Saw King (SK) is the region’s leading seller of the highest rated model of chain saws. Their challenge is that their present store site has space for only an inventory of 18 of the very large saws. SK is trying to develop a plan for how to make inventory replenishment decisions in that constrained environment. The company can place an order at the end of each week for goods to arrive at the beginning of the next, and the plan they are developing should cover the historically volatile coming 5 weeks. Placing an order costs $500 plus $900 per unit purchased, and inventory holding costs $95 per unit per week. If stock is available, sale of a saw produces revenue of $2000. Demand is uncertain, but the following table estimates the probability for different numbers of weekly sales assuming stock is available. If stock is not on hand, demand is lost.

(a) Taking states i = 0,

c, 18 for inventory on hand as each week begins, identify all the elements of definition 9.44 to formulate this inventory management challenge as a MDP maximizing the total expected undiscounted net profit of gross income from sales less ordering and holding costs.
Assume initial inventory in week 1 equals 18, and that the order the end of week 5 will restore that max inventory level.

(b) Sketch a digraph depicting your model of part

(a) over nodes for 5 stages and 18 states with arcs for possible transitions.
You need not insert all parameter details, but do show exemplars of all transition arcs with the decision to which they are attached, the reward they would realize, and the associated probability.

(c) Form the functional equations for all states and stages for your model of (a).

(d) Solve the equations of

(c) to determine an optimal inventory policy for SK.

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Dem Prob Dem Prob Dem Prob Dem Prob Dem Prob 0123 .020 .023 2 .027 .030 1567 4 .033 8 .046 12 .100 16 .041 .036 9 .060 13 .060 17 .038 .039 10 .100 14 .045 18 .036 .043 11 .130 15 .042 over 18 .051