DI 3.1-6. Use the graphical method to solve the problem: Maximize Z = 10x, + 20%). subject to =x, + 2m = 15 x + %<12 5m, + 3m = 45 and x) 20,20.341-7. The Whitt Window Company, a company with only three employees, makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. The company ‘eams $300 profit for each wood-framed window and $150 profit for each aluminum-framed window. Doug makes the wood frames and can make 6 per day. Linda makes the aluminum frames and can make 4 per day. Bob forms and cuts the glass and can make 48 square feet of glass per day. Each wood-framed window uses 6 square feet of glass and cach aluminum-framed window uses 8 square feet of glass. ‘The company wishes to determine how many windows of each type to produce per day to maximize total profit. (a) Describe the analogy between this problem and the Wyndor Glass Co. problem discussed in Sec, 3.1. Then construet and fill in a table like Table 3.1 for this problem, identifying both the activities and the resources. (b) Formulate a linear programming model for this problem. DJ (¢) Use the graphical method to solve this model. 1 @) A new competitor in town has started making wood-framed windows as well. This may force the company to lower the price they charge and so lower the profit made for each wood- framed window. How would the optimal solution change (if at all) if the profit per wood-framed window decreases from $300 to $200? From $300 to 100? (You may find it helpful to use the Graphical Analysis and Sensitivity Analysis procedure in JOR Tutorial.) 1 (e) Doug is considering lowering his working hours, which would decrease the number of wood frames he makes per day. How would the optimal solution change if he makes only 5 wood frames per day? (You may find it helpful to use the Graphical Analysis and Sensitivity Analysis procedure in IOR Tutorial.)33-2. Consider a problem with two decision variables, x, and x2, which represent the levels of activities 1 and 2, respectively. For each variable, the permissible values are 0. 1, and 2, where the fea- sible combinations of these values for the two variables are deter- mined from a variety of constraints. The objective is to maximize a a xy ° 1 2 0 ° 4 8 1 3 8 B 2 6 12 18 certain measure of performance denoted by Z. The values of Z for the possibly feasible values of (x, 22) ate estimated to be those given in the following table: Based on this information, indicate whether this problem com- pletely satisfies each of the four assumptions of linear program- ming. Justify your answers.34-9. Web Mercantile sells many household products through an online catalog. The vompany needs substantial warehouse space for storing its goods. Plans now are being made for leasing warehouse storage space over the next 5 months. Just how much space will be required in each of these months is known. However, since these space requirements are quite different, it may be most economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the additional cost for leasing space for additional ‘months is much less than for the first month, so it may be less expen- sive to lease the maximum amount needed for the entire 5 months. Another option is the intermediate approach of changing the total amount of space leased (by adding a new lease and/or having an old lease expire) at least once but not every month. The space requirement and the leasing costs for the various leasing periods are as follows: Required Month | Space (Sq. Ft.) Leasing Perlod | Cost per Sq. Ft. (Months) ‘The objective is to minimize the total leasing cost for meeting the space requirements. (a) Formulate a linear programming model for this problem. © (b) Solve this mode! by the simplex method.3.4-10. Larry Edison is the director of the Computer Center for Buckly College. He now needs to schedule the staffing of the cen- ter. It is open from 8 4.M. until midnight, Larry has monitored the usage of the center at various times of the day, and determined that the following number of computer consultants are required: Minimum Number of Consultants Required to Be on Duty Time of Day 8 am-noon 4 Noon-4 pm. 8 4 MB eM. 8 pm.-midnight ‘Two types of computer consultants can be hired: full-time and part-time, The full-time consultants work for 8 consecutive hours in any of the following shifis: morning (8 a.m—4 P.a.), afternoon (noon-8 pas), and evening (4 .st.—midnight), Full-time consultants are paid $40 per hour. Part-time consultants can be hired to work any of the four shifls listed in the above table. Part-time consultants are paid $30 per hour. ‘An additional requirement is that during every time period, there must be at least 2 full-time consultants on duty for every part- time consultant on duty. Larry would like to determine how many full-time and how many part-time workers should work each shift to meet the above requirements at the minimum possible cost. (a) Formulate a linear programming model for this problem. € (b) Solve this model by the simplex method.DJ 43-5. Work through the simplex method (in algebraic form) step by step to solve the following problem. Maximize Z= x + 2x) + 4x, subject to By + + 5x, < 10 Ut4e+ ms 8 a +2, 7 and 420, 220 520. 44-6. Consider the following problem. Maximize Z = 3x, + Sx + 6x3, subject to Qn t mt u<4 a +2mn+ sd M+ mt dys4 yt ot u<3 and M20 me 420. 4 (a) Work through the simplex method step by step in algebraic form. Dj (b) Work through the simplex method in tabular form, (€) Use a computer package based on the simplex method to solve the problem.94-2. The Childfair Company has three plants producing child push chairs that are to be shipped to four distribution centers. Plants 1, 2, and 3 produce 12, 17, and 11 shipments per month, respec- tively. Each distribution center needs to receive 10 shipments per ‘month. The distance from each plant to the respective distributing centers is given below: Distance Distribution Center 1 2 3 4 800 miles 1,300 miles 400 miles 700 miles 1 Plant 2 | 1,100 miles 1,400 miles 600 miles 1,000 miles 3 | 600 miles 1,200 miles 800 miles 900 miles ‘The freight cost for each shipment is $100 plus 50 cents per mile. How much should be shipped from each plant to each of the distribution centers to minimize the total shipping cost? (a) Formulate this problem as a transportation problem by con- structing the appropriate parameter table. (b) Draw the network representation of this problem, © (¢) Obtain an optimal solution.93-2. Four cargo ships will be used for shipping goods from one port to four other ports (labeled 1, 2, 3, 4). Any ship can be used for making any one of these four trips. However, because of differences in the ships and cargoes, the total cost of loading, transporting, and unloading the goods for the different ship-port combinations varies considerably, as shown in the following table: Port 1 2 3 4 1 3500 $400 $600 $700 sh 2 600 600 700 500 iP 3 700 500 700 600 4 500 400 600 600 ‘The objective is to assign the four ships to four different ports in such a Way as to minimize the total cost for all four shipments. (a) Describe how this problem fits into the general format for the assignment problem. € (b) Obtain an optimal solution. (c) Reformulate this problem as an equivalent transportation prob- Jem by constructing the appropriate parameter table. ba (@) Use the northwest comer rule to obtain an initial BF solu- tion for the problem as formulated in part (c).