Integer Programming

Definition of Integer Programming If requiring integer values is the only way in which a problem deviates from a linear programming formulation, then it is an integer programming (IP) problem. If only some of the variables are required to have integer values, this model is referred to as mixed integer programming (MIP). When distinguishing the all-integer problem from this mixed case, we call the former pure integer programming. IP problems that contain only binary variables are called binary integer programming (BIP) problems (or 0-1 integer programming problems).

12.1 PROTOTYPE EXAMPLE The California manufacturing company is considering expansion by new factory in either Los Angels or San Francisco, or perhaps even in both cities. It also is considering building at most one new warehouse, but the choice of location is restricted to a city where a new factory is being built. The data is given in the below table. Decision Number Yes-or-No Question Decision Variable Net Present Value Capital Required Built factory in LA Built factory in SF Built warehouse in LA Built warehouse in SF X 1 X 2 X 3 X 4 $9million $5million $6million $4million $6million $3million $5million $2million

The net present value (total profitability considering the time value of the money) of each of these alternatives is shown in above tables. The rightmost column gives the capital required (already included in the net present value) for the respective investments, where the total capital available is $10 million. The object is to find the feasible combination of alternatives that maximizes the total net present value.

Decision Variables All the decision variables have the binary form 0 if decision j is no 1 if decision j is yes = (j=1,2,3,4)

Objective function Let Z= total net present value of these decisions. If the investment is made to build a particular facility,

Constraints The rightmost column of table indicates that the amount of capital expended on the four facilities cannot exceed $10 million.

Constraints Because the last two decisions represent mutually exclusive alternatives (the company wants at most one new warehouse), we also need the constraint The company would consider building a warehouse in a city only if a new factory also were going there

Mathematical model of the problem Maximize Subject to And is integer, for j=1,2,3,4. Equivalently, the last three model can be replaced by the single restriction is binary, for j=1,2,3,4

Solving the problem Using Excel Solver

12.2 Some BIP applications Package Problem Fixed Investment Site Selection Job assignment

Package Problem A package box can hold maximal 50 Kg. There are three kinds of goods, and the weight and the value of each kind of goods are listed in the following table. Goods1Goods2Goods3 Weight (Kg/piece) Value (Dollars/piece) The question is: How to fill in the box to make the total value to be maximal?

Formulate Mathematical Model Let x 1, x 2, x 3 be the amount of the three kinds of goods, then Max Z= 17x 1 +72x 2 +35x 3 Subject to 10x 1 +41x 2 +20x 3 ≤50 x 1, x 2, x 3 are integer The solution is x 1 =1, x 2 =0, x 3 =2, Z=87 dollars

Fixed Investment The capital for investment is B, the number for investment projects is N, the capital needed and the expected profit of project j is a j and c j. There are other restrictions for the investment such as: – Must invest project m if invest project n – Must invest one project from project u and project v – (??invest at least on project from project u and project v) – Just invest two projects in project a, b, c How to arrange the investment to make the profit to be optimal ？

Formulate Mathematical Model Let Ｘ j= 1 invest project j 0 not invest project j s.t.

Site selection A company has two plants, A1 and A2. Now the management plans to build a new plant. There are two sites for the new plant, A3 and A4. There are four markets need the product, B1, B2, B3, and B4. The production capacity, market requirement and transportation cost are estimated in the following table. The production cost for A3 is 12 million dollars, and for A4 is 15 million dollars. Now the management must decide to build the new plant at A3 or A4, to make the total cost (including transportation and production cost) to be minimal.

c ij B1B2B3B4Capacity (KT Per year) A1 ２９３４４００ A2 ８３５７６００ A3 ７６１２２００ A4 ４５２５２００ Requirement （ KT per year ）

Let x ij = the products amount transported from Ai to Bj ， Ｚ = the total cost y= 1 Build new plant at A3 0 Build new plant at A4

Formulate Mathematical Model

Job Assignment A young couple, Eve and Steven, want to divide their main household chores (marketing, cooking, dishwashing, and laundering) between them so that each has two tasks but the total time they spend on household duties is kept to a minimum. Their efficiencies on these tasks differ ， where the time each would need to perform the task is given by the following table:

Time Needed per Week Marketing Cooking Dishwashing Laundry Eve4.5hours 7.8hours 3.6hours 2.9hours Steven4.9hours 7.2hours 4.3hours 3.1hours a) Formulate a BIP model for this problem. b) Display this model on an Excel spreadsheet. c) Use the computer to solve this model.

Let Marketing Cooking Dishwashing Laundry Eve x em x ec x ed x el Steven x sm x sc x sd x sl

Formulate Mathematical Model Minimize Z=4.5x em +7.8x ec +3.6x ed +2.9x el +4.9 x sm +7.2 x sc +4.3 x sd +3.1x sl Subject to x em +x ec +x ed +x el =2 x sm +x sc +x sd +x sl =2 x em +x sm =1 x ec +x sc =1 x ed +x sd =1 x el +x sl =1 and all x ij are binary

Exercise P

12.3 Innovative Uses of Binary Variables in Model Formulation Either-Or Constraints The Fixed-Charge Problem

Example 1 The R&D Division of the GOOD PRODUCTS COMPANY has developed three new products. Production Time Used for Each Unit Produced Production Time Available per Week Product 1 Product 2 Product 3 Plant 1 Plant 2 3 hours 4 hours 2 hours 4 hours 6hours 2 hours 30 hours 40 hours Unit profit (thousands of dollars) Sales potential (units per week)

If we let x1, x2, x3 be the production rates of the products, the model is formulated as: Maximize Z=5x 1 +7x 2 +3x 3 Subject to 3x 1 +4x 2 +2x 3 ≤30 4x 1 +6x 2 +2x 3 ≤40 x 1 ≤7 x 2 ≤5 x 3 ≤9

The management has imposed the following restriction. Restriction 1: From the three possible new products, at most two should be chosen. Restriction 2: Just one of the two plants should be chosen.

Restriction 2 means Either 3x 1 +4x 2 +2x 3 ≤30 Or 4x 1 +6x 2 +2x 3 ≤40 must hold Then we introduce a auxiliary binary variable y 4 y4=y4= 1 if 4x 1 +6x 2 +2x 3 ≤40 must hold 0 if 3x 1 +4x 2 +2x 3 ≤30 must hold 3x 1 +4x 2 +2x 3 ≤30+My 4 4x 1 +6x 2 +2x 3 ≤40+M(1-y 4 ) Introduce a large positive number M

Restriction 1 means three auxiliary binary variables (y 1, y 2, y 3 ) should be added yj=yj= 1 if x j >0 can hold (can produce project j) 0 if xj=0 must hold (cannot produce project j) Then we add the constraints x 1 ≤My 1 x 2 ≤My 2 x 3 ≤My 3 y 1 +y 2 +y 3 ≤2

Formulate Mathematical Model Maximize Z=5x 1 +7x 2 +3x 3 Subject to x 1 ≤7 x 2 ≤5 x 3 ≤9 3x 1 +4x 2 +2x 3 -My 4 ≤30 4x 1 +6x 2 +2x 3 +My 4 ≤40+M X 1 - My 1 ≤0 X 2 - My 2 ≤0 X 3 - My 3 ≤0 x 1 ≥0, x 2 ≥0, x 3 ≥0, y j is binary, for j=1, 2, 3, 4

Solve the problem Using Excel Solver Teamwork tasks P P ,12.3-4