Shadow Prices:
Definition and Explanation of Shadow Price:
The determination of the optimum mix to maximize the contribution margin or to minimize cost assumes a defined set of constraints.
It is useful to consider the sensitivity of the solution if a constraint is relaxed. This effect is often referred to as shadow price and simply shows the change in contribution margin (in a contribution margin maximization problem) or the change in cost (in a cost minimization problem) resulting from relaxing a constraint.
To present the idea of shadow price, the value of additional grinding or polishing hours (from the contribution margin maximization example) can be considered, i.e., the worth of additional grinding and polishing hours. If the machine shop had more grinding or polishing hours, the contribution margin could be increased by using more of each.
The index row of the third (optimum solution) simplex tableau (see contribution margin maximization example) shows the shadow prices in the slack variable columns, which is the location for both ≤ and ≥ constraints, while the artificial variable column is used for the = constraint, with the m value ignored. In this illustration, only ≤ constraints are encountered.
The coefficients under the s1 (grinding) and s2 (polishing) slack variable columns give the tradeoff in terms of product mix as the constraints are increased or decreased. Thus, one more hour of grinding time will increase the contribution margin by $0.625, computed as follows: as one more grinding hour is made available, 0.25 units of y (deluxe models) with a unit contribution margin of $4 (0.25 × $4 = $1) will replace 0.125 units of x (standard models) with a $3 unit contribution margin (0.125 × $3 = $0.375), for a net contribution margin of $0.625 ($1 – $0.375 = $0.625).
If additional grinding time could be obtained at no increase in unit variable cost, an increase of $0.625 contribution margin per grinding hour would result; and as much as $0.625 more than the present unit variable cost of grinding time could be incurred before reaching a point at which a zero per unit contribution margin would occur. Thus, overtime hours might be considered. The $0.4375 has the same meaning for each hour of polishing time, and in each case the observations assume that the sales price per unit remains unchanged.
The range of hours over which the shadow prices of $0.625 and $0.4375 for grinding and polishing hours are valid can be found as follows:
1. For the lower limit of range, divide each unit in the solution mix by the coefficient under the slack variable column, i.e., the s1 and s2 column. The smallest positive number that result in a column is the maximum decrease for that constraint:
Product  (1)Units  (2)Grinding  (3)Polishing  (4) s1 Grinding 1/2 
(5) s2 Polishing 1/3 
y  20  0.250  0.1250  80  160 
x  10  0.125  0.3125  80  32 
For the grinding constraint, the decrease is 80 hours. Since the original number of hours available was 120, the lower limit is 40 hours. For the polishing constraint, the decrease is 32 hours. Since 80 hours were originally available, the lower limit is 48 hours.
2. For the upper limit of the range, multiply each coefficient by 1 and repeat the step 1 process. The smallest positive number that result in a column is the maximum increase for that constraint:
(1)  (2)  (3)  (4) s1 
(5) s2 

Product  Units  Grinding  Polishing  Grinding 1/2 
Polishing 1/3 
y  20  0.250  0.1250  80  160 
x  10  0.125  0.3125  80  32 
For the grinding constraint, the maximum increase is 80; and since the original number of hours available was 120, the upper limit is 200 hours. For the polishing constraint, the increase is 160, and the upper limit is 240 hours (160 plus the original constraint of 80 polishing hours).
The limits occur for a constraint because increases and decreases beyond the limits will change the shadow price. In summary, the lower and upper limits for this example are:
Lower limit  Upper limit  
Grinding hours  40  200 
Polishing hours  48  240 
Both the constraints in this example are of the ≤ type. The same method for finding the lower and upper constraint range limits is used for the = type of constraint except that the coefficient under artificial variable column for the = constraint are used in the computations. For a ≥ constraint, the method for finding the lower and upper constraint range limits differs in that the signs of the coefficients under the slack variable column for the ≥ constraints are changed in step 1. With this exception, the procedure is the same.
When there is a zero shadow price (not the case in the above example), there is no defined upper limit for the ≥ type constraint because there is already more of this constraint used than is required.
The lower and upper limit computations apply, assuming that only one constraint is to be relaxed, and provided there is a unique solution to the linear programming problem. The limits for cases in which two or more constraints are relaxed simultaneously can be computed following a methodology that is beyond the scope of this discussion.
The described method is equally applicable to cost minimization problems. It should also be observed that the shadow price indicates the opportunity cost of using a resource (such as grinding or polishing hours) for some other purpose. For example, if an hour of grinding time could instead be used to produce some other product at a contribution margin greater than $0.625 per hour, then use of the grinding hour resource in producing the alternate product would be preferable.
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