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IRUPDQDJHULDODQGUHVHDUFKSXUSRVHV 5HODWHG5HIHUHQFHV &RPSLODWLRQRI5HIHUHQFHV $ERXWWKH&RQWULEXWRUV ,QGH[ &KDSWHU (VVHQWLDOVRI/LQHDU 3URJUDPPLQJIRU0DQDJHUV )URP6\VWHPRI,QHTXDOLWLHVWR 6RIWZDUH,PSOHPHQWDWLRQ Hossein Arsham Johns Hopkins University, USA M. Bardossy University of Baltimore, USA D. K. Sharma University of Baltimore, USA $%675$&7 This chapter provides a critical overview of Linear Programming (LP) from a manager’s perspective. The main objective is to provide managers with the essentials of LP as well as cautionary notes and defenses on common modeling issues and software limitations. The authors illustrate the findings by solving a simple LP directly on the original decision variables and constraints space without adding new variables or translating the model to fit a specific solution algorithm. The aims are the unification of diverse set of topics in their natural states in a manner that are easy to understand and providing useful information to the managers. The advances in computing software have brought LP tools to the desktop for a variety of applications to support managerial decision-making. However, it is already recognized that current LP tools, in ample circumstances, do not answer the managerial questions satisfactorily. For instance, there is a costly difference between the mathematical and managerial interpretations of sensitivity analysis. LP software packages provide one-change-at-a-time sensitivity results; the authors develop the largest sensitivity region, which allows for simultaneous dependent and/or independent changes, based on the optimal solution. The procedures are illustrated by numerical examples including LP in standard-form and LP in non standard-form. DOI: 10.4018/978-1-4666-4506-6.ch007 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV ,1752'8&7,21 approach is known as the Big-M method, (see Arsham 2006, 2007). On the other hand, using the Linear programming (LP) has been a fundamental Dual Simplex method has its own difficulties. For topic in the development of managerial decision- example, when some coefficients in the objective making. The subject has its origins in the early function are not dual feasible, one must introduce work of L. B. J. Fourier (1820s) on attempting an artificial constraint. Handling equality (=) to solve systems of linear inequalities. However, constraints by the dual simplex method is tedious the wide spread use and acceptance of LP had to because of the introduction of two new variables wait until the invention of the Standard Simplex for each equality constraint: one extraneous slack Method. variable and one surplus variable. Also, one may not be able to remove some equality (=) constraints 6WDQGDUG6LPSOH[0HWKRG by elimination at the outset, as this may violate the non-negativity condition introduced when Since World War II, LP has been used to solve constructing the “Standard Form.” problems of various dimensions in almost all dis- Arsham (2013) proposes a tabular interior ciplines. The most popular solution algorithm is boundary approach that intents to address these the Simplex method that is implemented by most concerns and solves the original managerial LP LP software packages. model, without any need of “Standard Form”, The Simplex algorithm can be considered as a artificial variables, artificial constraints, and sub-gradient directional method, jumping from an Big-M. While these variants of the simplex suc- initial feasible vertex to a neighboring vertex of the cessfully handle an array of constraint forms, they feasible region until it arrives at an optimal vertex. impose a burden and mathematical sophistication The Simplex method requires that the model be on manager that make difficult the success of LP expressed in a special format called “Standard applications. Form”, that is, some constraints and variables must be transformed. Consequently, when the 7KH&RVWO\'LIIHUHQFH manager obtains a solution through the Simplex %HWZHHQWKH0DQDJHULDODQG method, she/he is left with the task of interpreting 0RGHOHU,QWHUSUHWDWLRQ and transforming the Simplex solution back to the original managerial problem. However, this may Koltai and Terlaky (2000) state that managerial not be an easy task. questions are not answered satisfactorily with For instance, solving LP problems in which the mathematical interpretation of sensitivity some constraints are in (≥) or (=) form with analysis. Software packages provide sensitivity non-negative right-hand side (RHS) has raised results focused on the optimality of a basis and difficulties. With that purpose, the simplex method not on the optimality of the values of the decision requires a feasible stating solution. When such variables. The implementation of the misunder- starting solution is not readily at-hand, alterna- stood shadow prices and their range of validity tive methods are necessary. One version of the may coax managers to take poor decisions with Simplex method, known as the two-phase method, considerable financial losses and strategic con- introduces an artificial objective function, which sequences. Another source of confusion is the is the sum of artificial variables (see Arsham similarity of terms used in operations research (i.e., 1997a, 1997b). Another version adds penalty re-search, as a process), and other related fields. terms, which are the sum of artificial variables For example, “shadow price” and “opportunity with very large, positive coefficients. The latter cost” have somewhat different meanings in the LP (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV and Economics literature. The “opportunity cost” unrestricted variables. In this research work, the of an action in economics can be interpreted as algorithm adjusts itself to the original model and the “shadow price” of that action on the budget. solves it, instead of changing the model. The deci- The LP software packages provide sensitivity sion maker easily understands the result, because results about the optimality of a basis and not our method does not change the structure of the about the optimality of the values of the deci- original model. For example, while using Simplex sion variables and the shadow prices that are of approach the need to transform constraints to interest to the manager. In Section 6 we develop standard form changes the manager’s LP model to the largest sensitivity region based on the optimal something that is not his and therefore the optimal vertex, and allows for simultaneous dependent/ solution must be transferred back to the manager’s independent changes. problem. This can confuse the managers and lead to costly misinterpretations. One must understand 7KH&RQIXVLRQ%HWZHHQWKH the differences that could creep-in when a model 0RGHODQGWKH6ROXWLRQ$OJRULWKP is formulated and when it is solved using a solver’s solution algorithm; unfortunately the current state Naturally, a modeler, reflecting the modeler’s deci- of art of LP software do not differentiate and sion problem, creates the LP model. On the other causes confusions. hand, the solution algorithm is designed by, for There exists a dichotomy between the reality example, a systems analyst to solve LP problems. the managers have to deal with, the problems that Because of the complexity of Simplex method, they face and the type of strategic decisions they confusion could creep-in when the modeler’s need to make, and the methods that modelers have model is developed into a tool using the software access to solve those problems. Research and solution algorithm. For instance, the algorithm technology have brought great advances to the used in QM software and Management Scientists decision-making process. However, more ad- make all variables non-negative. This may create vances are needed to address directly the manag- confusion as many LPs may contain negative or ers’ decision problems. Figure 1. Manager’s world vs. modeler’s world (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV The main issue is that most software pack- model, (5) managerial interpretation, and (6) ages make assumptions about the structure of the the implementation of the solution. The authors problem, constraints have a certain direction; the believe that in most cases we have under-utilized decision variables take certain sign-constraints. our knowledge (study) and limited the application These assumptions are without loss of generality; of the OR process. There have been extremely few however, they do have implications that modify the studies and applications of OR that satisfies the original model and complicate the interpretation systems point of view, Arsham (2012). of the solution for implementation. As a result, The remainder of this chapter is organized the conceptual models from the manager’s envi- in thirteen sections. Almost every section can ronment have to be translated into mathematical stand by itself, independent of previous sections. models, gets translated and later into solution Section 2 introduces linear programming models models to fit software packages. Afterwards, these from model builders and decision makers. Some solutions must be translated back into the real prob- oddities are included. Section 3 describes the de- lem. This translation is often cumbersome for the cision maker’s environment explicitly in order to general user. While translating decision variable enable them to understand the decision problem. might be doable with some effort, interpreting Section 4 explores the feasible region by using sensitivity analyses from a modified model might algebra that is understandable by the decision be out of reach. In that regard the mathematical maker. The algebraic approach to solve linear model should represent to the conceptual model. programs is given in Section 5, which includes We devoted Sections to show how to build the computation of slack and surplus variables. The largest sensitivity region based on the “optimal methodology solves LP model in an “as is” state, solution”. These sections are independent of all without any need of “Standard form”, artificial previous sections; in that one needs the optimal variables, artificial constraints, the Big-M, and solution in any way it is obtained. This develop- slack/surplus. Section 6 constructs the largest ment differs significantly from the traditional sensitivity region for the right hand side of the Simplex’s tableau approach that limits itself to the constraints based on the optimal solution. The “Preservation of the Basis”. No one expects the shadow prices are byproducts of the sensitivity Manager to understand and interpret the Basis cor- analysis. The dual problem formulation is included rectly, but it is expected that a Manager should be in Section 7, which is general and easy to apply. able to understand and interpret the optimal values Section 8 considers the general sensitivity region of his decision variables and the consequences of for the cost coefficients of the objective function. any changes to them. The Dual generated from the Throughout this chapter we have used LPs of “Standard Form” does not look like the Manager’s two dimensions to save space but this method can dual problem of his primal model. be extended to higher dimensions. Even for two The Operations Research (OR) processes must dimension problems we do not use the graphical be considered from the viewpoint of General method because it is limited to two-dimensional Systems Theory. The authors assert that without problems. Another reason to avoid graphical a whole systems perspective OR can neither be methods is that in some textbooks the graphical understood nor applied effectively. Typically, OR methods for finding the cost sensitivity range, and problems can be broken down into five components interpretation of the shadow prices are misleading. viz: (1) the “reality” of the problem, (2) the con- We use the algebraic approach throughout this ceptual model of the problem, (3) the conversion chapter to overcome these limitations. The solu- of conceptual model to a mathematical model, tion to a non-standard form problem is outlined (4) solution determination of the mathematical in Section 9. Software implementation and their (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV limitations are examined in Section 10. The sensi- We do not deal with trivial cases, such as where tivity analysis of the degenerate optimal vertex to A = B = D = 0 (no constraints), or a = b = d = remain optimal is developed in Section 11. Section 0 (all boundaries pass through the origin point), 12 does sensitivity analysis of multiple optimal or having LP with a single point feasible region. solutions. The existence of a curious property; We assume the LP has a no redundant constraints. known as More-For-Less, or Less-For-More is covered in Section 13. Concluding remarks are :KDW&DQ*R:URQJLQWKH summarized in Section 14. 3URFHVVRI%XLOGLQJD/LQHDU 3URJUDPPLQJ /3 0RGHO" /,1($5352*5$00,1*02'(/ At the modeling stage potential pitfalls exist which affect any LP application; therefore, the In general terms, a linear programming model is a decision-maker and the modeler should be cog- mathematical approach for determining the set of nizant of modeling phase of LP, Arsham (2008). decisions necessary to achieve the best outcome Thus, potential problems exist which affect any given a set of resource limitations and requirements linear programming application. Optimal solution represented as a system of linear equations. In may be infeasible or unbounded, or there may be business, these can be the outsourcing conditions multiple solutions. Degeneracy may also occur. and quantities to attain the lowest possible total Figure 2 presents a classification of LP for model- cost, or the production plan to get the maximum ing validation process: profit, or the marketing mix to reach the highest market penetration. 'XDO3UREOHP&RQVWUXFWLRQ /LQHDU3URJUDPPLQJ0RGHO Associated with each (primal) LP problem there is a companion problem called its dual problem. We define the LP model in the most general way. Table 1 is a classification of the decision vari- able constraints is useful and easy to remember Problem P: in construction of the dual. Max (or Min) CX Subject to AX ≤ a, *HQHUDO3ULQFLSOHV BX ≥ b, DX = d, • If the primal is a maximization problem, Xi ≥ 0, i = 1,..., j then its dual is a minimization problem Xi ≤ 0, i = j+1,…, k (and vice versa). Xi unrestricted in sign, i = k+1,…, n • For each constraint there is a dual decision variable (and vice versa). where matrices A, B, and D have p, q, and r rows, • Use the variable type of one problem to find respectively with n columns and vectors c, a, b, and the constraint type of the other problem. d have appropriate dimensions. Therefore, there • Use the constraint type of one problem to are m = (p + q + r + k) constraints and n decision find the variable type of the other problem. variables. It is assumed that m ≥ n. Note that the • The RHS elements of one problem become main constraints have been separated into three the objective function coefficients of the subgroups. Without loss of generality, we assume other problem (and vice versa). that all RHS elements, a, b, and d are non-negative. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Figure 2. A classification of LP for modeling validation and sensitivity analysis Table 1. A One-to-one correspondence between the primal and the dual LP problems. The Dual Problem Construction Objective: Max (e.g. Profit) Objective: Min (e.g. Cost) Constraint types: Constraint types: ≤ a Sensible constraint ≥ a Sensible constraint = a Restricted constraint = a Restricted constraint. ≥ an Unusual constraint ≤ an Unusual constraint. Variables types: ≥ 0 a Sensible condition unrestricted in sign ≤ 0 an Unusual condition • The matrix coefficients of the constraints • It may be more efficient to solve the dual of one problem are the transpose of the ma- LP problem than the primal Problem. trix coefficients of the constraints for the • The dual solution provides important eco- other problem. That is, rows of the matrix nomical interpretation such as shadow become columns and vice versa. prices. • If a constraint in the primal is non-binding, $SSOLFDWLRQV i.e., the LHS value is not equal to the RHS value, then the associated variable in the One may use duality in a wide variety of applica- other problem is zero. tions including: (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Table 2. Possible combinations of primal and dual properties Condition Primal Problem Implies Dual Problem Feasible; bounded objective ↔ Feasible; bounded objective Feasible; unbounded objective → Infeasible Infeasible ← Feasible; unbounded objective Multiple solutions ↔ Degenerate solution Degenerate solution ↔ Multiple solutions • If a decision variable in the primal problem 8QERXQGHG6ROXWLRQV is not zero, then the associated constraint in the dual problem is binding. ,GHQWLILFDWLRQ • To obtain the sensitivity range of the RHS If the shadow prices of the primal problem do of primal problem from the sensitivity not satisfy the dual constraint, then the primal range of the cost coefficient in the other problem is unbounded. problem (and vice versa). 2FFXUUHQFH Table 2 implies some possible combinations of primal and dual problems. An unbound optimal solution means the con- straints do not limit the optimal solution and the 3UREOHPVZLWK/33DFNDJHV feasible region is also unbounded, it effectively extends to infinity. Most LP software solvers have difficulties in 5HVROXWLRQ recognizing the last two cases in the above table namely multiple optimal solutions and degenerate In real life, this is not possible. Check the objective optimal solution. Most LP software do not alert function. Possibly, the problem is a minimization the user about them yet produce sensitivity ranges problem and maximization was selected instead. that are not valid. One must make ensure that the Check the formulation of the constraints. Possibly, solution is unique, and non-degenerate before one or more constraints are missing. Check also the analyzing and applying the sensitivity ranges. constraints for any incorrect specification in the direction of the inequality, and numerical errors. (QVXULQJWKH)RUPXODWLRQ ,V&RUUHFW 2GG/30RGHOV ,QIHDVLELOLW\ Developing LP models with oddities can assist ,GHQWLILFDWLRQ in the process of ensuring that the formulation is There is no feasible solution, i.e., there is no correct. However, almost all LP software solvers feasible vertex. The solution algorithm produces have difficulties in recognizing these dark sides no solution. of LP and/or giving any suggestions as remedies. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV 2FFXUUHQFH even though it remains challenging for organiza- An infeasible solution means the constraints are tions in the public sector or not-for-profit orga- too limiting and have left no feasible region. That nizations, because the performance measures are is, no solution satisfies all the constraints of the the drivers for the optimal strategy and determine problem. the direction for the organization. The decision-making environment elements 5HVROXWLRQ are further discussed in the context of the follow- ing numerical example. Check the constraints for any incorrect specifica- tion in the direction of inequality constraints, and $Q,OOXVWUDWLYH1XPHULFDO([DPSOH numerical errors. If no error exists, then there are conflicts of interests. Consider the following mixed products LP prob- These and other odd situations, including lem of a price-taker small manufacturer producing multiple solutions and in particular degeneracy, table and chairs. His objective is to maximize are rare (except in network models since they net profit: often have a redundant constraint, causing de- generacy), and often result at the modeling stage. Maximize P(X) = 5X1 + 3X2 The decision-maker sets the objective function, Subject to 2X + X ≤ 40, while the constraints come from the decision- 1 2 X1 + 2X2 ≤ 50, maker environment, consequently, it is rare to get X ≥ 0, X ≥ 0. multiple solutions, i.e. the objective function has 1 2 the coefficient proportional to those of at least Figure 3 analyzes and depicts his decision one binding constraint. In these cases, rounding environment. in the coefficient(s) could be the source of the The performance measure is important for the multiple solutions. decision maker as a measuring tool for success and identifying the decision problem. Other ele- ments of the decision-maker’s environment are '(&,6,210$.(5¶6 classified as follows: (19,5210(17 • Uncontrollable Inputs: These come from In the process of conceptualizing and building a the decision maker’s external environment. mathematical model of the business reality, it is Uncontrollable inputs often create the important to identify the most important elements problem and constrain the actions. and categorize them into performance measures, • Parameters: Parameters are the constant uncontrollable inputs, parameters, controllable elements that do not change during the inputs and strategies. time horizon of the decision review. These Understanding the problem requires criteria for are the factors partially defining the prob- grouping entities of the decision model in the same lem. Strategic decisions usually have lon- category. Performance measure (or indicators), ger time horizons than both the tactical that is, measuring business performance, should and the operational decisions, at different be at the top of the decision-making process and levels of organization. should align with the management goals. The • Controllable Inputs: The collection of development of effective performance measures all possible courses of action the decision is increasingly important for many organizations, maker might take. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Figure 3. Decision-maker’s environment in a mixed product LP problem • Interactions among these Components: problems, adding new uncontrollable in- These are logical, mathematical functions puts, and /or new options. One must iden- representing the cause-and-effect rela- tify and anticipate these new problems and tionships among inputs, parameters, and recognize the cycle of such problems. outcomes. There are some constraints that are made up 62/9,1*7+(6<67(02) of all the parameters. Therefore, they do not need /,1($5,1(48$/,7,(6 to be treated separately. The widely used Simplex solution to an LP prob- • Actions: Action is the ultimate decision lem is strictly based on the theory and solution and is the best course of strategy to achieve of a system of linear inequalities (Arsham 2009). the desirable outcome. Decision-making The list of constraints in an LP model constitutes involves the selection of a course of action a system of linear inequalities that define the in the pursuit of one’s objective. feasible region of the problem. The basic solu- • Controlling the Problem: Few prob- tions to a linear program are the solutions to the lems in life, once solved, stay that way. systems of equations consisting of constraints at Changing conditions tend to create new binding positions. Not all basic solutions satisfy problems from the problems that were pre- all the constraints. Only those that do meet all the viously solved. Their solutions create new constraint restrictions are called the basic feasible (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV solutions. The basic feasible solutions correspond Step 4. Check feasibility of each basic solution precisely to the vertices of the feasible region. obtained in Step 3 by using all the constraints For illustration purposes we are going to follow and identify the basic feasible solutions the Improved Algebraic Method (IAM) to solve (BFS). the system of linear inequalities. This method is simple to understand and does not require the for- The coordinates of vertices are the BFSs of the mulation of an auxiliary LP problem like it might systems of equations obtained by setting some require the Simplex method. This methodology of the constraints at binding (i.e., equality) posi- can also be extended to solve systems of linear tion. For a bounded feasible region, the number inequalities of higher dimensions. of vertices is at most m!/[n!(m-n)!] where m is We are interested in finding the vertices of the number of constraints and n is the number of the feasible region of Problem P, expressed as a variables. Therefore, a BS is obtained by taking system of linear equalities and inequalities. any set of n equations then solving them, simul- taneously. By plugging this BS in the constraints Feasible Region S: of other equations, one can check for feasibility AX ≤ a, of the BS. If it is feasible, then this solution is BX > b, a BFS that provides the coordinates of a corner DX = d, point of the feasible region. 1XPHULFDO([DPSOH)RU where some Xi > 0, i = 1,..., j, Xi < 0, i = j+1,…, 6ROYLQJ6\VWHPRI,QHTXDOLWLHV k and Xi unrestricted in sign, i = k+1,…, n. Ma- trices A, B, and D as well as vectors a, b, and d have appropriate dimensions. We provide a feasible region problem; FR1 to Therefore, the optimization problem can be illustrates the Improved Algebraic Method steps: expressed as: Feasible Region FR1: Problem P: 12X1 + 10X2 ≥ 12000 (1) Max (or min) C(X) 10X1 + 15X2 ≤ 15000 (2) Subject to: X ∈ S X1 ≥ 0 (3) X2 ≥ 0 (4) 6WHSVRIWKH,PSURYHG $OJHEUDLF0HWKRG ,$0 Because of constraints 2, 3 and 4, are enclos- ing feasible region, therefore feasible region is Step 1. Convert all inequalities into equalities bounded with countable vertices. There will be including any variable restricted in sign. 4 equations and 2 variables yielding 6 possible Step 2. Calculate the difference between the combinations. The IAM provides six basic solu- number of variables (n) and the number of tions for the system of linear inequalities as shown equations (m). in Table 3. Step 3. Determine the solution to all square sys- There are 3 basic feasible solutions to the tems of equations, i.e., basic solutions. The system of inequalities namely: maximum number of systems of equations to be solved is: m!/[n!(m-n)!]. A= (X1, X2) =(1000, 0), B = (X1, X2) = (1500,0), C = (X1, X2) =(375, 750). (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Table 3. The basic solutions of the numerical example FR1 Binding Constraints Solution (X1, X2) Feasible? Why? 1 and 2 (375, 750) Yes, all satisfied 1 and 3 (0, 1200) No, constraint 2 is not satisfied 1 and 4 (1000, 0) Yes, all satisfied 2 and 3 (0, 1000) No, constraint 1 is not satisfied 2 and 4 (1500, 0) Yes, all satisfied 3 and 4 (0, 0) No, constraint 1 is not satisfied $OJHEUDLF3UHVHQWDWLRQ software become apparent. There are special cases RI)HDVLEOH5HJLRQ when using IAM: Using the scalar parameters λ1, λ2, and λ3 for the • If there is no feasible vertex, then the prob- 3 vertices, respectively, we obtain the following lem is infeasible, that is feasible region is convex combination of vertices which is a paramet- empty. ric representation of the bounded feasible region: • If the shadow price of the primal does not satisfy the constraints of the dual problem X1 = 1000λ1 + 1500λ2 + 375λ3 then the problem is unbounded. • If there are two distinct optimal vertices, X2 = 750λ3 then the problem has multiple solutions. In this case the dual has degenerate optimal for all parameters λ1, λ 2, λ 3 such that each is solution. The reverse is also true. non-negative, and λ1+ λ2 + λ3 = 1. Note that the above representation is valid if Notice that these results are not applicable to the feasible region is bounded. For example the trivial cases, such as a problem with no vertices, feasible region X1 + X2 ≤ 1, X1 ≥1, X2 ≥0, is such as: Maximize X1 + X2, subject to: X1+X2 unbounded contains one vertex only. = 1, all variable are unrestricted in sign. For Moreover if feasible region is bounded it must more oddities in LP modeling, visit the website: have more than one vertex. The feasible region The Tools for Modeling Validation Process: The of X1 + X2 = 1, X1 - X2 = 1, X1 ≥ 0, X2 ≥ 0, has Dark Side of LP http://home.ubalt.edu/ntsbarsh/ feasible region with single point (1, 0). In the case opre640A/partv.htm of unbounded feasible region one must includes the needed rays, for numerical examples, see Arsham (2008). 62/9,1*/,1($5352*5$06 6RPH8VHIXO&RPPHQWRQ Consider the following linear program P1: $OJHEUDLF6ROXWLRQ0HWKRG ,$0 Problem P1: The main purpose of presenting the IAM is for Minimize C(X) = 18X1 + 10X2 the manager to understand, use, gets useful infor- Subject to: 12X1 + 10X2 ≥ 12000 mation, the major software use a refine version 10X1 + 15X2 ≤ 15000 of IAM. Therefore, for large problems based on X1, X2 ≥ 0 (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Substituting the parametric version of the fea- Fundamental Linear Programming Result: if sible region into the objective function, we obtain: the optimal solution to an LP is bounded, then it occurs at one of vertices of the feasible region. C(λ) = 18X1 + 10X2 = 18000λ1 + 27000λ2 + 14250λ3 From the above proposition, one might found the tight bound for the objective function, if it The optimal solution occurs when λ3 = 1 and exists. For our numerical example we have the all other λi’s are set to 0, with a minimum value following tight bound: of 14250. The optimal solution is (X1 = 375, X2 = 750), the vertex corresponding to λ3. 14250 ≤ 18X1 + 10X2 ≤ 27000 Let the terms with the largest (smallest) coeffi- &RPSXWLQJ6ODFNDQG6XUSOXV cients in C(λ) be denoted by λL and λS respectively. Then the following proposition holds: Given that the right-hand-side (RHS) of a con- Proposition 1: The maximum (minimum) solution straint is non-negative, the slack is the leftover of an LP with a bounded feasible region corre- amount of a resource (≤) constraint, and surplus spond to the maximization (minimization) of the is the access over a requirement (≥) constraint. parametric objective function C(λ). Moreover, These quantities represent the absolute values of since C(λ) is a linear convex combination, the the difference between the RHS value and the optimal value of C(λ) is obtained by setting λL or LHS (Left Hand Side) evaluated at an optimal λS equal to 1 and all other λi = 0. point. Having obtained an optimal solution, one can compute the slack and surplus for each Proof: Proof follows from applying the IAM constraint at optimality. Equality constraints are to always binding with zero slack/surplus. Since this numerical example is a two-di- Max (Min) C(λ) mensional LP, one expects (at least) to have two Subject to: ∑ λi =1, and all λi ≥ 0. binding constraints. The binding constraints at optimality are equations 1 and 2: The optimal solution can be found by enu- merating all basic feasible solutions by applying 12X1 + 10X2 = 12000 the IAM method to this optimization problem. 10X1 + 15X2 = 15000 Since this feasible region is bounded it has a fi- nite number of vertices, then this result suggests With surplus and slack value of zero, respec- that optimum is associated with the BFS yielding tively, i.e., S1 = 0 and S2 = 0 which are the “Reduce the largest or smallest objective value, assuming Cost” of the dual problem. the problem is of maximization or minimization type, respectively. The maximum and minimum &216758&7,212)7+( vertex of an LP correspond to λL = 1 and λS = 1, respectively. As a result, since C(λ) is a (linear) /$5*(676(16,7,9,7< convex combination of its coefficients, the optimal 5(*,216%$6('21237,0$/ 62/87,21127³%$6,6´ solution of C(λ) is obtained by setting λL or λS equal to 1 and all other λi = 0. As an application of this proposition, we have Suppose we obtained the optimal solution by the following useful result. any means, such as graphical (for two dimen- (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV sional problems), simplex, dual simplex or interior system of equations, we get the following para- methods. The following two sections construct metric solution: the largest sensitivity regions based on the given optimal solution. 12X1 + 10X2 = 12000 + r1 We develop a novel approach to post-optimality 10X1 + 15X2 = 15000 + r2 analysis for general LP problems given a unique non-degenerate optimal solution this approach Solving this parametric system of equation for provides a simple framework for the analysis of X1 and X2, we have: any single or simultaneous change of right hand side (RHS) or cost coefficients by solving the X1 = 375 + 3/16r1 -1/8r2 nominal LP problem with perturbed RHS terms. X2 = 750 -1/8r1 + 3/20r2 Assume that the LP is of size (mxn), i.e., it has n decision variables and m constraints including The solution can be verified by substitution. For the non-negativity conditions (if any). the higher dimension LP, the parametric solution can be obtained by using the JavaScript: &RPSXWDWLRQRI6KDGRZ3ULFHV http://www.mirrorservice.org/sites/home.ub- Step 1: Identify the n constraints that are binding alt.edu/ntsbarsh/Business-stat/otherapplets/ at optimal solution. If there is more than PaRHSSyEqu.htm n binding constraints, then the problem is degenerate. The degenerate case are dealt For large-scale problems one may use symbolic with in Section 10. software such as Maple, Matlab, or Mathematica. Step 2: Construct the parametric RHS of the The parametric optimal solution shows the constraints, excluding the non-negativity impact of changes on the RHS on the optimal conditions. decision variables. For example, if the RHS of Step 3: Solve the parametric system of equations constraint 1 increases the quantity of X1 will in- consisting of the binding constraints. This crease and X2 decreases. This is often a concept provides the parametric optimal solution. misunderstood or overlooked by managers. Step 4: Construct the simultaneous sensitivity Plugging the parametric solution into objective analysis by plugging-in the solution obtained function, we have: in Step 3, in all other remaining parametric constraints, including the non-negativity 18X1 + 10X2 = 14250 + 17/8r1 - 3/4r2 conditions (if any). Step 5: Plug in the parametric optimal solution The shadow prices are the coefficients of the into the objective function. The coefficient parametric optimal function, i.e., U1 = 17/8 and for each parameter is the shadow price for U2 = -3/4, for the RHS of constraints 1 and 2, that binding constraint. By definition, the respectively. They are the rate of change in opti- shadow price for a non-binding constraint mal value with respect to changes in the RHS of is always zero. each constraint. Following the above steps for our numerical example, one must first solve the following RHS parametric system of equations. By setting all inequalities in binding positions and solving the (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV &RQVWUXFWLRQRIWKH5+6 Similarly, the range for the RHS of the second 6HQVLWLYLW\5HJLRQ 0DLQWDLQLQJWKH constraint (RHS2) can be obtained by setting r1 9DOLGLW\RI&XUUHQW6KDGRZ3ULFHV = 0 in the inequality set (2). This implies that r2 ≤ 3000 and r2 ≥ -5000. Therefore the allowable Notice that the parametric objective function increase and decrease in the original value 15000 14250 + 17/8r1 – 3/4r2 is valid when the paramet- for RHS2 are 3000 and 5000, respectively, i.e., the ric solution satisfies all other (i.e., non-binding) current set of shadow prices remain valid as long as: constraints. In the current numerical example, the other constraints are: RHS1 = 12000 and 10000 ≤ RHS2 ≤ 18000. 6HQVLWLYLW\5DQJHIRUWKH X1 ≥ 0 and X2 ≥ 0. 1RQ%LQGLQJ&RQVWUDLQWV These produce the following conditions: In the numerical example P1, the non-binding con- 375 +3/16r1 – 1/8r2 ≥ 0, and 750 - 1/8r1 + 3/20r2 straints are the non-negativity conditions which are ≥ 0. not subject to sensitivity analysis. Consequently, for illustration purposes let us pretend “as if” there Notice that this is the largest sensitivity region is a non-binding constraint X1 + 100X2 ≥ 20000. for the RHS of the binding constraints that allows To find the sensitivity range (one-change-at-a- for simultaneous, dependent/independent changes. time), construct the parametric form and plug in This convex region is non-empty since it always the optimal solution. We get contains the origin (r1= 0, r2 = 0), with a vertex at (r1= -12000, r2 = -15000) where both right-hand X1 + 100X2 ≥ 20000 + r3 sides binding constraints. 375 + 100 (750) = 75375 ≥ 20000 + r3 Arsham (1990) gives detail treatments of the r3 ≤ 55375 most popular sensitivity analysis, including the 100% rule, and the tolerance analysis. The amount of increase is 55375 and decreas- ing amount is unlimited; i.e., (RHS3 ≤ 75375). 6HQVLWLYLW\5DQJHIRU(DFK The following proposition formalizes the shift- 5+6RIWKH%LQGLQJ&RQVWUDLQWV ing of parametric non-binding constraints. o o The above set of inequality sensitivity region Proposition 2: For any given point X = (X1 , o o can be used to find the ordinary sensitivity range X2 , ...... , Xn ) the parameter r value for any re- (one change-at a time) for the RHS values of the source/production constraint is proportional to constraints. The range for the RHS of the first the ordinary distance between the point Xo and constraint (RHS1) can be obtained by setting r2 = 0 the hyper-plane of the constraint. in the inequalities set (2). This provides r1 ≤ 6000 and r1 ≥ -2000. Therefore, the allowable increase Proof: Proof follows from the fact that the o o o o and decrease in the original value 12000 for RHS1 distance from point X = (X1 , X2 , ...... , Xn ) to are 6000 and 2000, respectively, i.e., the current any non-binding constraint, i.e. set of shadow prices remain valid as long as: o o o a1 X1 + a2 X2 +.....+ an Xn = b + r 10000 ≤ RHS1 ≤ 18000 and RHS2 = 15000. is (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV o o o Absolute [a1 X1 + a2 X2 +.....+ an Xn - b - r] / depending on whether the constraint is a 2 2 2 1/2 (a1 + a2 +.....+ an ) binding or non-binding constraint). For ≤ type constraint: The change is in the same This reduces to: direction. That is, increasing the value of RHS does not decrease the optimal value 2 2 2 1/2 Absolute r / (a1 + a2 +.....+ an ) . (rather, increases or has no change depend- ing on whether the constraint is a binding Therefore the parameter r value is proportional or non-binding constraint). to the distance with a constant proportionality 1 For = constraint: The change could be in either 2 2 2 1/2 / (a1 + a2 +.....+ an ) . This is independent of direction (see the More-for-less section). point Xo. In the above example, Xo is the optimal vertex. This completes the proof. *(1(5$/6(16,7,9,7<5(*,21 %HKDYLRURI&KDQJHVLQWKH )25&267&2()),&,(176 5+69DOXHVRIWKH2SWLPDO9DOXH 0$,17$,1,1*7+(9$/,',7< 2)&855(17237,0$/ To study the directional changes in the optimal 9(57(;127³%$6,6´ value with respect to changes in the RHS (with no redundant constraints present, and all RHS ≥ Knowing the unique optimal solution for the 0), we distinguish the following two cases: dual problem by using any LP solver, one may construct the simultaneous sensitivity analysis &DVH, 0D[LPL]DWLRQSUREOHP for all coefficients of the objective function of the primal problem as follows. Assume that the dual For ≤ constraint: The change is in the same problem has n decision variables and m constraints, direction. That is, increasing the value of including the non-negativity conditions (if any). RHS does not decrease the optimal value. It increases or remains the same depending 6WHSVLQ)LQGLQJ6HQVLWLYLW\ on whether the constraint is a binding or 5HJLRQIRUWKH2EMHFWLYH non-binding constraint. )XQFWLRQ&RHIILFLHQWV For ≥ constraint: The change is in the reverse direction. That is, increasing the value of Step 0: Construct and solve the dual problem. RHS does not increase the optimal value. Step 1: Identify the n constraints that are binding It decreases or remains the same depending at optimal solution (that is, at the shadow on whether the constraint is a binding or prices of the primal problem). If there are non-binding constraint. more than n constraints binding, then the For = constraint: The change could be in either primal problem may have multiple solutions. direction (see the More-for-less section). Step 2: Construct the parametric RHS of the constraints, excluding the non-negativity &DVH,, 0LQLPL]DWLRQSUREOHP conditions. Step 3: Solve the parametric system of equations For ≥ type constraint: The change is in the consisting of the binding constraints. This reverse direction. That is, increasing the provides the parametric optimal solution. value of RHS does not increase the optimal Step 4: Construct simultaneous sensitivity value (rather, it decreases or has no change region by plugging in the parametric solu- (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Table 4. The parametric presentations of the RHS of Primal Problem P1: Classifications the binding constraints are as follows: Minimize C(X) = 18X1 + 10X2 Minimization Problem 12U1+ 10U2 = 18 + c1 Subject to: 12X1 + 10X2 > 12000 Sensible Constraint 10U1 + 15U2 = 10 + c2, 10X1 + 15X2 < 15000 Unusual Constraint X > 0 Sensible Condition 1 Solving these parametric equations we obtain X2 > 0 Sensible Condition The Dual Problem of Problem P1 is: U1 = 17/8 +3/16c1 -1/8c2 Maximize 12000U + 15000U Maximization Problem 1 2 U2 = -3/4 -1/8c1 +3/20c2 Subject to: 12U1 + 10U2 < 18 Sensible Constraint Sensible Constraint 10U1 + 15U2 < 10 Plugging this solution into objective function U1 > 0 Sensible Condition of the dual problem, we obtain the parametric U2 < 0 Unusual Condition objective function is 14250 + 375c1 + 750c2 with optimal value of 14250, same as for the optimal value of the primal problem, as expected. Again, tion obtained in Step 3, in all non-binding this parametric optimal solution is subject to constraints, including the sign conditions satisfying the unused constraints; namely, U1 ≥ 0 such as non-negativity conditions (if any). and U2 ≤ 0. These produce the following largest sensitivity region for the objective function coef- We demonstrate the method in our example. The ficients of the primal problem P1, simultaneously: dual problem of our numerical example is con- structed in Table 4. 3/16c1 -1/8c2 ≥ -17/8 and -1/8c1 +3/20c2 ≤ 3/4. The components of the optimal solution to the dual problem are the shadow prices to the original Notice that this is the largest sensitivity region (primal) LP model, and vise versa. The shadow for cost coefficients, allowing for simultaneous, dependent/independent changes. This convex prices of the primal, i.e., (U2 = 17/8, U2 = -3/4), as calculated earlier (or one can solve it afresh region is non-empty because it contains the origin without referring to previous section). The optimal (c1= 0, c2=0), with a vertex at (c1= -18, c2 = -10) value for the dual 12000(17/8) + 15000(-3/4) = which is the negative of the both cost coefficients. 14250 that is equal to the optimal value of the The ordinary sensitivity range (one change at primal problem, as always expected. The prop- a time) of the coefficient of first decision vari- erty that let the primal and the dual optimal values able X1, currently at 18, can be found by setting to be equal is known as strong duality, i.e., there c2 = 0 in the above sensitivity region inequality is no gape between the dual and primal problem. set, yielding c1 > -6 and c1 > -34/3. Therefore the The slacks of the first two constraints are 12(17/8) allowable decrease for the coefficient of X1 is 6 + 10(-3/4) -18 = 0, 10(17/8) + 15(-3/4) -10 = 0, with unbounded (i.e., Big-M) allowable increase. which are the “Reduce Cost” of the primal prob- Similarly, the sensitivity range of the coefficient of lem. second decision variable X2, currently at 10, can To find the ranges for the objective function be found by setting c1 = 0 in the above sensitivity coefficients, one may use the RHS parametric region inequality set, yielding c2 ≤ 5 and c2 ≤17. version of the dual problem; Hence, c2 ≤ 5, therefore, the allowable increase for the coefficient of X1 is 5 with unbounded al- lowable decrease, (i.e., - Big-M). (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Figure 4. Problem P3 feasible region The optimal solution, (X1, X2) = (1, 1), is determined by the two main constraints. The iso- value objective function line and first constraint boundary line have negative slopes, while the other constraint has positive slope. Bracketing the objective function slope between the slopes of the two constraints gives -1 ≤ - C1/C2 ≤ 1, which is an incorrect range. Notice that, not even the cur- rent objective function slope -5/3 belongs to this range. The main reason is that in this problem to transition the slope from negative to positive it has to go through being very large and negative, then undefined, and then very large and positive, as opposed to go through zero. Then, the correct ranges are C ≥ 3, and -5 ≤ C ≤ 5, respectively, )LQGLQJWKH&RVW6HQVLWLYLW\ 1 2 5DQJHE\WKH*UDSKLFDO0HWKRG found by equating the ratio of coefficients from &RXOG%H0LVOHDGLQJ the objective function plus and from each of the constraints as follow: It is a commonly held belief that one can com- (5+c1)/1= 3/1 gives c1 = -2 pute the cost sensitivity range by bracketing the (5+c )/1= 3/-1 gives c = -8 slope of the (iso-value) objective function by the 1 1 slopes of the two lines resulting from the binding where c1 represents the largest possible change in constraints. This graphical slope-based method the coefficient. A positive number represents the to compute the sensitivity ranges is described largest possible increase, while a negative number in popular textbooks, such as Anderson et al., represents the largest possible decrease. (2007), Lawrence and Pasternack (2002), and Therefore, since c = -2 is more restrictive than Taylor (2010). However, this approach is only valid 1 c1= -8, you may decrease C1 = 5, by 2. There is when the objective function and the two binding no lower limit, thus C ≥ 3. constraints have all slopes with the same sign, 1 On the other hand, for C2 we obtain an upper that is, all three functions have negative slopes and lower bound or all three functions have positive slopes. The approach fails if one of the binding constraints (3+c2)/1= 5/1 gives c2 = 2 has a non-negativity condition or the binding (3+ c )/-1= 5/1 gives c = -8 constraints have opposite signs. The following 2 2 counterexample illustrating the later case. Therefore you may decrease C2 = 3, by 8, -5 ≤ C , and you may increase C = 3, by 2, i.e. C ≤ 5. Problem P3: 2 2 2 Maximize 5X1 + 3X2 6KDGRZ3ULFHVDVWKH Subject to: X1 + X2 ≤ 2 /DJUDQJLDQ0XOWLSOLHUV X1 - X2 ≤ 0 X ≥ 0, X ≥ 0 1 2 The dual solution provides important economical interpretation such as the marginal values of the RHS elements. The elements of the dual solution (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV are known as the Lagrangian multipliers because change in the optimal value caused by any allow- they provide (a tight) bound on the optimal value able increase or decrease in the RHS. of the primal, and vise versa. For example, con- Unfortunately, there are misconceptions re- sidering our numerical example the dual solution garding the definition of the shadow price. One can be used to find a lower tight bound for the such misinterpretation is, “In linear programming optimal value, as follow: Multiply each constraint problems the shadow price of a constraint is the by its corresponding dual solution and then add difference between the optimized value of the them up, we get: objective function and the value of the objective function, evaluated at the optional basis, when the 17/8 [12X1 + 10X2 ≥ 12000] right hand side (RHS) of a constraint is increased -3/4 [10X1 + 15X2 ≤ 15000] by one unit.” “Shadow Prices: The shadow prices ______for a Linear Programming problem are the solu- th 18X1 + 10X2 ≥ 14250 tions to its dual. The i shadow price is the change in the objective function resulting from a one-unit Notice that the resultant on the left side is the increase in the ith coordinate of b. A shadow price objective function of the primal problem, and this is also the amount that an investor would have to lower bound is a tight one, since the optimal value pay for one unit of a resource in order to buy out is 14250. In other words, the objective function the manufacturer.” lower bound is 14250, which is the optimal value Consider the following LP as a counterex- for the minimization of the objective function 18X1 ample: + 10X2. Moreover, the optimal value for both problems, dual and primal is always the same. This Problem P3: fact is referred to as equilibrium of economical Maximize X2 systems, and efficiency in Pareto’s sense between Subject to: X1 + X2 ≤ 2 the Primal and the Dual Problems. Therefore, there 2.5X1 + 4X2 ≤ 10 is no duality gap in linear programming. X1 ≥ 0, X2 ≥ 0 0LVLQWHUSUHWDWLRQRI This problem attains its optimal solution at (0, WKH6KDGRZ3ULFH 2) with an optimal value of 2. Suppose we wish to compute the shadow price of the first resource The Shadow Price tells us how much the objective that is the RHS of the first constraint. Changing function will change if we change the right-hand the RHS of the first constraint by increasing it by side of the corresponding constraint. This is often one unit results in: called the “marginal value”, “dual prices” or “dual value”, for the constraint. Therefore, the shadow Maximize X2 price may not be same as the “Market price, i.e., Subject to: X1 + X2 ≤ 3 real price”. 2.5X1 + 4X2 ≤ 10 For each RHS constraint, the Shadow Price X1 ≥ 0, X2 ≥ 0 tells us exactly how much the objective function will change if we change the RHS of the cor- The new problem has the optimal solution (0, responding constraint within the limits given in 2.5) with an optimal value of 2.5. the sensitivity range on the RHS’s. Therefore, for Therefore, it seems “as-if” the shadow price for each RHS value, the shadow price is the rate of this resource is 2.5 - 2 = 0.5. In fact the shadow (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV price for this resource is 1, which can be found must be careful because the upper limit and lower by constructing and solving the dual problem. limit must be rounded down and up, respectively. The reason for this error becomes evident if we notice that the allowable increase to maintain the validity of the shadow price of the first resource '($/,1*:,7+12167$1'$5' is 0.5. The increase by 1 is beyond the allowable )250/3352%/(06 change on the first RHS value. Now suppose we change the same RHS value &UHDWLQJWKH by, increasing say, by 0.1, which is permissible 1RQ1HJDWLYLW\&RQGLWLRQV (compared with its nominal value of 2), then the optimal value for the new problem is 2.1. There- By default, most LP software such as Management fore the shadow price is (2.1 -2) / 0.1 = 1. We Scientist (1999), and QM for Windows (2003), must be careful when calculating shadow prices assume that all variables are non-negative. using this approach the change must be within To achieve this requirement, convert any unre- allowable limits. stricted variable Xj to two non-negative variables If you wish to compute the shadow price of a by substituting y - Xj for every Xj. This increases RHS when its sensitivity range is not available, the dimensionality of the problem by only one you may obtain the optimal values for at least two (introduce one y variable) regardless of how many perturbations. If the rate of change for both cases variables are unrestricted. gives you the same values, then this rate is indeed If any Xj variable is restricted to be non- the shadow price. As an example, suppose we per- positive, substitute - Xj for every Xj. This reduces turb the RHS of the first constraint by +0.02 and the complexity of the problem. -0.01. Resolving the problem after these changes Solve the converted problem and then substitute using your LP solver, the optimal values are 2.02, these changes back to get the values for the original and 1.09, respectively. Since the optimal value for variables and optimal value is straightforward. the nominal problem (without any perturbation) However, to convert sensitivity analyses including is equal to 2, the rate of change for the two cases the shadow prices are not easy tasks. Consider the are: (2.02 - 2)/0.02 = 1, and (1.09 - 2)/(-0.01) = following LP problem: 1, respectively. Since these two rates are the same, we conclude that the shadow price for the RHS of Problem P4: the first constraint is indeed equal to 1. Maximize X1 +0X2 Subject To: 0DQDJHULDO5RXQG2II(UURUV X1 + X2 ≥ 0 2X1 + X2 ≤ 2 You must be careful whenever you round the value X1 ≥ 0 of shadow prices. For example, the shadow price X2 ≤ 0 of the resource constraint in the above problem is 8/3; therefore, if you wish to buy more of this Following the IAM solution algorithm, opti- resource, you should not pay additional price mal solution is at vertex X1 = 2, X2 = -2, which more than $2.66. Whenever you want to sell any is depicted in Figure 5. unit of this resource, you should not sell it at an Performing the sensitivity analysis based on additional price below $2.67. optimal vertex to maintaining the optimal vertex, A similar error might occur whenever you we construct the following parametric RHS of round the limits on the sensitivity ranges. One binding constraints: (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Figure 5. A typical feasible region for a non- Figure 6. The largest sensitivity region for the standard form LP RHS of first and second constraints X1 + X2 = 0 + R1 The sensitivity region is convex and non-empty, 2X1 + X2 = 2 + R2 containing the origin (0, 0) and its vertex is at (0, -2), as shown in Figure 4. The parametric optimal solution is: Figure 6 presents the largest sensitivity region for the RHS for which the shadow prices remain X1 = 2 - R1 + R2 unchanged, i.e., the optimal dual problem vertex X2 = -2 + 2R1 - R2 remains optimal. The ordinary sensitivity range is R1 ≤ 1 for We plug the parametric solution into objective the first RHS and R2 ≥ -2 for the second, as can function, we have: be verified from the largest sensitivity region for the RHS depicted in Figure 4. X1 = 2 - R1 + R2 We now perform sensitivity analysis on the objective function coefficient. The dual problem The shadow prices are the coefficients of the is constructed in Table 5. parametric optimal function, i.e., U1 = -1 and U2 = The optimal solution is the shadow prices (U1 1, for the RHS of constraints 1 and 2, respectively. = -1, U2 = 1), with optimal value of 0U1 + 2U2 They are the rate of change in optimal value with = 2, same as primal, as expected. respect to changes in the RHS of each constraint. The shadow prices remain valid as long as this U1 + 2U2 = 1 + C1 parametric optimal solution satisfy all the other U1 + U2 = 0 + C2 constraints, i.e., X1 ≥ 0 and X2 ≤ 0. This gives the largest sensitivity region for the RHS. The dual parametric solution is: 2 – R1 + R2 ≥ 0 U1 = -1 – C1 + 2C2 -2 + 2R1 – R2 ≤ 0 U2 = 1 + C1 – C2 (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Table 5. optimal vertex for the primal problem remains The Primal Problem: Classifications optimal. Maximize X +0X Maximization 1 2 2QH&KDQJHDWD7LPH LH2UGLQDU\ Subject To: 6HQVLWLYLW\5DQJH X1 + X2 ≥ 0 Unusual 2X1 + X2 ≤ 2 Sensible Set C2 = 0, then C1 ≥ -1, that is, the allowable X ≥ 0 Sensible 1 decrease is -1, and allowable increase is big-M X2 ≤ 0 Unusual (i.e., a large nonspecific positive number). Set C1 Therefore the dual formulation is: = 0, Then C2 ≤ 1/2, that is, the allowable increase The Dual Problem: is 0.5, and allowable decrease is -big-M. Minimize 0U1 + 2U2 Minimization The parametric objective function is: Subject To: U1 + 2U2 ≥ 1 Sensible 0U1 + 2U2 = 2 + 2C1 – 2C2 U1 + U2 ≤ 0 Unusual With shadow prices (X = 2, X = -2) which U1 ≤ 0 Unusual 1 2 is solution to the primal with optimal value of X1 U2 ≥ 0 Sensible +0X2 = 2, as expected. The solution must satisfy the other con- 62)7:$5(3$&.$*(6 straints: 6+$'2:35,&($1'6(16,7,9,7< $1$/<6,65(32576 -1 – C1 + 2C2 ≤ 0 1 + C – C ≥ 0 1 2 In the previous Sections we solved manually stan- dard form problem P1 and the non-standard form Figure 7 depicts the largest sensitivity region problem P2 and then compared the results with for the coefficients of objective function, i.e., the results obtained by some of the LP packages the optimal solution remains unchanged, i.e., the available in the market. Unfortunately several of these educational LP Figure 7. The largest sensitivity region for the software packages give misleading and incomplete coefficients of objective function results that could increase complexity for the man- ager. For instance, the widely used LINDO (2003) LP software does not provide any direct warnings about the existence of multiple nor degenerate optimal solutions. Or the other hand WinQSB (2003) might indicates “Note: Alternate Solution Exists!!” when in fact there might be none. As a result, the final report on sensitivity analysis might not be valid in these cases, Lin (2010). Similarly, several other LP software packages have their own limitations listed in Tables 6 and 7. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Table 6. Scope and limitations of some popular software Non-Negativity Shadow Prices Optimal Solution to Sensitivity Ranges Conditions non-standard form problems Lindo Imposed on all variables Non-consistent May need conversion May need conversion (2003) by default, but can be which is not an easy task changed to un-restricted WinQSB A variable must be Consistent May need conversion Unreliable (2003) either non-negative or unrestricted. QM for Windows All variables must be Non-consistent May need conversion Difficult to Convert (2003) non-negative Management Scientist All variables must be Non-consistent May need conversion May need conversion (1999) non-negative which is not an easy task Excel Solver All variables are non- Consistent Readily Available Reliable (20013) negative or un-restricted Table 7. The standard form numerical example P1 summary results as a base for comparison with popular software Optimal Objective Allowable Allowable Variables Name Solution Coefficient Increase Decrease X1 375.0 18 M 6 X2 750.0 10 5 M Shadow Slack Allowable Allowable Constraints Price Surplus Increase Decrease 12X1 + 10X2 >= 12000 2.125 0 6000 2000 10X1 + 15X2 <= 15000 - 0.75 0 3000 5000 LINDO produces the output displayed in List- 6KDGRZ3ULFHV0LJKW ing 1 for the first numerical example P1, which +DYH'LIIHUHQW6LJQ is in standard form. In this LINDO final report the dual prices are When reading and interpreting Shadow Prices reported to be U1 = - 2.125, and U2 = 0.75, while from software packages, managers must be es- we found earlier that the shadow prices are U1 = pecially attentive to the software description of 2.125, and U2 = - 0.75. Even the term Dual the analysis. Prices (i.e., Dual solution, are the shadow Prices) There are no common accepted practices in is confusing as the values U1 = - 2.125, and U2 = Sensitivity Reports. For example, LINDO, Excel 0.75 are not a solution to the dual problem. Solver and GAMS (2013) use different notation This misleading result is not limited to LINDO, in their reports. In general, the terms shadow unfortunately, at least two other LP software prices and dual prices are used interchangeable; packages namely Quantitative Methods (QM however, if we compare the reports by LINDO and 2003) and Management Scientist (1999) produce by hand-computation of numerical example P1, misleading results. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Listing 1. Input and output of LINDO software package for the first numerical example /,1'2,1387DQG287387 0LQLPL]H;; 6XEMHFWWR;;! ;; (QG /3237,080)281'$767(3 2%-(&7,9()81&7,219$/8( 9$5,$%/(9$/8(5('8&('&267 ; ; 52:6/$&.256853/86'8$/35,&(6 12,7(5$7,216 5$1*(6,1:+,&+7+(%$6,6,681&+$1*(' 2%-&2()),&,(175$1*(6 9$5,$%/( &855(17&2() $//2:$%/(,1&5($6( $//2:$%/('(&5($6( ; ,1),1,7< ; ,1),1,7< 5,*+7+$1'6,'(5$1*(6 52: &855(175+6 $//2:$%/(,1&5($6( $//2:$%/('(&5($6( we see that the dual prices from LINDO have the cost of providing one additional …in additional opposite sign of the shadow prices from Solver. salary expense.” “Dual prices are sometimes The Lindo (2013) Manual gives the following called shadow prices, because they tell you how justification and interpretation: much you should be willing to pay for additional units of a resource. Going back to the Friday con- …improve is a relative term. In a minimization straint, if we use the shadow price interpretation problem, such as our example, interpreting the of the dual price, we can also say that we should dual price requires some thought. The dual price be willing to pay …. to have an additional... If …, is ….-. This means raising the right-hand side …available for less than .., then we might want of the… constraint by one unit would cause the to consider this option.” “As with reduced costs, objective to “improve” by negative 100. That is, dual prices are valid only over a limited range. it would increase ... In other words, the marginal (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Notice that unfortunately Excel Manual (2013) on the vocabulary and structure on the output of also uses the terms like “unit change”, “unit in- optimization software packages. crease” while defining shadow prices. Using Excel LP Solver produces the output shown in Listing 3 for the non-standard form The dual value for a constraint is nonzero only numerical example. (see Table 8) when the constraint is equal to its bound. This is The results are comparable with our hand called a binding constraint, and its value was driv- computation. However one must be careful in any en to the bound during the optimization process. generalization, since this comparison is done on Moving the constraint left hand side’s value away a specific numerical example. from the bound will worsen the objective function’s WinQSB Software produces the following value; conversely, “loosening” the bound will outputs for the non-standard form numerical ex- improve the objective. The dual value measures ample. The results with all constraints in explicit the increase in the objective function’s value per form, and then selecting the unrestricted option unit increase in the constraint’s bound.…. In the are shown in Listing 4. case of linear problems, the dual values remain The sensitivity ranges are not correct. They constant over a range are differed from the results already obtained by hand computations. Notice that the change by one unit may not be within the range, see sub-section 7.4. 3RRU6FDOLQJDQG8QVWDEOH Just to add to the confusion, other packages such 6RIWZDUH6ROXWLRQV as GAMS report marginal’s without any informa- tion about their range of validity. (see Listing 2) When modeling a business problem, different Subtle discrepancies like this one are unfor- conditions might require us to use parameters or tunate. There is a pounding need for consensus inputs having very different order of magnitudes. For example, an LP from manufacturing industry Listing 2. Excerpt from GAMS output for the first numerical example that is in standard form /2:(5 /(9(/ 833(5 0$5*,1$/ (48FRQVWU ,1) (48FRQVWU ,1) Table 8. Non-standard form numerical example P2 summary results Optimal Objective Allowable Allowable Variables Name Solution Coefficient Increase Decrease X1 2 1 M 1 X2 -2 0 0.5 M Shadow Slack Allowable Allowable Constraints Name Price Surplus Increase Decrease Constraint 1: X1 + X2 >= 0 -1 0 1 M Constraint 2: 2X1 + X2 <= 2 1 0 M 2 (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV Listing 3. Excel result with all constraint in explicit form, and then selecting the unrestricted option Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease X1 2 0 1 1E+30 1 X2 -2 0 0 0.5 1E+30 Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease Constraint 3: X1 >= 0 2 0 0 2 1E+30 Constraint 4: X2 <= 0 -2 0 0 1E+30 2 Constraint 1: X1 + X2 >= 0 0 -1 0 1 1E+30 Constraint 2: 2X1 + X2 <= 0 2 1 2 1E+30 2 Listing 4. WinQSB Input and Its Combined report /3,QSXWLQ0DWUL[)RUPDW 9DULDEOH ; ; 'LUHFWLRQ 5+6 0D[LPL]H &RQVWUDLQW ! &RQVWUDLQW &RQVWUDLQW ! &RQVWUDLQW /RZHU%RXQG 0 0 8SSHU%RXQG 0 0 9DULDEOH7\SH ;8QUHVWULFWHG;8QUHVWULFWHG 'HFLVLRQ9DULDEOH 6ROXWLRQ9DOXH 8QLW3URILW $OORZDEOH0LQ $OORZDEOH0D[ &RQWULEXWLRQ F M F M ; ; 0 2EMHFWLYH)XQFWLRQ 0D[ 1RWH $OWHUQDWH6ROXWLRQ([LVWV &RQVWUDLQW 6KDGRZ3ULFH $OORZDEOH0LQ5+6 $OORZDEOH0D[5+6 &RQVWUDLQW 0 &RQVWUDLQW 0 &RQVWUDLQW 0 &RQVWUDLQW 0 (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV might include dollar amounts in millions and at Solver might report as the optimal solution X1 the same time might include quality based con- = 1, X2 = 10, and X3 = 0. This solution violates ditions in the range of nanometers (one billionth constraint (3). The correct solution is X1 = 1, X2 of a meter). If an LP is modeled using the such = 10, and X3 =1. Therefore, we have numerical units then the solvers such as Lindo and Excel instability caused by too large values. gives us warning about poor scaling but then also Excel recommends that to avoid such occur- gives some updated values. Frontline Solvers (the rences a model must only be allowed to vary by developers of Excel Solvers) admit to this limita- an order of magnitude of 3 (i.e. 1 to 1000). Excel tion and the website reports that: allows the use of “Use Automatic Scaling” feature for automatically rescaling the values but it cannot The effects of poor scaling in a large, complex rescale the problems “intelligently”. Hence if the optimization model can be among the most dif- automatic rescaling feature is used the outcomes ficult problems to identify and resolve. It can could unknowingly become worse rather than cause Solver to return messages such as ‘Solver better. Automatic rescaling can also slowly down could not find a feasible solution’, ‘Solver could the optimization process significantly (Winston not improve the current solution,’ or even ‘The and Albright, 2011). Similar limitations exist for linearity conditions required by this LP Solver are Lindo also (Martin 1999) and may exist for several not satisfied,’ with results that are suboptimal or other solvers as well. However these limitations otherwise very different from your expectations. are largely and conveniently ignored. (Frontline Solvers, 2013) One of the possible reasons for such ignorance could be due to the high expectations and the prom- Unfortunately, Solver does not always alert the ise of convenience offered by LPs. Using LPs, a user about any issues with the problem. The fol- manager can get an optimal solution even when the lowing example clearly is prone to scaling issues problem includes a huge variety of factors (inputs with a very large coefficient for X3 in the third and constraints) which could be coming from a constraint. (Depending on the computer a larger or wide variety of domains. For large and complex smaller coefficient yields the issue describe here). LPs, the managers and the coders would want to However, when the problem is attempted with reduce the complexity associated with LPs and Solver, Solver sometimes yields an error message would prefer not to change the units for the sake of as the ones described earlier, but in most cases, convenience. For example when developing LPs it does not give any error message and reports as for manufacturing sector a manager (and even the optimal an infeasible solution. coders) might prefer to use dollars values directly in the model instead of converting them to mil- Numerical Example: lions (which reduces the order of magnitude by Minimize X2 six but then increase the chance of getting optimal Subject to: dollar values in decimals) and use it with values 10X1 – X2 ≤ 0 of quality tolerance in a fraction of millimeter. X1 = 1 Such instances are common and widespread. The X1 – 100000 authors lament that majority of books violates the X3 ≤ 0 X3 ≤1 ideal requirement of order of magnitude in the X3 is positive integer discussions and leaves the mangers very much vulnerable to being unknowingly dependent on sub-optimal results. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV The best approach recommended to deal with This problem has a unique, non-degenerate opti- this situation is to use consistent scaling at the mal solution at (X1 = 2, X2 = 3). However, if one time of modeling. Managers should be aware of rewrites the equality constraint in the form of two this issue and use consistent units for products and inequalities, the equivalent problem is: constraints. For example, if working with multiple monetary constraints they all should be scale in Maximize X2 hundreds or thousands. Alternative if working Subject to: with multiple products similar units should be X1 + X2 ≤ 5 used for all the products. X1 + X2 ≥ 5 -X1 + X2 ≤ 1 X1 ≥ 0, X2 ≥ 0 0$,17$,1,1*$35,0$/ '(*(1(5$7(237,0$/ Clearly, the optimal solution (2, 3) now makes 9(57(;237,0$/ three constraints binding. Three is more than the dimension n=2 of this problem; therefore one Almost all research work in LP literature starts may incorrectly conclude that this solution is a with an assumption of a non-degenerate primal/ degenerate optimal solution. This simple numeri- dual LP. Degeneracy is the case of having multiple cal example serves the purpose of the necessary vertices at the same point. (but not a sufficient) condition for a degenerate optimal solution. The necessary condition for the existence of LP An occurrence of degeneracy can have a signifi- degeneracy: If at optimal point the number of bind- cant impact on the shadow prices and may cause ing constraints is more than n (i.e., the dimension oddities in the output provided by LP solvers. of the problem) then the solution to an LP might Moreover, the usual sensitivity analyses do not be degenerate. provide complete information in the degenerate case; that is, the information one obtains from Resolution: Double-check the coefficients of all most LP packages is a subset of the true sensitivity constraints including the RHS values. There could intervals. There are more effective approaches to have been rounding error. cope with this problem; however, they are com- putationally much more involved, Lin (2010). )DOVH'HJHQHUDWH Almost all software packages do not alert the user 2SWLPDO6ROXWLRQ that the problem is degenerate. Consider the following n=2 dimensional LP Numerical Example P4: problem: Maximize C(X) = X1 + X2 Subject to X1 ≤ 1 Numerical Example P2: X2 ≤ 1 Maximize X2 X1 + X2 ≤ 2 Subject to: X1 ≥ 0 X1 + X2 = 5 X2 ≥ 0 -X1 + X2 ≤1 X1 ≥ 0, X2 ≥ 0 The optimal solution is X1 = 1, and X2 = 1, at which all three constraints are binding. We are interested in finding out how far the RHS of (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV binding constraints can change while maintain- degenerate therefore the sensitivity and shadow ing the degenerate optimal solution. Because of prices are valid. degeneracy, there is dependency among these constraints at the binding position, that is: 0$,17$,1,1*7+(08/7,3/( X1 = 1 237,0$/9(57,&(6237,0$/ X2 = 1 X1 + X2 = 2 The necessary condition for the existence of LP multiple solutions: If in the “software final In order to maintain the degenerate vertex while report” the total number of zeros in the Reduced changing the RHS values, the RHS proportionately Cost, together with number of zeros in the Shadow must be changed to the coefficients of the decision Price columns, exceeds the number of constraints, variables. The resulting parametric RHS system then you might have multiple solutions. Sensitivity of equations is as follows: analysis is not applicable. That is, the sensitiv- ity analysis based on one optimal solution may X1 = 1 + 1r1 + 0r2 not be valid for the others, Lin (2010). Software X2 = 1 + 0r1 + 1r2 package like LINDO alert the user if a problem X1 + X2 = 2 + 1r1 + 1r2 has multiple solutions, while others like Excel LP Solver do not. The parameters on the RHS allow for the de- generate vertex to move maintaining its structure. Resolution: Check the coefficients in the objective Since there are two decision variables, any two function and the coefficients of the constraints. of the parametric equations can be used to find There could have been rounding error. the parametric degenerate optimal solution. For example, the first two equations provide (X = 1 1 Numerical Example P5: + r1, X2 = 1 + r2), which clearly satisfies the third one. As mentioned earlier, for larger problems Consider the following LP: one may use JavaScript software: http://home. ubalt.edu/ntsbarsh/Business-stat/otherapplets/ Maximize C(X) = 6X + 4X PaRHSSyEqu.htm 1 2 Subject to X1 + 2X2 ≤ 16 The perturbed optimal value is C(X) = X1 + 3X1 + 2X2 ≤ 24 X2 = 2 + r1 + r2. In order for this vertex to remain X1 ≥ 0 optimal, it must satisfy all other constraints that X ≥ 0 have not been used. For this numerical example, 2 the non-negativity conditions are the remaining This LP has two optimal vertices; namely, (X1 constraints; therefore, we obtain the following = 8, X = 0) and (X = 4, X = 6). Notice that conditions for r and r : 2 1 2 1 2 the existence of two solutions means we have in- numerable optimal solutions. For example, for the {r and r | r ≥ -1, r ≥ -1}. 1 2 1 2 above problem, for all parameter a, 0 ≤ a ≤ 1, the following convex combinations are also optimal: Notice that while the changes satisfy these conditions the degenerate solution remain degen- X1 = 8a + 4(1 - a) = 4 + 4a, erate, for any change outside of this set ensures X = 0a + 6(1- a) = 6 - 6a. the manager that the solution is not any longer 2 (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV The Dual LP is: manager that the problem does not any longer have multiple solutions therefore the sensitivity Minimize 16U1 + 24U2 and shadow prices are valid. Subject to U1 + 3U2 ≥ 6 2U1 + 2U2 ≥ 4 7+(025()25/(6625 U1 ≥ 0 /(66)25025(6,78$7,216 U2 ≥ 0 This dual LP has a degenerate optimal solu- A curious property called the more-for-less (MFL) tion (U1 = 0, U2 = 2). In order to maintain the or less-for-more (LFM) phenomenon is associated degenerate vertex while changing the RHS, the with some linear programs (LP). The existing RHS values must be changed in proportion to literature has demonstrated the practicality and the coefficients of the decision variables. The value of identifying cases where the paradoxical parametric RHS system of equations is: situation exists, Arsham (1996). For example, consider the following production LP. U1 + 3U2 = 6 + 1c1 + 3c2 2U1 + 2U2 = 4 + 2c1 + 2c2 1XPHULFDO([DPSOH3 U1 = 0 + 1c1 Maximize C(X) = X1 + 3X2 + 2X3 As before, any two of the parametric equa- Subject to X1 + 2X2 + X3 = 4 tions can be used to find the parametric degener- 3X1 + 2X3 = 9 ate optimal solution. For example, the first two X1 ≥ 0, X2 ≥ 0, X3 ≥ 0 equations provide U1 = c1 and U2 = 2 + c2, which clearly satisfies the third equation. For this vertex The optimal solution for this LP occurs at vertex to remain optimal, it must satisfy all other con- X1 = 1, X2 = 0, X3 = 3, with the maximum value straints that have not been used. For this example, of objective function as $7. the non-negativity condition, U2 ≥ 0, is the only Note that, if the RHS of second constraint is remaining constraint. Therefore, we obtain the increased to 12, then the new optimal solution gives following conditions for c1 and c2: the optimal value of C(X) = $4; i.e., a decrease in profit while working more hours. This situation {c1, c2 | c2 ≥ -2}. arises frequently in LP models and is known as the “more-for-less” paradox where further analysis Moreover, the perturbed cost coefficients (6 could bring significant reduction in costs. We ap- + c1)X1 + (4 + c2)X2 must be proportional to its ply the primal sensitivity algorithm to solve the parallel constraint 3X + 2X ≤ 24, i.e., (6 + c )/3 1 2 1 RHS perturbed equation constraints with X2 = 0, = (4 + c2)/2. This simplifies to 2c1 = 3c2. Putting together all these conditions, we obtain the largest X1 + X3 = 4 + r1 sensitivity set as follows: 3X1 + 2X3 = 9 + r2 {c , c | c ≥ -2, 2c = 3c }. 1 2 2 1 2 The parametric solution is X1 = 1 – 2r1 + r2, X2 = 0, and X3 = 3 + 3r1 - r2, with the optimal Notice that, while the changes satisfy these parametric solution as C(X) = 7 + 4r1 - r2. Plug- conditions, the problem has multiple solutions; ging the parametric solution into other non-binding for any change outside of this set ensures the constraints, we obtain (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV X1 = 1 – 2r1 + r2 ≥ 0 being determined by the market conditions and X3 = 3 + 3r1 - r2 ≥ 0 competition. The proposed method can be used to optimize LP problems with varying objective Looking at the parametric optimal function, no- function. Given a system of linear equalities and/ tice that the shadow price of the second constraint, or inequalities, the method provides all vertices coefficient of r2, is negative. To find out the best of the feasible region. By means of examples, we number of hours, one must work to maximize the have illustrated the use of the Improved Algebraic profit function 7 + 4r1 - r2, by setting r1 = 0 and Method to efficiently derive the slack and surplus finding the largest negative value for r2. Therefore, amounts for the resources of an LP. The parametric the constraints reduce to: representation quickly provides all dual prices to carry out analysis for desirability of obtaining ad- X1 = 1 + r2 ≥ 0 ditional resources. The parametric representation X3 = 3 - r2 ≥ 0. also allows one to study variability (i.e, uncertain- ties) of the coefficients of the objective functions The largest negative value is r2 = -1. This gives and the right-hand-side of constraints. the optimal solution of (X1 = 0, X2 = 0, X3 = 4) We provide a comprehensive managerial with the optimal value of C(X) = 8. Therefore, coverage of linear programming post-optimality the optimal strategy is to work 8 hours instead analysis. The collections of presented tools are of 9 hours. easy to understand, easy to implement (for small size problems), and provide useful information to Proposition 3: A necessary and sufficient condi- the manager. Therefore this collection could prove tion for the existence of a less-for-more (more-for- valuable to involve the manager throughout the less) solution to a maximization (minimization) decision-making process to understand therefore problem is the existence of an equality constraint being implemented. with a negative shadow price. It is the proposed IAM approach to solve a sys- tem of inequalities that provides a bridge between Theorem 1: We are in LFM or MFL situation the graphical method and the simplex method. iff at least one of ri’s in the above parametric LP The simplex method is an efficient computer formulation is negative [positive] implementation of algebraic methods and almost all the LP software use it. All the LP tools are covered within the deci- &21&/86,21 sion variable space; no additional variables such as slack/surplus/artificial variables or constraints A business environment is dynamic. A problem are added. solution is valid in a limited time window only The proposed approach provides useful infor- and is subject to revision in the next time win- mation for the manager such as shadow prices and dow. Generally, a constraint set is less subject to expands the sensitivity analysis scope by proving change as compared to the objective function. For sensitivity regions rather than ranges that LP soft- example, in production and transportation prob- ware provides. Under the proposed methods, the lems, the capacity constraints may remain rather elaborate fundamental LP theorems fall naturally stable over a period of time. On the other hand, out as by-products. Moreover, it can also be used profit coefficients of the objective function are to fill the gap between the graphical method of inversely related to the price, which may fluctuate, solving LP problems and the simplex method. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV $&.12:/('*0(17 Arsham, H. (2008). A validation and verification tool for optimization solvers. Journal of Informa- We are most appreciative to the reviewers for tion & Optimization Sciences, 29, 57–80. doi:10. their useful comments and suggestions on an 1080/02522667.2008.10699791 earlier version. Arsham, H. (2009). A simplified algebraic method for system of linear inequalities with LP 5()(5(1&(6 applications. Omega: International Journal of Management Science, 37, 876–882. doi:10.1016/j. Anderson, D., Sweeney, D., Williams, T., & Mar- omega.2008.06.001 tin, R. (2007). An introduction to management Arsham, H. (2012). Foundation of linear program- science: Quantitative approaches to decision ming: A managerial perspective from solving making. New York, NY: South-Western College system of inequalities to software implementation. Publisher. International Journal of Strategic Decision Sci- ences 3 Arsham, H. (1990). Perturbation analysis of , (3), 40–60. doi:10.4018/jsds.2012070104 general LP models: A unified approach to sen- Arsham, H. (2013). An interior boundary pivotal stivity parametric, tolerance, and more-for-less solution algorithm for linear programs with the analysis. Mathematical and Computer Modelling, optimal solution-based sensitivity region. Inter- 13, 79–102. doi:10.1016/0895-7177(90)90073-V national Journal of Mathematics in Operational Research. Arsham, H. (1996). On the more-for-less paradoxi- cal situations in linear programs: A parametric (2013). Excel: Basic Solve, Frontline Solvers. optimization approach. Journal of Information New York, NY: Author. & Optimization Sciences, 17, 485–500. doi:10.1 GAMS: A User’s Guide 080/02522667.1996.10699297 (2013). . Create Space Independent Publishing Platform. Arsham, H. (1997a). An artificial-free simplex al- gorithm for general LP models. Mathematical and Koltai, T., & Terlaky, T. (2000). The difference Computer Modelling, 25, 107–123. doi:10.1016/ between the managerial and mathematical inter- S0895-7177(96)00188-4 pretation of sensitivity analysis results in linear programming. International Journal of Produc- Arsham, H. (1997b). Initialzation of the simplex al- tion Economics, 65, 257–274. doi:10.1016/S0925- gorithm: An artifical-free approach. SIAM Review, 5273(99)00036-5 39, 736–744. doi:10.1137/S0036144596304722 Lawerence, J., & Pasternack, B. (2002). Applied Arsham, H. (2006). A big-m free soltuion algo- management science: Modeling, spreadsheet rithm for general linear programs. Internation analysis, and communication for decision making. Journal of Pure and Applied Mathematics, 32, New York, NY: Wiley. 549–564. Lin, C. (2010). Computing shadow proces/costs Arsham, H. (2007). A computationally stable of dengernerate LP problems with reduced sim- solution algorithm for large-scalelinear programs. plex tables. Expert Systems with Applications, Applied Mathematics and Computation, 188, 37, 5848–5855. doi:10.1016/j.eswa.2010.02.021 1549–1561. doi:10.1016/j.amc.2006.11.031 LINDO. (2003). LINDO users’s manual. New York, NY: Author. (VVHQWLDOVRI/LQHDU3URJUDPPLQJIRU0DQDJHUV (1999). Management Scientist. New York, NY: Taylor, B. (2009). Introduction to management South-Western Educational Publishing. science. New York, NY: Prentice Hall. Martin, K. (1999). Large scale linear and integer (2003). WinQSB Manual. New York, NY: Wiley. optimization: A unified approach. New York, NY: Winston, W. W. L., & Albright, S. (2011). Practi- Springer. doi:10.1007/978-1-4615-4975-8 cal management science. New York, NY: Cengage (2003). QM Windows Manual. New York, NY: Learning. Prentice Hall.