MIE 620 J.M. Smith Homework #1 Umass Fall 2021 1. A small job shop has one automatic screw machine which can be operated a maximum of 7.5 hours per day. They plan to manufacture both pins and bolts on the machine and they feel that they can sell any amount they can produce of either or both products. However, a maximum of 1000 bolts per day can be produced due to raw material shortages. Other relevant production and cost data are given in the table below: Part Pin Bolt Production Rate Pieces/Hour 300 200 Material Costs Cents/Piece 2.5 3.0 Labor & Overhead Cents/Hour 240 240 Selling Price Cents/Piece 5 7 It is desired to find the quantity of each part to produce per day in order to maximize profit. Please formulate this problem and give a graphical solution. Check with AMPL, LINGO, or whichever program is appropriate. TORA only runs under windows 7 and that is why I did not put on Moodle. 2. Fred has $2200 to invest over the next five years. At the beginning of each year, he can invest money in one- or two-year time deposits. The bank pays 8 per cent interest on one-year time deposits and 17 per cent (total) on two-year time deposits. In addition, West World Limited will offer three-year certificates starting at the beginning of the second year. These certificates will return 27 percent (total). If Fred reinvests his money available every year, formulate a linear program to show how to maximize his total cash on hand at the end of the fifth year. 3. A manufacturing firm would like to plan its production/inventory policy for the months of August, September, October and November. The product under consideration is seasonal, and its demand over the particular months is estimated to be 500, 600, 800 and 1200 units respectively. Presently, the monthly production capacity is 600 units with a unit cost of $25. Management has decided to install a new production system with monthly capacity of 1100 units with a unit cost of $30. However, the new system cannot be installed until the middle of November (hint: this timing restriction provides an important bounding criterion on the level of production activity for this month). Assume that the starting inventory is 250 units and that at most 400 units can be stored any given month. If the holding inventory cost per month per item is $3, find the production schedule that minimizes the total production and inventory cost using the AMPL code. Assume that demand must be satisfied and that 100 units are required in inventory at the end of November. i) Please formulate this problem. 1 ii) Also please solve with AMPL. This is suggested so that you become familiar with the style of inputs required of AMPL. 4. Consider the following problem: Maximize − x1 − x2 + 2x3 + x4 subject to : 2x1 + x2 + x3 + x4 ≥ 6 x1 + 2x2 − 2x3 + x4 ≤ 12 x1 , x2 , x3 , x4 ≥ 0 a. Introduce slack variables and draw the requirements space. b. Interpret feasibility in the requirements space. c. You are told that an optimal solution can be obtained by having at most two positive variables while all other variables are set to zero. Utilize this statement and the requirements space to find an optimal solution. 5. Construct a general solution of the system Ax = b where A is an m × n matrix with rank m. What is the general solution of the following system? x1 − 2x2 + x3 = 2 −x1 + 3x2 + 2x3 = 6           1 2 −1 5 −4 6. Let a1 = , a2 = , a3 = , a4 = and a5 = . Illustrate 0 3 4 −3 3 geometrically the collection of all convex combinations of these five points. 7. Find all the extreme points of the following polyhedral set. X = {(x1 , x2 , x3 ) : x1 − x2 + x3 ≤ 1, −x1 + 2x2 ≤ 4, x1 , x2 , x3 ≥ 0} Does X have any recession directions? Why? (AMPL, Matlab, Mathematica, Python, or Maple may be helpful here) These are due in two weeks for on-campus students. 3 weeks from today for off-campus students. 2
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