# CS代写｜CS524 – Problem Set #10

## 1 You Say To-MAY-To, I Say To-MAH-To

Teresa’s Terrific Tomatoes (TTT) is in the business of selling a set P of tomato-related products (e.g salsa, ketchup, tomato paste). In order to create these products, resources from a set R are required (e.g. tomatoes, sugar, labor, spices). To be specific, the parameter apr is the amount of resource r ∈ R that is required to produce one unit of product p ∈ P. There is a limit br on the amount of each resource r ∈ R that can be used in a production period. The company can pay for extra resources at a cost of gr,r ∈ R.

TTT management is trying to determine the optimal way to meet an estimate of its demand dpt for each of its products p ∈ P in each planning period t over a horizon T. The regular production costs for each product are cpt,p ∈ P,t ∈ T. Any surplus production of a product p ∈ P must be stored at a cost of αp per unit. TTT management also considers unmet demand important, so it imposes a penalty cost of βp for each unit of unmet demand for product p ∈ P. TTT carries no initial inventory of products.

• P = {Salsa, Ketchup, Tomato-Paste}.

• R = {Tomatoes, Sugar, Labor, Spices}.

and the instance data in Tables 1— 5.
Suppose that the product demands are not known for certain until after the production decisions

are made. Specifically, TTT has to commit to an entire production schedule for all time periods, before the true demands for each time period becomes known. To account for the uncertainty, TTT forecasts a number of demand scenarios S = {1, 2, . . . |S|} where dpts is the demand for product p ∈ P in time period t ∈ T under scenario s ∈ S. The probability of a scenario s ∈ S occurring is ρs.

### 1.1 Problem

Formulate and solve a linear programming deterministic equivalent that will tell TTT management how to minimize their expected costs under varying demand scenarios. Demand scenarios and prob- abilities are listed in Table 5. All other instance data is listed in Tables 1— 4. Be sure to think carefully about which variables are first stage and which variables are second-stage variables.

### 1.2 Problem

What is the Expected Value of Perfect Information (EVPI) for the solution to Problem 1.1?

### 1.3 Problem

What is the Value of the Stochastic Solution (VSS) VSS for the solution to Problem 1.1? Be sure to write out the VSS and EVPI in your gams .lst file.

## 2 Multi-product assembly model

A manufacturer produces n products using m parts which are ordered from the suppliers. In the first stage, the manufacturer decides how many units of each part to order at the cost of cj per unit of part j, and after the (random) demands of the products are observed in the second stage, it then decides how many units of each product to make and sell. The objective is to maximize the overall profit.

A unit of product i requires aij units of part j, where i = 1,…,n and j = 1,…,m. Of course, aij may be zero for some combinations of i and j. The demand for the products is modeled as a random vector d = (d1 , . . . , dn ). It costs additionally li to satisfy a unit of demand for product i, and the unit selling price of this product is qi. The parts which are not used are assessed salvage values sj < cj. The unsatisfied demand is lost.

### 2.1 Problem

Formulate a mixed integer program that assumes a fixed demand D and solve it for the data given below:

```   sets i Product   /i1*i2/
j Part      /j1*j5/;
```
```   Table Data(*,*)
j1      j2      j3      j4      j5   l     q     d
```
```   c         10      30      10      100     50
v         1       8       2       30      10
```
1. i1  10 5 5 1 1500120010
2. ```   i2         8       5       0       2       1    600   3000  20;
```
```   Parameters
a(i,j) how many part j needed to make a unit of product i
c(j)   cost of procuring a unit of part j
v(j)   salvage value of a unit of part j
l(i)   manufacturing cost of a unit of product i
q(i)   selling price of a unit of product i,
md(i)  mean demand of product i;
```
```   a(i,j) = Data(i,j);
c(j)   = Data(’c’,j);
v(j)   = Data(’v’,j);
l(i)   = Data(i,’l’);
q(i)   = Data(i,’q’);
md(i)  = Data(i,’d’);
```

### 2.2 Problem

Now assume that demand for i1 and i2 are (independently) randomly generated using the discrete distributions below: