Modeling Full Supply Chain Optimization-a mixed integer goal programming approach P.K Viswanathan [email protected]Adjunct Professor, Institute for Financial Management and Research, Chennai G. Balasubramanian [email protected]Professor, Institute for Financial Management and Research, Chennai
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Modeling Full Supply Chain Optimization-a mixed integer goal programming approach
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:
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
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
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
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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