- f(x) = x12 – x2, x0 = [1 1]T. Sketch your path, predict the outcome of further steps.
- Minimum cost Hard brick, Inc, has two kilns, Kiln I can produce 3000 grey bricks, 2000 red bricks, and 300 glazed bricks daily. For Kiln II the corresponding figures are 2000, 5000 and 1500. Daily
- Nutrition Foods A and B have 600 and 500 calories, contain 15g and 30g of protein, and cost $1.80and $2.10 per unit, respectively, Find the minimum cost diet of at least 3900 calories containing at
- Do Prob. 1 with the last two constraints interchanged.
- Maximize z = 300x1 + 500x2 subject to 2x1 + 8x2 < 60, 2x1 + x2 < 30, 4x1 + 4x2 < 60.
- Maximize the total output f = x1 + x2 + x3 (production figures of three different production processes subject to input constraints (limitation of machinetime)
- Using an artificial variable, minimize f = 2x1 – x2 subject to x1 > 0, x2 > 0, x1 + x2 > 5 – x1 + x2 < 1, 5x1 + 4x1 < 40.
- If one uses the method of artificial variables in a problem without solution, this non-existence will become apparent by the fact that one cannot get rid of the artificial variable. Illustrate this
- In Prob. 8 start from x0 = [1.5 1] T, show that the next even-numbered approximations are x2 = kx0, x4 = k2x0, etc., where k = 0.04.
- Apply the method of steepest descent to f(x) = 9x12 + x22 + 18x1 – 4x2, 5 steps, starting from x0 = [2 4]T.
- Show that the gradients in Prob. 13 are orthogonal. Give a reason.
- Maximize f = x1 + x2 subject to x1 + 2x2 < 10, 2x1 + x2 < 10, x2 < 4.
- A factory produces two kinds of gaskets, G1, G2, with net profit of $60 and $30, respectively, Maximize the total daily profit subject to the constraints (xj = number of gaskets Gj produced per day)
- Find the adjacency matrix of the graph in Fig 476.
- The graph in Prob. 8, incidence Matrix of a Digraph; Matrix B = [bkj] with entries find the incidence matrixof;
- Call the length of a shortest path s → v the distance of v from s. Show that if v has distance l, it has label λ(v) = l.
- Show that the length of a shortest postman trail is the same for every starting vertex.
- Uniqueness the path connecting any two vertices u and v in a tree is unique.
- If a graph has no cycles, it must have at least 2 vertices of degree 1 (definition in Sec. 23.1)
- A tree with n vertices has n – 1 edges (Proof by induction).
- A graph with n vertices is a tree if and only if it has n – 1 edge and has no cycles.
- Complexity show that Prim’s algorithm has complexity O(n2).
- For a complete graph (or one that is almost complete), if our data is n n x n distance table (as in Prob. 12, Sec. 23.4) show that the present algorithm [which is O (n2)] cannot easily be replaced by
- Show that in a network G with all cij = 1, the maximum flow equals the number of edge-disjoint paths s → t.
- Show that in a network G with capacities all equal to 1, the capacity of a minimum cut set (S, T) equals the minimum number q of edges whose deletion destroys all directed paths s ? t. (A directed
- Three factories 1, 2, 3 are each supplied underground by water, gas, and electricity, from poins A, B, C respectively. Show that this can be represented by K3,3 (the complete bipartite graph G = (S,
- Edge coloring the edge chromatic number Xe (G) of a graph G is the minimum number of colors needed for coloring the edges of G so that incident edges get different colors. Clearly, Xe (G) > max d(u),
- Edge coloring the edge chromatic number Xe (G) of a graph G is the minimum number of colors needed for coloring the edges of G so that incident edges get different colors. Clearly, Xe (G) > max d(u),
- Which of the following mathematical relationships could be found in a linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why.a.
- Find the solutions that satisfy the following constraints:a. 4A + 2B ≤ 16b. 4A + 2B ≥ 16c. 4A + 2B = 16
- Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:a. 3A + 2B ≤ 18b. 12A + 8B ≥ 480c. 5A + 10B = 200
- Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:a. 3A - 4B ≥ 60b. -6A + 5B ≤ 60c. 5A - 2B ≤ 0
- Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:a. A ≥ 0.25 (A + B)b. B ≤ 0.10 (A + B)c. A ≤ 0.50 (A + B)
- Three objective functions for linear programming problems are 7A + 10B, 6A + 4B, and -4A + 7B. Show the graph of each for objective function values equal to 420.
- Identify the feasible region for the following set of constraints:0.5A + 0.25B ≥ 30 1A + 5B ≥ 2500.25A + 0.5B ≤ 50A, B ≥ 0
- Identify the feasible region for the following set of constraints:2A - 1B ≤ 0-1A + 1.5B ≤ 200A, B ≥ 0
- Identify the feasible region for the following set of constraints:3A - 2B ≥ 02A - 1B ≤ 2001A ≤ 150A, B ≥ 0
- For the linear programMax2A + 3Bs.t.1A + 3B ≤ 65A + 3B ≤ 15A, B ≥ 0Find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal
- Solve the following linear program using the graphical solution procedure:Max 5A + 5Bs.t.1A ≤ 100 1B ≤ 802A + 4B ≤ 400A, B ≥ 0
- Consider the following linear programming problem:Max 3A + 3Bs.t.2A + 4B ≤ 126A + 4B ≤ 24A, B ≥ 0a. Find the optimal solution using the graphical solution procedure.b. If the objective function
- Consider the following linear program:Max 1A + 2Bs.t. 1A ≤ 5 1B ≤ 42A + 2B = 12A, B ≥ 0a. Show the feasible region.b. What are the extreme points of the feasible region?c. Find the
- Par, Inc., is a small manufacturer of golf equipment and supplies. Parâ€™s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard model, and a
- Suppose that Par’s management encounters the following situations:a. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag.b. A new low-cost
- Refer to the feasible region for Par, Inc., in Problem 14.a. Develop an objective function that will make extreme point (0, 540) the optimal extreme point.b. What is the optimal solution for the
- Write the following linear program in standard form:Max5A + 2Bs.t.1A - 2B ≤ 4202A + 3B ≤ 6106A - 1B ≤ 125A, B ≥ 0
- For the linear programMax4A + 1Bs.t.10A + 2B ≤ 303A + 2B ≤ 122A + 2B ≤ 10A, B ≥ 0a. Write this problem in standard form.b. Solve the problem using the graphical solution procedure.c. What are
- Given the linear programMax 3A + 4Bs.t-1A + 2B ≤ 81A + 2B ≤ 12 2A + 1B ≤ 16A, B ≥ 0a. Write the problem in standard form.b. Solve the problem using the graphical solution procedure.c. What
- For the linear programMax3A + 2Bs.t.A + B ≥ 43A + 4B ≤ 24A ≥ 2A – B ≤ 0A, B ≥ 0a. Write the problem in standard form.b. Solve the problem.c. What are the values of the slack and surplus
- Consider the following linear program:Max 2A + 3Bs.t.5A + 5B ‰¤ 400 Constraint1-1A + 1B ‰¤ 10 Constraint 21A + 3B ‰¥ 90 Constraint 3A, B ‰¥ 0Figure shows a
- Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning horizon. Constraints affecting
- Embassy Motorcycles (EM) manufacturers two lightweight motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The
- Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcherâ€™s model. The firm has 900 hours of production time available in its cutting and
- George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund
- The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising. Management decided that at least
- Blair & Rosen, Inc. (B&R) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past
- Tom’s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom’s, Inc., makes two salsa products: Western
- AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New
- A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used
- Consider the following linear program:Min3A + 4Bs.t.1A + 3B ≥ 61A + 1B ≥ 4A, B ≥ 0Identify the feasible region and find the optimal solution using the graphical solution procedure. What is the
- Identify the three extreme-point solutions for the M&D Chemicals problem Identify the value of the objective function and the values of the slack and surplus variables at each extreme point.
- Consider the following linear programming problem:MinA + 2Bs.t.A + 4B ≤ 212A + B ≥ 73A + 1.5B ≤ 21-2A + 6B ≥ 0A, B ≥ 0a. Find the optimal solution using the graphical solution procedure and
- Consider the following linear program:Min 2A + 2Bs.t.1A + 3B ≤ 123A + 1B ≥ 131A - 1B = 3A, B ≥ 0a. Show the feasible region.b. What are the extreme points of the feasible region?c. Find the
- For the linear programMin6A + 4Bs.t.2A + 1B ≥ 121A + 1B ≥ 101B ≤ 4A, B ≥ 0a. Write the problem in standard form.b. Solve the problem using the graphical solution procedure.c. What are the
- As part of a quality improvement initiative, Consolidated Electronics employees complete a three-day training program on teaming and a two-day training program on problem solving. The manager of
- The New England Cheese Company produces two cheese spreads by blending mild cheddar cheese with extra sharp cheddar cheese. The cheese spreads are packaged in 12-ounce containers, which are then sold
- Applied Technology, Inc. (ATI) produces bicycle frames using two fiberglass materials that improve the strength-to-weight ratio of the frames. The cost of the standard grade material is $7.50 per
- Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up
- Photo Chemicals produces two types of photographic developing fluids. Both products cost Photo Chemicals $1 per gallon to produce. Based on an analysis of current inventory levels and outstanding
- Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of
- Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain.Max 4A + 8Bs.t.2A + 2B ≤ 10 -1A + 1B ≥ 8 A, B ≥ 0
- Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain.Max 1A + 1Bs.t.8A + 6B ≥ 242B ≥ 4A, B ≥ 0
- Consider the following linear program:Max 1A + 1Bs.t.5A + 3B ≤153A + 5B ≤ 15A, B ≥ 0a. What is the optimal solution for this problem?b. Suppose that the objective function is changed to 1A +
- Consider the following linear program:Max 1A - 2Bs.t.-4A + 3B ≤ 31A - 1B ≤ 3A, B ≥ 0a. Graph the feasible region for the problem.b. Is the feasible region unbounded? Explain.c. Find the optimal
- The manager of a small independent grocery store is trying to determine the best use of her shelf space for soft drinks. The store carries national and generic brands and currently has 200 square
- Discuss what happens to the M&D Chemicals problem (see Section 7.5) if the cost per gallon for product A is increased to $3.00 per gallon. What would you recommend? Explain.
- For the M&D Chemicals problem in Section 7.5, discuss the effect of management’s requiring total production of 500 gallons for the two products. List two or three actions M&D should
- PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 85 full-time equivalent pharmacists
- Expedition Outfitters manufactures a variety of specialty clothing for hiking, skiing, and mountain climbing. They have decided to begin production on two new parkas designed for use in extremely
- English Motors, Ltd. (EML), developed a new all-wheel-drive sports utility vehicle. As part of the marketing campaign, EML produced a video tape sales presentation to send to both owners of current
- Creative Sports Design (CSD) manufactures a standard-size racket and an oversize racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite alloy that was invented by the
- Management of High Tech Services (HTS) would like to develop a model that will help allocate their technicians’ time between service calls to regular contract customers and new customers. A maximum
- Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large computer company and involves the
- Digital Imaging (DI) produces photo printers for both the professional and consumer markets. The DI consumer division recently introduced two photo printers that provide color prints rivaling those
- Better Fitness, Inc. (BFI) manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use
- Hart Venture Capital (HVC) specializes in providing venture capital for software development and Internet applications. Currently HVC has two investment opportunities: (1) Security Systems, a firm
- Consider the following linear program:Max3A + 2Bs.t.1A + 1B â‰¤ 103A + 1B â‰¤ 241A + 2B â‰¤ 16A, B â‰¥ 0a. Use the graphical solution procedure to find the optimal solution.b. Assume
- Consider the linear program in Problem 1. The value of the optimal solution is 27. Suppose that the right-hand side for constraint 1 is increased from 10 to 11.a. Use the graphical solution procedure
- Consider the following linear program:Min8X + 12Ys.t.1X + 3Y â‰¥ 92X + 2Y â‰¥ 106X + 2Y â‰¥ 18X, Y â‰¥ 0a. Use the graphical solution procedure to find the optimal solution.b. Assume
- Consider the linear program in Problem 3. The value of the optimal solution is 48. Suppose that the right-hand side for constraint 1 is increased from 9 to 10.a. Use the graphical solution procedure
- Refer to the Kelson Sporting Equipment problem. LettingR = number of regular glovesC = number of catcher€™s mitts leads to the following formulation:Max 5R + 8Cs.t.R + 3/2 C ‰¤
- Refer to the computer solution of the Kelson Sporting Equipment problem in Figure.THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING EQUIPMENT PROBLEMa. Determine the objective coefficient
- Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual
- Refer to Figure which shows the computer solutionTHE MANAGEMENT SCIENTIST SOLUTION FOR THE INVESTMENTADVISORS PROBLEMa. How much would the return for U.S. Oil have to increase before it would be
- Recall the Tomâ€™s, Inc., problem. LettingW = jars of Western Foods SalsaM = jars of Mexico City SalsaLeads to the formulation:Max1W + 1.25Ms.t.5W + 7M â‰¤ 4480 Whole tomatoes3W + 1M â‰¤
- Recall the Innis Investments problem. LettingS = units purchased in the stock fundM = units purchased in the money market fundLeads to the following formulation:Min8S + 3Ms.t.50S + 100M â‰¤
- Refer to Problem 10 and the computer solution shown in Refer FigureTHE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS INVESTMENTS PROBLEMa. Suppose the risk index for the stock fund (the value of CS)
- Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production
- Refer to the computer solution of Problem 12 in FigureTHE MANAGEMENT SCIENTIST SOLUTION FOR THE QUALITY AIR CONDITIONING PROBLEMa. Identify the range of optimality for each objective function
- Digital Controls, Inc. (DCI) manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller
- Refer to the computer solution to Problem 14 in FigureTHE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITALCONTROLS, INC., PROBLEMa. Interpret the ranges of optimality for the objective function

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