i −1) F − poi ) (20) b) Price: The limit level on price, limpn, indicates the price at which company wishes to reach an agreement after negotiation. As shown in Eq. (21), it consists of production cost (Prodn), and profit margin coefficient (λn). (21) (lim pn ) g = (Pr od n ) g (1 + λ n ) where, I I i =1 i =1 I F (Pr od n ) g = ∑ SU ni + qn RM n + qn ∑ pni Pni + (hnIP ) g ∑ ∑ ( ERn (i +1) f − CTnif ) + (hnFGI ) g i =1 f =1 F ∑{(Qdn ) g − CTnif } f =1 (22) The production cost of new order includes set up cost, raw material cost, cost of processing the order and the inventory holding costs. Here also, except in process and finished goods inventory holding cost all 399 S. Piya / International Journal of Industrial Engineering Computations 6 (2015) the other costs are fixed and is unaffected by the sequence g. Profit margin coefficient in Eq. (21), λn, adds some profit margin to the production cost. In the proposed research, it is assumed that the profit margin of company will be the function of expected workload in the system and the expected number of competitors. Equations for calculating profit margin coefficient are similar to Piya et al. (2009b) except for the calculation of expected workload. Eq. (23) indicates the summation, within a time frame of (ts, te), of workload of all the confirmed orders, expected workload of contingent orders and the total processing time of new orders to calculate the expected workload. (23) I I   I  Wl (t ) =  ∑ ∑ (CToiF − ERoi1 ) + A s ∑ ∑ (CToiF − ERoi1 ) + ∑  qo ∑ poi  o∈ci =1 o∈ki =1 o∈n i =1  ii) Negotiation margin (NMjn,): The negotiation margin provides company with an allowance to negotiate on the contested issues with the customer. Therefore, if the customer requested to reduce the value on the quoted due date and/or price, the company can do it without risking the chance of order being tardy and without reducing their desired profit margin. The concept of cumulative difference is utilized to calculate negotiation margin. As shown in Eq. (24), the cumulative difference along with the limit level on the given issue at sequence g is used to calculate the negotiation margin for the given issue.  cd jn  g ( NM jn ) g =   (lim jn ) ( 1 − ) cd  jn    Q jl − a jl   + (1 − α )cd cd jn = α  jl  Q jl    (24) (25) Cumulative difference is basically the cumulative information on the difference between the quoted and the agreed values on given issues related to the past orders. In equation (25), smoothing constant (α) represents the value company gives to latest information as compared to past information on agreement. 4. Numerical Analysis The aim of numerical analysis is to show the working mechanism and effectiveness of proposed method. Working mechanism: The analysis is carried out by considering 5 work centers with each center capable of performing specific task. Nine orders are generated with orders A, B and C as confirmed orders; orders D, E and F are contingent orders and orders G, H and I are new orders for which quotations are to be prepared. The agreed due dates and prices of confirmed orders and the information on contingent orders necessary to calculate its acceptance probability are selected randomly as shown in Table 1. Processing time, total units, set up cost, raw material cost, cost of processing the order and inventory holding costs are uniformly distributed within the value as shown in Table 2. The total number of negotiation rounds, current rounds of negotiation for entire contingent orders and the weights of manufacturer and customer for each issue are fixed at 5, 3 and 0.5 respectively. Table 1 Status of confirmed and contingent orders Order status Confirmed order Contingent order Order A B C D E F do 24 60 80 - po 15000 12000 13800 - Ydor 38 34 40 Ypor 13000 10000 12000 Table 2 Value of other parameters considered in the analysis poi qo SUoi RMo (Units) (Time unit) ($) ($) U(2, 5) U(2, 4) U( 1000, 1750) U( 2500, 3200) limdo 45 38 42 Poi ($) U( 100, 125) limpo 18000 12500 14000 Qdo 80 74 88 Qpo 26000 21000 23000 hoIP hoFGI ($) U( 50, 80) ($) U( 80, 120) 400 Step 1: Managing contingent orders The analysis will first show the method of selecting contingent orders and its sequence with confirmed orders such that the objective of step 1 is accomplished. To manage the contingent orders, at first, proposed method elaborates all the possible combinations and sequences of confirmed and contingent orders for the given problem. Satisfying constraints in Eq. (9) and Eq. (10) fourteen different sequences are possible which are as shown in Table 3. Next, the due date and the price that can be submitted as a new offer in the next round i.e., (Xdor, Xpor), is calculated for each contingent orders available in the given sequence. For example, for order D in the sequence A-B-C-D, the completion time will be equal to 51. The price for this completion time can be calculated by using Eq. (4). X por = 0.5(51 − 45)(26000 − 18000)  1  0.6(26000 − 18000) −  + 18000 = $23,028 0.5  (80 − 45)  Aspiration level for round 3 in the above equation i.e., 0.6 is obtained by using Eq. (3). The acceptance probability of this offer is then calculated by Eq. (6). −1 Aor  3 −1   3 −1      1−   1−  5  5        − − 51(45 38) 23028(18000 13000)     =0.725% + 0.5 1.0 exp 0.5 1.0    45(80)  18000(26000           Next, identifying fixed processing cost and inventory holding cost based on the given sequence, the expected contribution of an order is computed by Eq. (2). Then, summing the expected contribution of all the orders in the sequence will gives value of Zs as shown in Table 3. From the table, it is seen that the maximum expected contribution, Zmax, is obtained for the sequence A-B-C-E-D-F. The acceptance probability As in Table 3 is calculated by taking the product of acceptance probability of all the orders in the sequence. For the example A-B-C-D, the acceptance probability of this sequence is calculated as shown below. As= (1x1x1x0.725) =0.725%. As the unit of Zs and As is different, the value of Zs is first normalized by equating Zmax with 1.0 and minimum Zs with 0. Then, by considering normalized value of Zmax, Zs, Amax and As, the distance D is calculated by Eq. (1). From Table 3 it is evident that the minimum distance is obtained for the sequence A-B-C-F-D. Table 3 Results obtained from step 1 Sequence A-B-C A-B-C-D A-B-C-D-E A-B-C-D-E-F A-B-C-E A-B-C-E-D A-B-C-E-D-F A-B-C-E-F-D A-B-C-F A-B-C-F-E A-B-C-F-E-D A-B-C-D-F A-B-C-F-D A-B-C-E-F Zs ($) 115769 146620 169094 206573 121167 172344 209808 209490 146932 170568 207669 190005 190217 172223 Step 2: Generating quotes Zmax 209808 As (%) 1.0 0.725 0.554 0.466 0.756 0.545 0.46 0.46 0.85 0.65 0.46 0.61 0.63 0.64 Amax 1.0 Norm(Zs) 0 0.328 0.567 0.965 0.157 0.601 1.000 0.996 0.331 0.582 0.977 0.789 0.792 0.600 Distance D 0.762 0.725 0.621 0.535 0.973 0.604 0.540 0.541 0.685 0.544 0.540 0.443 0.424 0.537