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10.3 Goal Programming

In today’s business environment, profit maximization or cost minimization is not always the only objective that a firm sets forth. Often, maximizing total profit is just one of several goals, including such contradictory objectives as maximizing market share, maintaining full employment, providing quality ecological management, minimizing noise level in the neighborhood, and meeting numerous other noneconomic goals.

The shortcoming of mathematical programming techniques such as linear and integer programming is that their objective function is measured in one dimension only. It’s not possible for LP to have multiple goals unless they are all measured in the same units (such as dollars), a highly unusual situation. An important technique that has been developed to supplement LP is called goal programming.

Goal programming is capable of handling decision problems involving multiple goals. It began with the work of Charnes and Cooper in 1961 and was refined and extended by Lee in 1972 and by Ignizio in 1985 (see the Bibliography).

In typical decision-making situations, the goals set by management can be achieved only at the expense of other goals. It is necessary to establish a hierarchy of importance among these goals so that lower-priority goals are tackled only after higher-priority goals are satisfied. Since it is not always possible to achieve every goal to the extent the decision maker desires, goal programming attempts to reach a satisfactory level of multiple objectives. This, of course, differs from LP, which tries to find the best possible outcome for a single objective. Nobel Laureate Herbert A. Simon, of Carnegie-Mellon University, states that modern managers may not be able to optimize but may instead have to “satisfice” or “come as close as possible” to reaching goals. This is the case with models such as goal programming.

In Action Continental Airlines Saves $40 Million Using CrewSolver

Airlines use state-of-the-art processes and automated tools to develop schedules that maximize profit. These schedules will assign aircraft to specific routes and then schedule pilots and flight attendant crews to each of these aircraft. When disruptions occur, planes and personnel are often left in positions where they are unable to adhere to the next day’s assignments. Airlines face schedule disruptions from a variety of unexpected reasons such as bad weather, mechanical problems, and crew unavailability.

In the 1990s, Houston-based Continental Airlines (now part of United Airlines) began an effort to develop a system of dealing with disruptions in real time. Working with CALEB Technologies, Continental developed the CrewSolver and OptSolver systems (based on 0–1 integer programming models) to produce comprehensive recovery solutions for both aircraft and crews. These solutions retain revenue and promote customer satisfaction by reducing flight cancellations and minimizing delays. These crew recovery solutions are low cost while maintaining a high quality of life for pilots and flight attendants.

In late 2000 and throughout 2001, Continental and other airlines experienced four major disruptions. The first two were due to severe snowstorms in early January and in March 2001. The Houston floods caused by Tropical Storm Allison closed a major hub in June 2001 and left aircraft in locations where they were not scheduled to be. The terrorist attacks of September 11, 2001, left aircraft and crews scattered about and totally disrupted the flight schedules. The CrewSolver system provided a faster and more efficient recovery than had been possible in the past. It is estimated that the CrewSolver system saved approximately $40 million for these major disruptions in 2001. This system also saved additional money and made recovery much easier when there were minor disruptions due to local weather problems at other times throughout the year.

Source: Based on Gang Yu, Michael Arguello, Gao Song, Sandra M. McCowan, and Anna White, “A New Era for Crew Recovery at Continental Airlines,” Interfaces 33, 1 (January–February 2003): 5–22, © Trevor S. Hale.

How, specifically, does goal programming differ from LP? The objective function is the main difference. Instead of trying to maximize or minimize the objective function directly, with goal programming we try to minimize deviations between set goals and what we can actually achieve within the given constraints. In the LP simplex approach, such deviations are called slack and surplus variables. Because the coefficient for each of these in the objective function is zero, slack and surplus variables do not have an impact on the optimal solution. In goal programming, the deviational variables are typically the only variables in the objective function, and the objective is to minimize the total of these deviational variables.

When the goal programming model is formulated, the computational algorithm is almost the same as a minimization problem solved by the simplex method.

Example of Goal Programming: Harrison Electric Company Revisited

To illustrate the formulation of a goal programming problem, let’s look back at the Harrison Electric Company case presented earlier in this chapter as an integer programming problem. That problem’s LP formulation, you recall, is

\begin{array}{l} Maximize profit & = & $ 7 X_{1} & + & $ 6 X_{2} \\ subject to & 2 X_{1} & + & 3 X_{2} & \leq & 12 & (wiring hours) \\ 6 X_{1} & + & 5 X_{2} & \leq & 30 & (assembly hours) \\ X_{1}, X_{2} & \geq & 0 \end{array}

$\begin{array}{l} Maximize profit & = & $ 7 X_{1} & + & $ 6 X_{2} \\ subject to & 2 X_{1} & + & 3 X_{2} & \leq & 12 & (wiring hours) \\ 6 X_{1} & + & 5 X_{2} & \leq & 30 & (assembly hours) \\ X_{1}, X_{2} & \geq & 0 \end{array}$

where

\begin{array}{l} X_{1} & = & number of chandeliers produced \\ X_{2} & = & number of ceiling fans produced \end{array}

$\begin{array}{l} X_{1} & = & number of chandeliers produced \\ X_{2} & = & number of ceiling fans produced \end{array}$

We saw that if Harrison’s management had a single goal—say, profit—LP could be used to find the optimal solution. But let’s assume that the firm is moving to a new location during a particular production period and feels that maximizing profit is not a realistic goal. Management sets a profit level, which would be satisfactory during the adjustment period, of $30. We now have a goal programming problem in which we want to find the production mix that achieves this goal as closely as possible, given the production time constraints. This simple case will provide a good starting point for tackling more-complicated goal programs.

We first define two deviational variables:

\begin{array}{l} d_{1}^{-} & = & underachievement of the profit target \\ d_{1}^{+} & = & overachievement of the profit target \end{array}

$\begin{array}{l} d_{1}^{-} & = & underachievement of the profit target \\ d_{1}^{+} & = & overachievement of the profit target \end{array}$

Now we can state the Harrison Electric problem as a single-goal programming model:

\begin{array}{l} Minimize under-or overachievement of profit target & = & d_{1}^{-} + d_{1}^{+} \\ subject to $ 7 X_{1} + $ 6 X_{2} + d_{1}^{-} - d_{1}^{+} & = & $ 30 (profit goal constraint) \\ 2 X_{1} + 3 X_{2} & \leq & 12 (wiring hours constraint) \\ 6 X_{1} + 5 X_{2} & \leq & 30 (assembly hours constraint) \\ X_{1}, X_{2}, d_{1}^{-}, d_{1}^{+} & \geq 0 \end{array}

$\begin{array}{l} Minimize under-or overachievement of profit target & = & d_{1}^{-} + d_{1}^{+} \\ subject to $ 7 X_{1} + $ 6 X_{2} + d_{1}^{-} - d_{1}^{+} & = & $ 30 (profit goal constraint) \\ 2 X_{1} + 3 X_{2} & \leq & 12 (wiring hours constraint) \\ 6 X_{1} + 5 X_{2} & \leq & 30 (assembly hours constraint) \\ X_{1}, X_{2}, d_{1}^{-}, d_{1}^{+} & \geq 0 \end{array}$

Note that the first constraint states that the profit made, $$ 7 X_{1} + $ 6 X_{2}$ $$ 7 X_{1} + $ 6 X_{2}$ , plus any underachievement of profit minus any overachievement of profit has to equal the target of $30. For example, if $X_{1} = 3$ $X_{1} = 3$ chandeliers and $X_{2} = 2$ $X_{2} = 2$ ceiling fans, then $33 profit has been made. This exceeds $30 by $3, so $d_{1}^{+}$ $d_{1}^{+}$ will be equal to 3. Since the profit goal constraint was overachieved, Harrison did not underachieve, and $d_{1}^{−}$ $d_{1}^{−}$ will be equal to zero. This problem is now ready for solution by a goal programming algorithm.

If the target profit of $30 is exactly achieved, we see that both $d_{1}^{+}$ $d_{1}^{+}$ and $d_{1}^{−}$ $d_{1}^{−}$ are equal to zero. The objective function will also be minimized at zero. If Harrison’s management was concerned only with underachievement of the target goal, how would the objective function change? It would be as follows: minimize underachievement $= d_{1}^{−}$ $= d_{1}^{−}$ . This is also a reasonable goal, since the firm would probably not be upset with an overachievement of its target.

In general, once all goals and constraints are identified in a problem, management should analyze each goal to see if underachievement or overachievement of that goal is an acceptable situation. If overachievement is acceptable, the appropriate $d^{+}$ $d^{+}$ variable can be eliminated from the objective function. If underachievement is okay, the $d^{−}$ $d^{−}$ variable should be dropped. If management seeks to attain a goal exactly, both $d^{−}$ $d^{−}$ and $d^{+}$ $d^{+}$ must appear in the objective function.

Extension to Equally Important Multiple Goals

Let’s now look at the situation in which Harrison’s management wants to achieve several goals, each equal in priority.

Goal 1: to produce profit of $30 if possible during the production period
Goal 2: to fully utilize the available wiring department hours
Goal 3: to avoid overtime in the assembly department
Goal 4: to meet a contract requirement to produce at least seven ceiling fans

The deviational variables can be defined as follows:

\begin{array}{l} d_{1}^{-} & = & underachievement of the profit target \\ d_{1}^{+} & = & overachievement of the profit target \\ d_{2}^{-} & = & idle time in the wiring department (underutilization) \\ d_{2}^{+} & = & overtime in the wiring department (overutilization) \\ d_{3}^{-} & = & idle time in the assembly department (underutilization) \\ d_{3}^{+} & = & overtime in the assembly department (overutilization) \\ d_{4}^{-} & = & underachievement of the ceiling fan goal \\ d_{4}^{+} & = & overachievement of the ceiling fan goal \end{array}

$\begin{array}{l} d_{1}^{-} & = & underachievement of the profit target \\ d_{1}^{+} & = & overachievement of the profit target \\ d_{2}^{-} & = & idle time in the wiring department (underutilization) \\ d_{2}^{+} & = & overtime in the wiring department (overutilization) \\ d_{3}^{-} & = & idle time in the assembly department (underutilization) \\ d_{3}^{+} & = & overtime in the assembly department (overutilization) \\ d_{4}^{-} & = & underachievement of the ceiling fan goal \\ d_{4}^{+} & = & overachievement of the ceiling fan goal \end{array}$

Management is unconcerned about whether there is overachievement of the profit goal, overtime in the wiring department, or idle time in the assembly department and whether more than seven ceiling fans are produced: hence, $d_{1}^{+}, d_{2}^{+}, d_{3}^{−}$ $d_{1}^{+}, d_{2}^{+}, d_{3}^{−}$ , and $d_{4}^{+}$ $d_{4}^{+}$ may be omitted from the objective function. The new objective function and constraints are

\begin{array}{l} Minimize total deviation & = & d_{1}^{-} + d_{2}^{-} + d_{3}^{+} + d_{4}^{-} \\ subject to {7X}_{1} + 6 X_{2} + d_{1}^{-} - d_{1}^{+} & = & 30 (profit constraint) \\ 2 X_{1} + 3 X_{2} + d_{2}^{-} - d_{2}^{+} & = & 12 (wiring constraint) \\ 6 X_{1} + 5 X_{2} + d_{3}^{-} - d_{1}^{+} & = & 30 (assembly constraint) \\ X_{2} + d_{4}^{-} - d_{4}^{+} & = & 7 (ceiling fan constraint) \\ All X_{i}, d_{i} variable & \geq & 0 \end{array}

$\begin{array}{l} Minimize total deviation & = & d_{1}^{-} + d_{2}^{-} + d_{3}^{+} + d_{4}^{-} \\ subject to {7X}_{1} + 6 X_{2} + d_{1}^{-} - d_{1}^{+} & = & 30 (profit constraint) \\ 2 X_{1} + 3 X_{2} + d_{2}^{-} - d_{2}^{+} & = & 12 (wiring constraint) \\ 6 X_{1} + 5 X_{2} + d_{3}^{-} - d_{1}^{+} & = & 30 (assembly constraint) \\ X_{2} + d_{4}^{-} - d_{4}^{+} & = & 7 (ceiling fan constraint) \\ All X_{i}, d_{i} variable & \geq & 0 \end{array}$

Ranking Goals with Priority Levels

In most goal programming problems, one goal will be more important than another, which, in turn, will be more important than a third. The idea is that goals can be ranked with respect to their importance in management’s eyes. Lower-order goals are considered only after higher-order goals are met. A priority $(P_{i})$ $(P_{i})$ is assigned to each deviational variable—with the ranking that $P_{1}$ $P_{1}$ is the most important goal, $P_{2}$ $P_{2}$ the next most important, then $P_{3}$ $P_{3}$ , and so on.

Let’s say Harrison Electric sets the priorities shown in the following table:

GOAL	PRIORITY
Reach a profit as much above $30 as possible	$P_{1}$ $P_{1}$
Fully use wiring department hours available	$P_{2}$ $P_{2}$
Avoid assembly department overtime	$P_{3}$ $P_{3}$
Produce at least seven ceiling fans	$P_{4}$ $P_{4}$

This means, in effect, that meeting the profit goal (which has a priority of $(P_{1})$ $(P_{1})$ ) is infinitely more important than meeting the wiring goal $(P_{2})$ $(P_{2})$ , which is, in turn, infinitely more important than meeting the assembly goal $(P_{3})$ $(P_{3})$ , which is infinitely more important than producing at least seven ceiling fans $(P_{4})$ $(P_{4})$ .

With ranking of goals considered, the new objective function becomes

Minimize total deviation = P_{1} d_{1}^{−} + P_{2} d_{2}^{−} + P_{3} d_{3}^{+} + P_{4} d_{4}^{−}

$Minimize total deviation = P_{1} d_{1}^{−} + P_{2} d_{2}^{−} + P_{3} d_{3}^{+} + P_{4} d_{4}^{−}$

The constraints remain identical to the previous ones.

Goal Programming with Weighted Goals

When priority levels are used in goal programming, any goal in the top priority level is infinitely more important than the goals in lower priority levels. However, there may be times when one goal is more important than another goal, but it may be only two or three times as important. Instead of placing these goals in different priority levels, they would be placed in the same priority level but with different weights. When using weighted goal programming, the coefficients in the objective function for the deviational variables include both the priority level and the weight. If all goals are in the same priority level, then simply using the weights as objective function coefficients is sufficient.

In Action The Use of Goal Programming for Tuberculosis Drug Allocation in Manila

Allocation of resources is critical when applied to the health industry. It is a matter of life and death when neither the right supply nor the correct quantity is available to meet patient demand. This was the case faced by the Manila (Philippines) Health Center, whose drug supply for patients afflicted with Category I tuberculosis (TB) was not being efficiently allocated to its 45 regional health centers. When the TB drug supply does not reach patients on time, the disease becomes worse and can result in death. Only 74% of TB patients were being cured in Manila, 11% short of the 85% target cure rate set by the government. Unlike other diseases, TB can be treated with only four medicines and cannot be cured by alternative drugs.

Researchers at the Mapka Institute of Technology set out to create a model, using goal programming, to optimize the allocation of resources for TB treatment while considering supply constraints. The objective function of the model was to meet the target cure rate of 85% (which is the equivalent of minimizing the underachievement in the allocation of anti-TB drugs to the 45 centers). Four goal constraints considered the interrelationships among variables in the distribution system. Goal 1 was to satisfy the medication requirement (a 6-month regimen) for each patient. Goal 2 was to supply each health center with the proper allocation. Goal 3 was to satisfy the cure rate of 85%. Goal 4 was to satisfy the drug requirements of each health center.

The goal programming model successfully dealt with all of these goals and raised the TB cure rate to 88%, a 13% improvement in drug allocation over the previous distribution approach. This means that 335 lives per year were saved through this thoughtful use of goal programming.

Source: Based on G. J. C. Esmeria, “An Application of Goal Programming in the Allocation of Anti-TB Drugs in Rural Health Centers in the Philippines,” Proceedings of the 12th Annual Conference of the Production and Operations Management Society (March 2001), Orlando, FL: 50, © Trevor S. Hale.

Consider the Harrison Electric example, in which the least important goal is goal 4 (produce at least seven ceiling fans). Suppose Harrison decides to add another goal of producing at least two chandeliers. The goal of seven ceiling fans is considered twice as important as this goal, so both of these should be in the same priority level. The goal of two chandeliers is assigned a weight of 1, while the goal of seven ceiling fans is given a weight of 2. Both of these will be in priority level 4. A new constraint (goal) would be added:

X_{1} + d_{5}^{−} - d_{5}^{+} = 2 (chandeliers)

$X_{1} + d_{5}^{−} - d_{5}^{+} = 2 (chandeliers)$

The new objective function value would be

Minimize total deviation = P_{1} d_{1}^{−} + P_{2} d_{2}^{−} + P_{3} d_{3}^{+} + P_{4} (2 d_{4}^{−}) + P_{4} d_{5}^{−}

$Minimize total deviation = P_{1} d_{1}^{−} + P_{2} d_{2}^{−} + P_{3} d_{3}^{+} + P_{4} (2 d_{4}^{−}) + P_{4} d_{5}^{−}$

Note that the ceiling fan goal has a weight of 2. The weight for the chandelier goal is 1. Technically, all of the goals in the other priority levels are assigned weights of 1 also.

Using QM For Windows To Solve Harrison’s Problem

The QM for Windows Goal Programming module is illustrated in Programs 10.8A and 10.8B. The input screen is shown first, in Program 10.8A. Note in this first screen that there are two priority-level columns for each constraint. For this example, the priority for either the positive or the negative deviation will be zero, since the objective function does not contain both types of deviational variables for any of these goals. If a problem had a goal with both deviational variables in the objective function, both priority-level columns for this goal (constraint) would contain values other than zero. Also, the weight for each deviational variable contained in the objective function is listed as 1. (It is 0 if the variable does not appear in the objective function.) If different weights were used, they would be placed in the appropriate weight column within one priority level.

Excel sheet displays the inputs Harrison Electric’s Goal Programming Analysis. — Program 10.8A Harrison Electric’s Goal Programming Analysis Using QM for Windows: Inputs

Figure 10.8A Full Alternative Text

The solution, with an analysis of deviations and goal achievement, is shown in Program 10.8B. We see that the first two constraints have negative deviational variables equal to 0, indicating full achievement of those goals. In fact, the positive deviational variables both have values of 6, indicating overachievement of these goals by six units each. Goal (constraint) 3 has both deviational variables equal to 0, indicating complete achievement of that goal, whereas goal 4 has a negative deviational variable equal to 1, indicating underachievement by one unit.

Excel sheet displays the summary solution screen of Harrison Electric Company Solution. — Program 10.8B Summary Solution Screen for Harrison Electric’s Goal Programming Problem Using QM for Windows

Figure 10.8B Full Alternative Text

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10.3 Goal Programming