The case of similar products

A firm⁴ produces red and blue pens, whose unit production cost is 15 cents (including labor and raw material). The firm incurs a daily fixed cost, amounting to €1000, to run the plant, which can produce at most 8000 pens per day in total (i.e., including both types). Note that we are expressing the capacity constraint in terms of the total number of pens produced, which makes sense if resource requirements are the same for both products; in the case of radically different products (say, needles and air carriers), this makes no sense, as we shall see in the next section. We are not considering changeover times to switch production between the two different items, so the above information is all we need to know from the technological perspective.

From the market perspective, we need some information about what the firm might sell and at which price. The blue pens sell for 25 cents, whereas things are a tad more complicated for the red ones. On a daily basis, the first 5000 red pens can be sold for 30 cents each, but additional ones can be sold for only 20 cents. This may sound quite odd at first, but it makes sense if we think that the same product can be sold in different markets, where competition may be different, as well as general economic conditions. Such a price discrimination can be maintained if markets are separated, i.e., if one cannot buy on the cheaper market and resell on the higher-priced market.⁵ In general, there may be a complex relationship between price and demand. We will consider QMs to estimate and take advantage of this relationship.

The problem consists of finding how many red and how many blue pens we should produce each day. Note that we are assuming constant demand; hence, the product mix is just repeated each day. In the case of time-varying demand and changeover costs, there could be an incentive to build some inventory, which would make the problem dynamic rather than static.

1. The production manager, an ugly guy with little business background, decides to produce 5000 red and 3000 blue pens, yielding a daily profit of €50 (please, check this result). This may not sound too exciting, but at least we are in the black.
2. A brilliant consultant (who has just completed a renowned master, including accounting classes) argues that this plan does not consider how the fixed cost should be allocated between the two product types. Given the produced quantities, he maintains that €625 ( of the fixed cost) should be allocated to red pens, and €375 to blue pens. Subtracting this fraction of the fixed cost from the profit contribution by blue pens, he shows that blue pens are not profitable at all, as their production implies a loss of €75 per day! Hence, the consultant concludes that the firm should just produce red pens.

What do you think about the consultant’s idea? Please, do try finding an answer before reading further!

A straightforward calculation shows that the second solution, however reasonable it might sound, implies a daily loss:

It is also fairly easy to see that the simple recipe of the production manager is just based on the idea of giving priority to the item that earns the largest profit margin. Apart from that, we should realize that the fixed cost is not really affected by the decisions we are considering at this level. If the factory is kept open, the fixed cost must be paid, whatever product mix is selected. However, this does not mean that the fixed cost is irrelevant altogether. At a more strategic decision echelon, the firm could consider shutting the plant down because it is not profitable. The point is that any cost is variable, at some hierarchical level and with a suitably long time horizon.

From a formal point of view, what we have been trying to solve is a problem such as

In this mathematical statement of the problem we distinguish the following:

• Two decision variables, x_r and x_b, which are the amounts of red and blue pens that we produce, respectively.
• An objective function, π(x_r, x_b), representing the profit we earn, depending on the selected mix, i.e., on the value assigned to the two decision variables. Our task is maximizing profit with respect to decision variables.
• A set of constraints on the decision variables. We should maximize profit with respect to the decision variables, subject to (s.t. in the model formulation) this set of constraints. The first constraint here is an inequality corresponding to the capacity limitation. Further, we have included nonnegativity requirements on sold amounts. Granted, unless you are pretty bad with marketing, you are not going to sell negative amounts, which would reduce profit. Yet, from a mathematical perspective, manufacturing negative amounts of an item could be an ingenious way to create capacity for another item, which makes little sense and must be forbidden. Constraints pinpoints a feasible region, i.e., a set of solutions that are acceptable, among which we should find the best one, according to our criterion.The feasible region in our case is just the shaded triangle depicted in Fig. 1.1. If you have trouble understading how to get that figure, you might wish to refer to Section 2.3; yet, we may recall from high school mathematics that an equation like ax₁ + bx₂ = c is the equation of a line in the plane; an inequality like ax₁ + bx₂ ≤ c represents one of the two half-planes separated by that line. To see which one, the easy way is checking if the origin of the plane, i.e., the point of coordinates (0, 0) satisfies the inequality, in which case it belongs to the half-plane, or not.Fig. 1.1 The feasible set for the problem of red and blue pens.Fig. 1.2 Shifting a function up and down does not change the optimal solution.

Intuitively, since the firm makes money by selling whatever pen it produces, the capacity constraint should be binding at the optimal solution, which means that we should look for solutions on the line segment joining points of coordinates (0, 8000) and (8000, 0). We will see how one can maximize a profit function (or minimize a cost function) in simple cases; a more thorough treatment. For now, we may immediately see why the fixed cost should be ignored in finding the optimal mix. Assume, for the sake of simplicity, that we have just one decision variable and consider the objective function π(x) in Fig. 1.2. Let us denote the optimal solution of the maximization problem, max π(x), by x^*. We see that if the function is shifted up (or down) by a given amount K, i.e., if we solve max π(x) + K, the optimal solution does not change. Yet, the optimal value does, and this may make the difference between a profitable business and an unprofitable one. Whether this matters or not depends on the specific problem we are addressing.

Takeaways Even from a simple problem like this, there are some relevant lessons that deserve being pointed out:

• A simple decision problem consists of decision variables, constraints on them, and some performance measure that we want to optimize, such as minimizing cost or maximizing profit.
• Not all costs are always relevant; this may depend on the level at which we are framing the problem.
• The relationship between price and demand can be complex. In real life, data analysis can be used to quantify their link, as well as the uncertainty involved.