Homework 1 Fall 2000 ENM3 82.1

Transcription

1 Homework 1 Fall 2000 ENM Jensen and Bard, Chap. 2. Problems: 11, 14, and 15. Use the Math Programming add-in and the LP/IP Solver for these problems. 11. Solve the chemical processing example in Section 2.8 using a computer program. Investigate the effect of the following changes separately. The changes are not cumulative. a. Eliminate the upper bound on sales for product X. b. Eliminate the upper bound on the manufacture of X, but assume any amount produced over 5000 must be discarded at a cost of $1/lb. c. Eliminate the restriction on the amount of raw material B that can be purchased. d. Change the cost of raw material C to $5/lb. Base Total Part a Remove the upper limit on the production of X Total The limit on the mix amount becomes tight. 1

2 Part b. The solution of the base case results. It is not beneficial to make more X if it is thrown away for a cost $1. Part c. Remove the restriction on B Total The limit on the amount of X is constraining the solution. Part d. Change the unit cost on C to $5 Total The product amounts don't change, but the raw material mix changes. 2

3 14. A company has two manufacturing plants (A and B) and three sales outlets (I, II, and III). Shipping costs from the plants to the outlets are as follows. Outlet Plant I II III A B The company wants to plan production, shipping and sales for the next two periods. The data for the periods is shown below. The plants can store products produced in one period for sale in the next. The maximum storage at each plant is 50 and the inventory cost is $1 per unit. Find the solution that maximizes profit. Plant A Manufacturing data Plant B Period Unit cost ($) Capacity Unit cost ($) Capacity Selling price ($) Demand Data Maximum sales Period I II III I II III Answer: Network model. (1,50) [0,-100,-15] [0,175,8] (4) I1 (4) A1 (6) [0,-200,-20] A2 (8) (6) [0,150,10] II1 (8) [0,200,7] (7) [0,170,8] (7) [0,-150,-14] B1 (4) B2 (4) (3) III1 (3) (1,50) [0,-150,-18] I2 [0,-300,-17] II2 [0,-150,-21] III2 [Fixed, Slack, Cost] (Cost, Capacity) Use these variables PA1, PB1, PA2, PB2: production at A and B in the two periods S11, S21, S31: sales in period 1 at three locations S12, S22, S42: sales in period 2 at three locations 3

4 xa11, xa21, etc: shipments in period 1 xa12, xa22, etc: shipments in period 2 IA, IB: Inventory storage at A and B Max profit: Z = 15S S21 + sales revenue - 8PA1-7PB1 - production costs - 4xA11 - xa21 - transportation costs - 1IA - 1IB Conservation constraints and upper limits on variables. Maxprofit = Z =

5 15. A company is planning its aggregate production schedule for the next three months. Units may be produced on regular time or overtime. The relevant costs and capacities are shown in the table below. The demand for each month is also shown. There are three ways of meeting this demand: inventory, current production, and backorders. Units produced in a particular month may be sold in that month or kept in inventory for sale in a later month. There is a $1 cost per unit for each month an item is held in inventory. Initially, there are 15 units in inventory. Also, sales can be backordered at a cost of $2/unit/mo. Backorders represent production in future months to satisfy demand in past months, and hence incur an additional cost. Capacity (units) Production cost ($/unit) Period Regular time Overtime Regular time Overtime Demand a. Develop a model that when solved will yield the optimum production plan. Find the solution with a computer program. b. How would the model change if the inventory cost depends on the total time an item is stored. Let the cost be $1/unit for items kept in inventory for 1 month, $3 for items kept for 2 months, and $5 for items kept for 3 months. Assume that the initial inventory has been in storage for 1 month. Solve the model with this change. a. Solution Variables for part a R k = Regular production in month k O k = Overtime production in month k I k = Inventory stored in month k B k = items backordered from month k 5

6 15 Initial Inv. R1 (Regular prod.) O1 (Regular prod.) 1 (60) Demand Conservation of product R1 + O1 - I1 + B2 = R2 + O2 + I1 - I2 - B2 + B3 = 80 I1 B2 R3 + O3 + I2 - B3 = 140 R2 (Regular prod.) O2 (Regular prod.) 2 (80) Demand Limits on regular production R1 100, R2 100, R3 60 O1 20, O2 10, O3 20 R3 (Regular prod.) O3 (Regular prod.) I2 3 B3 (140) Demand Objective: Minimize cost Z = 14R1 + 17R2 + 17R3 (regular prod. + 18O1 + 22O2 + 22O3 (overtime prod.) + 1(I1 + I2) (inventory) + 2(B2 + B3) (backorder) Solution (Sheet Agg_16) z = The cost is $15 greater if initial inventory is charged. Period Regular Production Overtime Production Inventory to Next Period Backorder to Previous Period

7 b. Variables for part b We have all the variables except I k : we break the inventory into parts depending on how many months passed until they are used. I1 k is the number used after 1 month, I2 k is the number used after 2 months and I3 k is the number used after 3 months. 15 Initial Inv. R1 (Regular prod.) O1 (Regular prod.) 0 1_P I02 I01 1_D (60) Demand B2 R2 (Regular prod.) O2 (Regular prod.) 2_P 2_D (80) Demand B3 I03 R3 (Regular prod.) O3 (Regular prod.) 3_P 3_D (140) Demand Same solution as in part a. 7