# Expressing problems in canonical form for solving with simplex

The Picnic Hamper Company has a store containing 10,000kg of nuts, 4000 packs of smoked salmon, 2000 bottles of wine and 1500 Victoria sponges. It intends to use these goods to make up three different sorts of hampers with contents and price as follows: • Hamper A, priced at £50, contains: 6kg nuts, 3 smoked salmon, 2 bottles of wine, 1 Victoria sponge. • Hamper B, priced at £30, contains: 4kg nuts, 2 smoked salmon, 1 Victoria sponge. • Hamper C, priced at £25, contains: 8kg nuts, 3 bottles of wine. The question is, how many of each type of hamper should the company sell in order to maximise its revenue?

a. Write down the objective function, and the non-trivial constraints.

nuts<=10000

salmon <=4000

wine <= 2000

sponge <= 1500

money <= 50A + 30B + 25C (maximize function)

A = 6nuts + 3salm + 2wine + 1sponge

B = 4nuts + 2salm + 1sponge

C = 8nuts + 3wine

b. Express your model of this problem in canonical form. This is where i have a problem because i don't know which of these i need to use and add slack variables. I need to solve it through the simplex method so it cant be that i have to do introduce artificial and slack variables to all of these. So how can i express the object and constraints in canonical form?

First you should define the variables

A=Amount of Hamper A

B=Amount of Hamper B

C=Amount of Hamper C

Constraint for the weight of nuts (kg):

$6A+4B+8c\leq 10,000$

If you produces 1000 Hamper C, then you need 6000 kg of nuts.

Constraint for the packs of smoked salmon:

$3A +2B\leq 4,000$

Constraint for the bottles of wine:

$2A+3C\leq 2,000$

Constraint for the Victoria sponges:

$A+B \leq 1,500$

For each constraint you need a slack variable ($s_i$) to transform the inequalities to equalities. You have to add them. You do not need artificial variables, because you only have $\leq$-inequalities.

And finally $A,B,C \in \mathbb N_0$ and $s_1,s_2,s_3,s_4 \geq 0$