## Objectively conditioned estimates and their meaning

*The components of the optimal solution of the dual problem are called optimal (dual) estimates of the original problem. * Academician LV Kantorovich called them

*strong> .*

**objectively determined estimates**To clarify the meaning of these estimates, we return to the problem I about the use of resources and the dual II problem (see Problem 6.2). The components of the optimal solutions of these problems, given in Problem 6.5, are given in Table. 6.3.

*Table 6. * 3

Components of the optimal solution of the original problem I |
|||||

Number of items |
Residual resources, units |
||||

Exceeding the cost of resources over the implementation price |
Objectively determined resource estimates (conditional resource prices) |
||||

Components of the optimal solution of dual problem II |

In Table. 6.3 additional variables of the original problem I x3, * x * 4 *, x * 5 *, x * 6, representing, according to the expression (6.15), the difference between the resources k of resources 5 , * S * 2, 53, 54 and their consumption, express * resource balances, * and additional

the variables of the dual problem II * y * v * y * 6, representing, in accordance with the expression (6.16), the difference between the cost of resources for the production of a unit of output from them and prices c. products * P * v * P * 2, express * the excess of costs over the price. *

Resources 5., according to the optimal plan are fully used (), and objectively estimated estimates of these resources are non-zero: (). The resources * S * v 54 are not fully used in the optimal plan (), and the objectively determined estimates of these resources are zero: ().

Thus, objectively determined resource estimates determine the degree of resource scarcity: according to the optimal production plan, scarce (ie, fully used) resources receive non-zero estimates, and non-deficient - * bullet estimates. *

According to the optimal plan in the initial task, both types of products should be produced (img src="images/image398.jpg">) and the excess of costs for resources over the sale price is zero (,

). If the costs of resources exceeded the price of the products produced from them, for example products , i.e. if , then on the basis of the theorem on p. 109 the optimal value of the corresponding variable , and in this case, according to the optimal plan, the production should not have occurred.

So, in * the optimal production plan can get only cost-effective, unproductive types of products * (although the criterion of profitability here is peculiar: the price of production does not exceed the cost of resources consumed in its manufacture, but exactly equal to it) .

To find out what the numerical values of objectively determined resource estimates show, we prove the following theorem.

**The third duality theorem. ** The components of the optimal solution of the dual problem are equal to the values of the partial derivatives of the linear function by the corresponding arguments,

(6.22)

□ Let us prove this theorem with the example of problem I (see expressions (6.1) - (6.3) on the use of resources and its dual problem II (see expressions (6.4) - (6.6)).

In accordance with the expression (6.4), the optimal (minimum) value of the linear function

Change the margin of the * i * resource A by * Ab *, but the optimal solution of the dual problem will not change (in Figure 6.1 for the simplest case of two variables, the optimal solution corresponds to * B), * ie, the previous ones, but only the linear function * Δ * (in Figure 6.1 for the case the slope of the level line will change).

Then changing the minimum costs for resources

**Fig. 6.1 **

And since the first duality theorem or , the maximum profit (revenue) will increase by, from where

(6.23)

It is important that the latter relationship is true for a sufficiently small change in , otherwise the slope of the level line (see Figure 6.1) and the optimum can vary significantly, since the minimum of the linear function can move to another corner point, respectively, with another optimal solution

The notion of a sufficient small increment in the resource reserve can be formalized by requiring that tend to zero. When the relation (6.23) is obtained and the assertion (6.22) is proved. ■

From the relation (6.23) it follows that objectively determined resource estimates indicate how many monetary units the maximum profit (revenue) from the sale of production will change when the stock of the corresponding resource changes by one unit.

6.7. In task I of 6.2, clarify the meaning of the values of objectively determined resource estimates.

**Solution. ** In problem 6.5 objectively conditioned estimates of the resources of the problem are obtained: ; ie. with an increase (decrease) in the stock of resources A, or 52 by 1 unit, the maximum profit (revenue) will increase (decrease) by 4/5 and 3/5 rubles, respectively, and if the reserve of resources or - will not change. ►

Dual assessments can serve as a tool for analyzing and making the right decisions in a constantly changing production environment. For example, with * using objectively determined resource estimates, it is possible to compare the optimal conditional costs and production results. *

6.8. In problem I of 6.2, as a result of the solution of which the optimal output plan was obtained, * P * {and * P * 2, it was possible to produce one more product - * P. * A. Expenses per unit of production P3 are: a13 * = * 3 units of the resource 5, * a. * n * = * 2 units * S * 2, α33 = 4 units 53 and а43 = 1 unit of resource ⅝ unit price of production * Р. * л is equal to с3 = 3 rubles. To determine whether the addition of additional products * P * to the output plan will give profit. What should be the profit from the unit (price) of production * P * so that its production is profitable?

**Solution. ** You can include the products * P * 3 in the conditions of the problem and solve the problem again, but this will require new costs (labor, cost, time). If necessary, this is not the case, since objectively determined resource estimates are known. Indeed, we compare the additional costs of resources per unit of output with the price of its realization. The first value, equal to

rub., more than the product price c3 = 3 rubles. Therefore, the output of * P * 3 should not be included in the release plan, and the need for a repeat The solution of the problem in the changed conditions disappears. And that the production of * Р. * j is profitable, obviously, its price should be at least 3.6 rubles. ►

*Objectively determined resource estimates allow you to judge the affect of not just any, but only relatively small changes in resources. * With drastic changes, the estimates themselves can become different, which will lead to their inability to use for analysis of production efficiency.

6.9. For problem I of 6.2, find intervals of stability (invariance) of ambiguous estimates with respect to changes in the reserves of resources of each type. Will these estimates change if the reserves of each resource are increased by 10 units: a) separately; b) Simultaneously? Find the corresponding change in the maximum profit (revenue) from the sale of products.

**Solution. ** Assume that the resource stocks , originally 18.16.5 and 21 units, have changed accordingly to < img src="images/image431.jpg">. Then the cost of resources in accordance with the expression (6.4) will be

Replacing the variables * y * y and * y * 2 with their expressions through the non-primary variables of the optimal solution (see page 76), we get after the transformations

(6.24)

In the case that , i.e. the resource stocks are equal to the original values, the familiar expression (6.21) of the linear function Z would be obtained through the non-basic variables of the optimal solution. In order for objectively determined resource estimates to remain unchanged even with a change in resource stocks, i.e. the optimal solution of the dual problem

, it is sufficient that the coefficients for non-basic variables in expression (6.24) remain nonnegative, that is,

(6.25)

Suppose that only the stock of the resource is changed and the remaining resource stocks remain unchanged:

Then from the expression (6.25) we get that

and or

i.e. at invariance of objectively conditioned estimations of resources the stock of a resource 5, can vary in limits from 13 up to 20,5 units. Similarly, we can get that

Thus, if the reserve of only one of the resources 5 changes, in the range from 13 to 20.5 units, or 52 - in the range from 11 to 17 2/3 units, or * S * 3 - within not less than 4 units, or * S * A - within the limits of not less than 18 units the optimal solution of the dual task remains the same, i.e. * Y - * (4/5; 3/5; 0; 0; 0; 0).

On the basis of the foregoing it is clear that an increase of 10 resource units per unit of resource 5 (equal to 18 units) or 5. (16 units) will lead to a change in their objectively determined estimates, and a resource reserve * S * 3 (5 units) or * S * A (21 units) will leave estimates of these resources as former (equal to zero). As a result, with the help of the obtained optimal resource estimates, it is impossible to find the corresponding change in the maximum profit (revenue) Afmax for a given increase in the resource margin * S * t or * S * 2, and the problem must be solved anew.

If the reserves of all resources change simultaneously, then the study of the stability of objectively conditioned estimates becomes more complicated, since in this case it is necessary to find the polyhedron of solutions of the system of inequalities (6.25). However, it is always possible to check whether specific changes in resource stocks satisfy the system (6.25). So, in our task, while increasing the reserves of all resources by 10 units, i.е. at 10, all the inequalities of system (6.25) are valid, therefore, the optimal solution of the dual problem remains the same, that is, . Therefore, the change in the maximum profit (revenue), taking into account the expression (6.23), will be

*According to the ratio of objectively determined estimates, the calculated norms of the replaceability of resources, * under which observable changes within the bounds of the stability of dual estimates do not affect the efficiency of the optimal plan can be determined.

6.10. By the condition of problem I of 6.2, determine the norms of replaceability of resources 5, and * S * 2, in the implementation of which the maximum profit (gain) from production will remain.

**Solution. ** Let's compose the ratio of objectively determined resource estimates of 5, and S2, that is, 4: 3, i.e. to maximize the total revenue (revenue) every additional 3 resource units 5 are equivalent to the additional 4 resource units * S * 2

*The conclusion is true within the bounds of the stability of dual estimates, when changes in the resource stock and also satisfy the system (6.25). ► !!!*

**.**6.11. Solve problem I of 6.2 with resource changes using (if possible) dual resource estimates.

**Solution. ** Since the changes in the stock of resources are within the bounds of the stability of dual estimates (they satisfy the system (6.25)), the solution The dual problem remains the same: K '= (4/5; 3/5; 0; 0; 0; 0). On the basis of the theorem (p. 108) and the correspondence (6.14) to the positive values of the variables * y * = 4/5,

*3/5 of the optimal solution of the dual problem there correspond zero values of the variables of the original problem: X, = 0,*

**y**2**=***4*

**x***0. Therefore, the remaining components of the optimal solution of the original problem can be found directly from its constraint system, in which x3 = 0, x4 = 0, and on the right-hand sides, new resources are indicated:*

**=**whence . The corresponding value of the maximum of the linear function

So, the maximum size of the profit (revenue) will be rub. with an optimal production plan

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