Modeling Full Supply Chain Optimization-a mixed integer goal programming approach

P.K Viswanathan pkv@ifmr.ac.in

Adjunct Professor, Institute for Financial Management and Research, Chennai

G. Balasubramanian bala@ifmr.ac.in

Professor, Institute for Financial Management and Research, Chennai

Modeling Full Supply Chain Optimization-a mixed integer goal programming approach

P.K. Viswanathan G. Balasubramanian

Abstract: In the current business environment, supply chain management has gained considerable

importance due to the advancements in Information Technology (IT). IT offers low cost

integration of suppliers of raw materials, manufacturers, distributors, and customers.

Supply chain management envisages effective sharing of market information that would

result in the reduction of overall system costs through best inventory management. This

has given raise to supply chain optimization, which is a trade off between system level

and entity level optimization. Modeling the interests of the different stakeholders in the

supply chain should be attempted in such a manner that it will make it easier for the

managers to better understand the drivers of supply chain value and adjust resources

accordingly for the overall benefit of the customers. While there have been a number of

papers on supply chain optimization, a single holistic representation of the costs,

revenues and resources of all the entities in the supply chain can greatly facilitate

managerial decision making involving issues like where to stock, how much to stock,

how to distribute, where to increase resources etc. It will also help in better negotiation

and a collaborative non-zero sum game situation. This paper presents a weighted penalty

cost approach to priority structure goal programming [GP] model to find the optimum

value of supply chain variables, and performing sensitivity analysis for managerial

decision support.

Keywords: Optimization, Multi-criteria, Priority-structure, Goal programming

Introduction: The global competition, margin pressures and demand uncertainties have

driven firms to focus more on supply chain optimization than firm level optimization.

Managing a Supply Chain to meet an organization's objectives is a challenge to many

firms. It involves collaboration in multiple dimensions, such as cooperation, information

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sharing, and capacity planning. It is important for firms to understand the dynamic

relations among various factors, and provides guidelines for management to minimize the

impact of demand uncertainty on the performance of the supply chain. The manufacturer

has to determine the proper plant capacity and adopt the right level of delayed

differentiation strategy for its products. The potential gains of cooperation among

different members of the supply chain needs to be quantified. Using such knowledge, a

manufacturer can develop an appropriate incentive plan to motivate the retailers and

suppliers to collaborate, and to realize the potential of the entire supply chain.

Supply chain optimization requires development of models that integrate the process of

manufacturing goods and getting them to the consumer. It can be thought of as a decision

support system that treats acquisition of materials to produce products as well as

manufacturing, storing and shipping of finished products as an integrated system of

events rather than as stand-alone separate components of the process.

The overall objective of these models has been to minimize total system costs while

maintaining appropriate production levels and transporting needed quantities to the right

location in a timely and efficient manner. A Decision support model based on priority

structure GP is suggested in this paper.

Review of literature

Literature survey shows a number of approaches for modeling supply chain optimization

situations. Jonatan et al [2001], show an example of how expert systems techniques for

distributed decision-making can be combined with contemporary numerical optimization

techniques for the purposes of supply chain optimization. Multi agent modeling technique

was applied to simulate and control a simple demand driven supply chain network system

with the manufacturing component being optimized through mathematical programming

with the objective of reducing operating cost while maintaining high level of customer

order fulfillment.

Jen et al [2004], propose a hybrid approach for managing supply chain that incorporates

simulation, Taguchi techniques, and response surface methodology to examine the

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interactions among the factors, and to search for the combination of factor levels

throughout the supply chain to achieve the 'optimal' performance.

Dimitris et al [2004], suggest a general methodology based on robust optimization to

address the problem of optimally controlling a supply chain subject to stochastic demand

in discrete time. Optimal supply chain management has been extensively studied in the

past using dynamic programming.

Supply chain problems are characterized by decisions that are conflicting by nature. Pinto

[2003], says that modeling these problems using multiple objectives gives the decision

maker a set of pareto optimal solutions from which to choose. His paper discusses the use

of Multi-objective evolutionary algorithms to solve pareto optimality in supply chain

optimization problems using non-dominated sorting genetic algorithm-II.

Some of the early attempts to model an integrated supply chain were mostly with single

objective functions-(Cohen and Lee, 1998;Arntzen et al 1995).

Recently researchers have started developing models based on multi-objective

functions(Ashayeri and Rongen, 1997; Min and Melachrinoudis, 2000; Nozick and

Turnquiest, 2001).

GP has been used for individual problems like the vendor selection problem (Buffa and

Jackson, 1983) or the transshipment problem (Lee and Moore, 1973).

Elif Kongar et al [2003], illustrate a GP approach to the remanufacturing supply chain

model, in the context of environmentally conscious manufacturing. They present a

quantitative methodology to determine the allowable tolerance limits of

planned/unplanned inventory in a remanufacturing supply chain environment based on

the decision maker’s unique preferences, by applying an integer GP model that provides

an unique solution for the allowable inventory levels. The need for a multi-objective

based decision support model is emphasized in that paper due to the environmentally

conscious manufacturing set up, it is no longer realistic to use a single objective function

since the introduction of restrictive regulations makes the decision procedure more

complicated and mostly multi-objective. The need for a multi-objective decision

criterion, which is more flexible to changes in decision criteria and governmental

regulations, is emphasized in that paper. The model, while fulfilling an acceptable profit

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level should also be capable of satisfying additional goals simultaneously. GP approach is

especially appropriate for decision-maker centered cases.

Ulungu and Teghem [1994], in their survey report on applications of multi-objective

combinatorial optimization problems, have found that GP is the most commonly used

approach to solve multi-objective location problems, which are a part of supply chain

optimization problems.

Supply chain problems includes transportation and transshipment problems which is a

modified version of the transportation problem, where goods and services are allowed to

pass through intermediate points while going from original sources to final destinations.

The most commonly used techniques for solving transportation problem are linear

programming (LP) and generalized minimum cost network flow approach. These are

single objective optimization techniques used for cost minimization. Dinesh et al[2003],

developed a GP model with penalty functions for management decision-making in oil

refineries in the context of transshipment problems.

The GP approach In the context discussed above, GP assumes greater importance as a model capable of

handling multiple decision criteria. GP, one of the most widely used multi-objective

programming technique [Kasana and Kumar] [2002], is a special type of LP developed

and extended by Charnes et al[1961]. Large number of applications of GP are discussed

by Romero[1991], Tamiz et al[1998].Goals have a special meaning in GP. They refer to

management desires while constraints refer to the environmental condition under which

the management makes its decisions. Konstantina Pendaraki et al[2004], differentiate

the objective function in a GP, from the common objective function. The objective

function in a GP defines a search direction for the identification of acceptable solutions.

It is important to discuss GP in the context of LP as GP is an extension of LP. In LP

models, only one goal is incorporated into the objective function to be maximized of

minimized. Multiple goals are treated as constraints of the problem. For example,

Beranek [1963] illustrate how to optimize the liquidity and profitability objectives jointly

by using linear programming. Here, the profitability objective is incorporated into the

objective function while the liquidity objective is specified as a constraint on level of

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cash and quick ratio. This means that liquidity requirements must be satisfied before

profit objective. The computational procedure picks from the set of all solutions that

satisfy the constraints while maximizing or minimizing the objective function. One of the

serious problems with this approach is the restrictive assumption on constraint and in a

situation where there are incompatible multiple objectives specified as constraints, there

may not be any feasible solution. The liquidity constraint in the above case need not

necessarily be so strict. Management can desire to have a level of current assets and

might like to minimize large deviations from this goal either in the positive or negative

direction.

Representation of a Goal Programming problem The general GP model can be mathematically expressed as

m

Minimize Z=∑ (yi+ + yi

-) i=1 Subject to Ax-Iy+ + Iy-=b x,y+,y- >=0 where

b is an ‘m’ component column vector containing b1,b2,…. bm, the right hand side values

of the goal equations.

A is an m x n matrix of technological coefficients associated with the decision variables

x1,x2….xn.

x, the column vector represents decision variables x1,x2,…xn. y+ and y- are m-component column vectors representing deviations from goals.

I is an identity matrix of m x n order.

The manager must analyze each one of the ‘m’ goals considered in the model in terms of

whether over or under achievement of the goal is satisfactory. If over-achievement is

acceptable, y+ will not be represented in the objective function. If exact achievement of

the goal is desired, both y+ and y- must be represented in the objective function. The

deviational variables y+ and y- must be ranked according to their priorities, from the most

important to the least important. In other words, the decision maker must specify all

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relevant constraints that define the feasible solutions, express his/her goals as functions of

the decision variables, define the appropriate target values for the goals and specify the

deviations from the target values, which are relevant to the analysis. Objective function is

to minimize deviations from the set goals. Priorities can be established for each of the

goals in a number of ways. This is explained in the subsequent paragraphs.

The Weighted Penalty Cost Approach (WPCA) This GP model, by incorporating a weighted penalty cost function into the objective

function, obtains solution through a single LP run. It also considers concurrent

achievement of all goals as the entire priority structure is captured into a single model.

This approach envisages converting the multiple criteria objective function with

preemptive priorities and differential weights into a cardinal penalty cost function that

retains the priority structure of the goals intact. The objective function then becomes

minimization of the penalty cost subject to the goal constraints and other system

constraints if any. The cardinal function unifies disparate objectives into a single

objective and then uses the normal single-function optimization procedures [LP]. The

first advantage of this method is the solution is obtained by one singe LP run. The second

advantage is that it gives opportunity to interpret the shadow prices in a meaningful

fashion that will help in the direction of attaining the priority goals. Concurrent

achievement of goals improves the solution due to the possibility of multiple goal

interactions. Zeleny[1981] mentioned the difficulty for the decision maker to determine

goals for each objective, the inaccuracy surrounding these values as well as the

phenomenon of dominant solutions. To address these issues, Kasana and Kumar[2002],

proposed a technique called grouping algorithm which considers all goal and real

constraints together as one group with the objective function being the sum of all

weighted deviations and solves for the optimal solution using the simplex method and

sensitivity analysis. Here again, the algorithm solves a sequence of LPs each using the

optimal solution of the previous sub problems. Martel and Aouni[1990] introduced the

concept of satisfaction functions in order to explicitly integrate the decision maker’s

preferences and to remedy the incommensurability of the scales. The method suggested

in this paper, namely the cardinal penalty cost approach, takes care of all these issues by

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converting a multi objective problem into a single objective problem. Similarly, Tamiz

and Jones[1997] proposed an interactive framework for investigating GP models so that

the decision maker can explore through an interactive process, the feasible and efficient

regions for solutions. The cardinal penalty cost function approach facilitates efficient

exploration of solution space. Kasana and Kumar[2002] discuss a situation where the

decision maker wants to know the priorities to be assigned to the goals so that the

maximum number of goals is satisfied. This aspect is also automatically taken care by the

cardinal penalty cost function which incorporates the trade-off between violations of the

different objectives. It offers wider range of solution space for lucid decision-making

regarding trade-offs and queries raised by other interveners in the decision process.

Sometimes small sacrifices in one objective may lead to tremendous improvements in the

other. This approach helps in such analytics. Sartoris and Spruill, argue that managers

may or may not have a feel for the priority of each goal. In such situations, managers can

clearly see an entire range of possible priority situations and associated goal trade offs

before narrowing down on their choices. The penalty cost approach will greatly facilitate

exploration of the solution space through sensitivity analysis and help management in

fine-tuning its priority structure in a without losing sight of reality. Sartoris and Sruill

show the impact on liquidity and profitability under three different priority structures.

When the liquidity goals have much higher priorities than the profit goal, cash and quick

ratio are at their desired values. It was evident that increase in current liabilities is the

best way to achieve the desired level of current and quick ratios. Managers can study the

impact of their subject priority structures on the ultimate results through sensitivity

analysis as suggested in this model. The subjective priority structures can be approached

simultaneously. It is also possible that the whole problem can be fitted into a multi-period

model after adjusting for time value for money.

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Representation of the GP Model(WPCA) Minimize z= C1∑W1i(d1i

-+d1i+) + C2 ∑W2i(d2i

-+d2i+)+….. Cm ∑Wmi(dmi-

+dmi+)

Subject to a11x1 +a12x2+…….a1nxn+ d1

--d1+=b1 a21x1 +a22x2+…….a2nxn+ d2

-- d2+=b2 ……….. am1x1 +am2x2+…….amnxn+ dm

—dm+=bm Where C1, C2, C3 …. Cm are the penalty costs for not attaining the priority goals corresponding to the priority levels P1,P2…..Pm. They are so chosen that they satisfy C1> C2 > C3 >….Cm and C1 W1i >C2 W2i >……..Cm Wmi. [C1 is significantly larger than C2 which is significantly larger than C3 and so on] W1i,,W2i…..Wmi are the weights within the priority levels. d1i

-,d1i+, d2i

-,d2i+,….. dmi

-,dmi+ are the deviational variables within the priority

level i=1,2,3,…..kj if there are kj deviational variables within priority level Pj each for under and over achievement. Please note that j varies from 1 to m. x1, x2,…. xn are the decision variables aij’s for i=1 to m and j=1 to n are the technological coefficients associated with decision variables b1, b2,….. bm are the right hand side value of goal equations. Note: If any deviational variable is not present in the objective function, then the corresponding weight will be set =0

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The objective function combines penalty cost with priority levels and differential weights

within the priority levels. It then becomes a straight LP problem for which a solution can

be obtained from any package with complete information on sensitivity analysis and

shadow prices. The ordinal ranking of the goals are converted into cardinal penalty

function in the objective function.

This alternative model of GP, namely the WPCA, by implementing a single run LP with

the objective of minimizing the weighted penalty cost, provides a very solid base to study

the effect of varying the above discussed model parameters and improvement of the

solution. The model can be easily implemented. The availability of shadow price

information helps in improving the GP solution by studying the cost benefit trade off of

introducing additional resources to satisfy multiple goals. This is a dynamic shadow price

as it changes with every addition of resources and captures the impact of concurrent

achievement of multiple goals. It is also flexible enough to incorporate both non-

preemptive as well as preemptive goal programming problems just by altering the weight

coefficients of the penalty cost function.

Research question Supply chain optimization involves consideration of several entities and their interests in

the entire delivery chain. In essence, it involves procurement of raw materials,

manufacturing and distribution to customers. In today’s context, the financial flows are

also included to complete the fulfillment cycle. There are multiple trade-offs involved

while system level optimization is attempted to balance several costs that have inverse

relationship. It is in this context that managers need a model to explore the various

combinations and their trade-off effect, as decision making here is not always for

immediate optimization but for long term sustainability for which a firm is willing to

invest money. Given this background, whether the GP approach with WPCA gives

greater flexibility and better decision support for managers, in modeling supply chain

optimization situations, is the research question addressed in this paper. A simple

illustration is used to address this question.

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Illustration

1. A company manufactures two products namely Product 1(P1) and Product 2(P2)

2. P1 and P2 are produced in 4 plants namely, Plant 1, Plant 2, Plant 3 and Plant 4.

3. The manufactured items are shipped from the plants to three regional distribution

centers located in three cities namely, City-1, City-2 and City-3; from these

locations they are distributed nationwide.

4. Demand for the products is currently far less than the total of the capacities at its

four plants.

5. Each plant has a fixed operating cost, and, because of the unique conditions at

each facility, the production costs, production time per unit, and total monthly

production time available vary from plant to plant, as given below:

Table-1

Fixed cost/ Production cost Per unit

Production time(Hr/Unit)

Available hours

Plant Month(1000) P1 P2 P1 P2 per month Plant-1 40 10 14 0.06 0.06 640 Plant-2 35 12 12 0.07 0.08 960 Plant-3 20 8 10 0.09 0.07 480 Plant-4 30 13 15 0.05 0.09 640

6. P1 is sold at 22 and P2 at 28 per unit

7. Current monthly demand projections at each distribution center for both products

are:

Table-2 Monthly demand projections City-1 City-2 City-3 P1 2000 3000 5000 P2 5000 6000 7000

8. To remain viable in each market, the Company must meet at least 70% of the

demand for each product at each distribution center. The transportation costs

between each plant and each distribution center, which are the same for either

products, are given below:

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Table-3 Transportation cost per 100 units To From City-1 City-2 City-3 Plant-1 200 300 500 Plant-2 100 100 400 Plant-3 200 200 300 Plant-4 300 100 100

9. This illustration focuses on manufacture and delivery of finished products to

various distribution centers.

10. Prior to this, ordering of raw materials and scheduling of production process is

involved. This stage of the model is not included in this illustration

11. Subsequent links would involve storage and distribution to retail centers from the

distribution centers. This stage is also not shown in this illustration

12. The management is concerned with the following questions:

a. The number of P1 and P2 to be produced at each plant

b. The shipping pattern from the plants to the distribution centers

c. Achieve a targeted net profit

d. Achieve a targeted total production of P1

e. Achieve a targeted total production of P2

f. Ensure that each distribution center receives between 70% and 100% of its

monthly demand projections

g. Impact of increased capacity to reach higher levels of profits

Modeling the situation with profit maximization objective using LP Designing the Model for deciding the total number of P1 and P2 to be produced monthly

at each plant, the total number to be shipped to each distribution center and the shipping

pattern from plants to distribution centers, involves the following decision variables

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Table-4

Decision Variables Shipment of P1 To Total From City-1 City-2 City-3 Produced

Plant-1 P111 P112 P113 P1p Plant-2 P121 P122 P123 P1SL Plant-3 P131 P132 P133 P1No Plant-4 P141 P142 P143 P1D Total Received P1C P1KC P1SF P1

Table-5

Decision Variables Shipment of (P2) To Total From City-1 City-2 City-3 Produced

Plant-1 P211 P212 P213 P2P Plant-2 P212 P222 P223 P2SL Plant-3 P213 P232 P233 P2NO Plant-4 P214 P242 P243 P2D Total Received P2C P2KC P2SF P2

The variables representing total production and distribution of P1 and P2 to various

distribution centers are shown in table-4 and table-5.

The gross profit is defined as follows: 22 P1+ 28 P2 [22 and 28 are the selling prices and it is assumed everything is sold]

Minus

Total production cost and total transportation cost ……….I

Total production cost is defined as 10 P1P+12P1SL+8 P1NO+P1 GD+14 P2P+12 P2SL+10P2NO+15 P2D…………..II

Total transportation cost is defined as 2P111+3P112+5P113+1P121+1P122+4P123+2P131+2P132+3P133+3P141+1P142+1G43+2P211+3P212+ 5P213+1P221+1P222+4P223+2P231+2P232+3P233+3P241+1P242+1P243………….III

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Total Profit is defined as I-[II+III]-Fixed cost ……………………..IV Constraints Total production of P1 P1P+P1SL+P1NO+P1D=P1 P111+P112+P113=P1P P121+P122+P123=P1SL P131+P132+P133=P1NO P141+P142+P143=P1D Total production of P2 P2P+P2SL+P2NO+P2D=P2 P211+P212+P213=P2P P221+P222+P223=P2SL P231+P232+P233=P2NO P241+P242+P243=P2D

Total Shipments P1 P111+P121+P131+P141=P1C P112+P122+P132+P142=P1KC P113+P123+P133+P143=P1SF

Total Shipments of P2 P211+P221+P231+P241=P2C P212+P222+P232+P242=P2KC P213+P223+P233+P243=P2SF Production time at each plant Plant-1: .06P1P+.06P2P<=640 Plant-2: .07P1SL+.08P2SL<=960 Plant-3: .09P1NO+.07P2NO<=480 Plant-4: .05P1D+.09 P2D<=640

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Minimum Amount shipped to each Distribution center>=70%(Total demand) Maximum Amount Shipped to each Distribution center<=(total demand) Center Minimum Shipment Maximum Shipment City-1 P1C>=1400

P2C>=2500 P1C<=2000 P2C<=5000

City-2 P1KC<=2100 P2KC>=4200

P1KC<=3000 P2KC<=6000

City-3 P1SF>=3500 P2SF>=4900

P1SF<=5000 P2SF<=7000

Modeling the solution in spreadsheet

OPEN City-1 City-2 City-3 Total Produced

0 Plant-1 0 0 0 01 Plant-2 0 1143 0 11431 Plant-3 2000 804 0 28041 Plant-4 0 1053 5000 6053

Total Received 2000 3000 5000 10000DEMAND/Profit 2000 3000 5000 $22

OPEN City-1 City-2 City-3 Total Produced0 Plant-1 0 0 0 01 Plant-2 5000 6000 0 110001 Plant-3 0 0 3252 32521 Plant-4 0 0 3748 3748

Total Received 5000 6000 7000 18000DEMAND/Profit 5000 6000 7000 $28

FIXED PLANT Hrs HRSOPEN City-1 City-2 City-3 COSTS P1 P2 P1 P2 Availble USED

0 Plant-1 2 3 5 40000 10 14 0.06 0.06 640 01 Plant-2 1 1 4 35000 12 12 0.07 0.08 960 9601 Plant-3 2 2 3 20000 8 10 0.09 0.07 480 4801 Plant-4 3 1 1 30000 13 15 0.05 0.09 640 640

$266,114.91

INPUTS

TOTAL PROFIT

SHIPPING COST PER UNIT PROD. COST PROD HRS

P1 PRODUCTION/SHIPPING SCHEDULE

P2 PRODUCTION/SHIPPING SCHEDULE

Note: 1. The base model in spreadsheet is shown above

2. The model is designed for maximizing profit

3. The decision variables are the number of units to be produced and moved from

various sources to destinations that satisfied the demand requirements

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4. The model also decides on the plant from which the products have to be moved so

that the profit is maximum

5. This is done by declaring the cell which represents production as binary

6. Running a simple LP model in solver results in a profit of 266114

7. The optimal solution has dropped production from Plant-1 completely

Management’s priorities and GP The management of the company wants to explore the solution further and generate more

alternatives. It has a hierarchy of goals in a particular priority order and also wants to

know the plans for enhancing the profits with additional investment in capacity. In other

words, it requires a flexible structure in the model to change the priority structure and

generate a solution space that can show the profits at higher levels of investment in

capacity and how does it affect the priority goal structure.

Priority structure and questions

1. Achieve a profit of $210000

2. Achieve a production of 10000 in P1

3. Achieve a production of 18000 in P2

4. Up to what level of profits will be supported by the existing capacity?

5. What level of profits will be supported by additional investment in capacity?

Converting the LP model in to a GP model decision support based on WPCA The above points are addressed by the GP model by simply altering the earlier LP model

in the following manner:

1. Introduce two deviational variables for each of the goals. The two deviational

variables are for minimizing underachievement and overachievement of the target

goal.

2. Introduce a penalty cost weight for underachievement or overachievement or for

both which reflects the priority structure of the management

3. Introduce two additional sets of cells-Goal equation and Goal to achieve

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4. Goal equation cell captures the derivations of the target from the original model

and links it with the deviational variables to define the target clearly

5. The goal to achieve set of cells will form the input cells for the management to

input any target values for the goals

6. The goal to achieve cells also facilitate sensitivity analysis and solution

exploration later, which is done by substituting a series of input values for the

target goals

7. Derive the weighted penalty cost by multiplying and summing the deviational

variables and the penalty cost, in a single cell, which now becomes the target cell

8. The single cell, namely the weighted penalty cost cell, becomes to target cell to

minimize

9. The solution is achieved by minimizing the weighted penalty cost cell

WPCA GP Model Representation Minimize Z=P1d1

- + P2 (d2- + d2

+)+P3(d3- + d3

+) Where P1, P2 and P3 are the priority goals stated in the order of importance, namely, the profit goal, production goal for product 1 and production goal for product 2. Goal Equations Profit goal R1∑∑Xij+R2∑∑Yij- (∑∑XijCij + ∑∑YijCij)- (∑∑XijPCi+∑∑YijQCi) -HC(∑∑Xijhi + (∑∑Yijgi)- ∑Fi + d1

- - d1+ = ∏

Where R1 and R2 are the prices of products 1 and 2, Xij and Yij are the production transported from plant i to city j of products 1 and 2 respectively, jPCi and QCi are the unit cost of production of plant i for products 1 and 2 respectively, HC is the hourly cost of operating the machine, hi and gi) are the hours required to produce one unit of product 1 and 2 in plant i,

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Fi is the total fixed cost of plant i, d1

- and d1+ are the deviational variables corresponding to the under and over

achievement of the profit goal ∏ is the set target profit Production goal product 1 ∑∑Xij + d2

- - d2+ = Φ1

Where d2- and d2

+ are the deviational variables measuring the under and over achievements of the target production of product 1(Φ1)

Production goal product 2 ∑∑Yij + d3

- - d3+ = Φ2

Where d3- and d3

+ are the deviational variables measuring the under and over achievements of the target production of product 2(Φ2)

Constraints

Demand constraints

Maximum demand possible for product 1

∑Xij ≤ D1j(maximum possible demand for city j) i

Minimum demand to be satisfied for product 1 ∑Xij ≥ m1D1j Where m1 is the proportion stipulated to be achieved on D1j

Maximum demand possible for product 2

∑Yij ≤ D2j(maximum possible demand for city j) i

Minimum demand to be satisfied for product 2 ∑Yij ≥ m2D2j Where m2 is the proportion stipulated to be achieved on D2j

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Capacity Constraints ∑Xij hi + ∑Xij gi ≤ Hi(total hours available in plant i) j j

note: Here the model is conceptualized for i=1 to m and j=1 to n, meaning there are ‘m’ plants catering to ‘n’ cities

Illustration of the WPCA GP model Table-6

Under Over Goal equation Goal to achieve Penalty cost Goal AchievedProfit goal 0 0 210000 210000 10000 210000Production goal-1 0 0 10000 10000 100 10000Production goal-2 0 0 18000 18000 100 28

WPC 0

Priority Structure-1Higher Priority for Profit with equal priority for production

1. “Under” refers to underachievement of goal

2. “Over” refers to overachievement of goal

3. There are three goals considered, namely, profit goal, production goal 1[product

1] and production goal 2[product 2]

4. The goal equation cells, link the under achievement and over achievement cells

with the cells for profit, production etc, in the original model-for example, Goal

equation cell for profit goal is derived by adding the profit cell of the original

model to the cell representing underachievement and subtracting the cell

representing over achievement

5. Goal to achieve are the inputs

6. Penalty cost is an input cell for the management to differentiate and communicate

the management’s priorities to the model. The number 10000 represents that the

management gives higher priority to the profit goal and equal priority to the

production goal1 and production goal 2

7. The WPC cell is derived by multiplying the weights with the underachievement

and summing it across all the goals. This becomes the single cell for the model to

minimize

8. The goals achieved cell is just to capture the results

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Change in the model

1. The cells representing under achievement and over achievement are introduced as

decision variables (changing cells in the model) to capture the extent of under

achievement and over achievement for each goal. This will facilitate managerial

decision making by trade off analysis among the different goals and additional

costs of achieving all the goals

2. An additional constraint is introduced as the goals to achieve cells should equal

the goal equation cells

3. The objective of the model is changed from maximizing profit to minimize the

weighted penalty cost

Results of sensitivity analysis Table-7

Results at Different Levels of Profit for Priority Structure 1

Profit Profit Prod-1 Prod-2 Profit Prod-1 Prod-2 Penalty cost Profit Prod-1 Prod-2

200000 0 0 0 0 0 0 0 200000 10000 18000205000 0 0 0 0 0 0 0 205000 10000 18000210000 0 0 0 0 0 0 0 210000 10000 18000

215000 0 0 0 0 0 0 0 215000 10000 18000220000 0 0 0 0 0 0 0 220000 10000 18000225000 0 0 0 0 0 0 0 225000 10000 18000230000 0 0 0 0 0 0 0 230000 10000 18000235000 3429 0 0 0 0 0 34290000 231571 10000 18000240000 8429 0 0 0 0 0 84290000 231571 10000 18000245000 13429 0 0 0 0 0 134290000 231571 10000 18000250000 18429 0 0 0 0 0 184290000 231571 10000 18000

Actual AchievementUnder Achievement Over Achievement

1. The above table, generated by running a sensitivity analysis on the GP model with

varying levels of profits, shows that, at the given capacity levels, a profit of

230000 can be achieved with all the goals being satisfied

2. Beyond a profit of 230000, the profit goals are not satisfied though the production

goals are satisfied as the maximum profit potential is reached

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The management wants to further explore the model. It wants to increase the production

of P1 to 15000, as it is possible to achieve this within the existing capacity. The model

generates this solution space for higher levels of production of P1.

Table-8

Profit Profit Prod-1 Prod-2 Profit Prod-1 Prod-2 Penalty Cost Profit Prod-1 Prod-2

200000 0 0 0 0 0 0 0 200000 15000 18000210000 0 0 0 0 0 0 0 210000 15000 18000220000 0 0 0 0 0 0 0 220000 15000 18000230000 0 0 0 0 0 0 0 230000 15000 18000240000 0 0 0 0 0 0 0 240000 15000 18000250000 0 0 0 0 0 0 0 250000 15000 18000260000 7218 0 0 0 0 0 72180000 252782 15000 18000270000 17218 0 0 0 0 0 172180000 252782 15000 18000280000 27218 0 0 0 0 0 272180000 252782 15000 18000290000 37218 0 0 0 0 0 372180000 252782 15000 18000300000 47218 0 0 0 0 0 472180000 252782 15000 18000

Under Achievement Over Achievement Actual AchievementEnhancement in Production of P1[5000,5000,5000]

1. This table reveals that the management can achieve a profit of 250000 while

satisfying all the goals by increasing the production and sale of P1[prod-1] to

15000 from 10000. It is within the capacity of the company

The management wishes to explore possibilities for enhancing the profits further by

increasing the production of P2. The table below shows the results.

Table-9

Profit Profit Prod-1 Prod-2 Profit Prod-1 Prod-2 Penalty Cost Profit Prod-1 Prod-2

200000 0 0 0 0 0 0 0 200000 15000 18000210000 0 0 0 0 0 0 0 210000 15000 18000220000 0 0 0 0 0 0 0 220000 15000 18000230000 0 0 0 0 0 0 0 230000 15000 18000240000 0 0 0 0 0 0 0 240000 15000 18000250000 0 0 0 0 0 577 0 250000 15000 18577260000 0 0 0 0 0 1457 0 260000 15000 19457

Under Achievement Over Achievement Actual AchievementEnhancement of Production of P2[6000,7000,7000]

1. This table shows that a maximum profit of 260000 can be reached by increasing

the production of P2 to 20000 from 18000

2. It is still within the capacity

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The management wishes to enhance the profit by investing in additional capacity to 3000

hours from the current level of 2720 hours. The cost of additional capacity is factored

into the profit equation and the results are shown below:

Table-10 Enhancement of capacity to 3000

Profit Profit Prod-1 Prod-2 Profit Prod-1 Prod-2 Penalty Cost Profit Prod-1 Prod-2

260000 0 0 0 0 0 0 0 260000 15000 20000270000 0 0 0 0 0 0 0 270000 15000 20000280000 0 0 0 0 0 0 0 280000 15000 20000

Under Achievement Over Achievement Actual Achievement

1. The management could reach a profit level of 280000 with additional investment

but still there is unused capacity. It wants to increase the production of P2 to

22000. The results are shown in the next table.

Table-11

Profit Profit Prod-1 Prod-2 Profit Prod-1 Prod-2 Penalty Cost Profit Prod-1 Prod-2

280000 0 0 0 0 0 0 0 280000 15000 22000290000 0 0 0 0 0 0 0 290000 15000 22000300000 0 0 0 0 0 0 0 300000 15000 22000310000 1608 0 0 0 0 0 16080000 308392 15000 22000320000 11608 0 0 0 0 0 116080000 308392 15000 22000

Under Achievement Over Achievement Actual AchievementEnhancement in production of P2[7000,8000,7000]

By additional investment and enhancing the production of P2, it is able to reach a profit

level of 300000.

Conclusion: In this paper, we have shown that a GP approach with WPC, helps in modeling supply

chain optimization situations in a way that it helps managerial decision making,

particularly in the presence of conflicting goals. Changes in priority structure can be

easily accommodated in the model by simply changing the weights. The model facilitates

sensitivity analysis with great ease, which is the present approach to most of the

optimization problems, as the management is more concerned with exploring the

different levels of the solution space, rather than sticking to one optimization goal. The

reality is that multiple goals have to be satisfied simultaneously and the final solution

need not always be the optimal solution. Supply chain optimization necessarily involve

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multiple agents, with conflicting interests and is a perfect candidate for application of the

above model, especially in today’s global supply chain competition. In this paper, we

have shown only optimization at one segment of the supply chain consisting of the

manufacturing and distribution activities. It can be shown that the other segments of the

supply chain, namely the suppliers of raw materials and the customer end fulfillment

requirements, can also be easily modeled in this approach. By brining in activity based

costing at every stage, the scope for optimization can be further enhanced.

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