Introduction to Optimization

Abstract

Broadly, one may describe management science as an interdisciplinary study of problem solving and decision making in human organizations. Management science uses a combination of analytical models and behavioral sciences to address complex business and societal problems.

Appendix

1.1 I Spreadsheet Models and Excel Solver Reports

There are a few online tutorials available to understand how to input a LP model in Solver. The two commonly used websites are Solver⁵ and Microsoft support⁶ page. This section describes the various fields in the LP reports generated by Microsoft Solver and how to locate the information related to shadow prices, reduced costs, and their ranges after the model has been solved. We use the prototype example referred earlier to describe these reports.

1.2 I The Answer Report

Figure 11.19 shows the answer report generated by Excel Solver for our prototype problem. We describe the entries in this report.

Fig. 11.19

Answer report

Target Cell

The initial value of the objective function (to be maximized or minimized), and its final optimal objective value.

Adjustable Cells

The initial and final values of the decision variables.

Constraints

Maximum or minimum requirements that must be met, whether they are met just barely (binding) or easily (not binding), and the values of the slacks (excesses) leftover. Binding constraints have zero slacks and nonbinding ones have positive slacks.

1.3 I The Sensitivity Report

Figure 11.20 shows the sensitivity report generated by Excel Solver for our prototype problem. Below we describe the entries in this report.

Fig. 11.20

Sensitivity report (shadow prices and validity range)

Adjustable Cells

The decision variables, their cell addresses, names, and optimal values.

Reduced Cost

This relates to decision variables that are bounded, from below ( such as by zero in the nonnegativity requirement), or from above (such as by a maximum number of units that can be produced or sold). Recollect:

• 1. A variable’s reduced cost is the amount by which the optimal objective value will change if that bound was relaxed or tightened.

• 2. If the optimal value of the decision variable is at its specified upper bound, the reduced cost is the amount by which optimal objective value will improve (go up in a maximization problem or go down in a minimization problem) if we relaxed the upper bound by increasing it by one unit.

• 3. If the optimal value of the decision variable is at its lower bound, its reduced cost is the amount by which the optimal objective value will be hurt (go down in a maximization problem or go up in a minimization problem) if we tightened the bound by increasing it by one unit.

Objective Coefficient

The unit contribution of the decision variable to the objective function (unit profit or cost).

Allowable Increase and Decrease

The amount by which the coefficient of the decision variable in the objective function can change (increase or decrease) before the optimal solution (the values of decision variables) changes. As long as an objective coefficient changes within this range, the current optimal solution (i.e., the values of decision variables) will remain optimal (although the value of the objective function optimal objective value will change as the objective coefficient changes, even within the allowable range).

Shadow Price

Recollect:

• 1. The shadow price associated with each constraints measures the amount of change in the optimal objective value optimal objective value that would result from changing that constraint by a small amount.

• 2. In general, it is the increase in optimal objective value resulting from an increase in the right-hand side of that constraint.

• 3. Its absolute value measures the marginal (or incremental) improvement in optimal objective value (i.e., an increase in the maximum profit or a decrease in the minimum cost) if that constraint was relaxed (i.e., if the lower limit was reduced or the upper limit was increased) by one unit. Similarly, it is the marginal degradation in optimal objective value (i.e., if the lower limit was raised or the upper limit was reduced) by one unit. For example, if the constraint represents limited availability of a resource, its shadow price is the amount by which the optimal profit will increase if we had a little more of that resource and we used it in the best possible way. It is then the maximum price that we should be willing to pay to have more of this resource. Equivalently, it is the opportunity cost of not having more of that resource.

Allowable Increase and Decrease

Recollect:

• 1. This is the amount by which the constraint can be relaxed or tightened before its shadow price changes. if the constraint imposes an upper limit, and it is relaxed by increasing this limit by more than the “allowable increase,” the optimal objective value will still improve but at a lower rate, so the shadow price will go down below its current value. Similarly, if the upper limit on the constraint is decreased by more than the “allowable decrease,” the optimal objective value will degrade at an even higher rate and its shadow price will go up.

• 2. If the constraint imposes a lower limit and that constraint is relaxed by decreasing the limit by more than the “allowable decrease,” the optimal objective value will still improve but only at a lower rate and the shadow price will decrease. If , on the other hand, the lower limit is increased by more than the “allowable increase,” the constraint becomes tighter, the optimal objective value will degrade faster, and the shadow price will increase. Thus, there are decreasing marginal benefits to relaxing a constraint, and increasing marginal costs of tightening a constraint.

It should be noted that all of the information in the sensitivity report assumes that only one parameter is changed at a time. Thus, the effects of relaxing or tightening two constraints or changing the objective coefficients of two decision variables cannot be determined from the sensitivity report. Often, however, if the changes are small enough to be within the respective allowable ranges, the total effect can be determined by simply adding the individual effects.

In an Excel report degeneracy can be spotted by looking at the rhs values of any of the constraints. If the constraints (for the range over which the optimal shadow price is valid) have an allowable increase or allowable decrease of zero, then the LP is degenerate. One has to be careful while interpreting optimal solutions for degenerate LPs. For example:

• 1. When a solution is degenerate, the reduced costs may not be unique. Additionally, the objective function coefficients for the variable cells must change by at least as much (and possibly more than) their respective reduced costs before the optimal solution would change.

• 2. Shadow prices and their ranges can be interpreted in the usual way, but they are not unique. Different shadow prices and ranges may apply to the problem (even if the optimal solution is unique).