Introduction to Macroeconomics: Optimization Problems and Linear Programming, Summaries of Macroeconomics

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G. Mankiw,

Introduction to macroeconomics

MBUS529 – WEEK 1:

CHAPTER 2 – EX.2.23:

MODEL

EZ RIDER

MF TIME

FRAME AVAILABLE

INFINITY

ASSEMBLY

PROFIT

LADY SPORT 3 280 2.5 1800

LIMIT 2100 1000

OBJECTIVE: IS TO MAXIMIZE TOTAL PROFIT

KEY DECISION: DETERMINE HOW MANY UNITS AT EACH PRODUCT TYPE TO MAKE

CONSTRAINTS: STAY WITHIN RESOURCE LIMIT: LIKE MANF. AND ASSEMBLY TIME,

AND FRAME AVAILABILE

DECISION VARIABLE: X 1 = NUMBER OF EZ RIDE TO MAKE

X 2 = NUMBER OF LADY SPORT TO MAKE

MODEL:

MAX Z= 2400 X 1 + 1800 X 2 [ TOTAL PROFIT]

Subject too [ST] CONSTRAINTS LIMIT: o = 6X1 + 3X2 ≤ 2100 [MF TIME LIMIT] o = 2X1+2.5X2 ≤ 1000 [ASSEMBLY TEST TIME LIMIT] o = X2 ≤ 280 [ ASSEMBLY FRAME LIMIT] o X1 , X2 ≥ 0 [ NON NEGATIVE] INPUTS: n = NUMBER OF PRODUCT TYPES o mi = MF TIME / UNIT PRODUCT i o fi = FRAME AVAILABLE FOR PRODUCT TYPE i

Week2: 4. Supplier Comp 1 2 3 Demand 1 12 13 14 1000 2 10 11 10 800 Supplier 1 2 3 Cap. 600 1000 800

• Demand= 1000 from com1 and 800 from comp 2
• Objective : Minimize cost - Limit:
  1. don’t exceed any supplier capacity
  2. meet all demand
• Key decision : How many comp. of each type to source from each supplier? which mean we have 6 plans comp1 for 3 supplier and comp 2 from 3 supplier - Inputs:
  1. n = number of component
  2. m= number of supplier
  3. Di= the demand for component type i
  4. Kj= total capacity for supplier j
  5. Cij= Cost/ unit for compo. (i) from supplier (j) - Decisions variable : 1. Xij =Number of compo. Type (i) to purchase from supplier (j) MODEL: m n

I. Minimize Z= [total cost] ¿∑ ∑ Cij^ Xij

i = 1 j = 1 m

II. ST ¿∑ Xij^ ≤ Kj for j=1,…,n [ Capacity Constraints ]

i = 1 n

III. ST ¿∑ Xij^ ≥ Di for i=1,…m [ meet all demand]

j = 1 IV. Xij ≥ 0 for i=1,..,m and j=1,..,n

************Now go excel *************** Ex4.

• Info.: Timeslot # of officers needed 8 > noon 5 Noon > 4 6 4 > 8 PM 10 8 > midnight 7 Midnight > 4 4 4 > 8 AM 6
• Objective : Minimize total number officers needed - Constraints: I. 8 hr. shift II. Ensure each timeslot is fully covered - Key Idea : I. total available timeslot (i) = II. # starting timeslot (i-1) + III. # starting timeslot (1) - Inputs: I. n = number of timeslots II. Di= officer demand timeslot (i) - Decisions Variable: I. Xi= number of officers starting in timeslot (i) - Model: n

I. Minimize Z (total Number of officers ) ¿∑ Xi

i = 1 II. (^) ST :

Ii ≤ K for I =1,…,n (inventory capacity) In = If (ensure target ending inventory) Xi, Yi, Ii ≥ 0 for i=1,..,n ( no neg.) **************now on Excel *******