Explanation Given: Smallville has three Chinese restaurants: Asia Gardens, Chef Chao’s, and Chung King Village. Chung King Village recently mailed a coupon to all residents of Smallville offering two dinners for the price of one. As a result, 67% of those who normally eat at Asia Gardens plan on trying Chung King Village within the next month, and 59% of those who normally eat at Chef Chao's plan on trying Chung King Village. Also, all of Chung King Village’s regular customers will return to take advantage of the special. And 30% of the people who normally eat at Asia Gardens will eat there within the next month because of its convenient location. Chef Chao’s has a new chef who isn’t doing very well, and as a result, only 15% of the people who normally eat at there are planning on returning next time. Approach: Find the equilibrium Matrix L for more than two unknowns. 1. Find the transition matrix T and the equilibrium matrix L . 2. Simplify the system of equations from L T = L and discard any redundant equations and include the equation x + y + ... + z = 1 . 3. Solve the resulting system of equations. Calculation: From the information provided, the transition matrix T is given below. T = [ 0.30 0.03 0.67 0.26 0.15 0.59 0 0 1.00 ] There are three variables, therefore, the equilibrium matrix L is given below. L = [ x y z ] The system of equation L T = L is given below. [ x y z ] [ 0.30 0.03 0.67 0.26 0.15 0.59 0 0 1.00 ] = [ x y z ] The system in the form of linear equations is given below. { 0.30 x + 0.26 y = x 0.03 x + 0.15 y = y 0.67 x + 0.59 y + z = z The system of equations can be written as shown below. { − 0.70 x + 0.26 y = 0 0.03 x − 0.85 y = 0 0.67 x + 0.59 y = 0 Multiply each element of the system by 100. { − 70 x + 26 y = 0 3 x − 85 y = 0 67 x + 59 y = 0 Discard the redundant equation 67 x + 59 y = 0 with the equation x + y + z = 1 . { − 70 x + 26 y = 0 3 x − 85 y = 0 x + y + z = 1 Solve the equations − 70 x + 26 y = 0 and x + y + z = 1

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Chapter 11.5, Problem 6E

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A long-range prediction of the ultimate market shares of the three restaurants, assuming that current trends continue.

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Consider the cones K = = {(x1, x2, x3) | € R³ : X3 ≥√√√2x² + 3x² M = = {(21,22,23) (x1, x2, x3) Є R³: x3 > + 2 3 Prove that M = K*. Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula (AL)* = (AT)-¹L*, for any cone LC R³ and any 3 × 3 nonsingular matrix A with real entries, where AL = {Ax = R³ : x € L}, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix A.

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A long-range prediction of the ultimate market shares of the three restaurants, assuming that current trends continue.

Solution Summary: The author explains that Chung King Village recently mailed a coupon to all residents of Smallville offering two dinners for the price of one.

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To find: A long-range prediction of the ultimate market shares of the three restaurants, assuming that current trends continue.

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To find:

A long-range prediction of the ultimate market shares of the three restaurants, assuming that current trends continue.