Optimal Offer Proposal Policy in an Integrated Supply Chain

Ind. Eng. Chem. Res. 2005, 44, 7405-7419

7405

PROCESS DESIGN AND CONTROL Optimal Offer Proposal Policy in an Integrated Supply Chain Management Environment Gonzalo Guille´ n, Cristina Pina, Antonio Espun ˜ a, and Luis Puigjaner* Department of Chemical Engineering, Universitat Polite` cnica, de Catalunya, ETSEIB, Avda. Diagonal, 647, E-08028 Barcelona, Spain

In this work, we present a novel approach that provides decision support in making optimal offer proposals during the negotiation process between customers and suppliers that takes place in chemical industry supply chains (SC). The proposed approach takes into account the tradeoff between the quality of the offers made to customers, i.e., the level of satisfaction perceived by the client, and the expected profit to be achieved in the short-term operation of the SC. Therefore, a two-stage stochastic formula is derived that considers the uncertainty associated with reactions to future demand in an attempt to compute a set of Pareto-optimal solutions to the proposed problem. Each of these solutions comprises an SC schedule and a set of values for the parameters of the offers. Through comparison of the Pareto curve and the solution that would be obtained without negotiation, a set of offers representing contracts that are desirable from the supplier’s perspective is obtained. This set of values may be offered by the supplier to reach an agreement with the customer during the negotiation procedure. The proposed approach facilitates a rational negotiation, in the sense that it enables the negotiator to simultaneously process many more data related to production and transport plans and customer preferences, thus avoiding having to rely exclusively on the negotiator’s beliefs and interests. 1. Introduction The concept of the supply chain (SC), which first appeared in the early 1990s, has recently been the focus of much interest, as the possibility of providing an integrated management of an SC can reduce the propagation of unexpected and/or undesirable events throughout the network and can markedly improve the profitability of all the parties involved. A supply chain (SC) can be defined as a network of business entities (i.e., suppliers, manufacturers, distributors, and retailers) who work together in an effort to acquire raw materials, transform these raw materials into intermediate and finished products, and distribute these final products to retailers. Different issues have forced businesses to invest in, and focus attention on, their SCs: the fierce competition in today’s global markets, the variability in demand, the introduction of products with short life cycles, the rapid development of new products, the wide variety of supply alternatives for similar products and services, the heightened expectations of customers, etc. These issues, together with continuing advances in communication technologies, have caused SCs and the techniques used to manage them to grow continuously more complex. At present, SCs are highly dynamic environments.1 Supply chain management (SCM) aims to integrate plants with their suppliers and customers so that they * To whom correspondence should be addressed. E-mail: [email protected]. Telephone: +34 934 016 678. Fax: +34 934 010 979.

can be managed as a single entity and to coordinate all input/output flows (of materials, information, and funds) so that products are produced and distributed in the right quantities, to the right locations, and at the right time.2 Therefore, SCM implies the handling of flows throughout the entire SC, from suppliers to customers, while encompassing warehouses and distribution centers (DCs), and usually including after-sales services, returns, and recycling.3 The main objective is to achieve acceptable financial returns together with the desired consumer satisfaction levels. The SCM problem may be considered at different levels depending on the strategic, tactical, and operational variables involved in decision making.4 Therefore, a large spectrum of a firm’s strategic, tactical, and operational activities are encompassed by SCM: The strategic level concerns those decisions that will have a long-lasting effect on the firm. It is focused on SC design, and entails determining the optimal configuration for an entire SC network, including the design of the embedded plants. The tactical level encompasses long- to medium-term management decisions, which are typically updated at a rate ranging between once every quarter and once every year. These include overall purchasing and production decisions, inventory policies, and transport strategies. The operational level refers to day-to-day decisions such as scheduling, lead-time quotations, routing, and lorry loading. Most of the works reported in the literature address SCM problems from a strategic or tactical point of

10.1021/ie0493208 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/11/2005

7406

Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005

view.1,5-8 They identify the distribution of production facilities or DCs, the flow of materials, and the inventory levels that optimize a certain measure of performance. From an operational perspective, and because of the complexity associated with the interdependencies between the production and distribution tasks in the network, the act of scheduling in detail the processes involved in these SCs has been left as a local decision. Along these lines, Reklaitis et al.1 highlight the importance of coordinating the activities of the different entities involved, each of which has its own performance function. Special emphasis is placed on the enterprise level, which requires aspects concerning logistics and manufacturing to be integrated with strategic business and financial decisions. The complexity of an SC is increased by the high degree of uncertainty brought about by external factors, such as continuously changing market conditions and customer expectations, and internal parameters, such as product yields, qualities, and processing times. Models addressing uncertainty have been developed mainly in the area of design and planning.9-15 Recently, Gupta et al.14 proposed a tactical planning model based on a stochastic programming approach for incorporating demand uncertainty into the decision-making process. Negotiation in an SC Environment. In a competitive environment, products constantly become more homogeneous and customers are better informed about prices and characteristics, and a firm’s competitive advantage does not merely lie in reducing production costs but also in gaining customer loyalty. Therefore, procurement in SCM has evolved from traditionally sporadic transactions into a more complex activity that adds value to the SC. Giunipero and Brand16 developed the following framework, which describes the evolution, in four stages, of the role played by procurement or purchasing in SCs: traditional, emphasizing supplier selection and the lowest possible price; relational, building closer relations with suppliers to reduce total cost and to minimize risk through an atmosphere of trust; operational, coordinating material and information flows to improve quality; and strategic: applying flexible business processes to a given situation and thereby achieving speed, flexibility, and a competitive advantage in the marketplace. This work is focused on the relational level. At this stage, both parties express their wish to establish longterm relationships to reduce uncertainty in their SCs. Long-term relationships enhance collaboration and fluent communication, and generally speaking reduce financial risks for all parties involved, thus creating value.17,18 However, there is not necessarily a perfect sharing of information between the enterprises. We consider a situation in which customer and supplier are self-interested but do not want to divulge full information regarding their purposes and strategies, and therefore, their interests may be opposed. Thus, it is necessary for both parties to make a tradeoff. Achieving any higher level of collaboration involves somehow integrating the supplier in the SC. In a perfect market, the prices and quantities of the goods and services that are bought and sold are determined by market forces. A perfect market (or perfect competition) exists when a product is homogeneous in nature, there are large numbers of buyers and sellers, sellers are free to enter and exit the market, and buyers

and sellers have perfect information/foresight with respect to prices. When one or more of these conditions is absent, the market will be imperfect to some degree. Let us consider, say, that a supplier is able to differentiate his product from others in the market: this means that his product is not homogeneous, and the supplier in this case increases his influence over price and to some extent causes a monopolistic competition to arise. That is to say, although his product can be substituted with others’ products, the supplier still has the power to negotiate, as his product possesses certain characteristics that make it different. Consequently, it is assumed that an SC of this type will have two types of customers: a first group, corresponding to the traditional procurement stage, and a second group, corresponding to the relational stage. The first group places orders without any previous agreements regarding delivery time, prices, or quantities. In practice, this means that the supplier has no contractual obligations to these customers, as their relationships are principally based on sporadic commercial transactions. Future demand and the future prices of products are in this case gauged by market research results. Such orders represent the main source of uncertainty in the problem. In industrial environments, forecasting tools are commonly used to provide forecasts and the associated confidence intervals for these sorts of uncertain orders. The main purpose of such techniques is therefore to generate the input data for planning models, which are used to determine beforehand which production and distribution tasks are to be carried out throughout the entire network. However, there is another group of customers that may be interested in reaching an agreement with the suppliers to establish a collaborative relationship. From the point of view of the customer, the objectives are to improve quality and reliability and to reduce operational costs.18 The supplier also has the contractual obligation to reserve part of his capacity to produce and store the products upon which they agreed, and the customer then has a guarantee that he will receive the products he needs. Moreover, he will receive them under the conditions that have previously been negotiated. Furthermore, suppliers also stand to improve their situation, as a contract of sale reduces uncertainty in such events as designing a production plan or estimating transport costs. In addition, if demand decreases considerably, contracts become an important source of income. In this case, both parties, customers and suppliers, are interested in reaching an agreement, as both of them see the possibility of improving the conditions under which they buy and sell. Designing an efficient policy for negotiating contracts with a business partner turns out to be a difficult task. To properly evaluate the different contractual alternatives, one must study the impact had by the offers made to customers in the day-to-day operation of the SC. From the supplier’s point of view, which is the focus of this paper, the decision maker must assess the feasibility of the plan that will enable the production and distribution of the amount of materials stipulated in the offers. Contracts lead to hard constraints (due dates) being imposed on the production and distribution activities carried out at different points on the SC, and for this reason, the SC manager must carefully evaluate the conditions involved in each of them. An acceptance of

Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7407

stringent conditions when closing contracts may lead to extra expenses due to unfeasible production and distribution plans, which may force the supplier to adopt drastic solutions (i.e., outsourcing) to meet the contractual commitment (i.e., due dates). Second, once the feasibility of the plan has been demonstrated, the supplier must evaluate the tradeoff between the expected profit and the quality of the offer made to a customer, i.e., the client’s perceived satisfaction with the offer. On one hand, offers that better satisfy customers’ requirements are desirable, as they may lead to greater earnings in the midterm as a consequence of good relationships with clients. Moreover, such offers will exhibit a higher probability of acceptance by customers. On the other hand, contracts with a higher satisfaction of demand usually result in smaller benefits, as they impose more stringent constraints on the production and distribution tasks involved in the SC. In this work, a novel strategy for evaluating offer proposals in production and distribution networks with embedded multiproduct batch plants is presented. The proposed approach represents a preliminary step that should be applied before the negotiation process starts, and suggests a set of values for delivery time and price that the supplier can offer when negotiating with a customer. Moreover, for each of these offer parameters, the SC schedule that optimizes the expected profit of the SC is also provided. Therefore, the solutions provided by our strategy weigh the expected profit and the quality of the offer, i.e., the customer satisfaction that the offer will provide the client. This is a tool for guiding negotiations to achieve a result that is satisfactory for both parties, by considering numerous aspects of the operational performance of an SC together with the customer’s preferences. After this first step, in which the proposed tool should be applied, a bargaining process can then be started, though of course a successful agreement cannot be one hundred percent guaranteed. This agreement may depend on how aggressive or willing to concede each party is in its negotiation strategy, but a probability of reaching an agreement will always exist if each party’s negotiation intervals (the difference between the minimum and maximum values that variables can take) overlap.19 To address these sorts of operational problems, which are commonly found in SC environments, a continuoustime formula for the short-term scheduling of chemical SCs with embedded batch plants that includes production, transport, and storage activities is introduced. Uncertainty associated with future demand and prices, which is represented by a set of scenarios with a given probability of occurrence, is also contemplated, and a two-stage stochastic formula is presented to address this issue. The problem is formulated mathematically as a mixed-integer linear programming (MILP) optimization model and is solved using branch-and-bound techniques. Therefore, the Pareto curve obtained by solving this multiobjective two-stage stochastic problem includes solutions that weigh expected profit and customer satisfaction. The decision maker may use this curve to make offers that are within the desirable range of values of the customer. In addition, as the supplier knows how much every offer contributes to the expected profit, he can construct his own preference scale using precise information regarding production and distribution plans and thus discarding those offers which do not improve

Figure 1. Supply chain structure.

the SC performance in terms of an increase in the expected profit or a decrease in risk. 2. Problem Statement According to the approach previously outlined, the proposed model helps to determine the best schedule for an existing SC along with the optimal parameters of the offers to be sent to customers, taking into account the decision maker’s preferences. This work considers a two-echelon SC, similar to those associated with smallbatch chemical industries. However, the model described in this paper could be easily adapted to more complex structures. The structure of this SC is depicted in Figure 1. It includes a set of multiproduct batch plants at which products are manufactured and stored prior to being sent to the DCs, which behave as final markets where products are available to customers. The overall problem can be formally stated as follows: Given Each cost parameter, such as manufacturing (νip), transport (µijp and λij), and inventory costs (θp) Manufacturing and transport data, such as batch sizes of products (BSip), processing times of products in stages (TOPipt and TTij), and capacity factors (Rip, CAPWi, βp, and CAPTli) Inventory data, such as initial inventory of products in warehouses and DCs (MWie0p and MDCjf0ps, respectively) and capacity factors (γjp and CAPDCj) Demand data, such as arrival times, amounts and prices of products included in future occasional demands, and their probability of occurrence (TDjdj, Demjdjps, PDjdjps, and Ps, respectively). Offers data. In this work we assume that the offer to be sent to the customer implies the fulfilment of a set of orders placed at the DCs of the SC. Therefore, the parameters of the offer encompass the earliest and the latest delivery times for the orders of the offer (TOmaxjoj and TOminjoj, respectively), the lowest and highest prices of the products involved in them (POminjojp and POmaxjojp, respectively), and the weights of the functions of value (ηPjoj and ηTjoj). Find Offer characteristics: delivery times and prices of the orders of the offer (POjojp and TOjoj, respectively) Number and type of batches to be produced (Xipbi) and initial and finishing times of all the operations involved in the production of the batches (TIibit and TFibit, respectively). Number and capacities of the transport batches to be sent from plants to DCs (TLIili, TLFili, and MLilip) Which maximizes the expected profit and customer satisfaction. To solve this operational problem, the following assumptions were made:

7408


The uncertainty associated with product demands and prices is represented by a set of scenarios with a given probability of occurrence, which must be provided as input data. In the event that the uncertain parameters follow a certain probability distribution, the scenarios may be generated by performing a Monte Carlo sampling. One production line with assigned equipment units and fixed batch sizes is considered for each product. This paper does not aim to discuss the different equipmenttask allocation possibilities that can be found at a batch plant. Nevertheless, this assumption can be easily relaxed. A zero-wait (ZW) transfer policy is adopted. Under this policy, an intermediate product must be immediately transferred to the next processing step after it is produced. Neither intermediate storage nor waiting times are available in the processing units. Once again, this paper does not aim to discuss the consequences of the different transfer policies that can be found at a general batch plant. This assumption could easily be modified so that other transfer policies can be considered. Each solution of the aforementioned problem exhibits an E[Profit] for a given CSat. This value of E[Profit] may change depending on the demand satisfaction target imposed by the supplier. Usually, solutions with higher E[Profit] values will perform better for lower CSat values; therefore, E[Profit] and CSat tend to be contradictory objectives, and the solution of the problem described above is a set of Pareto-optimal solutions. These solutions are obtained in this work by applying the -constraint method. This method is based on the maximization of one objective function, and by considering the other objectives as constraints bounded by some allowable levels o. Thus, the levels o may be altered to generate the entire Pareto-optimal set. The result of the model provides a set of Pareto solutions to be used by the decision maker in the day-to-day operation of the network. The model is described in the following section. 3. Multiobjective Stochastic Model A stochastic programming approach based on a recourse model with two stages is proposed in this work for incorporating the uncertainties associated with demand (amount of materials requested in future orders) and prices (prices of the products demanded in the future) in the scheduling process. In a two-stage stochastic optimization approach, the uncertain parameters in the model are considered to be random variables with an associated probability distribution, and the decision variables are classified into two stages. The first-stage variables correspond to those decisions that need to be made here and now, before the uncertainty materializes. The second-stage, or recourse, variables correspond to decisions made after the uncertainty is unveiled and are usually termed waitand-see decisions. After the first-stage decisions are made and the random events realized, the second-stage decisions are made in keeping with the restrictions imposed by the second-stage problem. Because of the stochastic nature of the performance associated with the second-stage decisions, the objective function consists of the sum of the first-stage performance measure and the expected second-stage performance (for an overview of stochastic techniques, refer to Birge and Louveaux20).

An MILP formula is derived on the basis of a continuous-time representation.21-23 Using this formula, the time horizon is viewed as a sequence of time points or events, each of which will be assigned to one particular activity. Therefore, a set of specific events is defined for each node, taking into account the production and distribution activities that take place inside it. An event can be the end or start of a task, the departure or arrival of a transport batch, or the delivery of material (either to satisfy contract requirements previously arranged with customers or to fulfill sporadic demands). In general, the number of event points to be set at each site can be either computed from the input data or fixed by the decision maker. Decision variables related to the number of batches of each product to be produced and the detailed production-distribution schedule, as well as the parameters of the offers to be sent to customers (Xipbi, TIibit, TFibit, MLilip, TLIili, TLFili, POjojp, and TOjoj), are regarded as first-stage decisions, as it is assumed that they have to be made at the scheduling stage before the demand uncertainty is unveiled. The overall SC schedule includes the detailed schedules of the multiproduct batch plants (the sequence of batches to be manufactured and the timing of the tasks involved in their production) as well as the starting and finishing times and loads of the transport batches. On the other hand, the sales and the inventory profiles in the DCs (QDjdjps and MDCjfjps, respectively) are regarded as second-stage variables. At the end of the scheduling horizon, different profit values are obtained for each particular demand uncertainty scenario. Customer satisfaction (CSat) does not depend on the scenario, as only first-stage variables are involved in determining it. The proposed model accounts for the maximization of the expected value of this profit distribution, and customer satisfaction. The mathematical formula includes three major sets of equations which are described in detail in the following section. 3.1. Production Scheduling Constraints. The scheduling constraints are derived on the basis of the continuous time representation. Using this formula, the time horizon is viewed as a sequence of Bi batches manufactured at plant i that can be assigned to only one particular product p. The maximum number of batches to be produced can either be estimated on the basis of capacity limitations or can be given by the decision maker. Sequence decisions are linked to a binary variable Xipbi, which represents the existence of a batch bi of product p manufactured at plant i and takes the value of 1, if bi belongs to p and 0 otherwise.

∑p Xipb e 1 ∀ i, p, bi

(1)

∑p Xipb e ∑p Xipb +1 ∀ i, bi < Bi

(2)

i

i

i

Constraint 1 states that each batch bi manufactured at plant i can belong to one sole product at most, while eq 2 enforces the condition by which the nonproduced batches must be located at the beginning of the schedule. Although constraint 2 is not necessary, it helps in making computations. Likewise, by fixing the position of the nonproduced batches, we obtain smaller branchand-bound trees and shorter computation times. The specific position of these nonproduced batches (at the


beginning of the schedule) is absolutely arbitrary, and is chosen for the sake of simplicity. To reduce the complexity of the formula, only the case of a ZW policy is considered here. Neither intermediate storage nor waiting times are available in the processing units. Under ZW policy, there can be no delay between the time a product batch bi finishes being processed on stage t at plant i (TFibit) and the time it begins to be processed on stage t + 1 (TIibit+1), as is stated by constraint 3:

TIibit+1 ) TFibit ∀ i, bi, t < T

(3)

of materials in these warehouses: the end of a production batch (output of material) and the departure (output of materials) or arrival (input of materials) of a transport batch. Two binary variables, Winibiei and Woutiliei, are defined to allocate activities to events. Such variables take a value of 1 if the activity being represented (the end of a production batch bi or the departure of a transport batch li) occupies event ei at plant i, and 0 otherwise.

∑b Winib e + ∑l Woutil e ) 1 ∀ i, ei

(9)

Winib e ) 1 ∀ i, bi ∑ e

(10)

Woutil e ) 1 ∀ i, li ∑ e

(11)

i i

The finishing time of stage t (TFibit) involved in the manufacture of batch bi at plant i is computed from the initial time of bi in t (TIibit) and the duration of stage t, which is given by the result of the product to which the batch belongs (TOPipt), as expressed by constraint 4.

TFibit ) TIibit +

∑p TOPiptXipb ∀ i, bi, t i

(4)

The ZW assumption that results in the aforementioned constraints (eqs 3 and 4) can easily be modified so that other transfer policies, such as unlimited intermediate storage and nonintermediate storage transfer policies (UIS and NIS, respectively), can be considered. For the sake of simplicity, it has also been assumed that only one production line with one assigned equipment is available per stage. Constraint 5 reflects the aforementioned policy by forcing stage t, involved in the manufacture of batch b′i at plant i, to start after the end of the same stage t performed in any previous batch bi. This policy could easily be modified to reflect other possibilities for equipment-task allocations.

TIib′it g TFibit ∀ i, b′i > bi, t

(5)

The initial and finishing times of any stage t of bi are constrained to be lower than the time horizon H provided that bi is produced at plant i (eqs 6 and 7). Although these constraints are not necessary, they help in making computations.

TIibit e H

∑p Xipb ∀ i, bi, t

(6)

∑p Xipb ∀ i, bi, t

(7)

i

TFibit e H

i

i i

i

i i

i

Therefore, eq 9 ensures that in each event an entrance or an exit of material must take place, while constraints 10 and 11 force all the inputs and outputs of materials to be represented by one event.

TWiei )

Winib e TSIib ∑ b i i

i

+ i

∑l Woutil e ∀ i, ei i i

(12)

i

TWiei e TWiei+1 ∀ i, ei < Ei

(13)

TWiei e H ∀ i, ei

(14)

Moreover, the time at which the event takes place is given by the time when the activity being represented occurs, as stated by constraint 12. In this equation, the time of the event (TWiei) takes the value of the initial storage time of batch bi at plant i (TSIibi) if the binary variable Winibiei is equal to 1. If Winibiei is equal to 0, it follows from constraint 9 that Woutiliei must take a value of 1, and therefore TWiei takes the value of the departure time of transport batch li (TLIili). Equation 13 is used to sort all the events by time, i.e., in order of occurrence. Equation 14 forces all the time points to be located within the time horizon.

MWiei-1p +

∑b Winib e Xipb BSpi - ∑l iWoutil e MLil p ) i i

i

i i

i

i

Finally, once a batch has been produced, it is stored in the plant’s final product warehouse. Therefore, the initial storage time of a batch bi at plant i (TSIibi) is equal to the finishing time of the last production stage of the batch (TFibit):

TSIibi ) TFibit ∀ i, bi, t ) T

i i

i

(8)

This constraint represents the link between production and inventory management tasks. 3.2. Final Product Warehouse Constraints. Material profiles in the warehouses are determined from the event-based time representation. The events required for the warehouse are derived from all the activities that result in input or output of materials. In the case being studied in this paper, there are two types of activities that can change the profile

MWieip ∀ i, ei, p (15) MWieip g 0 ∀ i, ei, p

(16)

The mass balance in the warehouses is enforced via constraint 15. That is, the initial amount of product p (MWiei -1p), plus the amount produced and minus the amount transported, must equal the hold-up. Here, the amount produced is computed using binary variables Winibiei and Xipbi. The first variable is used to allocate the final manufacturing time of batch bi at plant i to the time point ei during which this event takes place, while the second one is applied to add the batch size (BSpi) of the product p to the batch bi manufactured at plant i to which it belongs. Note that to compute the amount of p transported, one must use binary variable Woutiliei, which allocates the departure time of the transport batch to the corresponding event ei. Here, the continuous variable MLilip represents the amount of p in transport batch li. Equation 11 forces the variable

7410


MWieip to be greater than zero for all the event points.

∑p MWie pRip e CAPW ∀ i, ei i

i

(17)

Finally, constraint 12 imposes a capacity limit (CAPWi) on the amount of materials stored in the final product warehouse at plant i. In this equation, a capacity factor Rip has been applied to take into account the physical properties of the different materials. 3.3. Transport Constraints. This study supposes that the transport of products between nodes is performed in a discontinuous way; i.e., the transport batches sent from plants to DCs are considered. Each transport batch li is characterized by its departure time from plant i (TLIili), its arrival time to the DC (TLFili), and the amount of product p it transports (MLilip). A binary variable, Yilij, is defined, and takes a value of 1 if the transport batch li is sent from plant i to warehouse j and 0 otherwise. This variable comes into play in the following constraints, which enable the transport variables to be calculated:

∑p MLil pβp e CAPTl ∑j Yil j ∀ i, li i

i

i

i

i

(19)

All those transport batches that are not sent are located at the beginning of the sequence, as expressed by constraint 19. Constraint 19 is similar to eq 2, and it is added to help in making computations.

TLFili ) TLIili +

∑j TTijYil j ∀ i, li i

(20)

The arrival time of transport batch li sent from plant i to DC j (TLFili) is equal to its departure time (TLIili) plus the duration of the journey between nodes (TTij), as stated by eq 15

TLIili e H

DCinjil f ∑ i,l

+

i ij

i

∑o jDCout1jo f + ∑d DCout2jd f ) j j

j j

1 ∀ j, fj (23)

∑f DCinjil f ) 1 ∀ j, i, li

(24)

∑f DCout1jof ) 1 ∀ j, oj

(25)

∑f DCout2jdf ) 1 ∀ j, dj

(26)

i j

j

i

j

j

(18)

First, the total amount of product p sent from plant i to DCs j within the time horizon is constrained by the capacity of the transport batch (CAPTli), as expressed by constraint 18. Here, βp is a capacity factor that takes into account the physical properties of the products transported between nodes.

∑j Yil j e ∑j Yil +1j ∀ i, li < Li

Again, three binary variables are defined, DCinjilifj, DCout1jofj, and DCout2jdijfj, to allocate activities to events. DCinjilifj takes a value of 1 if the arrival of transport batch li sent from plant i to DC j takes place in event fj, and 0 otherwise. DCout1jofj is used to allocate the delivery time of the materials in order oj (of the offer sent to the customer) to the event fj, and is equal to 1 if the delivery time of the order oj takes place in fj, and 0 otherwise. Finally, DCout2jdjfj takes a value of 1 if sporadic demand dj occurs in event fj, and 0 otherwise.

j

Thus, eq 23 ensures that in each event an entrance or an exit of material must take place; i.e., at least one binary variable will represent either the arrival of a transport batch or the delivery of contracted materials, or the fulfilment of a sporadic demand must be equal to 1. Moreover, constraints 24-26 force all the activities involving inputs or outputs of materials to be represented by one event.

TDCjfi )

DCinjil f TLFil - ∑jDCout1jo TOjo ∑ i,l o i j

i

i

f

j

j

DCout2jd f TDjd ∀ j, fj ∑ d j j

j

(27)

j

The time when an event takes place is given by the time when the activity it represents occurs, as is stated by constraint 27.

TDCjfj e TDCj,fj+1 ∀ j, fj < Fj

(28)

TDCjfj e H ∀ j, fj

(29)

∑j Yil j ∀ i, li

(21)

All the events are sorted by time and must occur within the time horizon as expressed by constraints 28 and 29, respectively.

∑j Yil j ∀ i, li

(22)

MDCjfjps ) MDCj,fj-1ps +

TLFili e H

i

i

If a transport batch is not sent, its initial and finishing times are equal to 0, as stated in eqs 16 and 17, which are analogous to constraints 6 and 7. 3.4. Distribution Center Constraints. The profiles of the materials in a DC are determined in a manner similar to that used for those of the warehouses. Here, the equations that enable this calculation include variables that are scenario-dependent, which leads to different profiles according to whichever scenario is being considered. Two types of activities are considered in the DCs: the arrival of a transport batch (input of materials) and the delivery of materials either to fulfill sporadic demands or to satisfy contractual requirements (output of materials).

DCinil f MLil p ∑ i,l i j

i

i

DCout2jd f QDjd ps ∑ d j j

j

j

DCout1jo f Zjo QOjo p ∀ j, fj, p ∑ o j j

j

j

(30)

j

MDCjfjps g 0 ∀ j, fj, p, s

(31)

Equation 30 represents the mass balance in the DCs. The amount of p in DC j at event point fj in scenario s (MDCjfjps) must equal the initial amount of p in j (MDCj,fj-1,p,s) plus the amount arriving from plant i through transport batch li (MLilip) minus the amount of p fulfilling sporadic demands dj (QDjdjps) minus the amount of p delivered to satisfy the requirements


imposed by order oj of the offer (QOjojp). Note that here we use the three previously defined binary variables for allocating activities to event points. Moreover, with regard to the orders in the offer, a new binary variable Zjoj is introduced, which takes a value of 1 if the order oj of the offer is fulfilled at DC j, and 0 otherwise. If the order is not fulfilled, Zjoj takes a value of 0 and the associated term in the mass balance equation is equal to 0. On the other hand, if the order is accepted, Zjoj takes a value of 1 and the DC is forced to have enough products in stock to fulfill the requirements imposed by the offer at the time by which the materials in the order must be delivered. Note that constraint 30 must be satisfied in all the scenarios; that is, if an offer is made, the materials involved in the orders accepted must be delivered, independent of whatever sporadic demand eventually materializes. Equation 20 forces variable MWieip to be greater than zero at all the event points.

TOjoj e ZjojTOmaxjoj ∀ j, oj

(32)

POjoj e ZjojPOmaxjoj ∀ j, oj

(33)

The parameters of the orders of the offer (time and price of the orders) sent to customers are also constrained to be lower than upper bounds (TOmaxjoj and POmaxjoj), as is stated by constraints 32 and 33. Such bounds must be provided as input data and will be described in more detail in the following section. With regard to future orders that may arrive at the DCs, it should be pointed out that these may be the result of market research, which should provide a set of possible scenarios, each of which comprises a given demand and price for a product. On the basis of constraint 30, the model considers DCs to be able to cover future orders insofar as they can; that is to say, they will be able to deliver all the product they have in stock, even if that amount does not meet the total demand.

∑p MDCjf psγjp e CAPDC ∀ j, fj j

j

(34)

Finally, the amount of materials stored in a DC at each event must be lower than its capacity (CAPDCj), as stated by eq 34, where the capacity factor γjp was applied to take into account the physical properties of the different materials. 3.5. Objective Function. This model must attain two targets: maximizing expected profit E[Profit] and maximizing customer satisfaction CSat, which in turn may lead to offers with a higher probability of acceptance and may also increase future sales. Hence, there are two objective functions:

Max E[Profit] )

∑s Ps × Profits

Max CSat

(35) (36)

It should be mentioned that since CSat is evaluated considering only the orders, the second objective does not depend on the scenarios, because it is directly determined once the first-stage variables are fixed and without evaluating second-stage decisions. On the other hand, the expected profit depends on the second-stage variables that are computed for each scenario once the uncertainty that is sporadic orders is unveiled. There-

fore, this first term is calculated by determining the average of the profits obtained for all the scenarios in the study (Profits). 3.5.1. Expected Profit. This term must be computed according to the applicable rules (depreciation), legislation (taxes), etc., so this may lead to different formulas. In this paper, the profit associated with each possible plan is calculated for each scenario s as the difference between revenue (Revs) and the transport, production, and storage costs (TC, PC, and SCs, respectively), the aim of which is to reflect a general case, as stated by eq 26:

Profits ) Revs - TC - PC - SCs

(37)

Revenues. This term represents the inputs of cash, which are due to either fulfilments of sporadic demands (QDjdjps) or sales of products previously arranged contractually with customers (QOjojp), as stated by constraint 38. The price of the sporadic orders (PDjdjps) is an uncertain parameter, which depends on the scenario being considered. On the other hand, the price of the materials involved in the orders of the offer (POjojp) is a decision variable computed by the model to optimize the expected benefit and customer satisfaction.

Revs )

∑ PDjd psQDjd ps + j,o∑,pPOjo pZjo QOjo p ∀ s j

j,dj,p

j

j

j

j

j

(38)

Cost. We consider three different costs, namely, transport, production, and storage costs. It is assumed that transport costs are a linear function of the amount of materials transported between nodes. Therefore, computing them involves both a fixed term (λij) and a variable term (µijp), as expressed by constraint 39. i,li,j

TC )

∑i Yil j(λij + ∑p MLil pµijp) i

i

(39)

Production costs, which are caused by the tasks carried out at the plants, are directly proportional to the amount of materials manufactured at the different sites, as expressed by eq 40, in which νip represents the labor cost associated with product p at plant i.

PC )

∑ Xib pBSpiνip

i,bi,p

i

(40)

Note that transport and production costs depend only on first-stage variables, since they are directly determined from the SC schedule, which is regarded as having been determined before the uncertainty materializes. Storage costs include both the cost associated with keeping the inventory at every distribution node and the cost of having money invested in inventory instead of elsewhere. An average inventory of product p is computed within the time horizon H (Invps) in scenario s; considering the initial inventory of product p kept at the warehouses and DCs at time zero (MWie0ip and MDCjf0jps), the production rate and the sales of materials are as stated by eq 41.

7412


∑i MWie0 pH + ∑j MDCjf0 psH + XibipBSpi(H - TSIib ) - ∑ QDjd ps(H - TDjd ) ∑ i,b j,d

Invps )

i

j

i

i

j

j

j

QOjo p(H - TOjo )Zjo ∀ p, s ∑ j,o j

j

j

43). Any other type of function could be added to this model, but it would probably add complexity to the solution without yielding much more accuracy, since the scoring system is designed to be a simple method for systematizing preferences and evaluating offers.24

(41)

j

Finally, eq 42 computes the storage cost by penalizing (θp) the stocks kept at the nodes of the whole SC within the time horizon.

SCs )

∑p Invpsθp ∀ s

(42)

It should be mentioned that, unlike transport and production costs, storage costs are scenario-dependent, since second-stage variables are involved in their computation. 3.5.2. Customer Satisfaction. The function that enables customer satisfaction to be determined was developed bearing in mind the negotiation procedure carried out by suppliers and customers to reach an agreement that satisfies their aspirations. Let us consider the main issues discussed in reaching a buying and selling agreement. Quantity. This factor will not be negotiated. The quantity will be the one required by the customer. Time of Delivery. This factor can be negotiated. In this paper, it is supposed that there is an interval of time in which the customer would like to receive the merchandise. The customer will accept any of the days within this interval of time for delivery, although not all of them are equally satisfactory. Price. The price can also be negotiated. The supplier and the customer give their starting prices, which are their preferred prices for selling and buying, respectively. In the end, business deals are typically closed with a price ranging somewhere between the initial values. The situation described can be modeled as a process of two parties negotiating two issues. Both issues, price and delivery time, are continuous variables, and they constitute the parameters of the orders of the offer. We must determine the degree of satisfaction perceived by the customer. To this end, we use a scoring system conceived by the customer to assign scores to the different values of the issue, the highest score being given to the preferred value.17,24 We assume that the satisfaction perceived by a customer oj with a certain contract can be measured with the scoring function VPjoj. To begin the negotiation, suppliers and customers state their starting prices. The starting price of each party, POinijojp, is the price that provides him with the highest satisfaction. The reservation price of the customer, POmaxjojp, is also defined, in excess of which there will be no agreement. Hence, this can be summarized as follows:

if POjojp ) POinijojp, then VPojj ) 100 if POjojp ) POmaxjojp, then VPojj ) 0 For the sake of simplicity, we assume that the scoring function can be approximated by a linear function (eq

VPjoj )

POmaxjojp - POjojp

1 np

∑p POmax

jojp

- POinijojp

× 100

(43)

In eq 43, np represents the number of products requested and POjojp is the price of product p in order oj offered to the customer at DC j. To assign satisfaction values to different delivery times, an analogous function is used, as stated in eq 44. In this equation, TOmaxjoj represents the delivery time for order oj at DC j that provides the customer with the highest satisfaction, TOinijoj is the reservation delivery time to the customer beyond which point the agreement will no longer be valid, and TOjoj is the delivery time for order oj offered to the customer at DC j.

VTjoj )

TOmaxjoj - TOjoj TOmaxjoj - TOinijoj

× 100

(44)

Since these two issues are considered to be independent, that is to say, the way of scoring one issue does not affect the scores of the other, an additive model can be used to evaluate concession between issues.24 Finally, to obtain the final equation for determining customer satisfaction, we weigh both functions to consider the priorities of the customer as stated in constraint 45, where ηPjoj and ηTjoj are the weights associated with the price and delivery time scores, respectively.

CSat )

(ηPjo VPjo + ηTjo VTjo ) ∑ jo j

j

j

j

(45)

j

The overall SC problem would therefore be mathematically formulated as follows:

max{E[Profit];CSat}

(46)

subject to eqs 1-45. The mathematical model described herein is an MINLP (mixed-integer nonlinear programming) formula, as it presents products of integers and continuous variables. However, converting it into an MILP formula is made possible by replacement of the nonlinear terms with aggregated variables and using standard tricks for linearizing the products of integers and continuous variables, as described in the literature.25 3.6. Tradeoff between Objectives. Every SC schedule has an associated E[Profit] and a certain CSat. Usually, plans with a higher E[Profit] values lead to lower CSat values; therefore, E[Profit] and CSat tend to be contradictory objectives, and the solution of the problem described above results in a set of Paretooptimal SC solutions. The -constraint method, which was first introduced by Haimes et al.,26 is applied to generate such solutions in this study. This method is based on maximizing one (the most preferred) objective function and regarding the other objectives as constraints bounded by some allowable levels 0. Then, the levels 0 may be altered to generate the entire Pareto-


Figure 2. Event-oriented representation.

optimal set. Therefore, the following single MILP optimization formula is used to obtain the Pareto solutions.

f(xj*) ) max E[Profit]

Table 1. Batch Sizes product

BSi1p

P1 P2 P3

100 90 95

(47)

subject to CSat g 0 and eqs 1-45. Thus, if eq 31 is solved for all possible values of 0 and the resulting solutions (xj*) are unique, then these solutions represent the entire Pareto set of solutions of eq 47.27 If the solutions to eq 47 are not unique for some value(s) of 0, then the Pareto point(s) must be picked by direct comparison, i.e., by applying the Pareto definition. For the sake of simplicity, in this work we label as Pareto solutions those solutions that are computed using the -constraint method, even though it must be admitted that this method may yield some non-Pareto solutions if there are values of 0 that have not been explored.26 Nevertheless, the overall method described in this article is still valid, as any other multiobjective optimization method can be applied for computing Pareto solutions. Therefore, by changing the weights , a set of results can be obtained. Each of these results implies an SC schedule and a set of parameters included in the offer sent to the customer. The resulting Pareto solutions may be represented in a two-dimensional chart (E[Profit] and CSat). These are the expected profit-customer satisfaction Pareto points in the space of uncertain parameters. 4. Case Study To illustrate the performance and results of the model, a motivating example has been defined and solved. A relatively simple linear SC is considered (Figure 2). It entails a batch plant, which produces three different products. The manufacturer has a warehouse at which the products are stored when they leave the plant. As the plant has a limited capacity and is imagined to be next to the factory, no transport is

Table 2. Processing Times, TOPi1pt product

t1

t2

t3

P1 P2 P3

5 6 6

18 17 18

9 9 8

Table 3. Initial Stock and Capacity Factors warehouse

distribution center

product

MWi1e0p

Ri1,p

MDCj1f0p

γi1p

P1 P2 P3

15 5 5

1 1 1

10 15 10

1 1 1

Table 4. Transport Data CAPTl1i1 TTi1j1 βp

500 items 10 h 1

λi1j1 µi1j1p

200 um 10-4 um/unit

necessary. There is also a DC from which customers are served. Three occasional orders are met if the DC has some amount of the product requested in stock, provided that it is not planning to use that stock to satisfy contractual requirements. If an order has been only partially met, there is no penalization, but that particular sale will not be able to be carried out at a later point, when the merchandise reaches the DC. Moreover, the possibility of signing a contract with a customer is considered. All input parameters are listed in Tables 1-5. The prices and quantities of sporadic demands are expressed as normal probability distributions whose means and standard deviations are listed in Tables 6 and 7, respectively. Those parameters related to the offer that must be evaluated are listed in Tables 8 and 9. Although orders are subject to uncertainty, the time

7414


Table 5. Production Costs and Storage Costs product

production costs νi1p

storage costs θp

P1 P2 P3

1.4 1.2 0.6

0.015 0.009 0.008

Table 6. Forecasted Prices mean prices, MeanDPricej1dj1p

standard deviations, SDevDPricej1dj1,p

order

P1

P2

P3

P1

P2

P3

A1 A2 A3

7.7 7.7 7.7

6.6 6.6 6.6

6.1 6.1 6.1

0.05 0.05 0.05

0.10 0.10 0.10

0.05 0.05 0.05

Table 7. Forecasted Quantities mean quantities, MeanQDj,dj,p

standard deviations, SDevQDj,dj,p

order

P1

P2

P3

P1

P2

P3

A1 A2 A3

5 100 110

20 5 95

20 100 180

0.15 0.025 0.45

0.075 0.05 0.425

0.100 0.025 0.05

Table 8. Quantities of the Contract product

quantity, QOj1o1j1p

P1 P2 P3

5 90 2

Table 9. Other Parameters of the Contract TOminj1o1j1 ) 45 h TOmaxj1o1j1 ) 120 h

POminj1o1j1p ) {6, 4.8, 4.4} POmaxj1o1j1p ) {7, 5.8, 5.4}

of delivery can take place at only three specific moments during the week, at 40 h intervals. As has been mentioned before, the number of events to be set at each of the different nodes of the SC can easily be computed from the original problem data. For the final product warehouse at the plant, the procedure is as follows. First, the number of activities that can cause an input of materials to the warehouse is determined. Here, from the processing times given by the recipes of the products that can be manufactured by the SC and the time horizon, it can be concluded that the plant is capable of producing no more than five batches, and therefore, five inputs of products to the warehouse can occur at most. Second, the activities that cause an output of products are quantified. Here, as it is supposed that only three transport batches can be sent from the plant to the DC, the number of output events that can take place is three. Finally, these two values are added together to yield the total number of events to be considered at the warehouse (eight for the proposed example). The same procedure can be repeated for the DC, in which we have three inputs of materials (arrivals of transport batches) and four outputs (three occasional orders and one order), which yields a final figure of seven events. Figure 2 shows the example’s event-based structure. The five production batches cause inputs of materials, while the two transport batches lead to outputs of products from the warehouse. Moreover, there is one transport batch which is not used, and its associated event is thus located at the beginning of the sequence. For the DC, there are two inputs given by the transport batches and four outputs (three occasional orders and one contract). As occurred before, there is one null event located at the beginning of the sequence, which represents the transport batch that is not sent.

Figure 3. Deterministic Pareto curve.

The main objective of the proposed case study is to observe how most of the activities executed by an SC can be integrated using this model, rather than to study very complex structures involving many entities. Therefore, the following aspects of the problem have been focused on, which comprise most of the daily challenges that face SC managers: scheduling at multiproduct batch plants, managing uncertain demand and unknown prices and quantities, logistics of transport, lead times, offering or bidding for a contract of sale before uncertainty materializes, and storing products. 4.1. Deterministic Case. The model is initially solved as a deterministic case. This means that the behavior of demand for the coming days is assumed to be perfectly known. The prices and quantities for the sporadic demands used in the deterministic case are the mean prices and the mean quantities in Tables 6 and 7, respectively. The resulting mathematical formula comprises 2284 single equations, 738 continuous variables, and 139 binary variables. It was implemented in GAMS28 and solved using the CPLEX MIP solver (7.0). The computation time required to obtain solutions with a 0% integrality gap on an AMD Athlon 3000 computer ranges from 100 to 500 s, depending on the target for customer satisfaction (note that the main purpose of this study is to propose an SCM framework rather than to develop an algorithm for determining the most efficient solution). When the deterministic case is solved without constraints on the level of customer satisfaction, the best economic performance is reached without a contract agreement. This seems to be consistent with the fact that the most important reason for closing a contract is to reduce uncertainty, as will be discussed below. A Pareto curve is obtained (Figure 3), which forces customer satisfaction to be higher. It can be observed that as satisfaction increases, profit decreases. There is a tradeoff between these two objectives, which in this case becomes more appreciable as satisfaction reaches


Figure 4. Gantt charts of the deterministic Pareto curve.

values od more than 50%. Whenever satisfaction is greater than zero, no matter how small the value, this implies that there is an agreement, because the measure of customer satisfaction is associated with the signing of a contract. Every point on the curve represents the optimal solution (in terms of production and transport schedule) for attaining the imposed level of satisfaction. In the same figure, one can also see the values for profit when satisfaction is not constrained, and therefore equal to zero, as explained above. While this value should be depicted as a unique point, drawing a line makes it easier to discern the gap between the profit related to each level of satisfaction and the profit obtained without a contract. Figure 4 shows the optimal schedules of some Pareto solutions, i.e., the schedule chosen in the absence of a contract, the schedule for reaching a CSat of 80%, and finally the schedule for which CSat ) 100%, which occurs when the system is forced to accept all conditions imposed by customers. Note that the customer who aims to close the contract is primarily interested in product P2 (see Table 8), while the incoming orders request greater quantities of products P1 and P3, and a smaller amount of P2. Therefore, in Figure 4 one can see how the solution for which CSat ) 0% (no contract is closed), which yields a profit of 2593 mu, leads to a higher production of P1 and P3 (two batches of each) and smaller amounts of P2 (only one batch). The amount of P2 produced in this plan is used to meet the second sporadic demand and not to satisfy contractual requirements. However, to reach a customer satisfaction level of 80%, the model is forced to eliminate one batch of P3 from the original schedule to instead produce an extra batch of P2 to meet contractual requirements. Moreover, in this second solution, the first transport batch is sent earlier than in the previous solution, as products must be available to customers sooner in order to increase customer satisfaction. Accordingly, the profit drops to 2479 mu as the income arising from the sale of the extra

batch of product P2 is in fact lower than what would have been generated by the original batch of P3. Finally, the solution for which CSat ) 100% leads to a profit of 2309 mu. Here, since the transport batch must be sent very early to achieve a higher level of customer satisfaction, there is not enough time to finish the production of one batch of P1 and send it to the DC to meet the amount of P1 required by the second sporadic demand, which is expected to occur within 40 h. This batch of P1 is therefore replaced by one batch of P3, which is used to partially meet the amount of P3 requested by the last sporadic demand and thus leads to a lower profit. 4.2. Stochastic Case. Using the same input parameters, but taking into account the uncertainty associated with the amounts and prices entailed by sporadic demands, we solve the stochastic case. Therefore, it is assumed that the amount of products as well as the prices included in each sporadic demand follows a normal probability distribution. Their mean and standard deviation values are listed in Tables 6 and 7. Uncertainty is represented by 10 scenarios that are generated by performing a Monte Carlo sampling on the aforementioned probability distributions. Each scenario consists of three orders, each of which has a given amount of products and prices. In this case, the resulting mathematical formula has 5911 single equations, 1665 continuous variables, and 139 binary variables; it is also implemented in Gams28 and solved using the CPLEX MIP solver (7.0). The computation time required to obtain solutions with a 0% integrality gap on an AMD Athlon 3000 computer ranges from 3600 to 7200 s depending on the customer satisfaction target imposed. It is worth mentioning that the number of binary variables in this case is the same as in the deterministic formula, as they represent firststage decisions and are therefore not scenario-dependent. On the other hand, the number of continuous variables increases as some of them represent decisions that are scenario-dependent.

7416


Figure 5. Stochastic Pareto curve.

The Pareto curve obtained for the stochastic problem is depicted in Figure 5. The main difference in this case in comparison with its deterministic counterpart is that the model chooses to close a contract even when satisfaction is unconstrained, which shows how signing contracts with customers can improve SC performance when the environment is actually uncertain. For this reason, the Pareto stochastic curve begins with a CSat value of 26.67%. In Figure 5, the solid lines represent the expected, worst-case, and best-case profits. The expected profit is computed by taking an average of the different scenarios. The worst-case profit results from the worst scenario, low prices and demand. Finally, the best-case profit is obtained from the best possible scenario, high prices and demand. These lines are the Pareto solution resulting from the optimization of the expected profit, by imposing different customer satisfaction targets (0). Moreover, the dashed lines represent the maximum expected profit, and the associated worstcase and best-case profits that would be attained without closing any contract. From the intersection of both the dashed line and the Pareto curve, it is deduced that a certain value for satisfaction may exist that, from an economic point of view, there is no interest in surpassing, because not closing a contract with that customer will result in a higher expected profit. In this case, this point lies at approximately CSat ) 60%, but obviously, its location (or even its existence) depends on the parameters considered in each case. With this information, the supplier is supposed to prepare offers that reach satisfaction levels below 60%, since going beyond that point would result in an expected profit lower than the one achieved without negotiating. Moreover, by taking into account risk considerations, we can draw similar conclusions. A risk-averse manager would prefer to close a contract, since the profit variability is lower and the worst-case profit is higher than that achieved without closing any contract for values of CSat below 60%. In contrast, above this value of CSat, not only the worst case but also the expected values of the

Pareto solutions are lower than those obtained without closing any contract, which would make signing the contract inadvisable. Figure 6 shows three Gantt charts, which represent the production and transport schedules associated with three different points of the Pareto curve (CSat ) 26, 80, and 100%). As one can observe, in the solution for which CSat ) 26%, which exhibits an expected profit of 2419 mu, two transport batches are sent from the plant to the DC. The first one includes a batch of P2, which is used to satisfy contractual requirements, and two more batches, one of P1 and another of P3, which will meet the first sporadic demand. The second transport batch includes two batches of product P3, which will partially meet the last sporadic demand. In contrast, the amounts of P1 and P2 requested in this last demand are not covered, since their uncertainty is very high indeed. Moreover, one can observe how one batch of P3 should be eliminated from the previously described schedule, to produce an extra batch of P2 instead. This allows one to attain a customer satisfaction level of 80%. In this case, the first transport batch is sent earlier than in the previous case, since products must be available to customers sooner to increase customer satisfaction. For this reason, the first transport batch includes one batch of product P2 (contract) and only another batch of P1 and not two batches (P1 and P3), as occurred before. This is used to partially meet the second sporadic demand. Moreover, the second transport batch includes two batches of P3 and one of P2, which are used to cover the last sporadic demand despite the high uncertainty associated with P2. Finally, one can see that, in this case, producing one batch of P2 for the last order turns out to be more profitable than keeping the previous schedule and adding an extra transport batch to satisfy contractual requirements and cover the second sporadic demand for P3. With these changes, the profit drops to 2246 mu. Finally, the solution for which CSat ) 100% leads to a profit of 2165 mu. In this case, a transport batch is used exclusively to cover contractual require-


Figure 6. Gantt charts of the stochastic Pareto curve.

Figure 7. Deterministic vs stochastic solutions.

ments, and thus, a high level of customer satisfaction is achieved. Deterministic versus Stochastic Solutions. In the following section, a comparison is performed between the deterministic and the stochastic solution to measure the effect of not considering uncertainty in those situations in which the environment is actually uncertain. To carry out this comparison, a new non-Pareto stochastic curve is computed as follows. First, a customer satisfaction level is selected. Second, for the selected customer satisfaction level, the corresponding values of the variables of the deterministic Pareto solution (schedules and parameters of the offer to be

sent to customers) are computed by means of the deterministic formula solved for the mean value scenario. Finally, these variables are fixed as first-stage variables in the stochastic model, and the stochastic formula is then solved to compute the values of the second-stage variables for the given first-stage ones. Such a procedure is repeated until the entire set of deterministic Pareto solutions is evaluated through the stochastic formula. This procedure leads to a new curve (Det_10 curve) that is different from the original stochastic Pareto curve, as its points are not Paretooptimal in terms of expected profit and customer satisfaction.

7418


As can be observed in Figure 7, the stochastic Pareto curve lies above the Det_10 curve throughout the model. The difference between both curves for a CSat of 80% is approximately 160 mu, and remains more or less constant along the curves. Moreover, the worst-case curve associated with the stochastic Pareto solutions also lies above its corresponding deterministic counterpart. On the other hand, the deterministic Pareto solutions exhibit higher best-case scenario values than the stochastic solutions. In addition, the stochastic solution offers less profit variability. For instance, for the stochastic Pareto solution, the difference between the best-case and worst-case values for a CSat of 80% is approximately 450 mu (20%), while for the deterministic solution, this difference is equal to 1140 mu (55%). Therefore, the stochastic formula manages to minimize the impact of the uncertain environment by both increasing the expected profit and reducing the variability of the solution compared with that of the deterministic solution, which makes it more attractive from the perspective of the decision maker. Another important factor is the fact that the solution predicted by deterministic formulas with nominal demands represents the uncertain environment poorly; in other words, the plan obtained with the nominal parameters may be critically inefficient when other demands and/or prices arise. Indeed, when deterministic solutions are used to face uncertainty, the expected profit value drops nearly 15 compared with the deterministic benefit computed for the mean scenario. With regard to the schedules obtained using the two formulas, which are depicted in Figures 4 and 6, it should be mentioned that the stochastic solutions avoid meeting those demands with higher uncertainty. For instance, while the deterministic solution for which CSat ) 80% produces one batch of P1 to fulfill the amount of this product requested in the last sporadic demand, the corresponding stochastic counterpart with the same customer satisfaction level manufactures one batch of P3 instead, and thus takes advantage of the lower uncertainty associated with the last sporadic demand of P3 (see Table 7). Because of these changes, the stochastic solution is able to increase the expected profit by 160 mu (7.2%) in comparison with the deterministic solution. Similar conclusions can be derived from the other schedules. 5. Conclusions This work presents a novel approach to increasing SC competitiveness. The most remarkable contribution of this approach is the way in which it integrates the operational activities carried out in the SC with customer relationship management, which enables one to make production and distribution decisions at the same time as offers are proposed to customers. The proposed approach provides a set of Pareto solutions to be used by the decision maker during the negotiation procedure. The main advantages of this strategy have been demonstrated by means of a simple case study, which focuses on the tradeoff between the expected profit and the quality of an offer sent to a customer, i.e., the level of customer satisfaction attained by an offer, as well as the connection between customer relationship management (customer satisfaction) and production activities (schedules). Although a significant simplification was made to the particular case studied for illustrative purposes, it clearly shows how the integration of the

different SC activities is carried out. More complex cases in terms of intermediate storage policies, plant flexibility, number of entities in the SC network, etc., can be addressed using the same approach. In addition, since future predictions related to market behavior cannot be perfectly forecast, a number of the parameters in this scheduling problem, such as product demands and prices, were considered to be uncertain parameters. The two-stage stochastic formula developed in this work has allowed this situationscommonly found in practicesto be handled properly, which thus reduces the impact of the uncertainty on what profit is achieved in short-term planning. The usefulness of signing contracts as a way of reducing uncertainty has also been shown by means of the aforementioned stochastic formula. Finally, the stochastic solution was compared with the deterministic solution, and the advantages of the former have been demonstrated using a case study. The proposed strategy represents a method for facilitating rational negotiation, in the sense that it enables the negotiator to process a far greater amount of production, transport planning, and customer preference data simultaneously, and thus prevents one from exclusively relying on one’s beliefs and interests. Acknowledgment We express our gratitude for the financial support received from the Spanish Ministry of Education, Culture and Sport (through an FPU research grant to G.G.), the Spanish Ministry of Science and Technology (through Project DPI2002-00856), the Generalitat (government) of Catalonia (through Project I0353 and an FI research grant to C.P.), and the European Union (through Projects GIRD-CT-2000-00318 and GRD1-2000-25172). Nomenclature Indices bi ) index for production batches manufactured at plant i dj ) index for future sporadic demands at DC j ei ) index for events which take place at the final product warehouse of plant i fj ) index for events which take place at DC j i ) index for plants j ) index for distribution centers li ) index for transport batches to be sent from plant i oj ) index for the orders placed at DC j included in the offer p ) index for products s ) index for scenario t ) index for process stages Parameters BSpi ) batch size of product p at plant i CAPDCj ) capacity of DC j CAPTli ) transport capacity of batch li sent from plant i CAPW ) capacity of the final product warehouse of plant i H ) time horizon MDCjf0jps ) initial amount of product p at DC j in scenario s MWie0ip ) initial amount of product p at plant i PDjdjps ) price of p of dj in s POmaxjojp ) highest price of product p of order oj at DC j POminjojp ) lowest price of product p of order oj at DC j QDjdjps ) amount of product p included in occasional demand dj at DC j in scenario s QOjojp ) amount of product p included in order oj at DC j TDjdj ) arrival time of occasional demand dj at DC j TOminjoj ) earliest delivery time of order oj at DC j TOmaxjoj ) latest delivery time of order oj at DC j TOPipt ) processing time of task t of product p at plant i

Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7419 TTij ) transport time from plant i to DC j Variables CSat ) customer satisfaction DCout1jojfj ) binary variable, 1 if the delivery time of order oj occupies event fj at DC j, and 0 otherwise DCout2jdjfj ) binary variable, 1 if sporadic occasional demand dj occupies event fj at DC j, and 0 otherwise DCinjilifj ) binary variable, 1 if transport batch li sent from plant i to DC j occupies event fj, and 0 otherwise E[Profit] ) expected profit Invps ) average inventory of product p in scenario s MDCjfjps ) amount of product p in event fj at DC j in scenario s MLilip ) amount of product p in transport batch li at plant i MWieip ) amount of product p in event ei at plant i POjoj ) price of product p of order oj at DC j PC ) production cost Profits ) profit in scenario s Revs ) revenues in scenario s SCs ) storage cost in scenario s TC ) transport cost TDCjfj ) time of event fj at DC j TFibit ) finishing time of task t of production batch bi at plant i TIibit ) initial time of task t of production batch bi at plant i TLFilj ) finishing time of transport batch li sent by plant i TLIilj ) initial time of transport batch li sent by plant i TOjoj ) delivery time of order oj at DC j TSIibi ) initial storage time of production batch bi at plant i TWiei ) time of event ei at plant i VPjoj ) function of value of price of order oj at DC j VTjoj ) function of value of time of order oj at DC j Winibiei ) binary variable, 1 if the storage of bi is associated with ei at i, and 0 otherwise Woutiliei ) binary variable, 1 if the departure of li is associated with ei at i, and 0 otherwise Xipbi ) binary variable, 1 if bi manufactured at i belongs to p, and 0 otherwise Yilij ) binary variable, 1 if li is sent from i to j, and 0 otherwise Zjoj ) binary variable, 1 if oj at j is fulfilled, and 0 otherwise Greek Characters Rip ) capacity factor of product p at plant i βp ) transport capacity factor of product p 0 ) target imposed for customer satisfaction ηPjoj ) price weight of order oj at DC j ηTjoj ) time weight of order oj at DC j γjp ) capacity factor of product p at DC j λij ) fixed transport cost factor from plant i to DC j µijp ) variable transport cost factor of product p from plant i to DC j νip ) production cost factor of product p at plant i Ps ) probability of scenario s θp ) storage cost factor of product p

Literature Cited (1) Applequist, G. E.; Pekny, J. F.; Reklaitis, G. V. Risk and Uncertainty in Managing Manufacturing Supply Chains. Comput. Chem. Eng. 2000, 24, 2211. (2) Simchi-Levi, D.; Kamisky, P.; Simchi-Levi, E. Designing and Managing the Supply Chain. Concepts, Strategies, and Case Studies; Irwin McGraw-Hill: New York, 2000. (3) Silver, E. A.; Pyke, D. F.; Peterson, R. Inventory Management and Production Planning and Scheduling; John Wiley & Sons: New York, 1998. (4) Fox, M.; Barbuceanu, M.; Teigen, R. Agent-Oriented Supply Chain Management. The International Journal of Flexible Manufacturing Systems 2000, 12, 165-188.

(5) Chen, C.; Wang, B.; Lee, W. Multiobjective Optimization for a Multienterprise Supply Chain Network. Ind. Eng. Chem. Res. 2003, 42, 1879-1889. (6) Papageorgious, L. G.; Rotstein, G. E.; Shah, N. Strategic supply chain optimization for the pharmaceutical industries. Ind. Eng. Chem. Res. 2001, 40, 275-286. (7) Timpe, C. H.; Kallrath, J. Optimal planning in large multisite production networks. Eur. J. Oper. Res. 2000, 126, 422-435. (8) Tsiakis, P.; Shah, N.; Pantelides, C. C. Design of Multiechelon Supply Chain Networks under Demand Uncertainty. Ind. Eng. Chem. Res. 2001, 40, 3585-3604. (9) Balasubramanian, J.; Grossmann, I. E. Scheduling Optimization under Uncertainty: An Alternative Approach. Comput. Chem. Eng. 2003, 27, 469-490. (10) Guilleń, G.; Mele, F. D.; Bagajewicz, M.; Espunã, A.; Puigjaner, L. Multi-objective supply chain design under uncertainty. Chem. Eng. Soc. 2005, 60 (6), 1535-1553. (11) Guilleń, G.; Bagajewicz, M.; Sequeira, S. E.; Espunã, A.; Puigjaner, L. Management of pricing policies and financial risk as a key element for short-term scheduling optimization. Ind. Eng. Chem. Res. 2005, 44 (3), 557-575. (12) Bonfill, A.; Bagajewicz, M.; Espunã, A.; Puigjaner, L. Risk Management in Scheduling of Batch Plants under Uncertain Market Demand. Ind. Eng. Chem. Res. 2004, 43, 741-750. (13) Bonfill, A.; Espunã, A.; Puigjaner, L. Addressing Robustness in Scheduling Batch Processes with Uncertain Operation Times. Ind. Eng. Chem. Res. 2005, 44, 1524-1534. (14) Gupta, A.; Maranas, C. D. A two-stage modelling and solution framework for multisite midterm planning under demand uncertainty. Ind. Eng. Chem. Res. 2000, 39, 3799-3813. (15) Gupta, A.; Maranas, C. D. Managing demand uncertainty in supply chain planning. Comput. Chem. Eng. 2003, 27, 12191227. (16) Giunipero, L.; Brand, R. R. Purchasing’s Role in Supply Chain Management. The International Journal of Logistics Management 1996, 7 (1), 29-38. (17) Neubert, R.; Go¨rlitz, O.; Teich, T. Automated negotiations of supply contracts for flexible production networks. International Journal of Production Economics. 2004, 89 (2), 175-187. (18) Crespo, A.; Blanchar, C. The procurement of strategic parts. Analysis of a portfolio of contracts with suppliers using a system dynamics simulation model. International Journal of Production Economics 2004, 88 (1), 29-49. (19) Faratin, P.; Sierra, C.; Jennings, N. R. Negotiation Decision Functions for Autonomous Agents. Journal of Robotics and Autonomous Systems 1998, 24 (3-4), 159-182. (20) Birge, J. R.; Louveaux, F. Introduction to Stochastic Programming; Springer: New York, 1997. (21) Schilling, G.; Pantelides, C. C. A Simple Continuous-Time Process Scheduling Formulation and a Novel Solution Algorithm. Comput. Chem. Eng. 1996, 20, S1221-S1226. (22) Zhang, X.; Sargent, R. W. H. The Optimal Operation of Mixed Production Facilities: General Formulation and Some Approaches for the Solution. Comput. Chem. Eng. 1996, 20, 897904. (23) Mockus, L.; Reklaitis, G. V. Continuous Time Representation Approach to Batch and Continuous Process Scheduling. 1. MINLP Formulation. Ind. Eng. Chem. Res. 1999, 38, 197-203. (24) Raiffa, H. The Art and Science of Negotiation; Harvard University Press: Cambridge, MA, 1997. (25) Gjerdrum, A.; Shah, N.; Papageorgiou, L. G. Transfer Prices for Multienterprise Supply Chain Optimization. Ind. Eng. Chem. Res. 2001, 40, 1650-1660. (26) Haimes, Y. Y.; Lasdon, L. S.; Wismer, D. A. On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Transactions on Systems, Man and Cybernetics 1971, 1, 296-297. (27) Eschenauer, H. A.; Koski, J.; Osyczka, A. Multicriteria Design Optimization: Procedures and Applications; SpringerVerlag: New York, 1986. (28) Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A User’s Guide, GAMS Development Corporation: Washington, D.C., 1988.

Received for review July 30, 2004 Revised manuscript received April 27, 2005 Accepted July 8, 2005 IE0493208

Optimal Offer Proposal Policy in an Integrated Supply Chain

Recommend Documents