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- Second consulting is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and support personnel required for each project, and the expected profits for each project are summarized in the following table: Project 1 2 3 4 5 6 Engineers required 20 55 47 38 90 63 Support personnel required 15 45 50 40 70 70 Profit ($1,000,000s) 1.0 1.8 2.0 1.5 3.6 2.2 There are 175 engineers available and 150 support personnel available for these projects. Further, no more than 3 projects can be done. Formulate an integer program to solve this problem. Define the decision variables needed to model this problem. The objective is to maximize profit. Give the objection function. Write the constraints. What is the final answer? Explain in detail.Finco has the following investments available:Investment A For each dollar invested at time 0, we receive $0.10 at time 1 and $1.30 at time 2. (Time 0 =now;time= 1 one year from now; and so on.)Investment B For each dollar invested at time 1, we receive $1.60 at time 2.Investment C For each dollar invested at time 2, we receive $1.20 at time 3.At any time, leftover cash may be invested in T-bills, whichpay 10% per year. At time 0, we have $100. At most, $50can be invested in each of investments A, B, and C. Formulate an LP that can be used to maximize Finco’s cash onhand at time 3.Varma Investment Services must develop an investment portfolio for a new client. As an initial investment strategy, the new client would like to restrict the portfolio to a mix of two stocks: Stock Price/Share Estimated Annual Return (%) AGA Products $ 50 6 Key Energy 100 10 The client wants to invest $50,000 and established the following two investment goals. LO 1, 2
- What is true about the assignment problem if the solution has reached the table below? Project A Project B Project C Project D Expert W 0 100 200 0 Expert X 0 0 200 300 Expert Y 0 100 300 0 Expert Z 200 100 0 0 a. The minimum number of vertical and horizontal lines needed to cross out 0s is three. b. Optimal assignment of projects cannot be determined yet. c. Optimal assignment of projects can now be determined.A consulting firm has four projects to consider. Each project will require time (in days) in the next three months according to the table below. Project Time in first month Time in second month Time in third month Revenue A 6 8 6 22000 B 5 6 8 18000 C 8 7 6 16000 D 5 5 4 15000 Revenue from each project is also shown. There is 20 days’ time available in the first month, 22 days’ time available in the second month, and 22 days’ time available in the third month to do these projects. The management wants to select at most 3 projects. If project B is selected, then project C must also be selected. The objective of the firm is to maximize the total revenue. Formulate an integer optimization model for this problem by defining the decision variables, the objective function and all the constraints.…A goldsmith makes two types of jewelry. A model A ring is made with 1 g of gold and 1.5 g of silver and sells for 25 UM.Another model B ring sells for 30 UM and is made of 1.5 g of gold and 1 g of silver. If you only have 750 gof each metal, how many rings should be made of each type to obtain maximum profit?Requested:- Make Initial Table of the problem.- Obtain the Case Variables- Obtain the Objective Function- Get Restrictions- Create the Simplex Table- Obtain the Optimal Solution and the Slack Variables.Solve this operational research exercise.
- Fowle Marketing Research, Inc. is faced with the problem of assigning three available project leaders (agents) to each of three new research studies (tasks). Fowle’s management realizes, however, that the time required to complete each study will depend on the experience and ability of the project leader assigned to the study. Since the three studies have been judged to have approximately the same priority, the company would like to determine the assignments that will minimize the total number of days required to complete all three. If a project leader is to be assigned to one and only one client, what assignments should be made (optimal solutions)? How many days are required (objective function value).With three project leaders and three clients, there are total of nine possible assignment alternatives. The alternatives and the estimated project completion times in days are summarized in Table Client Project Leader 1 2…Varma Investment Services must develop an investment portfolio for a new client. As an initial investment strategy, the new client would like to restrict the portfolio to a mix of two stocks. Stock Price/Share($) Estimated AnnualReturn (%) AGA Products 1 50 6 Key Oil 2 100 10 The client wants to invest $38,000 and established the following two investment goals. Priority Level 1 Goal Goal 1: Obtain an annual return of at least 9%. Priority Level 2 Goal Goal 2: Limit the investment in Key Oil, the riskier investment, to no more than 60% of the total investment. (a) Formulate a goal programming model for the Varma Investment problem. (Let xi be the number of shares of stock ipurchased, dp i be the deviation variable which exceeds the value of goal i, dn i be the deviation variable which is less than the value of goal i, for i = 1, 2.) Min P1 + P2 s.t. Funds Available P1 Goal P2 Goal…Company Z is contemplating a product development program encompassing 6 major projects. The company is constrained from embarking on all of the projects at once by the number of available (budgeted) developers (60) and the budget allocated for project expenses (£200,000). The following table shows the resource requirements and the estimated profit for each project. Use EXCEL’s Solver to answer question 1. 1. Suppose that CEO of company Z decides that project 2 and project 5 are mutually exclusive when all other projects remain independent. What is the revised project portfolio and the revised maximum profit?
- Instruction: Formulate the dual problems for X2 the following linear programming model No need to compute/doing a tableau 1. Maximize Z = 2X1 + X2 Subject to: X2 < 10 2X1 + 5X2 < 60 X1 + X2 <18 3X1 + X2 < 44 2. Maximize Z = 2X1 + X2 +3X3 Subject X1 + X2 + 2X3 < 400 2X1 + X2 + X3 < 500 3. Maximize Z = 80X1 + 60X2 Subject X1 + X2 = 200 X1 < 50 X2 > 80The Bargain Hut has 2400 cubic feet of storage space for refrigerators. Large refrigerators come in 60-cubic-foot packing crates and small refrigerators come in 40-cubic-foot crates. Large refrigeratorscan be sold for a $250 profit and the smaller ones can be sold for $150 profit. How many of each typeof refrigerator should be sold to maximize profit and what is the maximum profit if:a) If the manager wants to sell at least 50 refrigerators must be sold each month, how many large refrigerators and how many smaller refrigerators should he/she order each month to maximize profit?b) At least 40 refrigerators must be sold each month.Vivian and Wayne are planning on going to Lamazenatural childbirth classes. Lamaze classes meet once a weekfor five weeks. Each class gives 20% of the knowledge neededfor “natural” childbirth. If Vivian and Wayne finish theirclasses before the birth of their child, they will forget duringeach week 5% of what they have learned in class. To maximizetheir expected knowledge at the time of childbirth, duringwhich week of pregnancy should they begin classes? Assumethat the number of weeks from conception to childbirthfollows the probability distribution given in Table 9. Week of Birth Probability36 .0537 .1539 .2040 .3041 .1542 .1043 .05