Mathematical Models for Managerial Decisions

Subject: Mathematical Models for Managerial Decisions (MBA Autumn 2015)

Assignment 1 Date:19/08/2015

Instructor: Kalyan K. Guin

Formulation 1.1

“Fresh and tasty” bakers make cakes and pastries. On a Sunday morning they have to make ‘plum cake” and “fruit pastries”. They have 59 units of flour and they use 5 units of flour per cake and 4 units per pastry. They also have 46 units of time in the oven that day and each cake requires 4 units in the oven and each pastry requires 3 units of time in the oven. They sell each cake for Rs 32 and each pastry for Rs 25. How many cakes and pastries they should make to maximize their sale?

Formulation 1.2

Sweet and softy Bakers make two products biscuits (in packets) and cakes. Their demand for biscuit packets and cakes are 20 and 5 respectively. They have 40 hours of regular time and it takes 2 hours to make a biscuit packet and 3 hours to make a cake. They wish to meet the demand and since the regular time available is not enough to meet the demand, they decide to employ overtime. To meet worker requirements they have to employ exactly 25 hours of overtime. The cost of making a biscuit packet by regular time and overtime it costs Rs 8 and 10 respectively. It also costs Rs 6 and 7 respectively to make a cake by regular time and overtime. How many units of biscuit packets and cakes do they produce by the two modes to minimize total cost of production? They also wish to talk to the workers to provide at least 10 hours of overtime.

Formulation 1.3

Consider two manufacturers (A and B) who are competitors for the same market segment for the same product. Each wants to maximize their profit market share and adopts two strategies. The gain (or pay off) for A when A adopts strategy i and B adopts strategy j is given by aij.

A 2  2 matrix of gains for A shown in Table 1.1 [pic 1]

Table 1.1 – Payoff matrix

During a given time period T, both A and B have to mix their strategies. If A plays only strategy 1, then B would play strategy 2 to gain, which A would not want. Each therefore wants to mix their strategies so that they gain maximum (or the other loses maximum).

Let us consider A’s problem of trying to maximize his return. Let us assume that A plays strategy 1 x1 proportion of times and plays strategy 2 x2 proportion of times. We have

x1 + x2 = 1

If B consistently plays strategy 1, then A’s expected gain is x1 – x2. If B consistently plays strategy 2, A’s expected gain is –3x1 + 2x2.

Player A also knows that B will not consistently play a single strategy but would play his (her) strategies in such a way that A gains minimum. So A would like to play his (her) strategies in proportions x1 and x2 to maximize this minimum return that B would allow.