6 Ch. 2 – 20(c)The binding constraints are the manufacturing time and theassembly and testing time.
7 Ch. 3 – 28(a)Let A = number of shares of stock AB = number of shares of stock BC = number of shares of stock CD = number of shares of stock DTo get data on a per share basis multiply price by rate ofreturn or risk measure value.
8 Solution: A = 333.3, B = 0, C = 833.3, D = 2500Risk: 14,666.7Return: 18,000 (9%) from constraint 2
9 Ch. 3 – 28(b)Solution: A = 1000, B = 0, C = 0, D = 2500Risk: 10A + 3.5B + 4C + 3.2D = 18,000Return: 22,000 (11%)
10 Ch. 3 – 28(c)The return in part (b) is $4,000 or 2% greater, but the riskindex has increased by 3,333.Obtaining a reasonable return with a lower risk is a preferredstrategy in many financial firms. The more speculative,higher return investments are not always preferred becauseof their associated higher risk.
15 Ch. 5 – 6(c)The original basis consists of s1, s2, and s3. It is theorigin since the nonbasic variables are x1, x2, and x3and are all zero.(d) 0x3 enters because it has the largest negative zj - cjand s2 will leave because row 2 has the only positivecoefficient.(e)(f)30; objective function value is 30 times 25 or 750.(g)Optimal Solution: x1 = 10 s1 = 20x2 = 0 s2 = 0x3 = 30 s3 = 0z = 800.
16 EMGT 501 HW #2 Due Day: Sep 27 Chapter 6 - SELF TEST 21
17 Ch. 6 – 21Consider the following linear programming problem:Write the dual problem.Solve the dual.Use the dual solution to identify the optimal solution to the original primal problem.Verify that the optimal values for the primal and dual problems are equal.
18 Ch. 6 – 22A sales representative who sells two products is trying to determine the number of sales calls that should be made during the next month to promote each product. Based on past experience, representatives earn an average $10 commission for every call on product 1 and a $5 commission for every call on product 2. The company requires at least 20 calls per month for each product and not more than 100 calls per month on any one product. In addition, the sales representative spends about 3 hours on each call for product 1 and 1 hour on each call for product 2. If 175 selling hours are available next month, how many calls should be made for each of the two products to maximize the commission?
19 Formulate a linear program for this problem. Formulate and solve the dual problem.Use the final simplex tableau for the dual problem to determine the optimal number of calls for the products. What is the maximum commission?Interpret the values of the dual variables.
21 One of the most important discoveries in the early development of linear programming was the concept of duality.Every linear programming problem is associated with another linear programming problem called the dual.The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways.
22 Primal and Dual Problems Primal ProblemDual ProblemMaxs.t.Mins.t.forforforforThe dual problem uses exactly the same parameters as the primal problem, but in different location.
23 In matrix notationPrimal ProblemDual ProblemMaximizesubject toMinimizesubject toWhere and are row vectors but and are column vectors.
26 Primal-dual table for linear programming Primal ProblemCoefficient of:RightSideCoefficientof:Objective FunctionCoefficients for(Minimize)Dual ProblemVIVIVIRightSideCoefficients forObjective Function(Maximize)
27 Relationships between Primal and Dual Problems One Problem Other ProblemConstraint VariableObjective function Right sidesMinimizationMaximizationVariablesConstraintsUnrestrictedConstraintsVariablesUnrestricted
28 The feasible solutions for a dual problem are those that satisfy the condition of optimality for its primal problem.A maximum value of Z in a primal problem equals the minimum value of W in the dual problem.
36 Question 1: Consider the following problem. (b) Work through the simplex method step by step in tabular form.(c) Use a software package based on the simplex method to solvethe problem.
37 Question 2:For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead. Explain.(a) Maximizesubject to(b) Maximizesubject toandand
38 Question 3:Consider the following problem.Maximizesubject toand(a) Construct the dual problem.(b) Use duality theory to show that the optimal solutionfor the primal problem has