Hints: Foreign Exchange Case

1. a_ij--the exchange rate between currencies i and j is denominated in the units of currency j. i.e., a₁₂=.591 is the number of pounds that one dollar will fetch.
2. The book has a typo: C_i should be 1/2(a_i1+1/a_1i). For instance the avearge "dollar value" of a pound is C₂ = 1/2(a₂₁ + 1/a₁₂)= 1/2(1.665+ 1/.591)= 1.6785.
3. b _i is the begining holding of currency in the denomination of that currency. e.g., b₂ is 5 million pounds. etc.
4. Likewise L _i is the final desired level of currency i. e.g., Final holding of FFrancs should be no less than L₃ = 7M FFrancs.

1. Decison variables X_ij measure the amount of currency i exchanged into a certain amount of currency j depending on the exchange rate between i and j. e.g., X₂₃ is the number of pounds (i = 2) exchaneged into certain number of francs (j = 3). How many? 9.12 * X₂₃ (each pound exchanged fetches 9.12 francs)
2. You can now write the problem as: Maximize C₁Y₁ (dollars) + C₂Y₂ (pounds) + etc. Notice that C_i are known numbers and Y_i are decison variables.
3. Subject to the constraints that the final holdings (Y_i) depend on the begining balance of currency i and the set of exchanges (X_ij) involving currency i. Y_i is defined as the begining amount of currency i plus all the other currencies exchanged into currency i; less total of currency i exchanged into other currencies. e.g., final pound holding (Y₂) will be 5M plus dollars converted into pounds (X₁₂) plus francs converted into pounds (X₃₂) plus etc. (Notice that X₁₂ and X₃₂ etc. are mesured in dollars, francs etc. resp., and thus must first be converted into pounds using appropriate exchange rates) less pounds used to buy dollars (X₂₁), pounds used to buy francs (X₂₃) etc. (Note that these later variables are already measured in pounds)
4. Finally subject to a set of constraints that insure that the final holdings of each currency satisfy the minimal reqirements. e.g., we must have at least 7M FFrancs i.e., Y₄ >= 7. etc.