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GOAL PROGRAMMING
This lecture is base on chapter 7 of the book “Linear Programming in Financial Planning” by Salkin and Kornbluth (1973).
Up to now we have been working with situations in which we had a single objective, although in the lecture on P-medians we introduced the idea of two objectives that had to be set one against the other. Now we are going to consider the situation of multiple objectives. Sometimes the objectives are not mutually compatible.
Consider the following problem which is summarised in the table below:
Product 1 Product 2 Available
Resource 1 3 2 12
Resource 2 5 0 10
Per unit profit 1 0.
This is a simple problem where we have to decide how much to make of product 1 (X (^) 1) and of product 2 (X2) in order to maximise profit while staying within available resources. The formulation presents no problem.
This is an easy problem to solve with Simplex
Basic X 0 X 1 X 2 S 1 S 2 Sol Ratio
X 0 1 -1 -0.5 0 0 0
S 1 0 3 2 1 0 12 4
S 2 0 5 0 0 1 10 2 ←
↑
X 0 1 0 -0.5 0 1/5 2
S 1 0 0 2 1 -3/5 6 3 ←
X 1 0 1 0 0 1/5 2
↑
X 0 1 0 0 1/4 1/20 3.
X 2 0 0 1 1/2 -3/10 3
X 1 0 1 0 0 1/5 2
We find that the optimal amount of profit that can be made is 3.5. This is achieved by producing 3 units of product 2, and 2 units of product 1.
Suppose now that the directors of the firm do not want to make a profit of 3.5. They might want to “smooth” earnings so that shareholders do not get used to large profits, or they may want to avoid taxes. It is, in fact, quite common to set a target for profits to be made. The firm can avoid making profits by creative use of depreciation, or by reinvesting extra earnings in equipment. Let the target for profits be 2.5. In fact, in the context of Linear Programming we do not talk about targets but about “goals”, this is why the technique I am introducing in this lecture is known as Goal Programming.
The problem of achieving the goal of 2.5 profits can be formulated as follows:
With both up and v (^) p taking positive values. The variables u (^) p and vp will never take a positive value at the same time. If they are both zero at the same time, the goal is exactly achieved. It is a “feasible goal”. If u (^) p is positive and vp is zero, the goal is underachieved. If up is zero and vp is positive, the goal is overachieved.
If you solve this problem you will find that x 0 takes the value 0. This means that the goal is attainable. The goal is attained by setting x 1 = 2 and x 2 = 1.
Now that the profit goal is attained and the shareholders are happy, the managers set themselves another target: they would like a fat bonus. Normally the bonus depends on the value of sales. In this case, product 1 sells at 5 Euros per unit, and product 2 sells at 2 Euros per unit. The managers have a goal of 10 Euros for sales. How do they do this? All that they need to do is to modify the problem formulation with an extra constraint.
When we solve this, we find that both goals are feasible, they can both be achieved at the same time by setting x 1 = 0 and x 2 =5, giving x 0 = 0. The goals are “compatible”.
Managers now get greedy, and they ask themselves, why not go for a higher volume of sales and get a higher bonus? Then they set the sales goal equal to 15. The problem is almost the same as before:
The solution now is x 0 = 0.75, with x 1 = 2, x 2 = 2.5, and v (^) p = 0.75. In fact, this means that the profit target has been overachieved by 0.75. It is possible to sell for a value of 15, but it is not possible to have a profit of 2.5 at the same time. The goals are not compatible; they form an unfeasible set of goals.
When the goals are not compatible, a new issue arises: are they equally important? We can, as we did in the P-median problem, highlight the importance of the two goals by means of a weighting factor. We will first consider the case when overachieving or underachieving a given goal is equally important. This can be formulated as follows:
Where the values of θ (^) p and θs are set so as to reflect the relative importance of the goals. This is not an unknown, it is a management decision.
In the most general case, underachievement and over- achievement have different importance, and we reflect this by means of different weights. The problem is now modified to read:
As I have insisted many times, we first think about decision variables. The decision relate to the quantities of the various financial instruments that can be used to pay for the company that is being acquired.
X1, the number of voting shares that will be offered for the company. These shares will entitle the shareholders of the acquired company to have a say in the management of the acquiring company.
X2, the number of non-voting shares to be offered. Shareholders of the acquired company will have a share in the profits of the merged company, but will not have a say in its management. X3, the number of debentures convertible into shares. These guarantee an interest rate for a given period, independently of the profits of the merged company, and at the end of this period they become voting shares. There might be several types of these instruments depending on maturity periods and interest rates offered. Here we will pretend there only one type in order to simplify the formulation for didactic reasons.
X4, the number of debentures not convertible into shares.
X5, a cash payment.
In this case we have constraints and goals. We will deal with goals only, as constraints can be formulated as goals.
First goal.
The purchasing company does not want to pay over the odds for the company being purchased. The purchasing company will estimate the maximum value of the company being bought and will not want to pay more that it is worth. Let this value be MV.
Where p1, p (^) 2, p 3 and p 4 are the market prices of the instruments. We do not have v (^) a in the equation, as we do not want to pay more than the maximum value.
Clearly, if the shares of the purchasing company increase in value prior to the purchase, it will be “a good thing” for the purchasing company. There was a famous merger between two alcoholic drinks companies, Guiness and Distillers. Guiness wanted to buy Distillers, and the chairman of Guiness bought his own shares in order to push their value up (increase the value of p 1 in the constraint). He was caught, there was a court case, and he ended up in prison. He did not stay there long, as he was released because he had developed Alzheimer’s disease while at prison. Not long after he left prison he recovered from the disease. He still remains the only person known to have had such a miraculous cure.
Second goal.
The purchasing company will also estimate a minimum value for the company being purchased, and will not want to pay less than this, because otherwise the deal will not be accepted by the shareholders of the company being purchased. Let this value be mV.
We do not contemplate paying less than the minimum value, and this is why there is no ub in the equation.
Third goal.
Companies publish their financial statements, and financial analysts calculate a series of financial ratios. There are values of these ratios that are considered to be “respectable” and the companies try to conform to them. One such ratio is Gearing, the total amount of debt divided by the total amount of capital. Established wisdom is that low values of the gearing ratio are “a good thing”, while high values of this ratio are “a bad thing”, since the company gets exposed to interest rate risk, may not be able to meet its future interest repayments and may fall into liquidation. Companies set themselves a target for this ratio, a goal.
Before the merger, the acquiring company has a market value of S (its total number of shares multiplied by their price in the stock market). After the merger, the capital will have increased by p1x1+p2x2, while the debt will have increased by p (^) 3x3+p (^) 4x4.
The value of the ratio after the merger will be (forget the company that has been acquired, after the merger it will not exist):
Rewrite this as
As a goal, this becomes:
Fourth goal.
Earnings per share must be guaranteed both to the shareholders of the purchasing company, as to the shareholders of the company being purchased.
Let Π 1 be the expected profit of the purchasing company, and Π 2 be the expected profits
of the company being purchased. Let Π 3 be the expected extra profit that the merger will bring. This extra profit can come from a monopolistic situation, from synergy, from horizontal integration, or from vertical integration. The purchase will also create interest payments that will eat into profits. Dealing with interest payments obliges us to think in terms of future years, but here only the coming year will be considered.
Where r is the interest rate paid on the debentures.
This can be reformulated as:
