# The Stochastic Goal Programming Model Business Essay

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Abstract: Supply Chain (SC) design problems are often characterized with uncertainty related to the decision-making parameters. The Stochastic Goal Programming (SGP) was one of the aggregating procedures proposed to solve the SC problems. However, the SGP does not integrate explicitly the Decision-Maker's preferences. The aim of this paper is to utilize the Chance Constrained Programming and the Satisfaction Function concept to formulate strategic and tactical decisions within the SC while demand, supply and total cost are random variables.

Keywords: Supply Chain, Stochastic Goal Programming, Chance Constrained Programming, Manager's Preferences, Satisfaction Functions.

## Introduction

A Supply Chain (SC) comprises the entities such as suppliers, manufacturers, warehouses, retailers, transporters, and customers. Supply Chain Management (SCM) includes planning and management of all activities involved in the above mentioned entities. The main objective of the SCM is to increase the organization's profit through maximizing the efficiency of the SC. Mathematical modeling is an effective, inexpensive and comprehensive approach to measure the efficiency of the SC under different input conditions. While most of the efforts for modeling and optimization of the SC design are based on precise and deterministic approach, real SC design problems are characterized with uncertainty especially when considering elements are beyond the scope of the company (Azaron et al., 2010). The Fuzzy Goal Programming (FGP) is one of the three Goal Programming (GP) variants deal with the fuzziness of the goal values (Cherif et al., 2008). GP with Intervals (GPI) and the Stochastic Goal Programming (SGP) with satisfaction functions are the other two variant approaches. Martel and Aouni (1998) underlined that FGP and GPI generally deal with situations that the membership and penalty functions are linear and symmetric and favor the central value of the goal values. They also argued that both FGP and GPI models emphasize more on the imprecision of the goals rather than Decision-Maker's (DM) preference modeling. Aouni et al. (2005) applied the satisfaction function concept introduced by Martel and Aouni (1990) into SGP to address the above mentioned shortcomings. The advantage of satisfaction function comparatively to the concepts of penalty functions (regret functions) membership functions is that it does not need to be necessarily symmetric and linear (Cherif et al., 2008).

The SGP has been applied in many fields; however the literature review shows that the SGP and the satisfaction functions have not been applied for modeling the SC problems under uncertainty. In this paper, the SGP with satisfaction function are used to formulate an SC design problem under condition that the goals for demand, supply and total cost are stochastic variables with Normal probability distribution.

In the following sections, first the existing literature regarding SCM and GP is examined, and then a general formulation of SC is presented. Next the deterministic model of SC is applied to formulate GP model of SC. After that, Chance Constrained Programming (CCP) and satisfaction function concepts are integrated to formulate a SGP model of SC. Finally a numerical example is presented and solved by LINGO software version 11. The research ends with a conclusion and suggestions for future researches.

## Supply Chain Management

A typical SC has different entities including Customer, Retailer, Distributor, Manufacturer and Supplier. According to Council of Supply Chain Management Professionals (CSCMP), Supply Chain Management (SCM) includes planning and management of all activities involved in sourcing and procurement, conversion, and logistics as well as coordination and collaboration with the entities. CSCMP describes the primary responsibility of SCM as integration of the major business functions and business processes within and across organizations into a solid and high performing business model. Based on the decisions frequency and their time frame of effectiveness, Chopra and Meindl (2001) divided the decisions involved in SCM into three categories of SC strategy (design), SC planning and SC operation. The strategic decisions address the location and capacity of the manufacturing facilities and warehouses, products to be produced or stored at different locations, transportation mode along different shipping legs and type of information system.

In planning phase companies define the operating policies that manage short-term operations, within the constraints established by the configuration set in the strategy phase. Planning usually starts with a forecast of the coming year of demand in different markets and includes deciding about which market will be supplied from which locations, the planned buildup of inventories, the subcontracting of production, the replenishment and inventory policies, the policies regarding backup locations in case of stock out and the timing and size of marketing promotions. In operation phase the decisions associated with individual orders, and the goal is to exploit the reduction of uncertainty and optimize performance within the constraint defined by the configuration and planning policies.

In order to optimize the SC, the organization should ensure that the entities are properly located, the capacity of the different entities is enough and the structure of SC is in line with the SC strategy. Mathematical modeling is an effective, inexpensive and comprehensive method for optimization of the performance of SC under different input conditions. Following a comprehensive review of the literature, Mula et al. (2009) listed the major mathematical programming models applied in SC related literature as: Linear Programming, Mixed Integer/ Integer Linear Programming, Non Linear Programming, Multi Objective Programming, Fuzzy Mathematical Programming, Stochastic Programming, Heuristics Algorithms and Meta-heuristics, and hybrid models. According to the review, linear programming-based modeling is the most frequently used approach in the literature. The review also pointed out that the research works have mainly focused on optimization of tactical (planning) rather than strategic or operational decisions.

## Goal Programming and Supply Chain Management

The GP is one of the most popular mathematical programming models for solving multi objective problems. The main objective of the GP is to minimize the deviations among the achievement and the aspirations levels of the objectives (Aouni et al. 2005). Since its introduction by Charnes et al. in 1955, the GP model application has stretched to different areas such as water management, waste management, accounting, stock management, marketing, quality control, human resource management, telecommunication, agriculture, forestry, aviation as well as Supply Chain Management (Aouni and Kettani 2001).

The SCM tries to harmonize the customer needs with materials flow from suppliers, to balance normally conflicting objectives of low unit cost, low inventory, low rejected rate, high customer service, and high flexibility (Stevens1989, Kumar et al. 2004). The issues of SCM have been extensively studied by the researchers, but most of the related research works have focused on a single component such as procurement, production, transportation, warehouses or scheduling, rather than the whole system as a single supply chain (Sabri and Beamon 2000). For example Wang et al. (2004) integrated Analytic Hierarchy Process (AHP) and Preemptive Goal Programming (PGP) to develop a multi-criteria decision-making methodology for supplier selection. Azadeh et al. (2010) integrated GP with computer simulation and Design of Experiment to solve a multi objective job shop scheduling problem. Leung and Chan (2009) developed a preemptive GP model to solve an aggregate production planning problem. Leung and lung Ng (2007) developed a PGP model to solve aggregate production planning for perishable products. Li et al. (2006) proposed a GP approach to formulate an Earliness-Tardiness Production Planning problem. Taylor and Anderson (1979) developed a GP algorithm to coordinate and integrate production and marketing and address conflicts involved in marketing-production planning decisions. Forza et al. (2005) applied GP approach to identify and formalize the potential tradeoffs among decisions across product design, process design, and supply chain design. Ho et al. (2008) devised an integrated multiple criteria decision making approach to optimize the facility location-allocation problem in the contemporary customer-driven Supply Chain. Finally, Zhou et al. (2000) applied GP to exert optimization on the whole supply chain from procurement to distribution for Continuous Process Industry. Their research is among the few that have considered the whole components as a single supply chain.

While most of the efforts for modeling and optimization of SC design are based on precise and deterministic approach, in practice SC design problem are characterized with uncertainty especially when considering elements are beyond the scope of the company (Azaron et al., 2010). The imprecision within SCM normally relates to the goal and constraint target values (Liang, 2009; Lotfi and Torabi, 2011; -zcan and Toklu, 2009), but could also relate to other aspects of the goal program such as the parameters ( Sinha et al. ,1988; Cherif et al., 2008), and the priority structure( Aköz and Petrovic, 2007).

The FGP and SGP are effective and comprehensive approaches to optimize the performance of SC under imprecise information condition and have been widely applied in SC related literature. However, Ignizio (1982) and Martel and Aouni (1998) highlighted some issues regarding the analytical form of the membership functions in the FGP. They underlined that FGP favors central values of the intervals and generally deal with situations that the membership are linear and symmetric. Besides, since the fuzzy goal values are expressed through intervals, the DM is assumed neutral regarding the solutions within the limits defining the interval (Cherif et al., 2008). Moreover, membership function does not explicitly incorporate the DM's preference into the problem formulation, which is expressed by the DM's satisfaction regarding the deviation of the achievement level from the aspired level (Martel and Aouni, 1996). In addition, in FGP it is assumed that the available information for decision making fuzzy which does not address the uncertainty that may be involved in decision making process due to stochastic information. The SGP variant can tackle this issue. In the following section SGP is discussed and it is explained how SGP and the satisfaction function can address the shortcomings of FGP model.

## Stochastic Goal Programming

The first formulation of the Stochastic Goal Programming (SGP) is provided by Contini (1968), where the proposed model maximizes the probability that the consequence of the decision will belong to a certain region around the uncertain goal. The most popular technique for solving SGP model is the Chance Constrained Programming (CCP) developed by Charnes and Cooper (1952, 1959, 1963). Although the literature review shows absence of application of the Chance Constrained Goal Programming model in SCM, it confirms wide employment of the concept in other applications. For example Bravo and Gonzalez (2009) applied SGP to optimize allocation of surface water among farmers to fulfill farm management and environmental impact goals. Bhattacharya (2009) proposed a model based on CCP to maximize the reach to the desired section of people through optimizing number and allocation of budget assigned to different advertising media. Liu (1996) developed a general formulation of Dependent-Chance Goal Programming (DCGP) which is an extension of SGP in a complex stochastic system and presented its application in water allocation and supply example. Ben Abdelaziz and Sameh (2001) utilized SGP to find appropriate releases from various water reservoirs in the system in order to satisfy multiple conflicting objectives, such as satisfaction of demands and minimization of the pumping cost. Yang and Feng (2007) incorporated expected value GP, CCP and DCGP into their approach to solve a bi-criteria solid transportation problem. Min and Melachrinoudis (1996) applied CCP approach to develop a multiple-period, multiple-plant, multiple-objective, and stochastic location model for formulating location strategies.

The CCP attempts to maximize the expected value of the objectives for given probability of realization of the different constraints. Depending on the treatment of the constraints two approaches for CCP can be adopted which are the unconditional CCP and the joint CCP. In the unconditional CCP the decision maker is interested in realization of the constraints independent from each other, but in the joint CCP the joint realization of the constraints is sought. The following programs demonstrate the general formulation of both forms.

## The unconditional CCP:

Program 1-1: Max E (f(x))

s.t.

xï‚³0

where αi (αi ïƒŽ [0 , 1]) is the threshold values that are defined by the decision maker.

## The joint CCP:

Program 1-2: Max E (f(x))

s.t. ,

xï‚³0.

Aouni et al. (2005) applied the unconditional CCP to reformulate the following goal program with stochastic goals into a deterministic format:

Program 1-3: Min

s.t. ,

## ,

where and are known

They also argued that since is equivalent to , where , and the second probability statement maximizes when ( ) approaches to zero, program 1-3 can be reformulated into the following deterministic equivalent program:

Program 1-4: Min

s.t.

In order to incorporate the decision maker's (DM) preference into the model, Martel and Aouni (1990), introduced the concept of satisfaction functions which allows the DM to express explicitly his/her satisfaction regarding the deviations of the achievement level from the aspired level goals. The general shape of the satisfaction functions has been depicted below:

ï¡id

ï¤i

ï¡iv

ï¡io

1

Fi(ï¤i)

Fig. 1. General shape of the satisfaction functions

where Fi(ï¤i) is the satisfaction function related to deviation ï¤i; ï¡id represents the indifference threshold; ï¡io represents the nil satisfaction threshold; ï¡iv represents the veto threshold.

According to this model, when the deviations ï¤i are within [0, αid], the decision maker is fully satisfied and the satisfaction level is 1. As the deviations exceeds αid, the satisfaction level falls till reaches zero at αio. If the deviation exceeds this level, the decision maker may still consider the solution. But the solution is rejected if it exceeds veto threshold αiv.

By using the concept of satisfaction functions, Aouni et al. (2005) reformulated program 1-4 as:

Program 1-5: Max

s.t.

xïƒŽ X

With application of Program 1-5 we can easily formulate the SGP model of the SC, but before that we first need to represent the general formulation of the SC. In the following sections the general formulation as well as GP and FGP models of SC are represented.

## General Formulation of Supply Chain

A typical Supply Chain network usually has several stages including suppliers, plants, warehouses, and markets. There may also be other middle facilities such as consolidation centers or transit points. Santoso et al. (2005) formulated a deterministic mathematical model for Supply Chain design problem that is described below.

Consider a Supply Chain network G=(N, A) which consists of the set of nodes N and the set of arches A. The set N comprises the set of suppliers S, the set of potential processing facilities P, and the set of customer centers C, i.e., N=SPC. The processing facilities consist of production centers M and warehouses W, i.e., P=MW, while K is the set of products distributed through the SC.

The strategic decisions involve deciding which of the processing centers to build and the tactical decisions include routing the stream of each product k ïƒŽ K from the suppliers to the customers.

A binary variable yi is associated to strategic decisions where yi equal to 1 if the processing facility i is built, and 0 otherwise. xijk denotes the flow of product k from node i to node j in the network where (ij) ïƒŽ A and zjk indicate shortfall of product k at customer center jïƒŽC when demand cannot be met. ejk represent capacity expansion at node jïƒŽP when the capacity should be increased.

A deterministic mathematical model for this SC design problem can be formulated as:

Program 2-1: Min 2a

s.t.

2b

2c

2d

2d

2f

2g

2h

2i

2j

where cj, djk ,sik ,rjk, mjk and ojk denote investment cost of building processing center j (jïƒŽP), demand of product k at node j (jïƒŽC), supply of product k at node i (iïƒŽS), per unit processing requirement for product k at node j (jïƒŽP), capacity of facility j ( jïƒŽP) for product k, and expansion limit of node j ( jïƒŽP) for product k, respectively.

In program 2-1, the objective function (2a) consists of minimizing the sum of investment, production/ transportation, shortage and expansion costs. Constraint (2b) enforces the flow conservation of product k across each processing node j. Constraint (2c) requires that the total flow of product k to a customer node j plus shortfall exceed the demand djk at that node. Constraint (2d) requires that the total flow of product k from a supplier node i stay below the supply sik at that node. Constraints (2e) and (2f) enforce capacity and expansion constraints of the processing nodes, respectively. Constraint (2g) enforces the binary nature of the configuration decisions for the processing facilities.

## Goal Programming Modeling of the Supply Chain

A generic GP model can be formulated as:

Program 3-1:

Min ï“ (ï¤i++ï¤i-)

s.t. i(x)- ï¤i++ï¤i-=gi (iïƒŽI) where I objectives are considered

xïƒŽXïƒŒRn

ï¤i+ , ï¤i- ï‚³0

Where gi represents the aspiration level (goal) associated with the objective fi(x), and X denotes the set of the feasible solutions. The variables ï¤i+ and ï¤i- designate the positive and negative deviations of the achievement level fi(x) from aspirated level gi.

The GP can be utilized to formulate the SC problem where gc, gd and gs denote goals for total cost, demand and supply and ï¤c+,ï¤n++ï¤n-,ï¤d-, ï¤s+,ï¤m+ and ï¤e+ designate the unwanted deviation variables associated to total investment cost, flow conservation in each processing node j, total demand, total supply, capacity limit and expansion limit, respectively.

Program 3-2: Min 3a'

s.t.

3a

3b

3c

3d

3e

3f

3g

3h

3i

3j

3k

## Stochastic Goal Programming modeling of Supply Chain

After explaining how to incorporate the satisfaction function into GP model in section 4 and formulating the linear and Goal Programming (GP) model of the Supply Chain (SC) in sections 5 and 6, now we are able to formulate the stochastic GP model of supply chain. Our assumption here is that the aspired goals for demand, supply and total cost are random variables with normal probability distribution with mean and standard deviation of (ïd , σd), (ïs , σs) and (ïc , σc), respectively. So the Stochastic Goal Programming (SGP) model of SC can be formulated as:

Program 4-1:

4a'

s.t.

4a

4b

4c

4d

4f

4h

4g

4h

4i

4j

4k

Then, by application of program 1-2 and replacing the random goals with their means (ïd , ïs and ïc ) in program 3-2, we can formulate the deterministic equivalent of program 4-1:

Program 5-1:

5a'

s.t.

5a

5b

5c

5d

5f

5h

5g

5h

5i

5j

5k

Now in order to incorporate decision maker's preference, we can apply Program 5-1 and transform the achievement function (5a') into the following statement:

Program 5-2

where ï¤c= Σï¤ci , ï¤s= Σï¤si and ï¤d=Σï¤di ( iïƒŽI) and F(ï¤c), F(ï¤s) and F(ï¤d) denote the DM's satisfaction regarding the deviation of achieved levels from the goals for total cost, supply and demand.

By formulating the achievement function, the last step to SGP modeling of SCM is accomplished. In the next section the program developed in this section is applied to solve a numerical SC problem.

## Numerical Example:

To explain our model we apply the Supply Chain (SC) network design problem was used by Azaron et al. (2010), however the goals in the problem are assumed to be stochastic variables with normal probability distribution in order to fit in our model. According to the problem a wine company wishes to design an AC with three customer centers placed in locations L,M and N , and four suppliers placed in locations A, B, C and D. To build the bottling plants four locations E, F, G and H are considered.

A

B

C

D

E

F

G

H

M

N

L

Wineries Bottling Plants Customer Centers

Fig. 2. SC network for numerical example

It is assumed that the shipping capacity, demands at customer centers and total cost (investment costs + transportation costs + production costs + shortage costs) goals are stochastic variables with normal probability distribution:

The shipping capacity of each winery A, B, C, and D:

{ (ïsi , σsi ): (375; 15), (187; 17), (250; 10), (150; 20)}

Demands at customer centers L, M and N:

{(ïdj , σdj ): (318; 62), (161; 17), (169; 27)}

Total cost :

{(ïc , σc ): (1,300,000; 10000)}

The value of the other deterministic parameters are listed below

(475,000, 425,000, 500,000, 450,000) are investment costs for building each bottling plant E, F, G, and H, respectively.

(65.6, 155.5, 64.3, 175.3, 62, 150.5, 59.1, 175.2, 84, 174.5, 87.5, 208.9, 110.5, 100.5, 109, 97.8) are the unit costs of production and transporting bulk wine from each winery A, B, C, and D to each bottling plant E, F, G, and H, respectively.

(200.5, 300.5, 699.5, 693, 533, 362, 163.8, 307, 594.8, 625, 613.6, 335.5) are the unit costs of production and transporting bottled wine from each bottling plant E, F, G, and H to each distribution center L, M, and N, respectively.

(10,000, 13,000, 12,000) are the unit shortage costs at each distribution center L, M, and N, respectively.

(315, 260, 340, 280) are the capacities of each bottling plant E, F, G, and H, respectively, if it is built.

(100, 80, 60, 50) are the unit capacity expansion costs at E, F, G and H. We cannot expand the capacity of these plants more than (40, 30, 50, 25) units in any situation.

The threshold values of the satisfaction function for each goal are:

Objective

ï¡ïi

ï³i

ïtotal

ï³total

ï¡id

ï¡io

ï¡iv

Total Cost

1,300,000

10,000

1,300,000

10,000

10,000

20,000

30,000

Demand

318,161,169

62,17, 27

648

70

70

140

210

Supply

375, 187,250 ,150

15, 17, 10, 20

962

32

32

64

96

Capacity

40

80

120

Expansion

20

60

60

Flow

40

80

120

Table 1: Figures of the stochastic variable

ï¡id

ï¤i

ï¡iv

ï¡io

1

Fig. 3. satisfaction function type applied for the numerical example

After solving the above problem by Lingo software version 11, we reached the following results:

Variable

Value

Variable

Value

Xij: number of units from winery i to bottling plant j

Xjk: number of units from bottling plant j to customer center k

Rj: Bottling plant capacity expansion

Yj: 1 if bottling plant j is made and 0 otherwise

Zk: unit shortage at customer center k

i ïƒŽ {A,B,C,D}, jïƒŽ{E,F,G,H}, kïƒŽ{L,M,N}

DM Satisfaction

1

XEM

0

XAE

205

XEN

226

XAF

0

XFL

0

XAG

140

XFM

0

XAH

0

XFN

0

XBE

0

XGL

189

XBF

0

XGM

161

XBG

0

XGN

0

XBH

0

XHL

0

XCE

0

XHM

0

XCF

0

XHN

0

XCG

250

YE

1

XCH

0

YF

0

XDE

150

YG

1

XDF

0

YH

0

XDG

0

ZL

0

XDH

0

ZM

0

XEL

129

ZN

0

Table 2: Results of the numerical example

## Conclusion

In this research we applied Chance Constrained Programming (CCP) to model Supply Chain (SC) under uncertainty. We also utilized Satisfaction Function concept to explicitly incorporate the Decision Maker's (DM) preferences into the model. The aspired level of demand, supply and total cost were assumed to be random variables with normal probability distribution. The model was then applied in a numerical example to decide site location and flow of the product flow in the SC network so that the decision make's satisfaction is maximized. The discussed model can noticeably simplify formulating SC design problems under uncertainty through generating a deterministic equivalent; however it has some limitations that need to be addressed. For example the model assumes that the random goals have normal probability distribution while in real situation the probability distribution may be different. Besides it assumes that only the goals are random while in reality the parameters on the left hand side as well as goal priority may also be stochastic. The other limitation of the model is that it only incorporates cost constraints while Quality and Delivery as well as environmental constraints can also influence the DM's preference. Finally, the model does not address the risk of unwanted deviation of achievement level from the aspired level which is very critical in stochastic environment.