Linear programming is one of a class of mathematical techniques designed to optimize results. Scarcity of resources is a prerequisite to the use of linear programming. A firm must have limited resources such as time, money, or materials, to profit by this tool. We are also interested in the techniques for its use in sensitivity analysis for tactical decisions. Sensitivity analysis allows us to deal with the issues of uncertainty we dealt with in earlier examples.

Broden Paint Co. produces two products, paint and stain. The contribution margin per gallon is $4 for paint and $5.10 for stain. Because of a material shortage, only 50,000 gallons of base mix are available. A half-gallon of base mix is required for each gallon of finished paint or stain. The plant capacity is 90,000 machine hours. It takes .75 hours to make a gallon of paint and 1 hour to make a gallon of stain. The formulation for this simple maximization problem is as follows:

Exhibit LP-1 shows the LINDO solution to this problem. The formulation and solution are deterministic, that is, no measures of risk or dispersion are included. In our discussion of sensitivity analysis, we will allow one item to vary at a time. This will suggest where we need to get additional data to quantify various uncertainties.

First, lets discuss the solution presented in Exh LP-1. At the optimal solution, Broden would produce 40,000 gallons of paint and 60,000 gallons of stain and would earn $466,000 in total contribution margin. The reduced cost column shows zeros, reflecting the fact that Broden would make some of each product. (A product will show a reduced cost if the optimal solution suggests that no units of that product should be made. The reduced cost column tells you the amount by which total CM would go down if one unit of that product had to be made, e.g. to satisfy the needs of a major customer.)

The SLACK OR SURPLUS column shows that Broden uses all of its available base mix (the row is labeled 2 since the base mix constraint is equation 2 in the problem formulation) and all of its available machine hours (row 3 for equation 3 in the problem formulation). The dual price column shows that Broden would be willing to pay $1.40 for an additional gallon of base mix or $4.40 for an additional hour of machine time. Why would they be willing to make these expenditures? Because with these additional resources, they would be able to change the production mix to increase total contribution margin by that much. By analyzing the TABLEAU, we will be able to specify exactly how this would happen -- more on that later.

Exhibit LP-1

The SENSITIVITY ANALYSIS SECTION shows provides information for two different types of sensitivity analysis. The first section, OBJ COEFFICIENT RANGES, provides data for analysis of changes in the contribution margin for either product. The first column, CURRENT COEF, shows what CMs are included in the problem formulation for each product. The ALLOWABLE INCREASE column shows how much each CM could increase before the optimal solution would change. For example, the CM of paint could increase by $1.10 (to $5.10) before Broden would alter its plans to make 40,000 gallons of paint and 60,000 gallons of stain. Consider what happens to Broden's plans if the contribution margin on paint increases by $1. Since this change is within the allowable increase, they would still make 40,000 gallons of paint and 60,000 gallons of stain. The only thing that would change would be the amount total CM earned. Since each gallon of paint would yield a CM of $5 instead of $4, Broden's total CM would increase by $40,000 to $506,000.

You can conduct the same analysis for decreases in CM on paint and for increases or decreases in CM on stain, as long as those changes are within the allowable ranges. If the CM on either product changes by more than the allowable increases or decreases, the solution becomes obsolete and the LINDO program needs to be rerun using the new CMs.

The second type of sensitivity analysis relates to changes in the amounts to the constraints available. The RIGHTHAND SIDE RANGES section provides data to analyze changes in the amount of the constraints. The first column, CURRENT RHS, shows the amounts of each constraint currently available, 50,000 gallons of base mix and 90,000 machine hours. (Again, the rows are identified by the numbers 2 and 3, indicating equations 2 and 3 in the problem formulation.) The analysis in this section is a little more complicated than the analysis of changes in CM above. Consider the information provided in the ALLOWABLE INCREASE column for base mix (the row labeled 2). The amount of base mix available can increase by 10,000 gallons to 50,000 gallons before the printout becomes obsolete. Similarly, the ALLOWABLE DECREASE column shows that the amount of base mix could decrease by 5,000 gallons to 45,000 gallons before the printout becomes obsolete.

If the amount of base mix available increases by more than 10,000 gallons or decreases by more than 5,000 gallons the solution is obsolete and the LINDO program needs to be rerun using the new constraint amounts. The same sensitivity analysis could be performed on the machine hours constraint.

In order to understand what Broden would do with additional base mix or machine time, we need to analyze the TABLEAU. I have two pieces of advice about reading the tableau. First, recognize that the tableau can be divided into a product side and a constraint side. The columns labeled P and S relate to the products and the columns labeled SLK 2 and SLK 3 relate to the constraint equations. The headings SLK 2 and SLK 3 refer to equations 2 and 3 in the problem formulation. I recommend drawing a vertical line between the S column and the SLK 2 column to indicate this distinction.

Second, I generally recommend approaching the tableau by reading down the columns. The only exception to this rule is Row 1 in the tableau labeled ART. If you read across this row, the amounts in the product columns (P & S) are the reduced costs for products P and S and the amounts in the constraint columns are the dual prices for base mix (SLK 2) and machine hours (SLK 3). The amount in the first row of the last column (the one without a heading) is the contribution margin earned at the optimal solution.

Here are two questions that can be answered from the tableau. If Broden could obtain more base mix, how much would they pay for it and what would they do with it? To answer these questions, locate the tableau column relating to base mix (SLK 2). The $1.40 in the first row is the dual price, or the amount that Broden would pay to acquire another gallon of base mix. Note that the second and third rows in the tableau are labeled S and P respectively. The -6.000 in the second row and the 8.000 in the third row tell you that Broden would use the additional base mix to produce an additional 8.0 gallons of paint but would reduce production of stain by 6.0 gallons. Consider the impact of these actions. First, by not producing 6 gallons of stain, Broden would free 3 gallons of base mix. (Recall that each gallon of stain requires .5 gallons of base mix.) Those 3 gallons of base mix and the one additional gallon could then be used to produce 4 gallons of paint at the rate of .5 gallons of base mix per gallon of stain.

Consider the impact on total contribution margin. By forgoing production and sale of 6 gallons of stain, total contribution margin will go down by $30.60 (6x$5.10). Production and sale of 8 gallons of paint would increase total contribution margin by $32 (8x$4). The difference between the CM foregone and the extra CM earned would be $1.40, which is the dual price for a gallon of base mix. Thus, the reason Broden would pay up to $1.40 for an additional gallon of base mix is because they could earn an additional CM of $1.40 if they had it.

Information in the SLK 3 column can be used in the same way to understand what Broden would pay for an additional hour of machine time and what they would do with it.

Additional examples of tableau interpretation will be provided in class.